Algebraic Curves over a Finite Field (Princeton Series in Applied Mathematics) 9780691096797

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Algebraic Curves over a Finite Field

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PRINCETON SERIES IN APPLIED MATHEMATICS

Edited by Ingrid Daubechies, Princeton University Weinan E, Princeton University Jan Karel Lenstra, Eindhoven University Endre S¨ uli, University of Oxford

The Princeton Series in Applied Mathematics publishes high quality advanced texts and monographs in all areas of applied mathematics. Books include those of a theoretical and general nature as well as those dealing with the mathematics of specific applications areas and real-world situations.

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Algebraic Curves over a Finite Field

J.W.P. Hirschfeld ´ G. Korchmaros F. Torres

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

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c 2008 by Princeton University Press Copyright Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Cataloging-in-Publication Data ISBN: 978-0-691-09679-7 (acid-free paper)

British Library Cataloging-in-Publication Data is available This book has been composed in LATEX The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

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vi For Lucca born 9th June 2006 JWPH For Adriana, Annachiara and Gabriella GK To the memory of Eduardo W. Torres Orihuela 6th March 1970 to 27th December 1989 FT

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Contents

Preface

xiii

PART 1. GENERAL THEORY OF CURVES

1

Chapter 1. Fundamental ideas

3

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Basic definitions Polynomials Affine plane curves Projective plane curves The Hessian curve Projective varieties in higher-dimensional spaces Exercises Notes

Chapter 2. Elimination theory 2.1 2.2 2.3 2.4 2.5

Elimination of one unknown The discriminant Elimination in a system in two unknowns Exercises Notes

Chapter 3. Singular points and intersections 3.1 3.2 3.3 3.4 3.5 3.6 3.7

The intersection number of two curves B´ezout’s Theorem Rational and birational transformations Quadratic transformations Resolution of singularities Exercises Notes

Chapter 4. Branches and parametrisation 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Formal power series Branch representations Branches of plane algebraic curves Local quadratic transformations Noether’s Theorem Analytic branches Exercises

3 5 6 9 13 18 18 19 21 21 30 31 35 36 37 37 45 49 51 55 61 62 63 63 75 81 84 92 99 107

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CONTENTS

4.8

Notes

Chapter 5. The function field of a curve 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13

Generic points Rational transformations Places Zeros and poles Separability and inseparability Frobenius rational transformations Derivations and differentials The genus of a curve Residues of differential forms Higher derivatives in positive characteristic The dual and bidual of a curve Exercises Notes

Chapter 6. Linear series and the Riemann–Roch Theorem 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

Divisors and linear series Linear systems of curves Special and non-special linear series Reformulation of the Riemann–Roch Theorem Some consequences of the Riemann–Roch Theorem The Weierstrass Gap Theorem The structure of the divisor class group Exercises Notes

Chapter 7. Algebraic curves in higher-dimensional spaces 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19

Basic definitions and properties Rational transformations Hurwitz’s Theorem Linear series composed of an involution The canonical curve Osculating hyperplanes and ramification divisors Non-classical curves and linear systems of lines Non-classical curves and linear systems of conics Dual curves of space curves Complete linear series of small order Examples of curves The Linear General Position Principle Castelnuovo’s Bound A generalisation of Clifford’s Theorem The Uniform Position Principle Valuation rings Curves as algebraic varieties of dimension one Exercises Notes

109 110 110 112 119 120 122 123 125 130 138 144 155 159 160 161 161 170 177 180 182 184 190 196 198 199 199 203 208 211 216 217 228 230 238 241 254 257 257 260 261 262 268 270 271

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CONTENTS

xi

PART 2. CURVES OVER A FINITE FIELD

275

Chapter 8. Rational points and places over a finite field

277

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10

Plane curves defined over a finite field Fq -rational branches of a curve Fq -rational places, divisors and linear series Space curves over Fq The St¨ohr–Voloch Theorem Frobenius classicality with respect to lines Frobenius classicality with respect to conics The dual of a Frobenius non-classical curve Exercises Notes

Chapter 9. Zeta functions and curves with many rational points 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12

The zeta function of a curve over a finite field The Hasse–Weil Theorem Refinements of the Hasse–Weil Theorem Asymptotic bounds Other estimates Counting points on a plane curve Further applications of the zeta function The Fundamental Equation Elliptic curves over Fq Classification of non-singular cubics over Fq Exercises Notes

277 278 281 287 292 305 314 326 327 329 332 332 343 348 353 356 358 369 373 378 381 385 388

PART 3. FURTHER DEVELOPMENTS

393

Chapter 10. Maximal and optimal curves

395

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12

Background on maximal curves The Frobenius linear series of a maximal curve Embedding in a Hermitian variety Maximal curves lying on a quadric surface Maximal curves with high genus Castelnuovo’s number Plane maximal curves Maximal curves of Hurwitz type Non-isomorphic maximal curves Optimal curves Exercises Notes

Chapter 11. Automorphisms of an algebraic curve 11.1 11.2 11.3

The action of K -automorphisms on places Linear series and automorphisms Automorphism groups of plane curves

396 399 407 421 428 431 439 442 446 447 453 454 458 459 464 468

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A bound on the order of a K -automorphism Automorphism groups and their fixed fields The stabiliser of a place Finiteness of the K -automorphism group Tame automorphism groups Non-tame automorphism groups K -automorphism groups of particular curves Fixed places of automorphisms Large automorphism groups of function fields K -automorphism groups fixing a place Large p-subgroups fixing a place Notes

470 473 476 480 483 486 501 509 513 532 539 542

Chapter 12. Some families of algebraic curves

546

11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14 11.15

12.1 12.2 12.3 12.4 12.5 12.6

Plane curves given by separated polynomials Curves with Suzuki automorphism group Curves with unitary automorphism group Curves with Ree automorphism group A curve attaining the Serre Bound Notes

Chapter 13. Applications: codes and arcs 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11

Algebraic-geometry codes Maximum distance separable codes Arcs and ovals Segre’s generalisation of Menelaus’ Theorem The connection between arcs and curves Arcs in ovals in planes of even order Arcs in ovals in planes of odd order The second largest complete arc The third largest complete arc Exercises Notes

Appendix A. Background on field theory and group theory A.1 A.2 A.3 A.4 A.5 A.6

Field theory Galois theory Norms and traces Finite fields Group theory Notes

546 564 572 575 585 587 590 590 594 599 603 607 611 612 615 623 625 625 627 627 633 635 636 638 649

Appendix B. Notation

650

Bibliography

655

Index

689

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Preface

BACKGROUND Algebraic curves defined over a finite field have been much studied in recent years, both for their intrinsic interest and for their applications in other areas, such as finite geometry, number theory, error-correcting codes, and cryptology. Conversely, applications in these areas have focused attention on particular problems for curves. This book is a self-contained introduction to this subject, including topics of current research interest. It is divided into three parts. The first is a general theory of algebraic curves over an algebraically closed field K of arbitrary characteristic p. To keep the treatment as simple as possible, Part 1 starts with the classical idea of a plane algebraic curve F as the set of zeros of a polynomial f (X, Y ) or a homogeneous polynomial F (X0 , X1 , X2 ), and develops its subject in the spirit of the geometric approach to the theory of algebraic curves, such as the treatises of Enriques, Lefschetz, Seidenberg, Semple and Kneebone, Semple and Roth, Severi, van der Waerden, and Walker. A rigorous treatment requires concepts and results from algebra; this is supplied by means of the function field K(F) obtained by adjoining to K(x) an element y satisfying f (x, y) = 0. Part 2 is the heart of the book. It deals to a great extent with the case where K is the algebraic closure Fq of a finite field Fq giving many properties of curves peculiar to finite fields. Part 3 collects together a number of more advanced results on curves over finite fields, on automorphism groups of curves, and on applications of curves in coding theory and in the combinatorics of finite projective spaces. It also contains a collection of the most important families of curves. The last section in each chapter consists of notes and references. Appendix B is a list of notation used in the book. Throughout the book numerous examples are worked out to illustrate the theory. The reader may also benefit from the exercises, the second last section in each chapter, and Appendix A reviewing basic facts required from algebra. The exercises are not supplied with hints for their solutions, but for many of them there are references in the notes. The algebraic side of the theory of curves can also be developed further with a modern approach demanding considerable background in algebra, in the style of the theory of compact Riemann surfaces or of the theory of algebraic functions of one variable.

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The book developed from a two-year course taught by the second author at the University of Basilicata and a series of lectures by the first author on the St¨ohr– Voloch Theorem at various summer schools. Advanced topics from Part 3 were also the subject of talks at international conferences, delivered by the authors.

OUTLINE OF THE BOOK The field K of coordinates is assumed to be algebraically closed except in the preliminary Section 1.1. This is not a restriction as every field is a subfield of an algebraically closed field. No assumption on the characteristic p of K is made. Hence either p = 0 or p ≥ 2. Since all curves considered in the book are algebraic, the adjective ‘algebraic’ is mostly omitted. Part 1 Classical algebraic geometry deals with plane projective curves in terms of their projective invariants. These are the properties of the curves which do not change under any projective transformation or, equivalently, under any change of the homogeneous coordinate system. For their study, the classical definition in its projective formulation is adequate: a plane projective curve F is the set of all points in the projective plane PG(2, K) satisfying a homogeneous polynomial F (X0 , X1 , X2 ) with coefficients in K; it is denoted by F = v(F (X0 , X1 , X2 )). If F is not the line v(X0 ), the notation F = v(f (X, Y )) is used, where f (X, Y ) = F (1, X, Y ) is the associated non-homogeneous polynomial. Calculations are often simplified by using the canonical form of a curve that goes back at least to Descartes in 1637 for conics, and to Newton in 1710 for cubics. The relevant projective invariants, notably that of degree, inflexion, k-fold point, ordinary singularity, intersection number, bitangents, are considered in Chapter 1. Many problems on curves ultimately reduce to questions about the intersections of two plane curves. So the most important projective invariant is the intersection number I(P, F ∩ ℓ) of a plane curve F and a line ℓ at a point P . It is defined in such a way that the number of common points, each counted according to the intersection number at that point, is exactly the degree of F, unless ℓ is a component of F. B´ezout’s Theorem extends this formula from ℓ to any plane curve G having no common component with F. Essentially, I(P, F ∩ ℓ) is generalised to I(P, F ∩ G), the intersection number of F with another plane curve G at P , such that the number of common points, each counted according to the intersection number at that point, is exactly the product of the degrees of F and G. Such a generalisation is not immediate. The method for calculating I(P, F ∩ G) requires knowing what condition the coefficients of two polynomials must satisfy in order that the polynomials have a common root of a given multiplicity. This is done by elimination theory relying on the resultant in Chapter 2. The geometric properties including the projective invariance of I(P, F ∩ G), together with the proof of B´ezout’s Theorem, are in Chapter 3. In investigating curves, an analysis of the singularities is essential. Singularities

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can be complicated and hence intractable for computations. Since projective transformations do not change the nature of the singularities, more general transformations are required. In Chapter 3, it is shown that locally quadratic transformations suffice to transform any plane curve, with possibly bad singularities, to one with only ordinary singularities. Then, certain properties of the curve can be retrieved from its transform, which is a plane curve with singularities in as simple a form as possible. In such an analysis, as in many other contexts in algebraic geometry, the idea of a branch of a plane curve F plays an important role. The centre of a branch of F is a point of F. There is exactly one branch of F centred at every non-singular point of F, but an ordinary r-fold point is the centre of exactly r branches of F. In general, an r-fold point of F is the centre of at least one and at most r branches of F. Also, two distinct irreducible plane curves F and G have no common branch. For any branch γ of G centred at a point P , the intersection number I(P, F ∩ γ) of F and γ is defined in such a way that I(P, F ∩ G) is the sum of the I(P, F ∩ γ) with γ ranging over all branches of G centred at P . This intersection number is suitable for use in B´ezout’s Theorem, and provides an efficient way for computing. The technical tools are formal power series rather than polynomials. Chapter 4 is dedicated to the theory of branches, including classical results on power series, such as Hensel’s Lemma and Weierstrass’ Preparatory and Division Theorems. Locally quadratic transformations are particular birational transformations. This suggests investigating the effect of birational transformations on a plane curve, and studying its unaffected properties, that is, its birational invariants. However, the intersection number, like other projective invariants considered in Chapter 1, is not a birational invariant. To understand why this happens, it is enough to observe that the nodal cubic and the cuspidal cubic are birationally equivalent and that any birational transformation mapping one to the other is not defined at singular points. So the classical idea of a plane curve is not sufficient to deal with birational transformations. A plane curve needs to be considered as the set of its branches rather than of its points. Then the idea of a generic point of a curve can be viewed as a straightforward generalisation of the idea of a branch. Another actor is the function field K(F) of an irreducible plane curve F, which is a field of transcendency degree 1 over K. Importantly, birationally equivalent curves have the same function field, up to a K-isomorphism, and the geometric idea of a branch of F can be translated into K(F) by the purely algebraic concept of a place. In particular, the valuation of an element ξ of K(F) at a place P leads to the consideration of the zeros and the poles of ξ. In Chapter 5, after a detailed presentation of these concepts, the most important birational invariant, namely the genus g of an irreducible plane curve F, is investigated. The genus is a non-negative integer which can be calculated from the equation of F when F has only ordinary singularities. If F is non-singular, then g = 21 (n − 1)(n − 2) where n = deg F. In particular, irreducible plane cubics are elliptic curves, that is, they have genus 1. Elliptic curves have many peculiar properties; their study requires a particular theory that is not included in the book. However, some of their structure is given in Chapter 6 and a summary of the isomorphism and projective equivalence classes is given in Chapter 8.

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An important formula to compute the genus g of any irreducible plane curve F is established in terms of differentials of K(F). Higher differentials are also considered, with some changes in the familiar definition of higher derivatives to avoid undesirable behaviour in positive characteristic. Such Hasse derivatives are the technical tool for the study of dual curves as well as in several contexts in later chapters. Other important birational invariants are the parameters of linear series, namely the order and dimension. A divisor is any formal sum of places; the principal divisor div ξ of ξ ∈ K(F), ξ 6= 0, is the sum of all zeros and poles of ξ, each counted with multiplicity. Two divisors A, B are equivalent if B = A + div ξ for some ξ ∈ K(F). A divisor A is effective or positive if no place appears in A with negative weight. A linear series is a set of certain effective equivalent divisors. A complete linear series |A| consists of all effective divisors equivalent to A. Chapter 6 develops the theory of linear series based on the idea of adjoint curves. For a complete linear series of order n and projective dimension r, the important Riemann– Roch Theorem states that r = n − g + i, where i is another birational invariant, the index of speciality. In particular, i = 0 when n > 2g − 2. The Riemann–Roch Theorem not only provides an alternate definition of the genus g, but is also the essential ingredient in many fundamental results on curves, such as the Reciprocity Theorem of Brill and Noether, Clifford’s Theorem, and Weierstrass’ Gap Theorem. The last concerns gaps at a place P, which are positive integers k such that kP is not the divisor of all poles of any function in K(F); it states that there exist exactly g gaps at any place of K(F). Chapter 6 also includes some basic facts on the zero divisor class group, the Picard group, and on its factor group, Pic0 (K(F)), together with an important result on the structure of certain subgroups of Pic0 (K(F)). The study of the divisor class groups is expanded in Part 3 for the case K = Fq . Even using birational transformations, an irreducible plane curve can rarely be transformed into a non-singular plane curve. To obtain a non-singular model for any irreducible plane curve, space curves need to be considered. The idea of a space curve in a higher-dimensional space as an algebraic variety of dimension one, that is, the set of common zeros of a finite set of polynomials, is not suitable for this. An alternative approach, developed in Part 2, provides a different but equivalent definition: an irreducible space curve Γ is the image of an irreducible plane curve F under a birational transformation ω. In particular, the function field K(Γ) of Γ is K(F). The approach is based on the idea of branches in a higher-dimensional space; those which are the images of the branches of F under ω are viewed as the branches of Γ. The points of Γ are the centres of its branches. Here, care is required, as some point of Γ may not be the image of a point of F under ω. This theory of space curves begins in Chapter 7. The geometry of a space curve Γ of PG(r, K) can be extracted from its intersections with hyperplanes, and studied in terms of the linear series gnr on Γ cut out by hyperplanes. Here n is the degree of Γ, that is, the number of common points P of Γ with a hyperplane H, each counted according to the intersection number I(P, Γ ∩ H). In particular, the singular points of Γ are those points P ∈ Γ for which I(P, Γ ∩ H) ≥ 2 for every hyperplane H through P . For sufficiently large n compared to r, no point of Γ is singular; this shows the existence of a non-singular

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model of Γ. For g > 1, the canonical curve K of Γ arising from the canonical linear g−1 series g2g−2 is a non-singular curve of PG(g − 1, K), provided that Γ is not hyperelliptic. Hyperelliptic curves occur as exceptions since their canonical linear series are composed of an involution of order two; in other words, K is doubly covered by Γ; equivalently, [K(Γ) : K(K)] = 2. Another important geometric idea in Chapter 7 comes from classical differential geometry. For a non-singular point P of Γ, there are exactly r + 1 Hermitian P -invariants; they are the non-negative integers j for which some hyperplane H intersects Γ at P with multiplicity j, that is, I(P, Γ ∩ H) = j. The Hermitian P -invariants written in increasing order j0 = 0 < j1 < · · · < jr define the order sequence (j0 , j1 . . . , jr ) of Γ at P . Let Li be the intersection of all hyperplanes H in PG(r, K) such that I(P, Γ ∩ H) ≥ ji+1 . Then, L0 is the point P , and L1 is the tangent line to Γ at P . The projective subspace Li is the i-th osculating space, and the flag L0 ⊂ L1 ⊂ · · · ⊂ Lr−1 is the algebraic analogue of the Frenet frame. For a singular point P , all this remains valid for each branch of Γ centred at P . Apart from finitely many points of Γ, the order sequence of Γ at P is the same and it defines the order sequence (ǫ0 , ǫ1 , . . . , ǫr ) of Γ. In zero characteristic, every curve is classical, the order sequence being (0, 1, . . . , r). This does not hold true for p > 0, although non-classical curves are rare and often have unusual behaviour. For instance, certain non-classical curves are projectively equivalent to their Gauss duals, and, up to a birational transformation, are the curves with many automorphisms. When K = Fq , these are the curves lying in PG(r, q) with many points. So non-classical curves are important and germane to several research areas of current interest, such as coding theory and cryptology. Chapter 7 contains the theory of non-classical curves. The existence of non-classical curves is a real obstacle to extending classical results from zero to positive characteristic. Nevertheless, Castelnuovo’s Bound, Halphen’s Theorem, and several other relevant results on curves do hold true for p > 0 under some reasonable extra conditions, notably the Linear General Position Principle and the Uniform Position Principle. This is the final topic in Part 2, together with the explanation of how the concept of a place, as used in this book, is related to the concept of a place arising in valuation theory. Also, the proof of the equivalence between the definition of an irreducible space curve and the concept of an algebraic variety of dimension one is described. Part 2 With K = Fq , the field K has a set of finite subfields containing Fq , Fq , Fq2 , . . . , Fqi , . . . , such that K = ∪i≥1 Fqi . So, for any r, the projective space PG(r, K) contains the finite projective spaces PG(r, q i ) with i ≥ 1. A plane curve F = v(f (X, Y )) is defined over Fq when f (X, Y ) ∈ Fq [X, Y ]. Irreducible plane curves defined over Fq are also called Fq -rational. This natural concept extends to higher dimensions: an irreducible space curve Γ is Fq -rational if it arises from an Fq -rational plane curve by an Fq -birational transformation ω.

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This means that ω is defined over the subfield Fq (F) of K(F), the Fq -rational function field of F. Given an Fq -rational irreducible curve Γ of PG(r, K), the aim is to extract useful information from the sequence M1 , M2 , . . . , Mi . . . , with Mi the number of points of Γ in PG(r, q i ). However, such a sequence may not be the same for another curve Fq -birationally equivalent to Γ. Again, the idea of an irreducible curve as a set of its branches is useful for placing the theory within birational equivalence: it is enough to count the Fqi -rational branches of Γ, and replace Mi by the number Ni of such branches, and consider the sequence N1 , N2 , . . . , Ni . . . . From the function field point of view, the corresponding concept of an Fq -rational branch is an Fq -rational place of K(F). It should be noted that a singular point of Γ lying in PG(r, q) may not be the centre of any Fq -rational branch of Γ. This shows that a bijection between the points of Γ lying in PG(r, q) and the Fq -rational places of K(Γ) can only be ensured for non-singular curves. If this is the case, points of Γ lying in PG(r, q) are the Fq -rational points. The foundation of the theory of irreducible curves over finite fields is given in Chapter 8. Gauss in 1801 considered cubic and quartic congruences in two variables modulo a prime, and so calculated the number of rational points for particular cubic and quartic curves. Here, a central result is the St¨ohr–Voloch Theorem, which highlights several peculiar geometric properties of curves with many Fq rational branches. Such curves are mostly non-classical. The St¨ohr–Voloch Bound derived from this theorem provides an accurate estimate of the number Sq of Fq rational branches for large families of curves. Chapter 8 explains all the background and details of the St¨ohr–Voloch Theorem, and gives an exposition of the theory of Frobenius non-classical curves. An elementary proof of the St¨ohr–Voloch Bound for non-classical plane curves is also presented. Another bound on Sq , depending only on g and q, is the famous Hasse–Weil Bound, proved by Hasse for g = 1 in 1934 and by Weil in 1948 for g > 1: √ √ q + 1 − 2g q ≤ Sq ≤ q + 1 + 2g q.

This bound is deduced, via zeta functions, from the Riemann hypothesis for function fields over finite fields. The Hasse–Weil Bound and its refinements, especially the Serre Bound, together with the St¨ohr–Voloch Bound, are widely used in investigating curves over finite fields, in various areas such as finite geometry, number theory, coding theory, and cryptology. In coding theory, the asymptotic behaviour of Nq (g)/g is important, where Nq (g) is the maximum number of Fq -rational points that a non-singular Fq -rational curve of genus g can have. The Hasse–Weil Theorem leads to other results on curves over finite fields, on their Picard groups, p-ranks and Frobenius linear series. These themes are treated in Chapter 9. A few results, such as the fundamental equation of a Frobenius linear series, are discussed without proofs but with an eye

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to further reading on algebraic number theory and abelian varieties. Modular curves are also important, but they are not included. Part 3 For q = pr , the Hermitian curve Hq = v(Y q − Y − X q+1 ) is a non-singular plane curve of genus g = 12 (q 2 − q) with exactly q 3 + 1 points in PG(2, q 2 ). So the number of Fq2 -rational points of Hq is equal to q 3 + 1 = q 2 + 1 + 2gq, the maximum allowed in the Hasse–Weil upper bound. Other important Fq2 -maximal curves are the DLS and DLR curves. Curves of genus zero are lines and so the lines of PG(2, q 2 ) are maximal. For each square q, there is also a maximal elliptic curve. Maximal curves are naturally used in algebraic-geometry codes. The theory of Fq2 -maximal curves is covered in Chapter 10. Various characterisations of Fq2 -maximality in both algebraic and geometric terms are given. An irreducible curve F defined over Fq2 is Fq2 -maximal if and only if one of the following holds. (i) The numerator in the zeta function of F, that is, the Lq2 (t)-polynomial of F, is (t + q)2g . (ii) Let X be a non-singular model of F, defined over Fq2 and embedded in PG(r, K). For any Fq2 -rational point P0 ∈ X , the Frobenius linear series |(q + 1)P0 | has fundamental equation, qP + Φ(P ) ≡ (q + 1)P0 , satisfied by all points P ∈ X , where Φ is the Frobenius collineation of PG(r, K) fixing PG(r, q 2 ) point-wise. (iii) F is Fq2 -birationally equivalent to an irreducible curve Γ of degree q + 1 embedded in PG(r, K) and lying on a non-degenerate Hermitian variety of PG(r, q 2 ). The curve Γ in (iii) is a non-singular model defined over Fq2 such that the Frobenius linear series is cut out by hyperplanes of PG(r, q 2 ). Therefore Γ plays a central role in the study of the geometry of Fq2 -maximal curves. The orders at any Fq2 -rational point P ∈ Γ are determined by the Weierstrass non-gaps at P less than equal to q + 1. The same kind of computation also works for other points giving all but one of these orders. From these results and basic facts on Weierstrass semigroups, Γ is shown to be Frobenius non-classical. Another relevant property of maximal curves, originally pointed out by Serre, is related to covers: given an Fq2 -maximal curve F, every curve F ′ which is covered by F over Fq2 is also Fq2 -maximal. Notably, the quotient curve F ′ = F/G arising from any non-trivial Fq2 -automorphism group G of F is covered by F. Since the Hermitian, DLS, and DLR curves have very large automorphism groups, Serre’s result provides an efficient construction technique. From the current literature, just one Fq2 -maximal curve, the Garcia–Stichtenoth curve over F36 , is known not to be such a quotient curve of Hq .

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For a given q, the highest value of g for which an Fq2 -maximal curve F of genus g exits is 12 (q 2 − q) and equality holds if and only if F is the Hermitian curve, up to Fq2 -birational equivalence. The upper part in the spectrum of genera of Fq2 -maximal curves contains only few values. The classification of some of these curves and the known results are given. Although the whole spectrum depends on the nature of q, the highest three genera in the spectrum have the same positions independently of q. Particular Fq2 -maximal curves, such as those of Hurwitz type, are also investigated, and some non-isomorphic Fq2 -maximal curves with the same genus are considered. Chapter 10 also describes some examples of Fq -optimal curves, that is, curves defined over Fq whose number of Fq2 -rational points is Nq (g). By definition, Fq2 maximal curves are Fq2 -optimal. The DLS and DLR curves are both Fq -optimal, but the DLS curve is Fq4 -maximal and the DLR curve is Fq6 -maximal. It is shown that the DLS curve is characterised as the unique Fq -optimal curve with genus g = q0 (q − 1) for p = 2, q0 = 2s and q = 2q02 . A similar characterisation for the DLR curve may be possible. Chapter 11 gives the theory of automorphisms of an irreducible curve. To establish the concept of an automorphism as a birational invariant, the theory begins with the concept of a field automorphism applied to a function field Σ of transcendency degree one over K or, equivalently, to the function field K(F) of an irreducible plane curve F. Here, only automorphisms fixing every element in K are admitted. Such a K-automorphism α of Σ defines a K-birational transformation of Σ, from which a geometric interpretation of α in terms of F can be deduced. The theory develops on the algebraic side with the study of the K-automorphism group AutK (Σ) of Σ = K(F) in terms of its abstract structure and its actions on the sets of places and divisors. Nevertheless, K-automorphism is a truly geometric concept. In fact, F is birationally equivalent to an irreducible non-singular curve X of some PG(r, K), and X can be chosen in such a way that every Kautomorphism of Σ is represented as a linear collineation of PG(r, K) preserving X . Then AutK (Σ) is the K-automorphism group of X , and the term ‘Kautomorphism group of Γ’ is adopted for any curve Γ birationally equivalent to X. The major result is the finiteness of AutK (Σ) for curves of genus g > 1, which is a consequence of the finiteness of the stabiliser of any place P of Σ. This result on stabilisers also holds for elliptic curves. Let G be a finite non-trivial subgroup of AutK (Σ). The elements in Σ fixed by every K-automorphism in G is the subfield ΣG of Σ. Importantly, ΣG is larger than K and Σ/ΣG is a Galois extension over K with Galois group G. In particular, ΣG = K(G) is the function field of a quotient curve G of F, defined up to birational equivalence. As a matter of fact, G inherits important properties from F: for instance, Fq2 -maximality as considered in Chapter 10. Derived from Hurwitz’s Theorem, linking g to the genus g ′ of ΣG , is the important equation, 2g − 2 = |G|(2g ′ − 2) + d,

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where d is the degree of the different divisor D(Σ/ΣG ). To compute d, two cases are distinguished according as G is tame or non-tame (wild). In the tame case, p ∤ |GP | for every place P of Σ; this occurs for instance for p = 0 and for p ∤ |G|. If G is tame, the classical argument due to Hurwitz works and gives a formula for d depending only on the lengths of the short G-orbits of places of Σ. From this, the bound, |G| ≤ 84(g − 1), follows for g ≥ 2. If G is non-tame, then |G| and hence AutK (Σ) may be much larger, even exceeding g 4 . A major result is the classification of all curves with |AutK (Σ)| ≥ 8g 3 . The long and delicate proof not only uses Hurwitz’s work with Hilbert’s different formula and Serre’s results on ramification groups but also exploits many results on finite permutation groups, especially the classification of all 2-transitive groups whose 2-point stabilisers are cyclic. There are four such curves, namely two hyperelliptic curves, the Hermitian curve and the DLS curve. On the other hand, |G| is bounded by 4g + 4 for abelian groups and p-subgroups of curves with positive p-rank, and by 24g 2 for certain soluble groups of curves with zero p-rank. Also, Hurwitz’s bound holds for ordinary curves with p-rank equal to g. There remains the problem of understanding better to what extent the Hurwitz bound holds in positive characteristic. Chapter 12 is dedicated to curves defined over finite fields with unusual properties that a complex curve cannot have. Artin–Schreier curves and, in particular, Hermitian curves are of this type. A family of such plane curves arises from separated polynomials. It consists of curves F = v(A(Y ) − B(X)) where p ∤ m with m = deg B(X) ≥ 2 and A(Y ) is any additive separable polynomial. The main properties of F are extracted from the local analysis of its unique singular point P∞ . The exposition describes the genus, the Weierstrass gap sequence at P∞ and the ramification groups of its translation automorphism group fixing P∞ . The full K-automorphism group of F fixes P∞ except in two cases, namely, when n n F is the Hermitian curve v(Y p − Y − X p +1 ) or the curve, n

F = v(Y p + Y − X m )

with m < pn , and pn ≡ −1 (mod m).

For p > 2 and m = 2, the latter curve is hyperelliptic. Notably for p > 2, these hyperelliptic curves and the Hermitian curves are the only curves whose Kautomorphism groups have order larger than 8g 3 . Deligne–Lusztig varieties of dimension one provide other examples of significant curves over finite field, namely the DLS curves of Suzuki type and the DLR curves of Ree type. They are characterised by their genera and K-automorphism groups. For p = 2, the Hermitian curves, the DLS curves, and the hyperelliptic h curves F = v(Y 2 +Y +X 2 +1 ) are the only curves with K-automorphism groups of order larger than 8g 3 . A further interesting example worked out in Chapter 12 is defined over F2 and attains the Serre Bound for q = 211 . The proof illustrates how the theory presented in Chapter 9 may be used to investigate sporadic curves. Chapter 13 looks at applications of curves in coding theory and in the combinatorics of finite geometry. A brief exposition of the main construction due to Goppa and a few illustrative examples are presented. Coding theory is also connected with curves via finite geometry, since complete arcs in PG(r, q) are the geometric

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counterpart of maximum distance separable (MDS) codes, which are linear codes correcting the greatest number of errors with respect to the field size and their dimension. An account of this subject is given, especially of the Main Conjecture for MDS codes.

ACKNOWLEDGEMENTS We would like to thank the following colleagues: Massimo Giulietti for sharing the writing of Chapter 11 and also for proof-reading; Marien Abreu, Vincenzo Giordano, Lucia Indaco for proof-reading Part 1; Angelo Sonnino and Fabio Pasticci for technical help. Discussions on a miriad of aspects of algebraic curves with colleagues in many countries were of great assistance and we are most grateful to all of them; they are Miriam Abd´on, Angela Aguglia, Antonio Campillo, John Cannon, Antonio Cossidente, Rainer Fuhrmann, Arnaldo Garcia, Paolo Maroscia, Nicola Melone, Ruud Pellikaan, Alessandro Siciliano, Henning Stichtenoth, Karl St¨ohr, Felipe Voloch, and Mike Zieve. The Mathematics Departments of the Universities of Basilicata, Campinas, Essen, Perugia, Rome ‘La Sapienza’, Sussex, Sydney and Valladolid, as well as IMPA in Rio de Janeiro and the Abdus Salam International Centre for Theoretical Physics in Trieste have supported us. Finally, this book could not have been produced without the love and support of our families.

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PART 1

General theory of curves

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Chapter One Fundamental ideas In this chapter, basic facts about curves are presented. The exposition also highlights some of the peculiarities that occur for positive characteristic, such as the existence of strange curves, that is, curves whose tangent lines at non-singular points have a point in common.

1.1 BASIC DEFINITIONS Over the real numbers, R, consider the parabola F given by F = Y − X 2 ; its points form, as in Figure 1.1, the set {(t, t2 ) | t ∈ R}. p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

Figure 1.1 The parabola y = x2 in the real affine plane

However, there are two other types of points associated with F , namely, (a) those at infinity and (b) those with coordinates in C, the algebraic closure of R. For example, regarding (b), the line with equation y + 1 = 0 meets F in the two points (i, −1), (−i, −1), where i2 = −1. Regarding (a), if F is homogenised to F ∗ = X0 X2 − X12 , with X = X1 /X0 , Y = X2 /X0 , then the line with equation X0 = 0 meets the corresponding projective curve F ∗ at the point (0, 0, 1). All these ideas need to be considered for a general curve and a general field. First, some notation and fundamental definitions for the spaces that appear are explained. D EFINITION 1.1 (i) For a field K, let K n = {(x1 , x2 , . . . , xn ) | xi ∈ K}, the n-fold Cartesian product of K.

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(ii) Let V (n, K) be n-dimensional vector space over K, which may be regarded as (K n , +, ·), where, for xi , yi , λ ∈ K, (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn ) = (x1 + y1 , x2 + y2 , . . . , xn + yn ), λ(x1 , x2 , . . . , xn ) = (λx1 , λx2 , . . . , λxn ). (iii) The affine plane AG(2, K) = A2 (K) is a pair (P, L), where P = {P = (x, y) | x, y ∈ K},

L = {ℓ = aX + bY + c | a, b, c ∈ K, (a, b) 6= (0, 0)},

and a point P = (x, y) lies on a line ℓ = aX + bY + c if ax + by + c = 0. (iv) More generally, affine space of n-dimensions is AG(n, K) = An (K) with points x = (x1 , x2 , . . . , xn ) and r-dimensional subspaces x + S, for rdimensional subspaces S of V (n, K). (v) The projective plane PG(2, K) = P2 (K) is a pair (P, L), where  P = P = (x, y, z) = (λx, λy, λz) | (x, y, z) ∈ K 3 \{(0, 0, 0)}, λ ∈ K\{0}} ,

L = {ℓ = aX + bY + cZ = λaX + λbY + λcZ | a, b, c, λ ∈ K, (a, b, c) 6= (0, 0, 0), λ 6= 0} ,

and a point P = (x, y, z) lies on a line ℓ = aX +bY +cZ if ax+by+cz = 0. (vi) More generally, projective space of n-dimensions is PG(n, K) = Pn (K) with points, x = (x0 , x1 , x2 , . . . , xn ) = (λx0 , λx1 , λx2 , . . . , λxn ), (x0 , x1 , x2 , . . . , xn ) 6= (0, 0, 0, . . . , 0), λ 6= 0, and r-dimensional subspaces S, for (r + 1)-dimensional subspaces S of V (n + 1, K). In each type of space, it is important to consider the structure-preserving transformations. D EFINITION 1.2 (i) A linear transformation T : V (n, K) → V (n, K), is given as follows: T (x) = x′

where t x′ = A t x for a suitable non-singular matrix A,

with x = (x1 , x2 , . . . , xn ), x′ = (x′1 , x′2 , . . . , x′n ), and t x the transpose of x. The linear transformations of V (n, K) constitute the the general linear group GL(n, K). A semilinear transformation T : V (n, K) → V (n, K), is given as follows: T (x) = x′ ,

where t x′ = A t σ(x) for a suitable non-singular matrix A,

with σ(x) = (σ(x1 ), σ(x2 ), . . . , σ(xn )) for some automorphism of K. The semilinear transformations of V (n, K) constitute its general semi-linear group ΓL(n, K).

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(ii) An affine transformation S : AG(n, K) → AG(n, K), is given as follows: S(x) = x′ = T (x) + b,

where T is a linear transformation and b = (b1 , b2 , . . . , bn ) The affine transformations of AG(n, K) constitute its affine group GL(n, K). An affine collineation S : AG(n, K) → AG(n, K), is given as follows: S(x) = x′ = T (σ(x)) + b.

(iii) A projectivity T : PG(n, K) → PG(n, K) is given as follows: T (x) = x′ ,

t ′

x = A t x,

where

with x′ = (x′0 , x′1 , . . . , x′n ),

x = (x0 , x1 , . . . , xn ),

and A a suitable non-singular matrix. It is also called a projective transformation or linear collineation. The projectivities of PG(n, K) constitute its projective general linear group PGL(n + 1, K). A collineation T : PG(n, K) → PG(n, K) is given as follows: T (x) = x′ ,

where

t ′

x = A t σ(x),

for a suitable non-singular matrix A. The collineations of PG(n, K) constitute its projective semilinear group PΓL(n + 1, K). (iv) When K contains the finite field Fq , the mapping Φ : Fq → Fq , x 7→ xq ,

is the Frobenius automorphism. The n-th Frobenius automorphism of K is n the map Φn that takes x ∈ K to xq ∈ K. Then, Fqn consists of all elements in K which are fixed by Φn . The Frobenius collineation associated to the Frobenius automorphism is the collineation of PG(n, K) with σ = Φ; that is, x 7→ x′ ,

t ′

x = A t Φ(x),

with Φ(x) = (xq0 , xq1 , . . . , xqn ), for some non-singular matrix A. When K = Fq , it is customary to replace K by q in the notation for all the spaces and groups; so V (n, q) means V (n, K), and similarly for AG(r, q), GL(r, q), AΓL(r, q), PG(r, q), PGL(r, q), PΓL(r, q).

1.2 POLYNOMIALS D EFINITION 1.3 (i) A polynomial f in the ring K[X1 , X2 , . . . , Xn ] of polynomials in the indeterminates X1 , X2 , . . . , Xn is reducible if there exist nonconstant f1 , f2 in K[X1 , X2 , . . . , Xn ] with f = f1 f2 ; otherwise, f is irreducible.

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(ii) The degree of a monomial X1r1 X2r2 . . . Xnrn is r1 + r2 + · · · + rn . (iii) A polynomial is homogeneous if all its terms have the same degree. (iv) The degree of a polynomial f is the largest degree of all its terms; write deg f .

1.3 AFFINE PLANE CURVES In the first instance, a curve is associated both to a set of points and to a polynomial. Let K be an algebraically closed field, and let F ∈ K[X, Y ]. Then an affine curve is viewed as a set of points. D EFINITION 1.4

(i) The plane affine curve

F = va (F ) = {P = (x, y) ∈ AG(2, K) | F (x, y) = 0}. (ii) The degree of F, written deg F, is deg F . Any affine transformation sends an affine curve to another having the same degree. Therefore deg F of an affine curve F is an affine invariant. D EFINITION 1.5 (i) A component of the affine curve F = va (F ) is an affine curve G = va (G) such that G divides F . (ii) The affine curve F = va (F ) is irreducible when it has no proper component, that is, when F is irreducible. Components are covariant, that is, the diagram below is commutative for any affine transformation T of AG(2, K). F = v(F ) component of F

T-

F ′ = va (F ′ )

component of F ′

? G = va (G)

T-

? G ′ = vα (G′ )

Any line containing at least n + 1 points from an affine curve F of degree n is a component of F. To show this, ℓ = va (Y ) may be assumed by covariance. Let F = va (F (X, Y )). Then |ℓ ∩ va (F )| ≥ n + 1 implies that F (X, 0) has more than n roots. Therefore F (X, 0) = 0, and hence X divides F (X, Y ). Let F = va (F ) be an affine curve with deg F = d, and let ℓ = bX −aY +c be a line containing the point P0 = (x0 , y0 ) on F. Then, for any point P = (x, y) ∈ ℓ, bx − ay = bx0 − ay0 ,

b(x − x0 ) = a(y − y0 ) = abt, x = x0 + at, y = y0 + bt

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for some t ∈ K. Then

F (x, y) = F (x0 + at, y0 + bt) = G(t) = G0 + G1 t + G2 t2 + · · · + Gd td = Gm tm + · · · + Gd td , (1.1)

with Gm 6= 0, Gd 6= 0.

L EMMA 1.6 The two irreducible curves F1 = va (F1 ) and F2 = va (F2 ) are the same if and only if F2 = λF1 for some λ ∈ K\{0}. Proof. This is a consequence of Theorem 2.10.

2

D EFINITION 1.7 If F ∈ K[X, Y ] satisfies

F = F1n1 F1n2 · · · Fsns

with each Fi irreducible, then F = va (F ) has components Fi = va (Fi ) with multiplicity ni for i = 1, . . . , s. The multiplicity of a component is an affine invariant. D EFINITION 1.8 Let ℓ be a line which is not a component of F. (i) The integer m of (1.1) is the intersection number of ℓ and F at P0 : write m = I(P0 , ℓ ∩ F);

(ii) if m = 1 for some line ℓ through P0 , then P0 is a simple or non-singular point of F; (iii) if m ≥ 2 for all lines ℓ through P0 , then P0 is a singular or multiple point of F; (iv) if m0 = min{m | ℓ a line through P0 }, then m0 is the multiplicity of P0 on F, or P0 is an m0 -fold point of F, and write m0 = mP0 (F) = mP0 (F );

(v) if m > m0 for a line ℓ, then ℓ is a tangent to F at P0 . The intersection number and the multiplicity of a point are affine invariants. D EFINITION 1.9 If mP (F) = 2, then P is a double point of F. A double point P with two distinct tangents to F at P is a node, and with only one tangent to F at P is a cusp. If mP (F) = 3, then P is a triple point of F. R EMARK 1.10 Let M be a subfield of K and suppose that F is defined over M , that is, F = v(f (X, Y )) with f (X, Y ) ∈ M [X, Y ]. If P is a double point with two distinct tangents, neither of them defined over M , then P is an isolated double point over M . L EMMA 1.11 If P0 is a simple point of F, then, in (1.1), ∂F ∂F G1 = a + b. ∂X ∂Y P0

P0

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C OROLLARY 1.12 The tangent to F at a simple point P = (x, y) is ∂F ∂F (X − x) + (Y − y). ℓP = ∂X P ∂Y P

Note the meaning of this corollary: the line ℓP has intersection multiplicity at least 2 with F at P . D EFINITION 1.13 A non-singular point P of F is a point of inflexion of F if I(P, ℓP ∩ F) ≥ 3. Here, P is also called an inflexion or, in some sources, a flex; the tangent ℓP at P is the inflexional tangent. Tangents and inflexional tangents are covariant. R EMARK 1.14 The behaviour of P = (0, 0) for an affine curve F = va (F ) follows simply from the form of F . Write F (X, Y ) = Fm + Fm+1 + · · · + Fd , where Fi is homogeneous of degree i in X and Y , and Fm 6= 0. Then (i) if m > 0, the point P lies on F; (ii) if m = 1, the point P is simple and F1 is the tangent at P ; Q (iii) if m ≥ 2, the term Fm = ℓi , where ℓ1 , . . . , ℓm are the tangents at P ; (iv) if ℓ1 , . . . , ℓm are distinct, then P is an ordinary multiple point.

Y − X3

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Figure 1.2 Plane cubics

E XAMPLE 1.15 (i) If F = Y − X 3 , then F = va (F ) has no singular points but (0, 0) is an inflexion.

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(ii) If F = Y 2 − X 3 , then F = va (F ) has a singular point (0, 0) of multiplicity 2 with the repeated tangent Y ; so, it is a cusp. See Figure 1.2. D EFINITION 1.16 An affine curve F = va (f (X, Y )) is rational if there exist rational functions a(T )/c(T ) and b(T )/c(T ) with a(T ), b(T ), c(T ) ∈ K[T ] such that f (a(T )/c(T ), b(T )/c(T )) = 0. Lines and conics, that is, curves of degree 1 and irreducible curves of degree 2, are rational curves.

1.4 PROJECTIVE PLANE CURVES Let K be an algebraically closed field. For any polynomial F ∈ K[X, Y ] of degree d, associate the homogeneous polynomial F ∗ ∈ K[X0 , X1 , X2 ], given by X = X1 /X0 ,

Y = X2 /X0 ,

F ∗ (X0 , X1 , X2 ) = X0d F (X1 /X0 , X2 /X0 ).

Similarly, for any homogeneous polynomial F ∈ K[X0 , X1 , X2 ], associate the polynomial F∗ ∈ K[X, Y ], given by F∗ (X, Y ) = F (1, X, Y ). D EFINITION 1.17 Given F ∈ K[X, Y ], the projective plane curve of affine equation F (X, Y ) = 0, or homogeneous equation F ∗ (X0 , X1 , X2 ) = 0, is F = v(F ) = v(F ∗ ) = {(x0 , x1 , x2 ) ∈ PG(2, K) | F ∗ (x0 , x1 , x2 ) = 0}. This definition implies that the projective curve consists of the affine points plus the points at infinity; that is, v(F ) = {(1, x, y) | (x, y) ∈ va (F )}∪{(0, x, y) = (0, λx, λy) | F ∗ (0, x, y) = 0}. R EMARK 1.18 (i) For a linear form L = a0 X0 + a1 X1 + a2 X2 , the corresponding line is indicated both by L and v(L). (ii) The notions of irreducibility, component, and degree extend in a natural way from affine curves to projective curves. It also follows, for any projective curve F = v(F (X, Y )) not containing the line v(X0 ) as a component, that F is irreducible if and only if the affine curve F ′ = va (F (X, Y )) is irreducible. D EFINITION 1.19 singular if

(i) With F homogeneous, a point P = (x0 , x1 , x2 ) of F is ∂F ∂F ∂F = = =0 ∂X0 ∂X1 ∂X2

at P .

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(ii) Otherwise, P is simple and the tangent at P is ∂F ∂F ∂F X0 + X1 + X2 . ∂X0 P ∂X1 P ∂X2 P Throughout, write

X∞ = (0, 1, 0),

Y∞ = (0, 0, 1),

O = (1, 0, 0),

E = (1, 1, 1).

Here, X∞ is the point at infinity on the X-axis, Y∞ is the point at infinity on the Y -axis, and O is the origin. In Example 1.15 (i), Y∞ is a cusp on F and F ∗ = Y Z 2 − X 3 ; in (2), Y∞ is an inflexion and F ∗ = Y 2 Z − X 3 . So, the curves are projectively equivalent. In general, to determine the behaviour of a point, translate it to the origin, and use Remark 1.14. Sometimes, it is convenient to use the notation, U0 = (1, 0, 0),

U1 = (0, 1, 0),

U2 = (0, 0, 1),

U = (1, 1, 1).

R EMARK 1.20 The properties of projective curves such as the degree, multiplicity of singular points, multiplicity of contact of a tangent are covariant, that is, invariant under projective transformations. Let F = v(F (X0 , X1 , X2 )) be a projective plane curve of degree d. D EFINITION 1.21 For any point P = (x0 , x1 , x2 ), if G(X0 , X1 , X2 ) =

∂F ∂F ∂F x0 + x1 + x2 ∂X0 ∂X1 ∂X2

(1.2)

is not the zero polynomial, the plane curve G = v(G(X0 , X1 , X2 )) of degree d − 1 is the polar curve of P with respect to F. Otherwise, the polar curve of P is vanishing. T HEOREM 1.22 The polar curve is covariant. Proof. If (X0′ , X1′ , X2′ ) is a new homogeneous coordinate system, the link between P2 the old frame and the new one is given by a linear substitution Xi = j=0 aij Xj′ , i = 0, 1, 2, such that the matrix (aij ) is non-singular. Under this change, the image of F is the curve F ′ = v(F ′ ) with P2 P2 P2 F ′ (X0′ , X1′ , X2′ ) = F ( j=0 a0j Xj′ , j=0 a1j Xj′ , j=0 a2j Xj′ ), P2 P2 P2 while the image of P is the point P ′ = ( j=0 a0j xj , j=0 a1j xj , j=0 a2j xj ). Covariance means that the diagram below is commutative, which is now shown. F = v(F ) polar of P ? G = v(G)

(aij )−1

- F ′ = v(F ′ )

polar of P ′ ? - G ′ = v(G′ )

(aij )−1

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By the usual rules of the derivatives of composite functions, 2

Therefore

X ∂F ′ ∂F aik , = ∂Xk′ ∂X i i=0

k = 0, 1, 2.

2 2 X X ∂F ∂F ′ ′ xi . x = ∂Xk′ k i=0 ∂Xi

k=0

2

T HEOREM 1.23 Suppose that the polar curve F ′ of P with respect to F is not vanishing. (i) If A is a non-singular point of F, then A ∈ F ′ if and only if the tangent to F at A passes through P . (ii) Let A be an r-fold point of F, with r > 1. Then (a) A is at least an (r − 1)-fold point of F ′ ;

(b) if there are infinitely many points P for which A is at least an r-fold point of F ′ , every tangent to F at A has multiplicity divisible by p. Proof. Let A = (a0 , a1 , a2 ). The tangent line to F at A is ∂F ∂F ∂F X0 + X1 + X2 , ∂X0 ∂X1 ∂X2

where the partial derivatives are evaluated at (a0 , a1 , a2 ). Thus, (i) is a consequence of the definition of a polar curve. To show (ii), note that the covariance of the polar curve allows A to be mapped to the origin. Write F = Φ0 X0d + Φ1 X0d−1 + · · · + Φi X0d−i + · · · + Φd , where Φi is a homogeneous polynomial in X1 and X2 of degree i. Since A is an r-fold point, Φ0 = . . . = Φr−1 = 0, but Φr 6= 0. Then, with G as in (1.2), G = X0d−r G1 + G2 ,

where G1 =

∂Φr ∂Φr x1 + x2 , ∂X1 ∂X2

and where the terms in G2 have degree at least r in X1 and X2 . Since G1 has degree at least r − 1, so (a) follows. If there are infinitely many points P for which A is an s-fold point of F ′ with s ≥ r, then both ∂Φr /∂X1 and ∂Φr /∂X2 vanish. This can occur only when Φr is a p-th power of some polynomial Ψr . But then every tangent to F at A has multiplicity divisible by p. 2 T HEOREM 1.24 If F is irreducible of degree d, then there is at most one point P with vanishing polar curve F ′ .

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Proof. Assume, on the contrary, that the polar curves of two distinct points are vanishing. By the covariance of polar curves, let these points be X∞ and Y∞ . Then ∂F/∂X1 and ∂F/∂X2 are both zero. Hence the general term in F (X0 , X1 , X2 ) is d−(i+j)

aij X0

X1i X2j

with both i and j divisible by p. Then p divides d, as otherwise X0 divides F , contradicting the irreducibility of F. So, p also divides d − (i + j). Now, define the coefficients bij by bpij = aij , and the polynomial, P (d−(i+j))/p i/p j/p L= bij X0 X1 X2 . Then, F = Lp and F is reducible, a contradiction.

2

Points with vanishing polar curves are characterised by a purely geometric property. T HEOREM 1.25 If F is irreducible, then the polar curve of P is vanishing if and only if the tangents to F at non-singular points pass through P . Proof. By the covariance, suppose that P is the point Y∞ . With F in its inhomoP geneous form F (X, Y ) = aij X i Y j , the polar curve of P is vanishing if and only if aij = 0 whenever j 6≡ 0 (mod p). This occurs if and only if there is no X2 -term in the form of the tangent to F at any non-singular point. 2 This condition can be weakened. T HEOREM 1.26 If F is irreducible, then the polar curve of P is vanishing if and only if infinitely many tangents to F pass through P .

Proof. If the polar curve F ′ of P is vanishing, the assertion follows from Theorem 1.25. If F is not vanishing, then the weak form of B´ezout’s Theorem 3.13 applied to F and F ′ implies that there are finitely many common points. Since F has a finite number of singular points, P lies only on finitely many tangents to F. 2

As becomes apparent later, vanishing polar curves can cause serious difficulties in certain situations, especially when the point lies on the curve. Here, some examples are given after formalising these concepts. D EFINITION 1.27 (i) A point is the nucleus of a projective irreducible curve F if it is the common point of all tangents to F at non-singular points. (ii) A curve F is strange if it admits a nucleus N . Following this definition, Theorem 1.25 states that a point P is a nucleus of F if and only if the polar curve of P with respect to F is vanishing. Theorem 1.24 has the following corollary. T HEOREM 1.28 An irreducible curve has at most one nucleus. The simplest example of a strange curve is an irreducible conic in characteristic 2. Note that the nucleus does not lie on the conic.

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E XAMPLE 1.29 An example of a strange curve F whose nucleus lies on the curve is the following. Let p = 2 and q = 2r . For an integer k < q with gcd(k, q−1) = 1, let k = 2s u with 1 < s < r, u > 1 and u odd. Let F be the irreducible curve v(f (X, Y )), where (X k + 1)(Y + 1) + (Y k + 1)(X + 1) = f (X, Y )(X + 1)(Y + 1)(X + Y ).

(1.3)

To show that N = (1, 1, 1) is a nucleus of F, write (1.3) as follows: s

s

s

(X u )2 Y + (Y u )2 X + (X u + Y u )2 + X + Y = f (X, Y )(X + 1)(Y + 1)(X + Y ).

(1.4)

Using this form, the tangent to F at a non-singular point A = (a0 , a1 , a2 ) of F is (ak1 + ak2 )X0 + (ak0 + ak2 )X1 + (ak0 + ak1 )X2 .

This shows that the tangent passes through the point N . E XAMPLE 1.30 An example in odd characteristic is the following. Let p > 2, and F = v(Y m − g(X)) with m 6≡ 0 (mod p). Then F is strange if and only if one of the following hold: (i) g(X) = f (X p ) + cX, where f ∈ K[X] and c ∈ K with c = 0 for m 6= 1; (ii) g(X) = (X + a)r f (X p ) where a ∈ K, f ∈ K[X] and 1 ≤ r < p with r ≡ m (mod p). Strange curves have exceptional behaviour with respect to duality; this is treated in Section 5.11. They also cause difficulties in the resolution of singularities; see the last part of the proof of Theorem 3.27.

1.5 THE HESSIAN CURVE Let F = v(F (X0 , X1 , X2 )) be a projective curve of degree d. Write Fi =

∂F , ∂Xi

Fij =

∂2F . ∂Xi ∂Xj

D EFINITION 1.31 If the determinant,

F00 H(X0 , X1 , X2 ) = F01 F02

F01 F11 F12

F02 F12 F22

(1.5) ,

is not vanishing, then the projective curve H = v(H(X0 , X1 , X2 )) is the Hessian curve of F; it has degree 3(d − 2). Otherwise, the Hessian curve is vanishing. Strange curves have vanishing Hessian; see Exercise 3. Other relevant examples of curves with vanishing Hessian are the Hermitian curves. T HEOREM 1.32 The Hessian curve is covariant.

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Proof. Arguing as in the proof of Theorem 1.22, for 0 ≤ i, j ≤ 2, 2 X 2 X ∂2F ′ ∂2F = a a . ki mj ∂Xi′ ∂Xj′ ∂Xk ∂Xm m=0 k=0

This shows that H ′ (X0′ , X1′ , X2′ ) = H(X0 , X1 , X2 ) · det(aij )2 . Since det(aij ) 6= 0, the assertion follows.

2

T HEOREM 1.33 If d ≡ 1 (mod p), then the Hessian curve is vanishing. Proof. Let G ∈ K[X0 , X1 , X2 ] be a homogeneous polynomial of degree m. By Euler’s formula, ∂G ∂G ∂G X0 + X1 + X2 = mG. ∂X0 ∂X1 ∂X2 When G = F , this becomes ∂F ∂F ∂F X0 + X1 + X2 = d F, ∂X0 ∂X1 ∂X2

(1.6)

while for G = ∂F/∂Xi with i = 0, 1, 2, ∂2F ∂2F ∂F ∂2F X0 + X1 + X2 = (d − 1) . ∂Xi ∂X0 ∂Xi ∂X1 ∂Xi ∂X2 ∂Xi Then, with the notation of (1.5),

(d − 1)F0 X0 H(X0 , X1 , X2 ) = (d − 1)F1 (d − 1)F2

F01 F11 F12

Therefore H(X0 , X1 , X2 ) = 0 when d ≡ 1 (mod p).

F02 F12 F22

(1.7)

.

2

The fundamental property of the Hessian curve is stated in the following theorem. T HEOREM 1.34 Assume that H is not vanishing. (i) Let p 6= 2. Then a non-singular point of F is an inflexion if and only if it is a common point of F and H. (ii) Every singular point of F lies on H. Proof. Let P be a non-singular point of F. Again by the covariance, suppose that P is the origin and that the tangent line ℓ to F at P is the X-axis. Then F (X0 , X1 , X2 ) = X0d−1 X2 + X0d−2 (a20 X12 + a11 X1 X2 + a02 X22 ) + · · · ,

where the other terms are powers of X0 with exponent at most d − 3. A straightforward calculation shows that 3(d−2)

H(X0 , X1 , X2 ) = −2(d − 1)2 a20 X0

+ ··· ,

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FUNDAMENTAL IDEAS

where the other terms are powers of X0 with exponent less than 3(d − 2). Since p 6= 2 and d 6≡ 1 (mod p) by Theorem 1.33, H passes through P if and only if a20 = 0; that is, P an inflexion. If P is singular, then the above computation shows that no term containing only X0 appears in H(X0 , X1 , X2 ). Hence P ∈ H. 2 In affine form, when F = v(f (X, Y )), write fX =

∂f ∂2f ∂f ∂2f ∂2f , fXY = . , fY = , fXX = , fY Y = 2 ∂X ∂Y ∂X ∂X∂Y ∂Y 2

L EMMA 1.35 Let d 6≡ 1 (mod p). If F = v(f (X, Y )), then H = v(h(X, Y )), where 2

h(X, Y ) = (fX ) fY Y + (fY )2 fXX − 2fX fY fXY


−d(d − 1)−1 fXX fY Y − (fXY )2 f.

Proof. As in the proof of Theorem 1.33, equations (1.6) and (1.7) are used. Substituting in H(X0 , X1 , X2 ), the expression reduces to the following: (d − 1)d F F1 F2 F11 F12 . X02 H(X0 , X1 , X2 ) = (d − 1)2 F1 (d − 1)2 F2 F12 F22 In inhomogeneous coordinates, this becomes the determinant (d − 1)d f fX fY fXX fXY h(X, Y ) = (d − 1)2 fX (d − 1)2 fY fXY fY Y

.

Expansion of the determinant and division by −(d − 1)2 give the result.

2

When studying the inflexion points of F or, more generally, the intersection of H and F, the last term of the polynomial in Theorem 1.35 can be omitted. So, it is also possible to define the Hessian curve as H′ = v(h′ (X, Y )), with 2

h′ (X, Y ) = (fX ) fY Y + (fY )2 fXX − 2fX fY fXY .

(1.8)

An advantage is that the Hessian curve in this form is no longer necessarily vanishing when F has degree d ≡ 1 (mod p); see Exercise 2. Therefore curves with such degrees can be considered in the study of inflexion points. With this approach, just two remarks are needed. (1) Theorem 1.32 holds true if H is given by (1.8). (2) Rewording the proof of Theorem 1.34 in terms of inhomogeneous coordinates, h(X, Y ) = −2a20 + · · · , which shows that condition d 6≡ 1 (mod p) depending on Theorem 1.33 disappears. Therefore the following result is established. T HEOREM 1.36 Let p 6= 2. If either (1.8) is identically zero or the Hessian H of F as in (1.8) contains all points of F, then every non-singular point of F is an inflexion. The converse also holds.

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R EMARK 1.37 The form (1.8) of the Hessian curve is inadequate to deal with the even characteristic case. What is actually needed is to modifyPthe definition of the second partial derivatives, as suggested by Hasse. If f = aij X i Y j then the Hasse second partial derivatives are as follows:     ∂ (2) f (X, Y ) X ∂ (2) f (X, Y ) X i j i−2 j X Y , X i Y j−2 . = aij = aij (2) (2) 2 2 ∂X ∂Y Put f = f (X, Y ) for brevity. If p 6= 2, then

∂2f ∂2f ∂ (2) f ∂ (2) f , . = 2 = 2 ∂X 2 ∂Y 2 ∂X (2) ∂Y (2) Now, the Hessian curve H = v(e h), with inhomogeneous form is defined to be the curve with 2 (2) 2 (2)   ∂ f ∂ f ∂f ∂f ∂f ∂ 2 f ∂f e . (1.9) h(X, Y ) = + − ∂X ∂Y ∂Y (2) ∂X (2) ∂X ∂Y ∂X∂Y

With this definition, H is still covariant, and Theorem 1.34 holds true for p = 2.

R EMARK 1.38 The vanishing of the Hessian is related to exceptional behaviour of the dual curve in positive characteristic. This is treated in Section 5.11. The Hasse derivatives are useful in several other contexts; see Sections 5.10 and 7.6. E XAMPLE 1.39 Let n be a positive integer which is not divisible by p. The curve F = v(X0n + X1n + X2n ) is the Fermat curve of degree n. When K = Fq with q a power of p and n = q + 1, then the Fermat curve is the Hermitian curve and denoted by Hq ; that is, Hq = v(X0q+1 + X1q+1 + X2q+1 ).

The properties of the Hermitian curve are further developed in Section 12.3. The Fermat curve is the set of its inflexion points if and only if (a) n ≡ 1 (mod p) for p 6= 2 or (b) n ≡ 1 (mod 22 ) when p = 2. In particular, this holds for the Hermitian curve Hq when p 6= 2. E XAMPLE 1.40 Let p 6= 0. Choose a power q of p such that q ≡ −1 (mod 3). Let F = v(F ) be the plane curve with homogeneous form, F (X0 , X1 , X2 ) = X0q X2 + X1q X0 + X2q X1 − 3(X0 X1 X2 )(q+1)/3 .

It is first shown that F is an irreducible curve with only ordinary singularities. Each of the fundamental lines meets F in only two points. If G is a component of F then G meets each fundamental line in at least one point, and so at least two of the vertices of the fundamental triangle are on G. If G1 is another component of F, then it also passes through two of the vertices of the fundamental triangle. So, one of these points is common to G and G1 , and hence is a singular point of F. However, this is not the case, since a first partial derivative of F is not zero at each of these points. More precisely, FX0 (0, 1, 0) = FX1 (0, 0, 1) = FX2 (1, 0, 0) = 1 6= 0.

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FUNDAMENTAL IDEAS

Next, it is shown that F has exactly (q 2 − q + 1)/3 singular points, each of which is an ordinary double point. This requires some typical computations involving polynomials and their partial derivatives. Since no vertex of the fundamental triangle is a singular point of F, it suffices to consider affine points. So, put X = X1 /X0 , Y = X2 /X0 ; then F = v(f (X, Y )) with f (X, Y ) = Y + X q + XY q − 3(XY )(q+1)/3 . Let A = (a, b) be a singular point of F. Then ab 6= 0 and fX (a, b) = fY (a, b) = 0. A direct calculation shows that this implies that 1 = a(q−2)/3 b(1−2q)/3 , 1=a (q−2)/3 (1−2q)/3

Thus, a b plies the following:

=a b(q

(q+1)/3 (q−2)/3

b

(q+1)/3 (q−2)/3

b

2

−q+1)/3

= 1,

(1.10)

.

(1.11)

, whence a = b

1−q

. Now, (1.11) im-

2

a = b−q .

Conversely, if a, b ∈ K satisfy these conditions, then fX (a, b) = fY (a, b) = 0, 2 2 showing that every point A = (a, b) such that b(q −q+1)/3 = 1, a = b1−q = b−q is a singular point of F. Also, fXX (a, b) = fY Y (a, b) = fXY (a, b) =

2 (q−5)/3 (q+1)/3 b , 3 a 2 (q+1)/3 (q−5)/3 b , 3 a 1 (q−2)/3 , 3 (ab)

fXX (a, b)fY Y (a, b) − fXY (a, b)2 =

1 3

(ab)(2q−4)/3 .

It follows that, if p 6= 2, then A is a double point, more precisely, a node. The same holds true for p = 2, but the proof requires Hasse partial derivatives, as explained in Remark 1.37. Finally, the Hessian H of F is v((X0 X1 X2 )(2q−4)/3 ), which shows that H splits into three linear components, each counted (2q − 4)/3 times. E XAMPLE 1.41 For a divisor k ≥ 2 of q − 1, and u, v ∈ K with uv 6= 1, the plane curve F = v(vX k Y k − X k − Y k + u) is irreducible and has the following properties. (i) F has two singular points, namely X∞ and Y∞ , both ordinary k-fold points. (ii) The Hessian H = v(H) of F has

H = k 3 X k−2 Y k−2 (X k u − 1)(Y k u − 1)(2X k Y k u + (k − 1)(X k + Y k )).

In particular, F is not the locus of its inflexions. (iii) For each a ∈ K with ak = u, the points A1 = (1, 0, a) and A2 = (1, a, 0) of F are inflexions. For i = 1, 2, the tangent to F at Ai is ℓi = v(Xi − aX0 ). Also, I(Ai , F ∩ ℓi ) = k.

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1.6 PROJECTIVE VARIETIES IN HIGHER-DIMENSIONAL SPACES D EFINITION 1.42 (i) Given a homogeneous F ∈ K[X0 , X1 , . . . , Xn ], the projective hypersurface F = v(F ) = {(x0 , x1 , . . . , xn ) ∈ PG(n, K) | F (x0 , x1 , . . . , xn ) = 0}. When deg F = 1, the hypersurface is a hyperplane. (ii) Given homogeneous F1 , . . . , Fr ∈ K[X0 , X1 , . . . , Xn ], the projective variety F = v(F1 , . . . , Fr ) = {x = (x0 , x1 , . . . , xn ) ∈ PG(n, K) | F1 (x) = . . . = Fr (x) = 0} .

When n = 2, a projective hypersurface is just a projective plane curve. Varieties can also be curves in a higher-dimensional space; this is explained in Section 7.17.

1.7 EXERCISES 1. Let p > 2 and let C = v(F ), where

F (X0 , X1 , X2 ) = X0q+1 + X1q+1 + X2q+1 + X0q−1 X1 X2 .

Show that the Hessian curve H of C in its inhomogeneous form (1.8) is nonvanishing. 2. Let K have characteristic 2. Show that the Hessian curve H of the Hermitian curve H2 in its inhomogeneous form (1.9) is non-vanishing. 3. If p > 3, show that the Hessian of a strange curve vanishes. 4. Prove that a plane curve F of degree k ≥ 2 is irreducible if it has both the following properties. (a) F contains a point A such that I(A, F ∩ ℓ) = k for the tangent ℓ of F at A when A is a simple point, and for at least one tangent ℓ to F at A when A is a singular point. In the latter case, ℓ is required to have multiplicity 1. (b) F contains no linear component through the point A. 5. Prove that an irreducible plane cubic curve is rational if and only if it is singular. 6. Prove that an irreducible plane curve of degree n with an (n − 1)-fold point is rational. 7. Prove that any plane curve of degree at least 3 whose tangent lines at collinear points are concurrent is either strange or projectively equivalent to the Hermitian curve.

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8. With F as in Example 1.29, show that the necessary and sufficient condition for N to lie on F is that s > 1. Also, show that the singular points of F are as follows: (a) N is an ordinary (2s − 2)- fold point with tangents, (m + 1)X0 + mX1 + X2 ,

2s −1

(b) (c) (d) (e)

for m = 1 but m 6= 1; for each b with bu = 1 but b 6= 1, the point P = (1, b, 1) is a (2s − 1)fold point with a single tangent; for each c with cu = 1 but c 6= 1, the point P = (1, 1, c) is a (2s − 1)fold point with a single tangent; for all b, c with bu = cu = 1 but b 6= 1, c 6= 1, b 6= c, the point P = (1, b, c) is a 2s -fold point with a single tangent; for b with bu = 1 but b 6= 1, the point P = (1, b, b) is a (2s − 1)-fold point with a single tangent.

1.8 NOTES There are many works on the elements of algebraic curves: Abhyankar [8], Baker [26], Bertini [49], Coolidge [85], Enriquez [122], Fischer [128], Fulton [135], Lefschetz [301], Hartshorne [193], Hilton [212], Kirwan [269], Reid [368], Salmon [380], Segre [399], Seidenberg [400], Semple and Kneebone [402], Semple and Roth [403], Severi [408], Walker [497]. For the historical development of algebraic geometry, see Coolidge [86] and Dieudonn´e [104, 105]. Example 1.30 comes from [230]. For Exercise 7, see [236]. For strange curves, as in Section 1.4, see Hartshorne [192, Chapter IV]. In [37], strange curves invariant under a cyclic projective group fixing a triangle are investigated. A survey paper on Fermat curves in positive characteristic is [429]. In the classical literature, polar curves and their generalisations, the m-ic polar curves, together with Hessian curves, are used to establish the Pl¨ucker formulas and their generalisations. These are equations linking numerical parameters of curves, such as the number of nodes, cusps, inflexion points, singularities of higher multiplicities, and other constants; see [85, Chapter 9]. Polar curves in positive characteristic are studied in [202]; see also [203]. Hessian curves are also important in investigating the number of points of curves defined over a finite field. For instance, let F = v(F ) be a projective irreducible curve of degree d, where F ∈ Fq [X0 , X1 , X2 ]. If its Hessian curve is nonvanishing, then the number of points of F lying in PG(2, q) is bounded above √ by 12 d(q + d − 1); see Theorem 8.41. For q square and d = q + 1, this upper 1√ √ bound is 2 q( q + 1)2 . On the other hand, the Hermitian curve H√q has vanishing Hessian and has √ q q + 1 points lying in PG(2, q); see Example 8.67. Also, the polar curve of any

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CHAPTER 1

√ point with respect to H√q is reducible, being a line counted q times. See Chapter 12 for more examples of plane curves in positive characteristic with properties that a complex algebraic curve cannot have.

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Chapter Two Elimination theory The intersection number I(P, ℓ ∩ F) of an algebraic plane curve F and a line ℓ at a point P was used in Chapter 1 for the study of local projective invariants, such as singularities and inflexions. In this and subsequent chapters, a natural extension of I(P, ℓ ∩ F) to the intersection number I(P, G ∩ F) of F and another algebraic plane curve G at P is investigated. The main outcome, B´ezout’s Theorem 3.14, is an equation linking the degrees of F and G to the intersection numbers at their common points. Consider F = v(f (X, Y )), ℓ = v(uY − X) and P = (0, 0), such that ℓ is not a component of F. In this special situation, the intersection number I(P, ℓ ∩ F) is obtained by eliminating X from the system, f (X, Y ) = 0, uY − X = 0,

that is, I(P, ℓ ∩ F) = m if D(Y ) = F (uY, Y ) = Dm Y m + · · · + Dd Y d with Dm 6= 0. When ℓ is replaced by G = v(g(X, Y )), the elimination of X or Y is no longer straightforward; the technical tool to deal with it is the resultant, treated in Sections 2.1 and 2.2. From Section 2.3, the polynomial D(Y ) obtained by elimination of X in the system, f = 0, g = 0, is the resultant RX (f, g) of f and g. Therefore the natural definition of the intersection number of F and G at P = (0, 0) is the degree of the smallest non-zero term in D(Y ). That this definition is satisfactory from a geometric point of view is shown in Section 3.1.

2.1 ELIMINATION OF ONE UNKNOWN The resultant decides whether or not two given polynomials have a root in common. Two polynomials f (X) and g(X) with coefficients in a field L, not necessarily algebraically closed, have a root in common in L or in its closure L if and only if they have a factor in common. Consider the two polynomials: f (X) = a0 X n + a1 X n−1 + · · · + an−1 X + an , g(X) = b0 X s + b1 X s−1 + · · · + bs−1 X + bs .

(2.1)

Let α1 , . . . , αn be the roots of f (X) and let β1 , . . . , βs be the roots of g(X); they may lie in L if L is not algebraically closed.

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CHAPTER 2

D EFINITION 2.1 The resultant of the polynomials f (X) and g(X) is Qn Qs R(f, g) = (a0 s b0 n ) i=1 j=1 (αi − βj ).

It is a symmetric polynomial in the two sets of indeterminates αi , βj .

L EMMA 2.2 The resultant R(f, g) is zero if and only if there exists a common root of the polynomials f (X) and g(X). L EMMA 2.3 The following properties hold: Qn (i) R(f, g) = a0 s i=1 g(αi ); Qs (ii) R(f, g) = b0 n j=1 f (βj ); (iii) R(f, g) = (−1)ns R(g, f );

(iv) R(f1 f2 , g) = R(f1 , g) · R(f2 , g). Proof. (i) By definition, R(f, g) = a0 s b0 n Therefore

Qn

i=1

Qs

j=1

(αi − βj ).

g(X) = b0 (X − β1 ) · · · · · (X − βs ) = b0 Now, substitute in place of X the value αi :

Qs

g(αi ) = b0 (αi − β1 ) · · · · · (αi − βs ) = b0 Then Qn

i=1

So

j=1

Qs

j=1

(X − βj ). (αi − βj ).

g(αi ) = b0 (α1 − β1 ) · · · · · (α1 − βs )b0 (α2 − β1 ) · · · · · (α2 − βs ) · · · · b0 (αn − β1 ) · · · · · (αn − βs ) Qs n Qn = b0 i=1 j=1 (αi − βj ). Qn

i=1

Multiplying both sides of

g(αi ) = bn0

Qn

i=1

Qs

j=1

(αi − βj ).

this equation by as0 and using Q R(f, g) = as0 ni=1 g(αi ).

Definition 2.1 shows that

(ii) The proof of this is analogous. (iii) This follows immediately. (iv) Let deg g = s, deg(f1 f2 ) = m + n, with m = deg f1 and n = deg f2 . The leading coefficient of f1 f2 is a10 a20 , where a10 and a20 are the leading coefficients of f1 and f2 . The roots of f1 f2 are given by x1 , . . . , xn and y1 , . . . , ym , which are just the roots of f1 and f2 ; write these roots as αi for i = 1, . . . , m + n. Then Qm+n R(f1 f2 , g) = (a10 a20 )s i=1 g(αi ) Qn Qn+m = as10 i=1 g(αi ) · as20 i=n+1 g(αi ) = R(f1 , g) · R(f2 , g). 2

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ELIMINATION THEORY

C OROLLARY 2.4 The product rule for resultants is as follows: Qn Qn R ( i=1 fi , g) = i=1 R(fi , g).

The resultant R(f, g) can be expressed as a polynomial in the indeterminates a0 , . . . , an , b0 . . . , bs . To show this, the theory of symmetric polynomials is used. From Definition 2.1, Q Q R(α1 , . . . , αn , β1 , . . . , βs ) = (a0 s b0 n )−1 R(f, g) = ni=1 sj=1 (αi − βj )

is a symmetric polynomial with respect to both the set {α1 , . . . , αn } and the set {β1 , . . . , βs }. Let σ1 , . . . , σn be the elementary symmetric polynomials in the indeterminates α1 , . . . , αn and let τ1 , . . . , τs be the polynomials in β1 , . . . , βs . By Newton’s theorem on symmetric functions, R(α1 , . . . βs ) can be expressed in terms of the σi and τj . Up to sign, σ1 , . . . , σn and τ1 , . . . , τs are the coefficients of the polynomials −1 a−1 0 f (X) and b0 g(X). So R(α1 , . . . , βs ) can be expressed as a polynomial in the indeterminates, a1 an b 1 bs ,..., , ,..., . a0 a0 b 0 b0 The factor a0 s b0 n makes the denominator in R(α1 , . . . , βs ) disappear. Now, it is shown that R(f, g), as given by Definition 2.1, is equal to a determinant D of order n + s. T HEOREM 2.5 Given f (X) = a0 X n + a1 X n−1 + · · · + an−1 X + an , g(X) = b0 X s + b1 X s−1 + · · · + bs−1 X + bs ,

then R(f, g) = D, where a0 a1 . . . 0 a0 a1 .. .. . . 0 0 . . . D = b b 1 0 0 b0 b1 .. . 0 ... 0

an ... 0 ...

0 an a0

0 0 a1 bs

... .. . b0

b1

... ... .. . 0 bs

0

... ... ... .. . ...

an . 0 0 bs 0 0

   

         

s rows (2.2) n rows

Proof. Let α1 , . . . , αn be the roots of f (X) and β1 , . . . , βs the roots Consider another determinant of order n + s: n+s−1 n+s−1 n+s−1 n+s−1 β1 · · · αnn+s−1 1 n+s−2 β2n+s−2 · · · βsn+s−2 αn+s−2 β β2 · · · βs α1 · · · αnn+s−2 1 .. .. .. .. .. . . . . . M = 2 2 2 2 2 · · · α α β β · · · β n 1 1 2 s β β · · · β α · · · α 1 2 s 1 n 1 1 ··· 1 1 ··· 1

of g(X). . (2.3)

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The product (a0 s b0 n )DM is now calculated in two different ways. First, M is a Vandermonde determinant , whose value is given by the following formula: Qs Qn Q Q M = 1≤ i < j ≤s (βi − βj ) j=1 i=1 (βj − αi ) 1≤ i < j ≤n (αi − αj ). Now,

as0 bn0 DM = D as0 bn0

Qs

j=1

= D · R(g, f )

Qn

i=1

Q

(βj − αi ) ·

1≤ i < j ≤s

Q

1≤ i < j ≤s

(βi − βj )

Q

Q (βi − βj ) 1≤ i < j ≤n (αi − αj )

1≤ i < j ≤n

(αi − αj ).

(2.4)

The product of the matrices associated to the determinants D and M is the matrix   B 0 N= , 0 A where



  B= 

β1s−1 f (β1 ) β2s−1 f (β2 ) β1s−2 f (β1 ) β2s−2 f (β2 ) .. .. . .

f (β1 ) f (β2 )  n−1 α1 g(α1 ) · · ·  αn−2 g(α1 ) · · ·  1 A= ..  . ··· g(α1 )

···

··· ··· ··· ···

βss−1 f (βs ) βss−2 f (βs ) .. .

f (βs )  αnn−1 g(αn ) αnn−2 g(αn )   . ..  .



  , 

g(αn ).

Expanding the determinant det N of N by Laplace’s method and taking out column and row factors, and calculating the corresponding Vandermonde determinants gives the following: (a0 s b0 n ) det N Q Q Q Q = (as0 bn0 ) sj=1 f (βj ) ni=1 g(αi ) 1≤k m;  Fm , Gm , if n = m, s = 0, r > 0; Dm = (3.23)  α0 Fm + β0 Gm , if n = m, r = s = 0. Now, the proof can be completed. If mO (H) > n, then (3.22) implies that Gn = X n−m Fm , so that each tangent to F at O is also tangent to G at O. By (3.18), I(O, F ∩ G) ≥ mO (D) · mO (H) > mn.

If mO (H) = n, then mO (D) · mO (H) = mn, and

Hn = −X n−m Fm + Gn 6= 0.

(3.24)

It remains to show that F and G have tangents in common if and only if D and H have; in other words, Hn and Dm have a non-constant divisor if and only if Fm and Gn have. But this follows immediately from the fact that Hn and Dm are linear combinations of Fm and Gn with coefficients in K[X], and vice versa; see (3.24), (3.23), (3.20). So the result is true for mO (H) = n. It remains to show that the assumption that Y divides neither Fm nor Gn is unnecessary. Given a, b, c, d ∈ K with ad − bc 6= 0, define a new function ¯ F¯ ∩ G) ¯ = I(P, F ∩ G), I(P, where F¯ (X, Y ) = F (aX + bY, cX + dY ), ¯ G(X, Y ) = G(aX + bY, cX + dY ). ¯ F¯ ∩ G) ¯ also satisfies the postulates (I 1),. . . ,(I 7). It follows immediately that I(P, Now, choose a, b, c, d ∈ K so that as a sum of homogeneous polynomials, F¯ (X, Y ) = F¯m (X, Y ) + F¯m+1 + . . . , ¯ ¯ n (X, Y ) + G ¯ n+1 + . . . , G(X, Y )=G ¯ n are not divisible by Y . All the same, F¯m and G ¯ n have a the terms F¯m and G ¯ it follows that common divisor cX + dY . By the preceding result applied to I,

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¯ ¯ > mO (F¯ )mO (G). ¯ Since mO (F¯ ) = mO (F ), mO (G) ¯ = mO (G), it I(O, F¯ ∩ G) also follows that I(O, F ∩ G) > mO (F )mO (G), as required. 2 The results so far for I(P, F ∩G) have P = O, the origin. Now they are extended to the case that P 6= O. With P = (x0 , y0 ), the translation, τ : (X, Y ) 7→ (X − x0 , Y − y0 ),

sends P to the origin. The same τ sends the curves F and G to the curves F ′ and G′ , where F ′ (X, Y ) = F (X − x0 , Y − y0 ), G′ (X, Y ) = G(X − x0 , Y − y0 ).

¯ F ∩ G) by Define the function I(P, ¯ F ∩ G) = I(O, F ′ ∩ G′ ). I(P,

¯ F ∩ G) satisfies all of (I 1),. . . , (I 7). ConThen it follows immediately that I(P, versely, if a function I(P, F ∩ G) satisfies these properties for a fixed point P , then ¯ also the function I(O, F ′ ∩ G′ ) = I(P, F ∩ G) enjoys the same properties with respect to the origin. By Propositions 3.4 and 3.6, the following theorems are finally obtained. T HEOREM 3.7 I(P, F ∩ G) ≥ mP (F ) · mP (G), with equality if and only if the curves F and G have no common tangent at P . T HEOREM 3.8 (Uniqueness Theorem) For two curves F and G, there is at most one function I(P, F ∩ G) that satisfies the seven postulates (I 1),. . . , (I 7). The following result completes the theory of the intersection number. T HEOREM 3.9 (Existence Theorem) There exists a function I(P, F ∩ G) that satisfies the seven postulates (I 1),. . . , (I 7). Proof. Suppose that F and G have no common component. let RY (F, G) be their resultant when F and G are considered as polynomials in Y . Then RY (F, G) is a polynomial D(X) ∈ K[X]. Now, choose coordinates so that  (i) Y∞ = (0, 0, 1) is not on F or G;  (ii) Y∞ is not on any line through two distinct points of F ∩ G; (3.25)  (iii) Y∞ is not on the tangent to F or G at any point of F ∩ G.

It is now shown that, for P = (x0 , y0 ) ∈ F ∩ G, Let

I(P, F ∩ G) = max{k ∈ N | (X − x0 )k divides RY (F, G)}. D(X) = c

Q

i

(X − αi )ri ,

c ∈ K;

then, for each point P = (x0 , y0 ) such that F (P ) = G(P ) = 0, the abscissa x0 is one of the roots αi of D(X). In accord with this, put   ∞, if F, G have a common factor H and H(P ) = 0; I(P, F ∩ G) = 0, if F (P ) 6= 0 or G(P ) 6= 0;  ri , if F (P ) = G(P ) = 0 and x0 = αi . (3.26)

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45

The truth of postulates (I 1). . . (I 7) is a consequence of results on the resultant and elimination theory, as can now be verified. The conditions (I 1),(I 2),(I 3),(I 5) follow from the definition. For (I 4), let F = A1 (X − x0 ) + B1 (Y − y0 ), G = A2 (X − x0 ) + B2 (Y − y0 ). Then RY (F, G) = (B2 A1 − B1 A2 )(X − x0 ), with B2 A1 − B1 A2 6= 0. So I(P, F ∩ G) = 1. For (I 6), note that the resultant RY (F, G + AF ) in the form of (2.2) equals RY (F, G), since the former is obtained from the latter by adding linear combinations of the first m rows to the last n rows, where n = deg F, m = deg G. It follows from Lemma 2.3 that RY (F, GH) = RY (F, G)RY (F, H), from which (I 7) follows. Thus (I 1),. . . , (I 7) are satisfied. 2 R EMARK 3.10 For another proof of this theorem in terms of the local ring at P , see Exercise 6. These results lead to the following geometric and covariant concept. D EFINITION 3.11 For the affine curves, F = vα (F ), G = vα (G), and the point P = (a, b), the intersection number is I(P, F ∩ G) = I((a, b), F ∩ G). ´ 3.2 BEZOUT’S THEOREM For a projective curve, local properties are the same as for the corresponding affine curve. D EFINITION 3.12 For the projective curves, F = v(F ), G = v(G), and the point O = (1, 0, 0) the intersection number is I(O, F ∩ G) = I((0, 0), F∗ , G∗ ). where F∗ and G∗ are the polynomials associated to F and G as in Section 1.4. Intersection numbers of F = v(F ) and G = v(G) at another point P are calculated, as in the affine case, by using covariant properties; that is, a projectivity is applied to change P to Y∞ . The covariance of the intersection number of projective curves at a point can be shown by using Theorem 3.9 and (3.26). This important property also follows from the proof of B´ezout’s Theorem 3.14 below. Now, some global results are obtained. T HEOREM 3.13 If the projective plane curves F and G have degrees m and n, and no common component, then they have at most mn points in common.

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Proof. Suppose, on the contrary, that the curves have more than mn points in common. Choose any mn+1 of the common points and join every pair by a line. There is a point P neither on any of these lines nor on F or G, and there is a line ℓ through P distinct from these lines. Take a frame for which P = U1 and ℓ = U1 U2 . Then every one of the mn + 1 points chosen has a different value for y, since the join of any two does not contain U1 . Let F = v(f ), G = v(g). Then the system of equations = a0 (Y )X m + a1 (Y )X m−1 + · · · + am−1 (Y )X + am (Y ), = b0 (Y )X n + b1 (Y )X n−1 + · · · + bn−1 X + bn (Y ) (3.27) has at least mn + 1 solutions (x, y) with different values for y. As explained in Section 2.3, this implies that the polynomial D(Y ) = RX (f, g) has at least mn + 1 distinct roots. From Theorem 2.21, RX (f, g) is identically zero. By Theorem 2.20, F and G have a common factor, a contradiction. 2 f (X, Y ) g(X, Y )

T HEOREM 3.14 (B´ezout’s Theorem) If the projective plane curves F = v(F ) and G = v(G) have degrees m and n, and no common component, then P I(P, F ∩ G) = mn.

Proof. By Theorem 3.13, F and G have at most nm common points. Let h ≥ 0 denote the exact number of such points. Let R = (ξ0 , ξ1 , ξ2 ) and Q = (η0 , η1 , η2 ) be two distinct points, and let ℓ be the line through R and Q. A point P = (x0 , x1 , x2 ) lies on ℓ if and only if there exist λ, µ ∈ K such that P = (λξ0 + µη0 , λξ1 + µη1 , λξ2 + µη2 ). A necessary and sufficient condition for P to be a point of F is F (λξ0 + µη0 , λξ1 + µη1 , λξ2 + µη2 ) = 0.

Since F (λξ0 + µη0 , λξ1 + µη1 , λξ2 + µη2 ) can be viewed as a homogeneous polynomial F ′ (λ, µ) in the indeterminates λ and µ, write F ′ (λ, µ) = a0 λm + a1 λm−1 µ + · · · + am−1 λµm−1 + am µm . To calculate as , let F (X0 , X1 , X2 ) = Then as =

X

X

u+v+w=m−s i+j+k=m

P

i+j+k=m

aijk X0i X1j X2k .

    i j k aijk (ξ0u ξ1v ξ2w )(η0i−u η1j−v η2k−w ). u v w

It may be noted that as is a homogeneous polynomial of degree m in the indeterminates ξ0 , ξ1 , ξ2 , η0 , η1 , η2 . If η0 , η1 , η2 are considered to be constants, then as is a homogeneous polynomial of degree m − s in the indeterminates ξ0 , ξ1 , ξ2 . Similarly, if ξ0 , ξ1 , ξ2 are constants, then as is a homogeneous polynomial of degree s in the indeterminates η0 , η1 , η2 . The non-trivial roots of F ′ (λ, µ) correspond to the common points of the curve F with the line ℓ.

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A similar result holds for the curve G. The line ℓ through R and Q meets G in the points P such that (λ, µ) is a non-trivial root of the polynomial G′ (λ, µ) = b0 λn + b1 λn−1 µ + · · · + bn−1 λµn−1 + bn µn . Therefore F and G have a non-trivial common root; equivalently, R(F ′ (λ, µ), G′ (λ, µ)) = 0,

if and only if F and G have a common point (i)

(i)

(i)

Pi = (x0 , x1 , x2 ) on the line through R and Q. It follows that R(F ′ (λ, µ), G′ (λ, µ)) = 0 if and only if at least one of the determinants ξ0 ξ1 ξ2 η1 η2 D(ξ, η, x(i) ) = η0 (3.28) x(i) x(i) x(i) 0 1 2

is zero. Now, fix the point R and consider Q as a variable point; that is, leave the homogeneous triple (ξ0 , ξ1 , ξ2 ) fixed and consider η0 , η1 , η2 as indeterminates. Then the determinant D(ξ, η, x(i) ) in (3.28) is a linear homogeneous polynomial while the resultant R(F ′ (λ, µ), G′ (λ, µ)) is a homogeneous polynomial of degree mn. Every root of D(ξ, η, x(i) )) is also a root of R(F ′ (λ, µ), G′ (λ, µ)). By Study’s Theorem 2.10, D(ξ, η, x(i) ) divides R(F ′ (λ, µ), G′ (λ, µ)). Conversely, every root of R(F ′ (λ, µ), G′ (λ, µ)) is a root of some D(ξ, η, x(i) ). Therefore, for certain positive integers ri , Q (3.29) R(F ′ (λ, µ), G′ (λ, µ)) = c ki=1 D(ξ, η, x(i) )ri , where c is a non-zero constant and k ≤ h. This implies that Pk nm = i=1 ri .

(3.30)

As a consequence, k ≥ 1 and hence h ≥ 1. Next, it is shown that ri is covariant. To do this, the notation in the proof of Theorem 1.22 is used. It is a straightforward to show that D(ξ, η, x(i) ) = D(ξ ′ , η ′ , (x′

(i)

)) det(aij ),

which shows the covariance of D(ξ, η, x(i) ). Also, from the definition of F ′ (λ, µ) and G′ (λ, µ), it follows that their resultant is a covariant. Therefore F and G may be assumed to have no common point on the line v(X0 ). Then their intersection is in the affine plane with coordinates (x, y) where, as usual, x = x1 /x0 and y = x2 /x0 . So, take the curves in their the affine forms, with F = v(F∗ (X, Y )) and G = v(G∗ (X, Y )). It may also be supposed that no two such common points lie on the same line v(X − c). If R = (u, 0) and Q = Y∞ , then D(ξ, η, x(i) ) = X − ui ,

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where ui is the abscissa of the point Ri . Also, R(F ′ (λ, µ), G′ (λ, µ)) is the resultant RY (X) of F∗ (X, Y ) and G∗ (X, Y ), both considered as polynomials in the indeterminate Y . Now, (3.29) reads as follows: Qk RY (X) = c i=1 (X − ui )νi . This shows that, if Pi = (ui , vi ) is a common point of the curves F and G, then ri is equal to the multiplicity of the factor X − ui in the resultant RY (X). Finally, ri = I(Pi , F ∩ G), (3.31) from Theorem 3.9 and (3.26). 2 E XAMPLE 3.15 With K = C and (X, Y, Z) homogeneous coordinates, let F = Y 5 − X(Y 2 − XZ)2 , G = Y 4 − X 2 Z 2 + Y 3 Z. Then ZF − Y 2 G = −XZ(Y 2 − XZ)2 − Y 2 (Y 2 − XZ)(Y 2 + XZ) = (Y 2 − XZ)(−Y 2 XZ + X 2 Z 2 − Y 4 − Y 2 XZ) = (Y 2 − XZ)(X 2 Z 2 − 2Y 2 XZ − Y 4 ).

If F = v(F ), G = v(G) and P = (x, y, z) ∈ F ∩ G, then y 2 − xz = 0 ⇒ y = 0 ⇒ P = P1 or P2 with P1 = (1, 0, 0), P2 = (0, 0, 1);

y 4 − x2 z 2 = −2y 2 xz ⇒ −2y 2 xz + y 3 z = 0 ⇒ y = 2x ⇒ 16x4 − x2 z 2 + 8x3 z = 0 ⇒ 16x2 + 8xz − z 2 = 0

⇒ P = P3 or P4 √ √ √ √ with P3 = (−1 + 2, −2 + 2 2, 4), P4 = (−1 − 2, −2 − 2 2, 4). The points P3 and P4 are simple on both the curves F and G, which intersect transversally at both points, whence I(P3 , F ∩ G) = I(P3 , F ∩ G) = 1. For P2 = (0, 0, 1), put f (X, Y ) = F (X, Y, 1), g(X, Y ) = G(X, Y, 1). Then f = Y 5 − X(Y 2 − X)2 = −X 3 + 2X 2 Y 2 − XY 4 + Y 5 , g = Y 4 − X 2 + Y 3 = −X 2 + Y 3 + Y 4 , h = f − Xg = 2X 2 Y 2 − XY 3 − 2XY 4 + Y 5 = Y 2 (2X − Y )(X − Y 2 );

I(P2 , F ∩ G) = I(P2 , G ∩ F ) = I(P2 , g ∩ f ) = I(P2 , g ∩ h)

= I(P2 , g ∩ Y 2 ) + I(P2 , g ∩ (2X − Y )) + I(P2 , g ∩ (X − Y 2 )) = 4 + 2 + I(P2 , (X − Y 2 ) ∩ Y 3 ) = 4 + 2 + 3 = 9.

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For P1 = (1, 0, 0), put f (Y, Z) = F (1, Y, Z),

g(Y, Z) = G(1, Y, Z).

Then f = −Z 2 + 2Y 2 Z − Y 4 + Y 5 ,

g = −Z 2 + Y 3 Z + Y 4 , h = g − f = −2Y 2 Z + Y 3 Z + 2Y 4 − Y 5 = −Y 2 (2 − Y )(Z − Y 2 );

I(P1 , F ∩ G) = I(P1 , f ∩ g) = I(P1 , f ∩ h) = I(P1 , f ∩ Y 2 ) + I(P1 , f ∩ (2 − Y )) + I(P1 , f ∩ (Z − Y 2 )) = 4 + 0 + I(P1 , (Z − Y 2 ) ∩ f ) = 4 + I(P1 , (Z − Y 2 ) ∩ (f + (Z − Y 2 )2 )) = 4 + I(P1 , (Z − Y 2 ) ∩ Y 5 ) = 4 + 5 = 9.

Hence P4

i=1

I(Pi , F ∩ G) = 1 + 1 + 9 + 9 = 20 = deg F · deg G.

3.3 RATIONAL AND BIRATIONAL TRANSFORMATIONS D EFINITION 3.16 Given two projective planes π and π ′ over K, then the mapping ω : π → π ′ is a rational transformation if there exist independent homogeneous polynomials ϕ0 , ϕ1 , ϕ2 ∈ K[X0 , X1 , X2 ] of the same degree n and, for any point P = (a0 , a1 , a2 ) ∈ π, ω(P ) = P ′ = (b0 , b1 , b2 ), ρb0 = ϕ0 (a0 , a1 , a2 ), ρb1 = ϕ1 (a0 , a1 , a2 ), ρb2 = ϕ2 (a0 , a1 , a2 ),

(3.32)

where ρ ∈ K. Also, n is the order of the transformation. Denote by ζ the net of curves v(µ0 ϕ0 + µ1 ϕ1 + µ2 ϕ2 ). As P ′ traverses the line v(λ0 Y0 + λ1 Y1 + λ2 Y2 ) of π ′ , its pre-image P describes in π the curve, v(λ0 ϕ0 (X0 , X1 , X2 ) + λ1 ϕ1 (X0 , X1 , X2 ) + λ2 ϕ2 (X0 , X1 , X2 )). Here, to a point P ′ = (η0 , η1 , η2 ), centre of a pencil of lines, corresponds the set of base points of a pencil of curves ϕ of the net ζ. Therefore to the point P ′ correspond the points common to three curves: (1) v(ϕ1 , ϕ2 ), for (η0 , η1 , η2 ) = (c, 0, 0), c 6= 0; (2) v(η0 ϕ1 − η1 ϕ0 , ϕ2 ), for η1 6= 0, η2 = 0;

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(3) v(η2 ϕ0 − η0 ϕ2 , η2 ϕ1 − η1 ϕ2 ), for η2 6= 0. To ensure that ω is bijective, n2 − 1 of these points must be fixed: denote them P1 , . . . , Pn2 −1 . Then the n2 -th point P = (ξ0 , ξ1 , ξ2 ) is the pre-image of P ′ , and X0 , X1 , X2 can be obtained as rational functions of Y0 , Y1 , Y2 : ω −1 : π ′ → π, ρX0 = f0 (Y0 , Y1 , Y2 ), ρX1 = f1 (Y0 , Y1 , Y2 ), ρX2 = f2 (Y0 , Y1 , Y2 ).

(3.33)

Here f0 , f1 , f2 are independent homogeneous polynomials of the same degree. In this case, the transformation ω is birational; it is also called a Cremona transformation. Then through the points P0 , P1 , . . . , Pn2 −1 pass all curves of the net ζ. This is therefore a homaloidal net; that is, every two of its curves have a single variable intersection. P ROPOSITION 3.17 (i) The transformation ω changes the curves ϕ of degree n of the homaloidal net ζ passing through the points P0 , P1 , . . . , Pn2 −1 in π into the lines of π ′ . (ii) In π, the points P0 , P1 , . . . , Pn2 −1 constitute the set of fundamental points of ω. (iii) The image P ′ of a point P other than a fundamental point is the image under ω of the unique pencil in ζ passing through P . (iv) When there are two curves C1 , C2 in ζ with a common component C, every P on C different from a fundamental point has the same image P ′ , which is a fundamental point for ω −1 . Of particular interest among birational transformations are those of order two, that is, the quadratic transformations. If ω is a quadratic transformation from π to π ′ , to the set L of lines of π correspond in π ′ the set of conics of a homaloidal net ζ ′ , that is, the set of conics passing through three fixed points, Q0 , Q1 , Q2 , the fundamental points of ω in π ′ . Also the inverse transformation ω −1 is quadratic, since to the lines of π ′ correspond in π the conics passing through three fixed points P0 , P1 , P2 , the fundamental points in π, and constitute the homaloidal net ζ. In the general case, the points P0 , P1 , P2 are distinct and not collinear, as are the points Q0 , Q1 , Q2 . If the reference system is chosen so that P0 = Q0 = U0 , P1 = Q1 = U1 , P2 = Q2 = U2 , ω(U) = U, then ω : π → π′ , ρY0 = X1 X2 , ρY1 = X0 X2 , ρY2 = X0 X1 .

(3.34)

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The transformation ω is not defined at the points P0 , P1 , P2 . The lines joining pairs of these points are the exceptional lines. Every non-fundamental point on one of these lines v(Xj ) is transformed to the point (δ0j , δ1j , δ2j ), where δij is the Kronecker symbol. So, for example, under ω, the line P1 P2 goes to the point Q0 , while, under ω −1 , the point Q0 dilates to P1 P2 . Similarly, ω takes the lines P0 P2 and P0 P1 to the points Q1 and Q2 , and ω −1 reverses these. If P is any point off the exceptional lines, its image P ′ is also off the exceptional lines of ω −1 , and ω −1 (P ′ ) = P . Therefore, with ∆ = {P = (x0 , x1 , x2 ) ∈ π | x0 x1 x2 6= 0},

∆′ = {Q = (y0 , y1 , y2 ) ∈ π ′ | y0 y1 y2 6= 0},

the restriction ω|∆ : ∆ −→ ∆′ is bijective. The inverse ω −1 of ω has the following equation: ω −1 : π ′ → π, τ X0 = Y1 Y2 , τ X1 = Y0 Y2 , τ X2 = Y0 Y1 .

(3.35)

D EFINITION 3.18 The transformation ω is the standard quadratic transformation.

3.4 QUADRATIC TRANSFORMATIONS Given a projective plane curve F = v(F ) with F = F (X0 , X1 , X2 ), the points Q = (y0 , y1 , y2 ) on the image of F under ω lie on the curve G = v(G), with G(Y0 , Y1 , Y2 ) = F (Y1 Y2 , Y0 Y2 , Y0 Y1 ).

(3.36)

The curve G given by (3.36) is the algebraic transform of F. To better understand the geometric relation between F and G, consider first the case that F is a line with F = a0 X0 + a1 X1 + a2 X2 . There are three possibilities. (1) F does not pass through any fundamental point. In this case, a0 a1 a2 6= 0 and the algebraic transform of F is the conic, C2 = v(a0 Y1 Y2 + a1 Y0 Y2 + a2 Y0 Y1 ).

This conic C2 passes through each of the fundamental points in π ′ but has no further intersection with any exceptional line. The fundamental points Q0 , Q1 , Q2 correspond to the points of intersection S0 , S1 , S2 of F with the exceptional lines in π. So, ω establishes a bijection between F \{S0 , S1 , S2 } and C2 \{Q0 , Q1 , Q2 }. (2) F is a line through P0 other than an exceptional line. Hence F = X1 + λX2 , and its algebraic transform under ω splits into two distinct lines. They are the exceptional line v(Y0 ) and the line F ′ = v(F ′ ) with F ′ = Y2 + λY1 . As in (1), ω induces a bijection between F and F ′ . (3) F coincides with one of the exceptional lines. Now, the algebraic transform of F splits into two exceptional lines; they are v(Y1 ) and v(Y2 ). Every point of F

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for which ω is defined has as its image the point Q0 that can be considered as the transform of F. Given a curve F = v(F (X0 , X1 , X2 )) containing no exceptional line as a component, let G = v(G(Y0 , Y1 , Y2 )) = v(F (X1 X2 , X0 X2 , X0 X1 )) be its algebraic transform. Then G(Y0 , Y1 , Y2 ) = m(Y )F ′ (Y0 , Y1 , Y2 ), where m(Y ) = Y0 r0 Y1 r1 Y2 r2 but F ′ (Y0 , Y1 , Y2 ) is not divisible by any Yi . D EFINITION 3.19 The curve F ′ = v(F ′ ) is the geometric transform of F under ω. T HEOREM 3.20 (i) If F ′ is the geometric transform of F for ω, then F is the geometric transform of F ′ for ω −1 . (ii) Apart from a finite number of points of F and F ′ , the remaining points of F and F ′ are in bijective correspondence. Proof. By definition, F (Y1 Y2 , Y0 Y2 , Y0 Y1 ) = M1 (Y )F ′ (Y0 , Y1 , Y2 ), F ′ (X1 X2 , X0 X2 , X0 X1 ) = M2 (X)F ′′ (X0 , X1 , X2 ),

(3.37)

where F ′′ = v(F ′′ ) is the geometric transform of F ′ for ω −1 . Put Yi = Xj Xk in (3.37) for {i, j, k} = {1, 2, 3}: F (X02 X1 X2 , X0 X12 X2 , X0 X1 X22 ) = M3 (X)F ′ (X1 X2 , X0 X2 , X0 X1 ) = M4 (X)F ′′ (X0 , X1 , X2 ), where M3 , M4 are monomials in X0 , X1 , X2 . Since F has degree n, it follows that F (X02 X1 X2 , X0 X12 X2 , X0 X1 X22 ) = (X0 X1 X2 )n F (X0 , X1 , X2 ). Since no Xi divides F or F ′′ , so F = F ′′ . Further, both F and F ′′ have a finite number of points in common with the exceptional lines, and the bijection between the components of F and F ′′ is a consequence of the formulas that define ω and ω −1 . 2 Now consider the behaviour of a singular point of a plane curve under a quadratic transformation. Here, use the following correspondence between the non-exceptional lines through P0 and the non-fundamental points of the line v(Y0 ): v(X1 + λX2 ) ←→ (0, 1, −λ).

As P = (1, −λu, u) describes the line ℓ on π, the image P ′ = (−λu, 1, −λ) describes the line v(Y0 − uY2 ) on π ′ . As P 7→ P0 , that is, the parameter u 7→ 0, then P ′ 7→ Q = (0, 1, −λ). T HEOREM 3.21 Let F = v(F (X0 , X1 , X2 )) be a curve of degree d such that (a) the fundamental point Pi has multiplicity mi on F, 0 ≤ i ≤ 2; (b) no exceptional line through a fundamental point is tangent to F at that point.

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Then (i) the algebraic transform G of F admits the fundamental line v(Yi ) as a component of multiplicity mi ; (ii) the geometric transform F ′ has degree 2d − m0 − m1 − m2 ; (iii) there is a multiplicity-preserving bijection between the tangents to F at Pi and the intersections of F ′ with the exceptional line v(Yi ), apart from the two fundamental points of v(Yi ); (iv) I(R, F ′ ∩ v(Yi )) = j, where R is the point corresponding to a line counted j times among the tangents to F at Pi ; (v) the curve F ′ has a point of multiplicity d − mj − mk at the fundamental point Qi , where {i, j, k} = {1, 2, 3}; (vi) the tangents to F ′ at Qi are not exceptional lines and correspond to the intersections of F with the line v(Xi ), apart from the fundamental points. Proof. (i),(ii) Here the discussion is limited to considering what happens to the point P0 . The polynomial F of degree d is written in the form P F (X0 , X1 , X2 ) = dj=m0 X0d−j Fj (X1 , X2 ),

where Fj is a homogeneous polynomial of degree j in X1 , X2 and Fm0 , Fd are both non-zero. The algebraic transform of F is G = v(G) with G(Y0 , Y1 , Y2 ) = F (Y1 Y2 , Y0 Y2 , Y1 Y2 ) P = dj=m0 (Y1 Y2 )d−j Fj (Y0 Y2 , Y0 Y1 )

= Y1d−m0 Y2d−m0 Y0m0 Fm0 (Y2 , Y1 ) + · · · + Y0d F (Y2 , Y1 ).

So Y0m0 divides G but Y0m0 +1 does not, and similarly for Y1m1 and Y2m2 . Therefore G = Y0m0 Y1m1 Y2m2 F ′ , where F ′ is the geometric transform. It follows that F ′ has degree 2d − (m0 + m1 + m2 ). (iii) On the one hand, the tangents to F at P0 correspond to the linear factors of Qk=n Fm0 (X1 , X2 ) = k=1 (µk X1 + νk X2 )αk , with α1 + · · · + αn = m0 ; on the other hand, I(Q, F ′ ∩ v(Y0 )) at points Q is given by the multiplicity of the roots of the polynomial Y1d−m0 −m1 Y2d−m0 −m2 Fm0 (Y2 , Y1 ) = Y1d−m0 −m1 Y2d−m0 −m2 with α1 + · · · + αn = m0 . (iv) Note that

Qk=n k=1

(µk Y2 + νk Y1 )αk ,

F ′ = Y1d−m0 −m1 Y2d−m0 −m2 Fm0 (Y1 , Y2 ) + · · · + Y0d−m0 Y1−m1 Y2−m2 Fd (Y2 , Y1 );

then the multiplicity of F ′ at Q0 = (1, 0, 0) is given by the degree of B(Y1 , Y2 ) = Y1−m1 Y2−m2 Fd (Y1 , Y2 ),

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and this is d − m1 − m2 . (v) The tangents to F ′ at Q0 are given by the factors of B(Y1 , Y2 ). If one of these were Y1 or Y2 , then Fd (Y2 , Y1 ) would be divisible by a power m > m1 of Y1 or m > m2 of Y2 . But, if that happens, then I(P1 , F ∩ v(X0 )) = m or I(P2 , F ∩ v(X0 )) = m. Therefore one of the tangents to F at P1 or at P2 would be the exceptional line v(X0 ), a contradiction. (vi) Applying (v) to ω −1 gives the result. 2 T HEOREM 3.22 (i) To each point P external to the fundamental triangle of π corresponds a point Q external to the fundamental triangle of π ′ . (ii) If such a point P is an m-ple point of the curve F, then its image Q is an m-ple point of the geometric transform F ′ of F. Proof. Take P = (1, 1, 1), and so Q = (1, 1, 1). Since the algebraic transform G and the geometric transform F ′ differ by a component not passing through P , it suffices to consider G instead of F ′ . In π, take a new system of coordinates: ρZ0 = X0 , ρZ1 = X1 − X0 , ρZ2 = X2 − X0 .

(3.38)

The point P is now (1, 0, 0), and

F (X0 , X1 , X2 ) = F (Z0 , Z0 + Z1 , Z0 + Z2 ) = F¯ (Z0 , Z1 , Z2 ) = Z d−m F¯m (Z1 , Z2 ) + · · · + F¯d (Z1 , Z2 ) =

0 X0d−m F¯m (X1

− X0 , X2 − X0 ) + · · · + F¯d (X1 − X0 , X2 − X0 ).

Therefore the algebraic transform G = v(G(Y0 , Y1 , Y2 )) is given by G = (Y1 Y2 )d−m F¯m (Y0 Y2 − Y1 Y2 , Y0 Y1 − Y1 Y2 ) + · · · + F¯d (Y0 Y2 − Y1 Y2 , Y0 Y1 − Y1 Y2 ).

In π ′ , use a similar change of coordinates:

ρW0 = Y0 , ρW1 = Y0 − Y1 , ρW2 = Y0 − Y2 .

(3.39)

This takes Q = (1, 1, 1), the image of P under ω, to Q′ = (1, 0, 0) in π ′ . Also, ¯ 0 , W1 , W2 ). G = G(Y0 , Y1 , Y2 ) = G(W0 , W0 − W1 , W0 − W2 ) = G(W

Now, ¯ 0 , W1 , W2 ) G(W

= [(W0 − W1 )(W0 − W2 )]d−m F¯m (W0 W1 − W1 W2 , W2 W0 − W1 W2 ) +[(W0 − W1 )(W0 − W2 )]d−m−1 F¯m−1 (W0 W1 − W1 W2 , W2 W0 − W1 W2 ) + · · · + F¯d (W0 W1 − W1 W2 , W2 W0 − W1 W2 )

= W02d−m F¯m (W1 , W2 ) + · · · + W02d−m−1 F¯m+1 (W1 , W2 ) + · · · + W0d F¯d (W1 , W2 ) + · · · ,

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in descending powers of W0 . The highest power of W0 appears in the first term. Therefore F¯d (W1 , W2 ) determines the configuration of the tangents to G at Q′ , and the theorem is proved. 2

3.5 RESOLUTION OF SINGULARITIES This section is devoted to the classical result that every projective plane curve is birationally equivalent to a curve with only ordinary singularities. Here, a point P of a curve is ordinary if the tangents to the curve at P are distinct. D EFINITION 3.23 If C n is an irreducible projective plane curve of degree n, its virtual genus is P1 g ∗ (C n ) = 12 (n − 1)(n − 2) − (3.40) 2 r(r − 1), where the sum is over all singular points P of C n , with r the multiplicity of P . L EMMA 3.24 The virtual genus of a curve is non-negative. Proof. Let P1 , P2 , . . . , Pk be the singular points of the curve C n , with multiplicities r1 , r2 , . . . , rk . First, it is shown that the integer Pk s = 12 (n − 1)(n + 2) − i=1 21 ri (ri − 1) (3.41) is non-negative. If C n = v(F (X0 , X1 , X2 )), then the polar curve of C n with respect to the point U0 = (1, 0, 0) is C0n = v(∂F/∂X0 ). When p > 0, it is possible that ∂F/∂X0 is the zero polynomial. In that case, Theorem 1.24 ensures that ∂F/∂X1 is not the zero polynomial. Thus it may be assumed that C0n exists. A point P such that mP (C n ) = r satisfies mP (C0n ) ≥ r − 1. By B´ezout’s Theorem 3.14, Pk i=1 ri (ri − 1) ≤ n(n − 1). Since n(n − 1) < (n − 1)(n + 2), so s = 0 for n = 1 and s > 0 otherwise. Now, choose s points Q1 , . . . , Qs on C n . First, it is shown that there exists a curve C n−1 of degree n − 1 passing through each point Pi with multiplicity at least ri − 1 and through each Qj . If G = v(G) is a general plane curve of degree n − 1, then G has n(n + 1)/2 coefficients. Its passage through an (r − 1)-ple point P requires at most 21 r(r − 1) linear conditions on its coefficients. In fact, if P = U0 P and G = gi (X1 , X2 )X0n−1−i , the condition that P is an (r − 1)-ple point is that g0 = g1 = . . . = gr−2 = 0;

1 2 r(r − 1)

these terms contain coefficients. It follows that the passage through the k Pk singular points requires at most i=1 12 r(r−1) linear conditions on its coefficients. Therefore the number of conditions on C n−1 to pass through the Pi and Qj is at most Pk s + i=1 12 ri (ri − 1) ≤ 21 (n − 1)(n + 2),

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by the definition of s in (3.41). Hence, such a C n−1 exists. Now apply B´ezout’s Theorem 3.14 to the curves C n and C n−1 : P n(n − 1) = I(P, C n ∩ C n−1 ) P P ≥ I(Qj , C n ∩ C n−1 ) + I(Pi , C n ∩ C n−1 ) Pk = s + i=1 ri (ri − 1) Pk Pk = 21 (n − 1)(n + 2) − i=1 21 ri (ri − 1) + i=1 ri (ri − 1) Pk = n(n − 1) − 21 (n − 1)(n − 2) + i=1 21 ri (ri − 1) = n(n − 1) − g ∗ (C n ). Hence g ∗ (C n ) ≥ 0.

2

D EFINITION 3.25 The index of the irreducible plane projective curve F = v(F ) is the non-negative integer P P (3.42) I(F) = I(F ) = P T (mT − 1),

where the outer sum is over all singular points P of F and the inner sum is over all distinct tangents T to F at P , with mT the multiplicity of the tangent. It is worth noting explicitly the following property of the index.

L EMMA 3.26 The index I(F) is zero if and only if each singular point of F is ordinary. Now, a start is made on the proof of the famous theorem on the resolution of singularities. T HEOREM 3.27 Through a finite sequence of quadratic transformations, an irreducible projective plane curve can be transformed into a curve with only ordinary singularities. Proof. Given a finite sequence of quadratic transformations, ω1 , . . . , ωr , if F is an irreducible plane curve, the symbol F (i) indicates the geometric transform of F (i−1) under ωi , for i = 1, . . . , r − 1 with F (0) = F. Now, let F be an irreducible plane projective curve of degree d with a nonordinary singularity; it suffices to exhibit a finite sequence ω1 , . . . , ωr of quadratic transformations such that I(F (k) ) ≤ I(F (k−1) ) and, if equality holds, then g ∗ (F (k) ) < g ∗ (F (k−1) ).

If F (k) still has non-ordinary singularities, then, with a new sequence of quadratic transformations, another curve is obtained with strictly lower index or virtual genus. This procedure is repeated till a curve is obtained with index zero, which necessarily has only ordinary singularities by Lemma 3.26. To construct such a sequence, a particular system of coordinates is used that satisfies the following conditions for a given m-ple point of F: (i) the m-ple point is P0 = (1, 0, 0);

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(ii) the curve F does not pass through the other two fundamental points, and F intersects each of the two exceptional lines through P0 in d − m points other than P0 ; (iii) the exceptional line v(X0 ) meets F in d distinct points. The existence of such a system of coordinates does not always happen, for in positive characteristic there exist strange curves containing their own nucleus, for which condition (ii) does not hold. This possibility is treated below. 2 The utility of such a system of coordinates lies in the following result. L EMMA 3.28 If ω is the quadratic transformation relative to a triangle satisfying conditions (1), (2), (3) and, if F ′ is the geometric transform of F under ω, then the following properties hold: (i) g ∗ (F ′ ) ≤ g ∗ (F), I(F ′ ) ≤ I(F); (ii) if P0 is not an ordinary singularity, then I(F ′ ) = I(F) =⇒ g ∗ (F ′ ) < g ∗ (F). Proof. Denote by ℓ1 , . . . , ℓs the tangents to F at P0 , and by α1 , . . . , αs their respective multiplicities. From Theorem 3.22, every singular point of F other than P0 is transformed into a point of F ′ with the same multiplicity; an ordinary singularity of F external to the fundamental triangle is transformed into an ordinary singularity of F ′ , also external to the fundamental triangle. By part (v) of Theorem 3.21, F ′ has three ‘new’ singular points: Q0 of multiplicity d, and Q1 , Q2 both of multiplicity d − m. From part (iv) of the same theorem, it follows that F ′ has s points R1′ , . . . , Rs′ on the line v(Y0 ) distinct from the fundamental points, with ri′ the multiplicity of Ri′ ; then ri′ ≤ I(Ri′ , F ′ ∩ v(Y0 )) = αi

for 1 ≤ i ≤ s. Also suppose that F ′ has degree 2d − m as in (ii) of Theorem 3.21. To prove (i), it suffices to see that Ps g ∗ (F ′ ) = g ∗ (F) − j=1 21 rj′ (rj′ − 1). (3.43)

If F has k singular points other than P0 with multiplicities r1 , . . . , rk , then

g ∗ (F ′ ) = 12 (2d − m − 1)(2d − m − 2) − 21 d(d − 1) − (d − m)(d − m − 1) P P − ki=1 12 ri (ri − 1) − sj=1 12 rj′ (rj′ − 1).

This expression can also be written in the form Pk 1 Ps 1 ′ ′ 1 1 2 i=1 2 ri (ri − 1) − j=1 2 rj (rj − 1) 2 (d − 3d + 2) − 2 m(m − 1) − P s = g ∗ (F) − j=1 21 rj′ (rj′ − 1),

as required. For part (ii), note that, if I(F) = I(F ′ ), then Ps Ps P ′ i=1 (αi − 1) − i=1 Ti (mTi − 1) = 0,

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where, in the second summation, Ti varies in the set of tangents to F ′ at Ri′ . It follows that Ps P ′ Ti mTi ] + S − s = 0, i=1 [αi − where S is the total number of tangents to F ′ at the points R1′ , . . . , Rs′ . Since P αi = I(Ri , F ′ ∩ v(Y0 )) ≥ ri′ = Ti m′Ti

and S ≥ s, this implies that αi = ri′ and S = s. It follows that if, as in the proof of (ii) of Lemma 3.28, some αi > 1, then also ′ ri′ > 1. Note also that P T = s if and only if F has a unique tangent at each point 2 Ri′ . Hence the sum si=1 21 ri′ (ri′ − 1) is positive, which proves (ii).

By virtue of Lemma 3.28, the procedure described immediately after the enunciation of Theorem 3.27 carries over to an irreducible plane curve with only ordinary singularities once it is possible to find, for each curve F (i) of the sequence, a system of coordinates for which conditions (i),(ii),(iii) hold. To finish the proof of Theorem 3.27, it is necessary to construct the sequence F (1) = F ′ , F (2) , . . . in such a way that points not satisfying (ii) are avoided. To do this, use non-homogeneous coordinates X = X1 /X0 , Y = X2 /X0 , and let F = v(f (X, Y )). To see that condition (ii) can be satisfied, the intersection of F with a line ℓ = v(Y − tX), t ∈ K ∗ , must be found: f (X, Y ) = 0,

Y − tX = 0.

(3.44)

Write f (X, Y ) = Fm (X, Y ) + Fm+1 (X, Y ) + · · · + Fd (X, Y ),

where Fi (X, Y ) is a homogeneous polynomial of degree i with Fm 6= 0, Fd 6= 0. Now, (3.44) gives Fm (X, tX) + Fm+1 (X, tX) + · · · + Fd (X, tX) = 0, which becomes X m (Fm (1, t) + XFm+1 (1, t) + · · · + X d−m Fd (1, t)) = 0.

(3.45)

If ℓ is not a tangent to F at P0 , then Fm (t) 6= 0. Suppose this is the case, and put Fi (t) = Fi (1, t). Then (ii) is satisfied by the line ℓ since Pt (X) = Fm (t) + XFm+1 (t) + · · · + X d−m Fd (t)

has no multiple roots. This is equivalent to saying that Pt (X) and its derivative Qt (X) have no common root. Now, Qt (X) = Fm+1 (t) + 2XFm+2 (t) + · · · + (d − m)X d−m−1 Fd (t).

To see that this happens, distinguish two cases according as Qt (X) is identically zero or not. In the second case, since Fj (t) is a polynomial in t, so Pt (X) can be written as a polynomial in the indeterminates X, t as P (X, t). Similarly, define Q(X, t). Using the resultant, the system P (X, t) = 0,

Q(X, t) = 0

(3.46)

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has a finite number of solutions (x, λ) unless the two polynomials have a common factor. This possibility can be ruled out as follows. First, it is shown that P (X, t) is irreducible. Otherwise, P (X, t) = U (X, t)V (X, t), with U, V non-constant; therefore P (X, Y /X) = U (X, Y /X)V (X, Y /X). Now, if both sides are multiplied by X m , the left becomes f (X, Y ) and the right the product of X i U (X, Y /X) and X j V (X, Y /X) with i + j = m. But this is a contradiction since both are non-constant in that they contain Y ; otherwise, X divides P (X, t), which is impossible since Fm (t) 6= 0 for all t. To show that the two polynomials in (3.46) have no common factors, it suffices to show that deg P (X, t) > deg Q(X, t). To see this, note that, as Fj (t) has degree at most j, so 2i + m − 1 is an upper limit for the degree of Fm+i (t)X i−1 . This number reaches its maximum for the largest value of i allowed that equals d − m. Therefore the degree of Q(X, t) cannot be greater than 2(d − m) + m − 1 = 2d − m − 1. Since deg P (X, t) = 2d − m, the result is proved. So it has been shown that if Qt (X) is not identically zero, then (3.46) has a finite number of solutions (x, λ). Therefore there is an infinite number of values of λ for which (x, λ) is not a solution; so the line ℓ = v(Y − λX) satisfies condition (ii). Now, consider the case that Qt (X) is the zero polynomial. This can only happen when p is positive, and p | (d − m). Also p | i for each i, with 1 ≤ i ≤ d − m, for which Fm+i (t) is not identically zero. This gives a necessary and sufficient condition for Qt (X) to be zero. The following example shows that this can occur. E XAMPLE 3.29 Let f (X, Y ) = Y − X p+1 . The plane curve F = v(f (X, Y )) is non-singular. The intersection of F with ℓ = v(Y − tX), apart from P0 , is the single point P = (x, xp+1 ), where x is the unique root of X p − t. Therefore there is no line through P0 satisfying condition (ii). D EFINITION 3.30 The point P is a terrible point of F if, when it is transformed to P0 , it does not satisfy condition (ii). It follows that a point of F is terrible if and only if F is strange and P is its unique nucleus. Summarising so far, if none of the singular points of F is terrible, Theorem 3.27 is proved. Now it is shown how to get rid of the obstacle of terrible points. Suppose P is a terrible m-ple point of F; then, as above, d ≡ m (mod p). Take a triangle with sides not containing P and with vertices not lying on the curve if d 6≡ 0 (mod p), but with a single vertex on the curve at a non-singular point if d ≡ 0 (mod p). Apply the quadratic transformation ω relative to this triangle as fundamental triangle; it is shown that ω takes P to a non-terrible point Q of the geometric transform F ′ . Note that F ′ has degree 2d or 2d − 1 according

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as the vertices of the triangle are all external to F or as one is a non-singular point of F. If Q were terrible, in the first case d 6≡ 0 (mod p) and 2d − m ≡ 0 (mod p); in the second case, d ≡ 0 (mod p) and 2d − 1 − m ≡ 0 (mod p). But both cases are impossible in that they contradict d ≡ m (mod p). Therefore Q is not terrible. So it has been shown that, if there is a singular terrible point P on F, the strategy is to first apply the above quadratic transformation ω to give a new curve F ′ , the geometric transform of F, for which the corresponding point Q is no longer terrible. Using this, Theorem 3.27 is finally proved. E XAMPLE 3.31 Let q = 2s and let K = Fq . For u < 2s , let D be the irreducible plane curve v(g(X, Y )) such that s

s

(X u + 1)(Y 2 + 1) + (Y u + 1)(X 2 + 1) = g(X, Y )(X + 1)(Y + 1)(X + Y ).

(3.47)

Let g1 (X, Y ) = g(X + 1, Y + 1). Then s

s

((X + 1)u + 1)Y 2 + ((Y + 1)u + 1)X 2 = g1 (X, Y )XY (X + Y ).

(3.48)

The analysis of the singular points of D is as follows: (i) E = (1, 1) is an ordinary (2s − 2)-ple point, with the appropriately distinct s tangents v(Y + mX + m + 1), where m2 −1 = 1 but m 6= 1; (ii) for each b with bu = 1 but b 6= 1, the point P = (b, 1) is a (2s − 1)-ple point with a single tangent; (iii) for each c with cu = 1 but c 6= 1, the point P = (1, c) is a (2s − 1)-ple point with a single tangent; (iv) for all b, c with bu = cu = 1 but b 6= 1, c 6= 1, the point P = (b, c) is a 2s -ple point with a single tangent. Let ω be the standard quadratic transformation (X, Y ) 7→ (X −1 , Y −1 ). Then the geometric transform D′ of D is the curve v(g2 (X, Y )) of degree 2s − 2, where s

(X u−1 + . . . + 1)X 2

−u

s

+ (Y u−1 + . . . + 1)Y 2 = g2 (X, Y )(X + Y ).

′

−u

(3.49) ′

Then the curve D has only ordinary singularities. Also, the genus of D , and so of C, is (u − 2)(2s − u − 2). The analysis of the singular points of D′ is as follows: (1) O = (0, 0) is an ordinary (2s − u − 1)-ple point; (2) E = (1, 1) is an ordinary (u − 1)-ple point. No point at infinity is a singular point. So, if P = (b, c) is a singular point of D′ , then ∂g2 /∂X = 0 at P , and so s

(1 + b)u−1 b2

−u−1

= 0.

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Then b = 0 or 1; similarly, c ∈ {0, 1}. So neither P = (0, 1) nor Q = (1, 0) lies on D′ , whence D′ has only two singular points, O = (0, 0) and E = (1, 1). More precisely, O = (0, 0) is an ordinary (2s − u − 1)-ple point with tangents s v(Y + λX), where λ2 −u = 1 but λ 6= 1. To find the tangents to D′ at E = (1, 1), consider the translation, (X, Y ) 7→ (X + 1, Y + 1),

that takes E to O. Then D′ becomes D′′ = v(g3 (X, Y )), where s

X u (X + 1)2

−u

s

s

+ X 2 + Y u (Y + 1)2

−u

s

+ Y 2 = g3 (X, Y )(X + Y ).

This shows that E is a (u − 1)-ple point with tangents v(Y + λX + λ + 1), where λu = 1 but λ 6= 1. r E

Ar

L

D

r

r M

rC

r

N

r r r B

F

Figure 3.1 The Pascal hexagon

3.6 EXERCISES 1. Let F = v(X1 X02 − X23 ) and G = v(X1 X04 − X25 ). Show that the common points of F and G are P1 = (1, 0, 0), P2 = (1, 1, 1), P3 = (1, −1, −1), P4 = (0, 0, 1),

where

  3 when i = 1, 1 when i = 2, 3, I(Pi , F ∩ G) =  10 when i = 4.

2. (Pascal’s Theorem) Assume that the points of intersection L, M, N of the opposite sides of a hexagon ABCDEF in PG(2, K) are distinct. Show that these points, the diagonal points, are collinear if and only if the hexagon is inscribed in a conic. 3. (Maclaurin’s Theorem) Let F be an irreducible projective plane cubic curve, and let P be any non-singular point of F. If P is not an inflexion point of F, the tangential point P ′ of P is the unique point of F on the tangent line

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to F at P other than P . If P is an inflexion point, let P ′ = P . If P, Q, R are three non-singular collinear points of F, show that their tangential points P ′ , Q′ , R′ are also collinear. 4. (Lam´e’s Theorem) Let ℓ1 , ℓ2 , ℓ3 and r1 , r2 , r3 be two triples of distinct lines in PG(2, K). Assume that no line from one triple passes through the common point of two lines from the other triple. For 1 ≤ j, k ≤ 3, let Rjk denote the common point of the lines ℓj and rk . There are nine such common points. Show that if eight of these points lie on a cubic curve then the ninth also does. This is a particular case of the Theorem of the Nine Associated Points that two projective plane cubic curves with eight common points have a ninth common point; it is used in Section 9.9. 5. Generalise the preceding exercise. Let N = 12 n(n + 3), and consider a set S of N distinct common points of two projective plane curves of degree n. Then either S contains a point P and a subset S ′ of N − 2 points distinct from P such that any curve of degree n through S ′ also passes through P , or, for every point P ∈ S, any curve of degree n through S\{P } also passes through P . 6. For two polynomials F, G ∈ K[X, Y ] and a point P , the ideal generated by F and G in the local ring OP is (F, G) · OP . Show that the residue class ring OP /((F, G) · OP ) is a finite-dimensional vector space over K whose dimension dimK OP /((F, G) · OP ) satisfies the postulates (I 1), . . . , (I 7) studied in Section 3.1. Note that, by Theorem 3.8, dimK OP /((F, G) · OP ) = I(P, F ∩ G). 3.7 NOTES Section 3.1 is based on Scherk [382], which in turn is a variant of Fulton [135, Chapter 3]. See also the latter for more on the local ring approach as in Remark 3.10. The proof of B´ezout’s Theorem 3.14 is based on van der Waerden [484, Chapter 3]; see also [166, Chapter 2]. For other proofs, see [135], [269], [400], [497]. For Sections 3.3, 3.4, 3.5, see Fulton [135, Chapter 7] and Moreno [336].

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Chapter Four Branches and parametrisation From now on, only projective curves are considered, and the adjective ‘projective’ is omitted. Concepts such as the degree of a curve, the multiplicity of a point on a curve, and the number of singular points of a curve are all projective invariants. The resolution of singularities in Chapter 3 required a broader class of transformations, namely birational transformations: projective transformations were not sufficient. Any projective invariant is also a birational invariant. So the question arises of the existence of birational invariants; that is, non-negative integers that are constant under a birational transformation that does not necessarily fix a projective invariant. Such an invariant must have the property that some points in the plane have no image under a transformation. The aim of this chapter is to introduce the concept of a branch of a curve F and show that every branch of F is transformed to a branch of F ′ , the geometric transform of F under a birational transformation. So, this concept can be viewed as a birational invariant; but, more importantly, it leads to the idea of the genus of a curve, which is the most useful birational invariant; see Chapters 5 and 6. Every point of the curve defines exactly one branch, but finitely many branches can arise from a singular point. This, and other properties of branches, are investigated by means of formal power series rather than polynomials.

4.1 FORMAL POWER SERIES In this section, the relevant facts on formal power series are presented. Let K[[X, Y ]] be the set of elements of the form a00 + a10 X + a01 Y + a20 X 2 + · · · , aij ∈ K, in which two operations, the sum and the product, are defined as follows: (a00 + a10 X + a01 Y + · · · ) + (b00 + b10 X + b01 Y + · · · )

= (a00 + b00 ) + (a10 + b10 )X + (a01 + b01 )Y + · · · ; (a00 + a10 X + a01 Y + · · · ) × (b00 + b10 X + b01 Y + · · · ) = a00 b00 + (a00 b10 + a10 b00 )X + · · · .

Then (K[[X, Y ]], +, ×) is a commutative ring whose elements are called formal power series. The identity is represented by the element 1 + 0X + 0Y + · · · and the zero by the element 0 + 0X + 0Y + · · · . A formal power series can be written

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in the form F = F0 + F1 + F2 + · · · , where Fi is a homogeneous polynomial in X and Y of degree i. With this convention, if G = G0 + G1 + G2 + · · · , then F + G = (F0 + G0 ) + (F1 + G1 ) + · · · , F G = (F0 G0 ) + (F0 G1 + F1 G0 ) + · · · .

(4.1) (4.2)

If F = Fr + Fr+1 + · · · , with Fr 6= 0, then r is called the order or subdegree of F. The element 0 = 0 + 0X + 0Y + · · · does not have an order since there is no first non-zero term; the convention is to write ord 0 = ∞. Then, for F, G ∈ K[[X, Y ]], ord (F + G) ≥ min{ord F, ord G}, ord (F G) = ord F + ord G. Since K[[X, Y ]] is an integral domain, it has a quotient field K((X, Y )) = {F/G | F, G ∈ K[[X, Y ]], G 6= 0}, called the quotient field of formal power series. Pn Given F1 , F2 , . . . ∈ K[[X, Y ]], the definition of i=0 Fi follows P from the def∞ inition of the sum of two power series, (4.1). However, to define i=0 Fi , the following procedure suffices. Let Fi = Fi0 + Fi1 + · · · + Fij + · · · , and suppose that ord Fi → ∞ for i → ∞; then define P∞ P∞ P∞ i=0 Fi = i=0 Fi0 + i=0 Fi1 + · · · . P∞ This make sense since each sum i=0 Fij is well-defined. The following properties hold: P∞ P∞ P∞ (4.3) i=0 (Fi + Gi ); i=0 Gi = i=0 Fi + P∞ P∞ (4.4) F i=0 Gi = i=0 F Gi . T HEOREM 4.1 The units of K[[X, Y ]] are the formal power series of order zero.

Proof. Let F be a unit in K[[X, Y ]]. Then there exists G ∈ K[[X, Y ]] such that F G = 1; hence 0 = ord (1) = ord (F G) = ord F + ord G. Since both ord F and ord G are non-negative, then ord F = ord G = 0. To show the converse, let ord F = 0; then, with a00 6= 0, F = a00 + a10 X + a01 Y + · · ·   a10 a01 = a00 1 + X+ Y + ··· a00 a00 = a00 (1 − G(X, Y )), with ord G(X, Y ) > 0; that is, F = a00 (1 − G).

(4.5)

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Now, note if ord G = r, then ord Gi = ir and so ord Gi → ∞ for i → ∞. P∞that, i Hence 0 G makes sense. So, multiplying (4.5) by this gives the following: F (1 + G + G2 + · · · + Gi + · · · ) = a00 (1 − G)(1 + G + G2 + · · · + Gi + · · · ) P∞ = a00 (1 − G) i=0 Gi P∞ = a00 i=0 (1 − G)Gi P∞ = a00 i=0 (Gi − Gi+1 ) = a00 (1 − G + G − G2 + · · · ) = a00 .

Dividing by a00 gives F (a00 −1 (1 + G + G2 + · · · + Gi + · · · )) = 1;

that is, F is invertible with inverse, a00 −1 (1 + G + G2 + · · · + Gi + · · · ).

2

The preceding ideas and results for two indeterminates can be extended to any finite number of indeterminates, including the case of a single indeterminate. Some further properties of K[[t]] = {F (t) = ar tr + ar+1 tr+1 + · · · , ai ∈ K}

are now given. First, if ord F = r, write with E(t) invertible.

F = tr (ar + ar+1 t + · · · ) = tr E(t),

T HEOREM 4.2 The ring K[[t]] is a unique factorisation domain. Proof. Every element has the form f = tr E(t), with E(t) invertible and r ≥ 1. If t were reducible, then t = tr1 E1 (t)tr2 E2 (t) with r1 , r2 ≥ 1. But this implies that 1 = ord t = r1 + r2 , a contradiction. If f is irreducible, then r = 1; further, tE(t) and t are associates. Therefore every complete factorisation of f ∈ K[[t]] has exactly r = ord f factors of the form t or tE(t). Hence, the only complete factorisation of f is t · . . . · t · tF (t), for a unit F (t). 2 D EFINITION 4.3 The field, K((t)) = {F/G | F, G ∈ K[[t]], G 6= 0}  s  t E1 (t) = r = ts−r E(t) = tm E(t) , t E2 (t)

with E1 (t), E2 (t), E(t) invertible in K[[t]] and m an integer, is the field of rational functions of the formal power series in the indeterminate t. The definition of ord F extends to K((t)): when F = tm E(t) ∈ K((t)) with E(t) invertible in K[[t]], then ord F = m. Now, all the K-monomorphisms and K-automorphisms of K[[t]] are determined. Here, a K-homomorphism of K[[t]] is a mapping of the ring to itself preserving addition and multiplication, and fixing every element of K: it is a Kmonomorphism if it is injective, that is, it has zero kernel. A K-automorphism is a

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surjective K-monomorphism, that is, a bijective K-homomorphism. If τ ∈ K[[t]], τ 6= 0 with ordt τ ≥ 1, then the substitution t 7→ τ is the mapping K[[t]] → K[[t]], P∞ P∞ i i i=0 ci τ . i=0 ci t 7→

(4.6)

The order of the substitution is the order of τ . It is immediate that such a mapping is a K-monomorphism of K[[t]]. The following theorem shows that every Kmonomorphism of K[[t]] is of this type. All this extends to K((t)). T HEOREM 4.4 (i) The K-monomorphisms of K[[t]] are precisely the substitutions t 7→ τ . (ii) A K-monomorphism of K[[t]] is a K-automorphism if and only if the associated substitution has order 1. (iii) Any K-monomorphism of K((t)) induces a K-monomorphism of K[[t]], and conversely. Proof. It is first shown that, if the substitution t 7→ τ has order 1, then it is surjective. Given g = g0 + g1 t + g2 t2 + · · · in K[[t]], it is opportune to ensure the existence of an element f = f0 + f1 t + f2 t2 + · · · in K[[t]] such that f 0 + f 1 τ + f 2 τ 2 + · · · = g 0 + g 1 t + g 2 t2 + · · · .

Equivalently, with τ = d1 t + d2 t2 + d3 t3 + · · · , the coefficients f0 , f1 , · · · can be chosen with f0 = g0 and successively f2 d21

f1 d1 = g1 , + f1 d2 = g2 ,

f3 d31 + f2 2d1 d2 + f1 d3 = g3 , .. . . Since fi di1 appears in the i-th row, but there is no other term in this row containing fi , these equations determine the coefficients f0 , f1 , f2 , . . . uniquely, and so f is obtained. It remains to consider a K-monomorphism σ of K[[t]], and show that it is obtained by a substitution t 7→ τ , for a suitable element τ ∈ K[[t]] of order at least 1. First, note that σ transforms every invertible element of K[[t]] to another invertible element of K[[t]]. In fact, if f ∈ K[[t]] is invertible, there exists g such that f g = 1; then σ(f )σ(g) = σ(1) = 1, and so σ(f ) is invertible. The invertible elements have order zero; so σ sends elements of order zero to elements of order zero. Now, it is shown that σ(t) has positive order. If it were otherwise, put σ(t) = c0 + c1 t + · · · ; then σ(−c0 + t) = c1 t + · · ·

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with c0 6= 0, while σ(−c0 + t), being of order 0, must be invertible; this is a contradiction. It follows that σ sends non-invertible elements to non-invertible elements in K[[t]]; in other words, σ sends elements of positive order to elements of positive order. In fact, if f = tn E(t) with n ≥ 1 and E(t) is invertible, then σ(f ) = τ n F (t) with ordt τ ≥ 1, and F (t) = σ(E(t)) is invertible. More precisely, σ sends an element of order v to one of order v · ordt τ , and fixes the orders if and only if ordt τ = 1. This last observation also shows that, if σ is an automorphism, then σ(t) must have order 1. Now, since σ is a K-monomorphism, then, for every positive integer r, Pr Pr Pr Pr σ( i=0 ci ti ) = i=0 ci σ(ti ) = i=0 ci (σ(t))i = i=0 ci (τ )i . More must be shown, namely that P∞ P∞ P∞ σ( i=0 ci ti ) = i=0 ci σ(t)i = i=0 ci (τ )i . For this, write

P∞

i=0

P ci ti = ( ri=0 ci ti ) + Rr ,

P∞ with Rr = i=r+1 ci ti ∈ K[[t]] and ord Rr > r. Therefore P∞ Pr Pr σ( i=0 ci ti ) = σ( i=0 ci ti ) + σ(Rr ) = i=0 ci (τ )i + σ(Rr ).

(4.7)

Since ord σ(Rr ) ≥ ord Rr > r, it follows from (4.7) that
P Pr i i > r. ordt σ( ∞ i=0 ci t ) − i=0 ci τ Then, since r can be chosen arbitrarily, this implies that P∞ P∞ σ( i=0 ci ti ) − i=0 ci τ i = 0,

as required. To prove that every K-monomorphism σ of K((t)) induces a K-monomorphism of K[[t]], it suffices to add just a remark to the above proof, namely that σ(t) has non-negative order. Assume on the contrary that ν = ordt σ(t) < 0. Then ordt σ(t + c) = ordt (σ(t) + c) = ν. On the other hand, ordt σ(t + c) = 0 for c ∈ K\{0}, as t + c is invertible in K[[t]] for c ∈ K\{0}, and hence σ(t + c) is also invertible in K[[t]]. But this contradicts the hypothesis that ν < 0. The converse follows from (i) since any substitution defines a K-monomorphism of K((t)). 2 It is worth noting explicitly that the preceding proof also shows the following result. T HEOREM 4.5 Let f (t) ∈ K[[t]] and g(τ ) ∈ K[[τ ]]. If f (t) = g(τ ) for some τ ∈ K[[t]], then ordt f = ordτ g · ordt τ.

(4.8)

The following result relates polynomials and power series. T HEOREM 4.6 Let F (X, Y ) ∈ K[X, Y ] with F (0, 0) = 0. Let ∂F/∂Y 6= 0 at (0, 0). Then there is a unique f (X) = c1 X + c2 X 2 + · · · in K[[X]] such that F (X, c1 X + c2 X 2 + · · · ) = 0.

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Proof. Without loss of generality, assume that ∂F/∂Y = 1 at (0, 0), Then F (X, Y ) = Y + a10 X + · · · ∈ K[X, Y ], ∂F Z + ··· , F (X, Y + Z) = F (X, Y ) + ∂Y where the partial derivatives are calculated with respect to X and Y . Since ∂F = 1 + G(X, Y ) ∂Y with ord G(X, Y ) > 0, it follows that F (X, Y + Z) = F (X, Y ) + [1 + G(X, Y )]Z + G2 Z 2 + · · · .

(4.9)

First, the uniqueness of f is shown. Suppose that both the formal power series c1 X + · · · + ci−1 X i−1 + ci X i + · · · , c1 X + · · · + ci−1 X i−1 + di X i + · · · satisfy the conditions of the theorem. Substitute Y = c1 X + · · · + ci−1 X i−1 , Z = ci X i + · · · in (4.9): 0 = F (X, c1 X + · · · + ci−1 X i−1 + ci X i + · · · )

= F (X, c1 X + · · · + ci−1 X i−1 ) + [1 + G(X, c1 X + · · · + ci−1 X i−1 )] ×(ci X i + · · · ) + X i+1 H(X).

Hence it follows that ci is uniquely determined by the coefficients c1 , . . . , ci−1 . But this implies that ci = di for all i. For the existence of F , construct inductively for each i a polynomial, c1 X + · · · + c i X i ,

such that ord F (X, c1 X + · · · + ci X i ) > i. For i = 1, put c1 = −a10 . Suppose then that a polynomial c1 X + · · · + ci−1 X i−1 has been determined with ord F (X, c1 X + · · · + ci−1 X i−1 ) > i − 1.

Write F (X, c1 X + · · · + ci−1 X i−1 + ci X i ) = F (X, c1 X + · · · + ci−1 X i−1 ) + [1 + G(X, c1 X + · · · + ci−1 X i−1 )]ci X i +X i+1 H(X) = (dX + · · · ) + ci X i + X i+1 H1 (X). i

Now, choose ci = −d to ensure that F (X, c1 X + · · · + ci X i + · · · ) = 0. In fact, by (4.9), F (X, c1 X + · · · + ci−1 X i−1 + ci X i + · · · ) = F (X, c1 X + · · · + ci−1 X i−1 ) + X i H2 (X).

By the inductive hypothesis, ord F (X, c1 X + · · · + ci−1 X i−1 ) > i − 1, whence ord F (X, c1 X + · · · + ci X i ) > i. Since this is true for all i, the only possibility is that F (X, c1 X + c2 X 2 + · · · ) = 0. 2

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It may be noted that the condition in this theorem can be expressed geometrically: the algebraic curve F = v(F (X, Y )) has a non-singular point at the origin and the tangent there is not the Y -axis. Since Y = c1 X + c2 X 2 + · · · is a solution in K[[X]] of the equation F (X, Y ) = 0, it follows from Theorem 4.6 that F (X, Y ) = [Y − (c1 X + c2 X 2 + · · · )]F1 (X, Y ), with F1 (X, Y ) ∈ K[[X]][Y ]. Also, F1 (0, 0) 6= 0, since the origin is a simple point of the curve; that is, ord F1 = 0. It follows that

ord F (X, 0) = ord (c1 X + c2 X 2 + · · · ). So the following theorem has been proved. T HEOREM 4.7 Let F be a plane curve passing through the origin O. If O is a non-singular point of F and the tangent to F at O is not the Y -axis, then the intersection multiplicity of F and v(Y ) at O is I(O, F ∩ v(Y )) = ord (c1 X + c2 X 2 + · · · ).

This theorem is generalised as follows. T HEOREM 4.8 Let F = v(F (X, Y )) and G = v(G(X, Y )) be two plane curves with no common components, and suppose that F has a non-singular point at the origin with tangent different from the Y -axis. If c1 X + c2 X 2 + · · · is the unique formal power series such that F (X, c1 X + c2 X 2 + · · · ) = 0, then I(O, F ∩ G) = ord G(X, c1 X + c2 X 2 + · · · ).

When O is a singular point of F, the study of I(O, F ∩ G) is more involved. It is postponed to Section 4.4 because it requires the idea of a branch, or branch point, as developed in Section 4.2. Meanwhile, some more results on power series must be established; these are essential for the study of I(O, F ∩ G), as well. Interestingly, these technical results are also the main ingredients in the generalisation of Theorem 4.2 to any number of indeterminates; see Theorem 4.16. L EMMA 4.9 (Hensel) Let F (X, Y ) = Y n + a1 (X)Y n−1 + · · · + an (X) ∈ K[[X]][Y ]

be a monic polynomial of degree n > 0 in Y with coefficients a1 (X), . . . , an (X) in K[[X]]. Suppose that F (0, Y ) = G(Y )H(Y ), where G(Y ) = Y r + b1 Y r−1 + · · · + br ,

H(Y ) = Y s + c1 Y s−1 + · · · + cs , are monic polynomials of degree r > 0 and s > 0 in Y with coefficients in K such that gcd(G(Y ), H(Y )) = 1. Then there exist two unique monic polynomials G(X, Y ) = Y r + b1 (X)Y r−1 + · · · + br (X),

H(X, Y ) = Y s + c1 (X)Y s−1 + · · · + cs (X) in K[[X]][Y ] of degree r, s in Y with coefficients b1 (X), . . . , br (X) and c1 (X), . . . , cs (X) in K[[X]] such that G(0, Y ) = G(Y ), H(0, Y ) = H(Y ), F (X, Y ) = G(X, Y )H(X, Y ).

(4.10)

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Proof. Write F = F (X, Y ) as a power series in X whose coefficients are polynomials in Y : F = F0 (Y ) + F1 (Y )X + · · · + Fm (Y )X m + · · · , where F0 (Y ) = F (0, Y ) ∈ K[Y ] is monic with deg F0 = n and Fm (Y ) ∈ K[Y ] with deg Fm < n for all m > 0. It is necessary to find G = G0 (Y ) + G1 (Y )X + · · · + Gi (Y )X i + · · · ,

H = H0 (Y ) + H1 (Y )X + · · · + Hj (Y )X j + · · · , with G0 = G0 (Y ) = G(Y ) ∈ K[Y ] monic of degree r, H0 = H0 (Y ) = H(Y ) ∈ K[Y ] monic of degree s,

Gi = Gi (Y ) ∈ K[Y ] and deg Gi (Y ) < r for each i > 0, Hj = Hj (Y ) ∈ K[Y ] and deg Hj (Y ) < s for each j > 0,

such that F = GH. It should be noted that F = GH means that P Fm = i+j=m Gi Hj for each m ≥ 0.

The proof proceeds by induction on m. For m = 0, the result is that P F0 = i+j=0 Gi Hj ,

F0 = F0 (Y ) = F (0, Y ) = G(Y )H(Y ) = G0 (Y )H0 (Y ).

Let m > 0. Assume that there exist Gi , Hj ∈ K[Y ], with deg Gi < r and deg Hj < s for 1 ≤ i, j < m satisfying the equation (4.10) for values smaller than m. Then it is necessary to find Gm , Hm in K[Y ] with deg Gm < r, deg Hm < s satisfying P Fm = i+j=m Gi Hj . This equation may be written as follows:

G0 Hm + H0 Gm = Um ; here, Um = Fm −

P

i+j=m i,j 0,

H0 (Y ) = Y d and Hj (Y ) ∈ K[Y ] with deg Hj (Y ) < d for j > 0,

such that F (X, Y ) = G(X, Y )H(X, Y ). Now, F (X, Y ) = G(X, Y )H(X, Y ) means that P Fm (Y ) = i+j=m Gi (Y )Hj (Y ) for each m ≥ 0.

(4.11)

Proceed by induction on m. Since ord F (0, Y ) = d, with H0 (Y ) = Y d , there exists a unique G0 (Y ) ∈ K[[Y ]] with G0 (0) 6= 0 such that F0 (Y ) = F (0, Y ) = G0 (Y )H0 (Y ).

So the case that m = 0 has been established. Let m > 0, and suppose that there exist Gi (Y ) ∈ K[[Y ]] for 1 ≤ i < m and Hj (Y ) ∈ K[Y ] with deg Hj (Y ) < d for 1 ≤ j < m that satisfy (4.11). So, the task is to find Gm (Y ) ∈ K[[Y ]] and Hm (Y ) ∈ K[Y ] with deg Hm (Y ) < d that satisfy the equation P Fm (Y ) = i+j=m Gi (Y )Hj (Y ). This equation may be written as follows:

G0 (Y )Hm (Y ) + H0 (Y )Gm (Y ) P = Um (Y ) = Fm (Y ) −

i+j=m i,j 0, and Hj (Y ), with deg Hj (Y ) < d, that satisfy (4.11) have been calculated. To prove the uniqueness, again proceed by induction. Since G0 (Y ) is unique, suppose that Gi (Y ) and Hj (Y ) are unique for i < m and j < m. Then, by the construction of Gm (Y ) and Hm (Y ), their uniqueness follows. 2 A generalisation of Theorem 4.10 to more variables may be stated as follows.

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T HEOREM 4.11 If F (X1 , . . . , Xn ) ∈ K[[X1 , . . . , Xn ]] and F (0, . . . , 0, Xn ) = cν Xnν + cν+1 Xnν+1 + · · · , cν 6= 0, then there exists a unique unit G in K[[X1 , . . . , Xn ]] such that GF = Xnν + A1 (X1 , . . . , Xn−1 )Xnν−1 + · · · + Aν (X1 , . . . , Xn−1 ) (4.12) with Ai (X1 , . . . , Xn−1 ) ∈ K[[X1 , . . . , Xn−1 ]] and Ai (0, . . . , 0) = 0 for each i = 1, . . . , ν. An element of K[[X1 , . . . , Xn ]] that contains a term in which Xn appears is regular in Xn ; hence F is regular in Xn . A polynomial in Xn with form (4.12) is special. T HEOREM 4.12 (Weierstrass Division Theorem) Given F (X1 , . . . , Xn , Y ) and H(X1 , . . . , Xn , Y ) in K[[X1 , . . . , Xn , Y ]], assume that F (0, . . . , 0, Y ) 6= 0 and d = ordY F (0, . . . , 0, Y ). Then there exist unique A, B such that H = AF + B, with A ∈ K[[X1 , . . . , Xn , Y ]], B ∈ K[[X1 , . . . , Xn ]][Y ], and degY B < d. Proof. Proceed by induction on n. The case n = 1 coincides with Theorem 4.10. For n > 1, write H = H0 (X2 , . . . , Xn , Y ) + H1 (X2 , . . . , Xn , Y )X1 + · · · , A = A0 (X2 , . . . , Xn , Y ) + A1 (X2 , . . . , Xn , Y )X1 + · · · , F = F0 (X2 , . . . , Xn , Y ) + F1 (X2 , . . . , Xn , Y )X1 + · · · ,

B = B0 (X2 , . . . , Xn , Y ) + B1 (X2 , . . . , Xn , Y )X1 + · · · . Then the required equation H = AF + B is equivalent to the following: H0 = A0 F0 + B0 , H1 = A0 F1 + A1 F0 + B1 , .. . Hi = A0 Fi + · · · + Ai−1 F1 + Ai F0 + Bi , .. .. Since F (0, · · · , 0, Y ) 6= 0, so F0 (0, · · · , 0, Y ) 6= 0. Also, ordY F (0, . . . , 0, Y ) = d implies that ordY F0 (0, . . . , 0, Y ) = d. Since B(X1 , . . . , Xn , Y ) ∈ K[[X1 , . . . , Xn ]][Y ] and degY B < d, then also Bi (X1 , . . . , Xn , Y ) ∈ K[[X1 , . . . , Xn ]][Y ] and degY Bi < d for all i ≥ 0. By induction, Ai and Bi are found successively; so they exist and are unique by construction. Therefore the same is true for A and B. 2 The proof of Theorem 4.11 follows from Theorem 4.12 by taking H = Y d . The following corollaries are a consequence of the general Weierstrass Preparation Theorem 4.10.

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C OROLLARY 4.13 Given F ∈ K[[X1 , . . . , Xn ]] with

F (0, . . . , 0, Xn ) = cν Xnν + cν+1 Xnν+1 + · · · , cν 6= 0,

and A′ F ∈ K[[X1 , . . . , Xn−1 ]][Xn ] of degree < ν, with A′ ∈ K[[X1 , . . . , Xn ]], then A′ = 0. Proof. The proof follows from Theorem 4.12. Putting H = 0, it follows that A′ F = B ′ and F 6= 0 by the uniqueness of A and B. Hence A′ = 0. 2 C OROLLARY 4.14 Let F be a special polynomial in K[[X1 , . . . , Xn−1 ]][Xn ] of degree ν and let A′ ∈ K[[X1 , . . . , Xn ]]. If A′ F = B ′ ∈ K[[X1 , . . . , Xn−1 ]][Xn ] has degree ν, then A′ ∈ K[[X1 , . . . , Xn−1 ]]. Proof. Let B ′ = CXnν + · · · , with C ∈ K[[X1 , . . . , Xn−1 ]]. Then CXnν − A′ F = CXnν − B ′

has degree less than ν. Put H = CXnν . Then the relation H − AF = B is satisfied not only by A = A′ and B = CXnν − B ′ but also by A = C and B = H − CF. From Theorem 4.12, since A and B are unique, it follows that A′ = C. 2 T HEOREM 4.15 If F is special in Xn and irreducible in K[[X1 , . . . , Xn ]], then F is irreducible in K[[X1 , . . . , Xn−1 ]][Xn ]. Proof. Suppose that F is reducible in K[[X1 , . . . , Xn−1 ]][Xn ] with F = G1 G2 , where degXn Gi < degXn F. Since F is irreducible in K[[X1 , . . . , Xn ]], then one of the Gi is a unit in this ring, If it is G1 , then G−1 1 F = G2 and, by Corollary 4.13, G2 = 0, which is impossible. 2 T HEOREM 4.16 The ring K[[X1 , . . . , Xn ]] is a unique factorisation domain. Proof. For n = 1, the assertion is Theorem 4.2. The proof is by induction on n; so K[[X1 , . . . , Xn−1 ]][Xn ] isP assumed to be a unique factorisation domain. If F = G1 · · · Gm , then ord F = ord Gi . This shows that any non-unit F has a factorisation into irreducible non-units. Let E1 F1 · · · Fs = E2 G1 · · · Gt be two complete factorisations of F in K[[X1 , . . . , Xn ]], where E1 , E2 are units but Fi and Gi are non-units. Then the following occur. (i) Applying a linear homogeneous substitution X = A(X′ ) with X = (X1 , . . . , Xn ) 7→ X′ = (X1′ , . . . , Xn′ ),

A ∈ PGL(n + 1, K),

it may be assumed that Fi and Gi are regular in Xn . (ii) By Weierstrass’s Preparation Theorem 4.10 , and absorbing the extra units into E1 and E2 , it may be assumed that Fi and Gi are special in Xn . Suppose that F = G1 G2 with degXn Gi < degXn F. Since F is irreducible in K[[X1 , . . . , Xn ]], one of the Gi is a unit, say G1 . Then G−1 1 F = G2 and, by

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Theorem 4.13, G2 = 0. This is impossible, and hence the following has been shown. (iii) If F is special in Xn and irreducible in K[[X1 , . . . , Xn ]], then F is also irreducible in K[[X1 , . . . , Xn−1 ]][Xn ]. This has the following consequence. (iv) Multiplying both sides of E1 F1 · · · Fs = E2 G1 · · · Gt by E1−1 , it may be assumed that E1 = 1. Theorem 4.13 shows that deg F1 · · · Fs < deg G1 · · · Gt is not possible. On the other hand, E2−1 F1 · · · Fs = G1 · · · Gt shows that deg F1 · · · Fs > deg G1 · · · Gt is impossible, as well. Hence the degrees are equal, and E2 = 1 from Theorem 4.14. Now, the unique factorisation in K[[X1 , . . . , Xn−1 ]][Xn ] implies that s = t and that, up to relabelling of subscripts, Fi and Gi are associates in K[[X1 , . . . , Xn−1 ]][Xn ]. As the units in this ring are also units in K[[X1 , . . . , Xn ]], it follows that Fi and Gi are also associates in K[[X1 , . . . , Xn ]]. 2

4.2 BRANCH REPRESENTATIONS D EFINITION 4.17 A branch representation is a point of the plane PG(2, K((t))) not belonging to PG(2, K). Given a point (x0 (t), x1 (t), x2 (t)) of PG(2, K((t))), it is therefore a branch representation if ρ(t)xi (t) 6∈ K for at least one i and for all ρ(t) ∈ K((t))\{0}. −1 −1 If x0 (t) 6= 0, put x(t) = x1 (t)x0 (t) , y(t) = x2 (t)x0 (t) ; then the branch representation becomes the point (x(t), y(t)) of the affine plane AG(2, K((t))) = PG(2, K((t)))\ℓ∞ , where ℓ∞ is the line at infinity of AG(2, K((t)). Let (x0 (t), x1 (t), x2 (t)) be a branch representation and let m = min{ordt xi (t)}. Then, with ξi (t) = t−m xi (t), the branch representation (ξ0 (t), ξ1 (t), ξ2 (t)) has the order of one of its components zero and the other two non-negative; it is a special branch representation and its coordinates are also special. In this case, let xk (t) = ck + ck1 t + . . . + cki ti + . . . , k = 0, 1, 2; the point P = (c0 , c1 , c2 ) of PG(2, K) is the centre of the branch representation. Now, the further notion of the order of a branch representation is introduced. With (x0 (t), x1 (t), x2 (t)) already in special coordinates, the order is the positive integer min{ordt (a0 x0 (t) + a1 x1 (t) + a2 x2 (t))},

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for all points (a0 , a1 , a2 ) ∈ PG(2, K) with a0 c0 +a1 c1 +a2 c2 = 0. If x0 (t), x1 (t), x2 (t) are linearly independent over K, then ordt (a0 x0 (t) + a1 x1 (t) + a2 x2 (t)) is zero or positive according as a0 c0 + a1 c1 + a2 c2 is non-zero or zero. Also, it takes one of two positive values, j1 and j2 , where j1 < j2 ; the value is j1 or j2 according as a0 c01 + a1 c11 + a2 c21 is non-zero or zero. In fact, ordt (a0 x0 (t) + a1 x1 (t) + a2 x2 (t)) = j2 for a unique point (a0 , a1 , a2 ), in which case the line v(a0 X0 + a1 X1 + a2 X2 ) of PG(2, K) is the tangent of the branch representation. The triple (0, j1 , j2 ) is the order sequence of the branch representation and j1 is its order. When x0 (t), x1 (t), x2 (t) are linearly dependent over K, the same holds true provided that j2 is defined to be ∞. The notion of branch representation is projective in the sense that every projective transformation of PG(2, K) transforms a branch representation into a branch representation; also, special coordinates become special coordinates. Similarly, the notions of order and tangent are projective invariants. If one of the special coordinates is 1, then ξ0 (t) is invertible. Hence, every branch representation can be expressed in special affine coordinates (x(t), y(t)) with x(t) = u + u1 t + · · · ,

y(t) = v + v1 t + · · · ,

with centre the point (u, v). This notion is preserved under a linear transformation of the affine plane AG(2, K). In what follows, the branch representations mostly considered are those in special affine coordinates, as these simplify the exposition. D EFINITION 4.18

(i) Two branch representations in special affine coordinates (x(t), y(t))

and (ξ(t), η(t))

are equivalent if there exists a K-automorphism σ of K[[t]] such that x(t) = σ(ξ(t))

and y(t) = σ(η(t)).

(ii) Two branch representations are equivalent if they are so when put in the form of special affine coordinates. In other words, (x(t), y(t)) and (ξ(t), η(t)) are equivalent branch representations if there exists a substitution t 7→ τ of order 1 such that x(t) = ξ(τ ) and y(t) = η(τ ). D EFINITION 4.19

(i) A branch representation in special affine coordinates (x(t), y(t))

is imprimitive if there exists a branch representation in special affine coordinates (ξ(t), η(t)) such that x(t) = σ(ξ(t))

and y(t) = σ(η(t)),

with σ a K-monomorphism of K[[t]] that is not a K-automorphism.

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(ii) A branch representation is imprimitive if it is so when put in the form of special affine coordinates. This definition can also be stated in terms of substitutions: (x(t), y(t)) is imprimitive if there exists a branch representation in special affine coordinates (ξ(t), η(t)) such that x(t) = ξ(τ ) and y(t) = η(τ ) for a suitable τ ∈ K[[t]] with ordt τ > 1. If this is the case, then (x(t), y(t)) has ramification index at least ν = ordt τ > 1. The reason for this definition is that ξ(t), η(t) have, in general, simpler expressions than x(t), y(t). Now, if (ξ(t), η(t)) is also imprimitive, then a further branch representation (ξ1 (t), η1 (t)) with ramification index ν1 > 1 is obtained, and so on until a primitive branch representation is obtained. As this happens in a finite number of steps, the following result is reached. T HEOREM 4.20 Let (x(t), y(t)) and (ξ(t), η(t)) be two branch representations in special affine coordinates. If (a) x(t) = σ(ξ(t)), y(t) = σ(η(t)), (b) σ is a K-monomorphism of K[[t]], (c) ν = ordt σ(t), then ordt (x(t), y(t)) = ν · ordt(ξ(t), η(t)). Proof. If (u, v) is the centre of (x(t), y(t)), then, since σ is a K-monomorphism of K[[t]], the same point (u, v) is also the centre of (ξ(t), η(t)). Therefore, given a non-trivial triple (a0 , a1 , a2 ), with ai ∈ K satisfying a0 + a1 u + a2 v = 0, then (x(t), y(t)) has order ordt (a0 + a1 x(t) + a2 y(t)) and (ξ(t), η(t)) has order ordt (a0 + a1 ξ( t) + a2 η(t))). Now, ordt (a0 + a1 x(t) + a2 y(t)) = ordt (a0 + a1 σ(ξ(t)) + a2 σ(η(t))) = ordt σ(a0 + a1 ξ(t) + a2 η(t)). By Theorem 4.5, this is the required result.

2

Since every product of K-monomorphisms of K[[t]] is also such a K-monomorphism, for each imprimitive branch representation in special affine coordinates (x(t), y(t)) there exists a primitive branch representation in special affine coordinates (ξ(t), η(t)) such that x(t) = σ(ξ(t)), y(t) = σ(η(t)), where σ is a K-monomorphism of K[[t]] that is not a K-automorphism. To show the uniqueness, three further results are required that are interesting in themselves. T HEOREM 4.21 Let (x(t), y(t)) be a branch representation in special affine coordinates. (i) If τ is of positive minimum order in the subfield K(x(t), y(t)) of K((t)), then the order of each element z ∈ K(x(t), y(t)) is divisible by g = ordt τ .

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(ii) The element z can be expanded in a formal power series in τ with coefficients in K. (iii) In the particular case that (x(t), y(t)) is primitive, there is an element of order 1 in K(x(t), y(t)). Proof. (i) If µ is the order of z, the Euclidean algorithm gives integers q, r such that µ = qg + r and 0 ≤ r < g. Hence z/τ q is an element of order r belonging to K(x(t), y(t)). Since τ has minimum order, then r = 0, as required. (ii) Now, write z/τ q = c + z1 with c ∈ K, c 6= 0 and z1 ∈ K[[t]]; then z1 is also in K(x(t), y(t)). As in the previous argument, there is a positive integer q1 such that z1 /τ q1 = c1 + z2 . Hence z = cτ q + c1 τ q+q1 + τ q+q1 z2 . Continuing this procedure gives the required formal power series. (iii) In particular, the same coordinates x(t) and y(t) become formal power series in τ , so that, if (x(t), y(t)) is primitive, τ is necessarily of order 1. 2 T HEOREM 4.22 Given a branch representation (x(t), y(t)) in special affine coordinates, let (x1 (t), y1 (t)) and (x2 (t), y2 (t)) also be branch representations in special affine coordinates, of which the first is primitive. If the relations x(t) = x1 (τ1 ) = x2 (τ2 ),

y(t) = y1 (τ1 ) = y2 (τ2 ),

(4.13)

with τ1 , τ2 ∈ K[[t]], are satisfied, then τ1 can be written as a formal power series in τ2 . Further, if (x2 (t), y2 (t)) is also primitive, then ordt τ1 = ordt τ2 . Proof. Let τ be an element of minimum positive order in K(x1 (τ1 ), y1 (τ1 )). By Theorem 4.21 and the primitivity of (x1 (t), y1 (t)), this order must be 1. Therefore τ = cτ1 + · · · with c 6= 0, whence τ = σ(τ1 ) for some σ that is a K-automorphism of K[[t]]. Since σ −1 is also such an automorphism, so also τ1 = dτ + · · · with d 6= 0. Since τ is a rational function of the elements x1 (τ1 ) and y1 (τ1 ), there exist polynomials a(X, Y ), b(X, Y ) ∈ K[X, Y ] such that τ=

a(x1 (τ1 ), y1 (τ1 )) . b(x1 (τ1 ), y1 (τ1 ))

τ=

a(x2 (τ2 ), y2 (τ2 )) . b(x2 (τ2 ), y2 (τ2 ))

From (4.13), it follows that

Therefore τ = e τ2 i + · · · with i ≥ 0. Hence τ1 = de τ2 i + · · · ; that is, τ1 is a formal power series in τ2 of positive order. When (x2 (t), y2 (t)) is primitive, then their roles may be interchanged, giving that τ2 is a formal power series in τ1 . But this is possible if and only if τ1 is an element of order 1 in K[[τ2 ]]. 2 From this theorem and Theorem 4.20, the following result is obtained.

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T HEOREM 4.23 Let (x(t), y(t)) be an imprimitive branch representation in special affine coordinates. If (x1 (t), y1 (t)) and (x2 (t), y2 (t)) are branch representations in special affine coordinates satisfying the relations x(t) = x1 (τ1 ) = x2 (τ2 ),

y(t) = y1 (τ1 ) = y2 (τ2 ),

with τ1 , τ2 ∈ K[[t]], then (x1 (t), y1 (t)) and (x2 (t), y2 (t)) have the same order. Now the idea of ramification index can be introduced. D EFINITION 4.24 (i) If (x(t), y(t)) is an imprimitive branch representation in special affine coordinates and if (ξ(t), η(t)) is a primitive branch representation in special affine coordinates with x(t) = σ(ξ(t)), y(t) = σ(η(t)), then ν = ordt σ(t) is the ramification index of (x(t), y(t)). (ii) The ramification index of a branch representation is its ramification index in its special affine coordinate form. Theorem 4.23 thus has the following corollary. C OROLLARY 4.25 Equivalent imprimitive branch representations have the same ramification index. Further, the primitive representations to which they give rise are also equivalent. A sufficient condition for imprimitivity is reducibility, where a branch representation in special affine coordinates (x(t), y(t)), and any equivalent one, is reducible if, in Definition 4.19, σ(t) = tm ; that is, x(t), y(t) ∈ K[[tm ]] for an integer m > 1. In fact, the adjective ‘reducible’ signifies ‘imprimitivity for a substitution of type t 7→ τ with τ m = t’. A useful criterion for reducibility is given by the following theorem. T HEOREM 4.26 Let p = 0 or p > 0 with p ∤ n. A branch representation in special affine coordinates, where a1 6= 0 and 0 < n1 < n2 < · · · , x(t) y(t)

= u + tn , = v + a1 tn1 + a2 tn2 + · · · ,

(4.14)

is reducible if and only if gcd(n, n1 , n2 , . . .) > 1. Proof. That the condition is sufficient is immediate. To show the necessity, suppose that the representation is reducible. Then there exists a substitution t 7→ τ of order 1 such that x(τ ), y(τ ) ∈ K[[tm ]] with m > 1. Next it is shown that τ /t ∈ K[[tm ]]. Suppose, by way of contradiction, that τ = t(b0 + b1 tm + · · · + bh thm + cts + · · · )

with c 6= 0 but with m not dividing s. It follows that

x(τ ) = u + tn (b0 + b1 tm + · · · + bh thm + cts + · · · )n = u + tn (b0 + · · · + bh thm )n−1 + · · · .

Since x(τ ) ∈ K[[tm ]], it is immediate that m | n. Further, m | (n + s) since n 6≡ 0 (mod p). Hence m | s, a contradiction.

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Therefore write τ = zt with z = u0 + u1 tm + · · · ∈ K[[tm ]] and u0 6= 0. Now, suppose that at least one of the exponents n1 , n2 , . . . is not divisible by m, and let nk+1 be the smallest such exponent. Then y(t) − (v + a1 tn1 z n1 + · · · + ak tnk z nk ) = ank +1 tnk +1 (u0 + u1 tm + · · · )nk +1 + · · ·

= ank +1 un0 k +1 tnk +1 + · · · . This shows that the first expression, but not the second, belongs to K[[tm ]]. This is a contradiction, which shows that m must divide not only n but also each exponent n1 , n2 , . . .; this finishes the proof. 2 T HEOREM 4.27 Let p = 0 or p ∤ n. Every branch representation of order n has a special affine coordinate form of the type x(t) = u + tn ,

y = v + η(t)

(4.15)

with ordt η(t) ≥ n. Proof. Let x(t) = u + un tn + uk tk + · · · , y(t) = v + y1 (t), with un 6= 0, ordt y1 (t) ≥ n, be a branch representation. First, the equation τ n = u n tn + u k tk + · · · (4.16) is solved for τ ∈ K[[t]]. The case n = 1 already provides a solution; so, suppose that n > 1. Dividing both sides of (4.16) by tn gives the equation z n = un + uk tk−n + · · · (4.17) in the unknown z ∈ K[[t]]. Since p ∤ n, the polynomial Z n − un certainly does not have multiple roots, and therefore has at least two distinct roots. By Hensel’s Lemma 4.9, Z n − (un + uk tk−n + · · · ) is reducible in K[[t]][Z]; that is, Z n − (un + uk tk−n + · · · ) = (Z r + ρ(Z))(Z s + σ(Z)), where ρ(Z), σ(Z) ∈ K[[t]][Z]. Write

Z r + ρ(Z) = Z r + ρ1 (t)Z r−1 + · · · + ρr (t), ρi ∈ K[[t]]. The polynomial g(Z) = Z r + ρ1 (0)Z r−1 + · · · + ρr (0), with coefficients in K, divides Z n − un . Hence g(Z) has no multiple roots and so at least two distinct roots. Applying Hensel’s Lemma 4.9 iteratively shows that the polynomial

Z n − (un + uk tk−n + · · · ) splits into linear factors in K[[t]][Z]. Let Z − ζ(t) be one of these factors; note that ordt ζ(t) = 0. Then ζ(t)n − (un + uk tk−n + · · · ) = 0, n whence (ζ(t)t) = un tn + uk tk · · · . Putting τ = ζ(t)t gives x = u + τ n . Since ordt ζ(t)t = 1, the relation τ = ζ(t)t gives t; that is, there exists an element f (τ ) ∈ K[[τ ]] such that t = f (τ ). Finally, putting η(τ ) = y1 (f (τ )) gives x = u + τ n , y = v + η(τ ), with ordτ η(τ ) ≥ n, as required. 2

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T HEOREM 4.28 Let p = 0 or p ∤ n. A branch representation of order n is imprimitive if and only if it is reducible. Proof. It suffices to show that the imprimitivity implies the reducibility. By Theorem 4.27, the branch representation in special affine coordinates (x(t), y(t)) may be taken in the form (4.15). If (x(t), y(t)) is imprimitive, then there exists a representation (ξ(τ ), η(τ )) with ξ(τ ), η(τ ) ∈ K[[τ ]] such that, for a suitable element τ = ci ti + · · · , i > 1, x(t) = ξ(τ ),

y(t) = η(τ ).

By Theorem 4.20, the order of the primitive representation (ξ(τ ), η(τ )) is n/i, which is coprime to p; write m = n/i. Therefore, still by Theorem 4.27, it may be supposed that (ξ, η) is of type (4.15), that is, ξ(τ ) = u + τ m , Hence

η(τ ) = v + b1 τ m1 + b2 τ m2 + · · · . tn = τ m ,

a1 t

n1

+ a2 t

n2

+ · · · = b 1 τ m1 + b 2 τ m2 + · · · .

(4.18) (4.19)

From (4.18) it follows that (ti /τ )m = 1, whence τ = cti since c is an m-th root of unity; in particular c ∈ K\{0}. Now, (4.19) implies that nj = imj for every j. This shows that gcd(n, n1 , n2 , . . .) > 1; hence (x(t), y(t)) is reducible. 2 Finally, in this section, the definition of a branch is given; it is one of the central concepts in the following sections. D EFINITION 4.29 A branch is an equivalence class of primitive branch representations. The centre, the order, and the order sequence of a branch are the centre, order, and the order sequence of any branch representation.

4.3 BRANCHES OF PLANE ALGEBRAIC CURVES The next definition is the foundation for developing the study of algebraic plane curves based on the idea of branch; in fact, branches constitute an important tool for the local study of algebraic curves. D EFINITION 4.30 A branch of a plane curve F = v(F (X0 , X1 , X2 )) is a branch whose representations (x0 (t), x1 (t), x2 (t)) are zeros of F (X0 , X1 , X2 ). T HEOREM 4.31 The centre of a branch of a plane curve is a point of the curve. Proof. Let F = v(F ) be a plane curve and let (ξ0 (t), ξ1 (t), ξ2 (t)) be any special branch representation; then F (ξ0 (t), ξ1 (t), ξ2 (t)) = 0. Putting t = 0 gives F (ξ0 (0), ξ1 (0), ξ2 (0)) = 0. Therefore the point P = (ξ0 (0), ξ1 (0), ξ2 (0)), which is the centre of the branch representation, is a point of the curve F.

2

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T HEOREM 4.32 There exists a unique branch of a plane curve F centred at a simple point of F. Proof. Without loss of generality, let the point be the origin and let the tangent there be any line other than v(X). If F = v(F (X, Y )), then, by Theorem 4.26, the representation x = t,

y = c 1 t + c 2 t2 + · · ·

is primitive and represents a branch of F centred at the origin. To show the uniqueness, let (ξ(t), η(t)) be a primitive branch representation of F centred at the origin; then ord ξ > 0, ord η > 0 and F (ξ, η) = 0. If the origin O is a simple point of F and the tangent O to F is not v(X), by Theorem 4.6, there exists a unique formal power series c1 X + c2 X 2 + · · · such that F (X, c1 X + c2 X 2 + · · · ) = 0. Then F (X, Y ) factorises in K[[X]][Y ]: F (X, Y ) = (Y − (c1 X + c2 X 2 + · · · ))H(X, Y ), with ord H(X, Y ) = 0. In K[[t]], F (ξ, η) = (η − (c1 ξ + c2 ξ 2 + · · · ))H(ξ, η) = 0.

Hence η = c1 ξ + c2 ξ 2 + · · · . Since (ξ, η) is a primitive representation, ξ must be of order 1. The substitution σ : t → ξ is of order 1, and so, by Theorem 4.4, is an automorphism of K[[t]]; its inverse σ −1 is also an automorphism. Also, σ −1 sends ξ to t and c1 ξ + · · · to (t, c1 t + c2 t2 + · · · ). It follows that (ξ, c1 ξ + · · · ) is equivalent to the branch representation (x(t), y(t)) introduced at the beginning of the proof; so the uniqueness is established. 2 E XAMPLE 4.33 Let Hq = v(Y q + Y − X q+1 ), with q a power of p, be the Hermitian curve. Then the origin is a simple point and the tangent to H at the origin is v(Y ). A primitive representation of the unique branch of Hq at the origin is (x = t, y = y(t)) with P∞ i+1 i y(t) = i=1 (−1)i+1 tq +q . D EFINITION 4.34 Let G = v(G) be a plane curve and γ a branch centred at the point P . If (ξ0 (t), ξ1 (t), ξ2 (t)) is a representation of γ in special coordinates, then the intersection multiplicity is  ordt G(ξ0 (t), ξ1 (t), ξ2 (t)) if γ ∈ / G, I(P, G ∩ γ) = ∞ if γ ∈ G. Note that I(P, G ∩ γ) depends only on γ, and not on the representation chosen.

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T HEOREM 4.35 The intersection number I(P, G ∩ γ) is covariant. P Proof. For i = 0, 1, 2, let Xi = 2j=0 aij Xj′ be the equation of the change of the coordinate system (X0 , X1 , X2 ) into (X0′ , X1′ , X2′ ). If P2 P2 P2 G′ (X0 , X1 , X2 ) = G( j=0 a0j Xj′ , j=0 a1j Xj′ , j=0 a2j Xj′ ) P2 and xi (t) = j=0 aij x′j (t) for i = 0, 1, 2, then ordt G(x0 (t), x1 (t), x2 (t)) = ordt G′ (x′0 (t), x′1 (t), x′2 (t)).

If G(x0 (t), x1 (t), x2 (t)) = 0, then I(P, G ∩ γ) = ∞.

2

The following result shows how Definition 4.34 appears naturally in the study of the intersection multiplicity at a common point of two curves. T HEOREM 4.36 (i) If the branch γ of the curve F is centred at a simple point P of F, and G is any other curve, then I(P, G ∩ γ) = I(P, G ∩ F). (ii) More generally, if P is a singular point of the irreducible curve F and G is a plane curve not containing F as a component, then P I(P, G ∩ F) = γ I(P, G ∩ γ), where γ runs over all branches of F centred at P.

(iii) If P is an mP -fold point of F then mP is a bound on the number of branches of F with centre P . Before dealing with Theorem 4.36 it needs to be shown that every point of F is the P centre of at most a finite number of branches. This ensures that the definition γ I(P, G ∩ γ) is meaningful. The proof requires some more results on the behaviour of branches under local quadratic transformations. These results are stated in the next section after proving the following result. T HEOREM 4.37 Two distinct irreducible plane curves have no branches in common. Proof. Suppose, by way of contradiction, that the two curves C n and C m have a branch γ in common. Let (ξ(t), η(t)) be a branch representation; it is a point of the plane over K((t)) and so is a common point of the two curves regarded as curves of the plane over K((t)). Applying B´eP zout’s Theorem 3.14 over the field K, the two curves being irreducible, gives that I(P, C n ∩ C m ) = nm. Similarly, B´ezout’s Theorem 3.14 over K((t)) gives the same result. Hence, the two curves have all their intersections in the smaller field, contradicting that, over K((t)), they have the point (ξ(t), η(t)) in common. 2 C OROLLARY 4.38 Every branch (ξ, η) with ξ = 0 is a branch of the line v(X), but of no other irreducible curve.

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4.4 LOCAL QUADRATIC TRANSFORMATIONS The study of branches centred at a singular point of a plane curve is helped by local quadratic transformations, that is, transformations of the form σ(X, Y ) = (X ′ , Y ′ ), X ′ = X, Y ′ = Y /X.

(4.20)

It is not defined for points on the line v(X), but the point Y∞ = (0, 1, 0) is taken to be the image of these points, other than the origin. Let (ξ(t), η(t)) be a branch representation with centre the origin and ξ(t) 6= 0. Then the branch representation (ξ(t), η(t)/ξ(t))

is the image of the branch (ξ(t), η(t)) under σ, and so a branch representation corresponds to another branch representation. This gives the following result. T HEOREM 4.39 The local quadratic transformation σ given by (4.20) defines a bijective correspondence between branches centred at the origin with tangent not v(X) and those centred at any affine point on v(X). For a point P = (a, b), with a 6= 0, on a plane curve F = v(F (X, Y )), the image of P under σ is on the curve v(F¯ (X, Y )), where F¯ (X, Y ) = F (X, XY ). This gives the following idea. D EFINITION 4.40 The algebraic transform of a curve F = v(F (X, Y )) is the curve F¯ = v(F¯ (X, Y )), where F¯ (X, Y ) = F (X, XY ). A useful property is the following relation: given a branch γ centred at P and an algebraic plane curve G, let γ ′ and G¯ be their images under a local quadratic transformation. If P¯ is the centre of γ ′ , then I(P, G ∩ γ) = I(P¯ , G¯ ∩ γ ′ ). (4.21) ′ ′ ¯ To show (4.21), put G = v(G(X, Y )), G¯ = v(G(X, Y )), and let (x (t), y (t)) be a primitive representation of γ ′ . Then ¯ ′ (t), y ′ (t)), ¯ y(t)/x(t)) = ordt G(x ordt G(x(t), y(t)) = ordt G(x(t), showing (4.21). Every polynomial F (X, Y ) ∈ K[X, Y ] can be written in the form F (X, Y ) = Fr (X, Y ) + Fr+1 (X, Y ) + · · · , Fr 6= 0,

with Fi (X, Y ) homogeneous of degree i. The curve F = v(F (X, Y )) passes through the origin O if and only if r ≥ 1; in this case, O is a point of F of multiplicity r. Since F¯ (X, Y ) = F (X, XY ) = Fr (X, XY ) + Fr+1 (X, XY ) + · · · = X r (Fr (1, Y ) + XFr+1 (1, Y ) + · · · ),

so the algebraic transform of F has X r as a component. For this reason, it is appropriate to eliminate X r and introduce the notion of the geometric transform.

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D EFINITION 4.41 The geometric transform of a curve F = v(F (X, Y )) with an r-ple point, with r ≥ 0, at the origin is the curve F ′ = v(F ′ (X, Y )) given by F ′ (X, Y ) = F¯ (X, Y )/X r = F (X, XY )/X r .

Now the first, important property of the geometric transform is established. T HEOREM 4.42 If F is irreducible and is not the line v(X), then the geometric transform F ′ is also irreducible. Proof. Let F (X, Y ) be irreducible and different from cX, c ∈ K. It must be shown that F ′ (X, Y ) = F (X, XY )/X r is irreducible. Suppose that F ′ (X, Y ) = G(X, Y )H(X, Y ) with G(X, Y ), H(X, Y ) nonconstant polynomials. First, consider the possibility that H(X, Y ) = h(X). Then F (X, XY ) = X r h(X)G(X, Y ). Putting Y = Z/X gives F (X, Z) = X r h(X)G(X, Z/X). Let d be the minimum value for which X d G(X, Z/X) ∈ K[X, Z]; then X r−dh(X) ∈ K[X],

whence h(X) = h1 X d−r + h2 X d−r+1 + · · · . Therefore F (X, Z) = (h1 + h2 X + · · · )(X d G(X, Z/X)). Since F (X, Y ) contains Y , then F (X, Z) must contain Z and consequently is not a constant. The irreducibility of F (X, Z) therefore implies that h(X) = h1 X d−r . This means that F (X, XY ) = X d G(X, Y ). Then, from the definition of r, necessarily d = r; that is, h(X) is a constant, as required. Finally, the possibility that H(X, Y ) or G(X, Y ) contains Y must be excluded. If it were so, then F (X, Z) = X r G(X, Z/X)H(X, Z/X) = [X m G(X, Z/X)][(X r−mH(X, Z/X)], where both factors in square brackets are polynomials. But they are not constant polynomials since they contain Z and that contradicts the irreducibility of both F (X, Z) and F (X, Y ). 2 As a corollary, there is the following result. T HEOREM 4.43 Let F1 = v(F1 ), . . . , Fs = v(Fs ) be irreducible curves, with each Fi in K[X, Y ] and Fi 6= v(X) for all i. Then the geometric transform of m m F = v(F1m1 · · · Fsms ) is F ′ = v(F1′ 1 · · · Fs′ s ). Now, Theorem 4.39 may be made more precise.

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T HEOREM 4.44 If F is a plane curve such that the line v(X) is not tangent to F at O and F ′ is the geometric transform of F by the local quadratic transformation σ as in (4.20), then there exists a bijection between the branches of F centred at O and the branches of F ′ centred at an affine point of v(X). Proof. Considering Corollary 4.38, it follows from Theorem 4.37 that the branches of F and those of F ′ admit a representation (ξ(t), η(t)) with ξ( t) 6= 0. Further, if (ξ(t), η(t)) satisfies F (X, Y ) = 0, then (ξ(t), η(t)/ξ(t)) satisfies F (X, XY ) = 0 and also F (X, XY )/X r = 0; and conversely. This gives a bijective correspondence between the branches of F and F ′ . To show that the branches of F ′ obtained in this way are centred at affine points of v(X), it suffices to see that ord ξ(t) ≤ ord η(t); that is, ξ(t) = ai ti + · · · ,

η(t) = bj tj + · · ·

with i ≤ j. Suppose instead that j < i. The hypothesis that v(X) is not tangent to F at O implies that Y r is present in F (X, Y ), whence, with cr 6= 0, Fr (X, Y ) = c0 X r + c1 X r−1 Y + · · · + cr Y r .

So ord Fr (ξ(t), η(t)) = jr, but ord Fr+k (ξ(t), η(t)) ≥ (r + 1)j for k > 0. Therefore ord F (ξ(t), η(t)) = rj, which contradicts that F (ξ(t), η(t)) = 0. This finishes the proof. 2 T HEOREM 4.45 If O is an ordinary singular point on the curve F = v(F (X, Y )) of multiplicity r, then (i) it is the centre of exactly r branches of F; (ii) the branches are all linear, and their tangents are precisely the tangents to F at O. Proof. With O at the origin and no vertical tangent to F at O, Qr F (X, Y ) = i=1 (Y − mi X) + · · · .

Then, the geometric transform F ′ = v(F ′ (X, Y )) of F under the local quadratic transformation (4.20) is given by Qr F ′ (X, Y ) = i=1 (Y − mi ) + XH(X). The line v(X) meets F ′ in the points Qi = (0, mi ) for i = 1, . . . , r. Now, it is shown that these points are all simple on F ′ . Choose Q1 , say, and take it to (0, 0) via the translation (X, Y ) 7→ (X, Y − m1 ). The transformed curve is G = v(G(X, Y )) with G(X, Y ) = F ′ (X, Y + m1 ) Q = rj=1 ((Y + m1 ) − mj ) + XH1 (X)

= Y (Y − (m2 − m1 )) · · · (Y − (mr − m1 )) + XH2 (X) Q = (−1)r−1 Y ri=2 (mi − m1 ) + · · · .

Since all the mi − m1 are non-zero, Qr it follows that G(X, Y ) contains a single term of degree 1 given by (−1)r−1 Y i=2 (mi − m1 ); this shows that Q1 is simple for

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v(G), and so also for F ′ . From Theorem 4.32 it now follows that there are precisely r branches of F ′ with centre at an affine point on v(X). By the previous theorem, F has exactly r branches centred at the origin. By Theorem 4.32 the branch of F ′ centred at Qi has a representation (t, mi + · · · ). The corresponding branch of F therefore has a representation (t, mi t + · · · ). Since the tangent of such a branch is the line v(Y − mi X), the theorem is proved. 2 T HEOREM 4.46 Each point P of F is the centre of at least one and at most a finite number of branches of F. Proof. Since each branch of F = v(F ) must annul at least one irreducible factor Fi , the polynomial F may be supposed irreducible. In fact, let F = F 1 ∪ F2 ∪ · · · ∪ Fl

with each Fi = v(Fi ) an irreducible component of F; then

F (X, Y ) = F1 (X, Y ) · · · Fl (X, Y ).

Let γ = (ξ(t), η(t)) be a branch of F; so F (ξ(t), η(t)) = 0, or F1 (ξ(t), η(t)) · · · Fl (ξ(t), η(t)) = 0.

Hence Fi (ξ(t), η(t)) = 0 for at least one i and so γ is a branch of Fi for that i. It may also be supposed that P , of multiplicity r, is at the origin and that the line Q v(X) is not tangent to F at P. Then, F = ri=1 (Y − mi X) + · · · with the mi not necessarily distinct. Applying σ to F gives F ′ = v((F ′ (X, Y ′ )/X r ) with Qr F ′ (X, Y ′ )/X r = i=1 (Y ′ − mi ) + XH(X) = 0.

Therefore the line v(X) meets F ′ in the points Q′i (0, mi ), and the sum of the intersection multiplicities at these points is r; that is, Ps ′ ′ (4.22) i=1 I(Qi , v(X) ∩ F ) = r. If at least two of the mi are distinct, each point (0, mj ) of F ′ has multiplicity less than r. By an induction argument, with the assumption of the existence of at least one and at most a finite number of branches centred at the point (0, mj ), then F ′ has at least one and at most a finite number of branches centred at an affine point on v(X). By Theorem 4.44, F has at least one and at most a finite number of branches centred at O. This leaves the case that all the mi are equal: F (X, Y ) = (Y − m1 X)r + · · · , r > 1.

In this case, F ′ = v(F ′ (X, Y ′ )/X r ) with

F ′ (X, Y ′ )/X r = (Y ′ − m1 )r + XH(X).

Now, the translation X = X, Y = Y ′ − m1 , which takes the point (0, m1 ) to (0, 0), transforms F ′ to v(Y r + XG(X, Y )). Hence, F ′ may be assumed to have the singular point of multiplicity r at the origin. With a similar argument to the above, by induction on r, it may be supposed that the curve F ′ = v(F ′ (X, Y )) with F ′ (X, Y ) = (Y − m2 X)r + H(X, Y ).

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Then, applying σ gives a plane curve F (2) = v(F (2) (X, Y )), which may be taken to have a point of multiplicity r at (0, m2 ), whence F (2) (X, Y ) = (Y − m2 )r + XH(X, Y ). Is it possible that an infinite sequence of points of multiplicity r is obtained? The answer is, in fact, in the negative, as is shown below. Therefore this procedure gives rise to a point that is not of multiplicity r. Induction now finishes the proof. To show that the answer is negative, introduce m3 , m4 , . . . through the curves F (2) , F (3) , . . . exactly as m1 and m2 were obtained in relation to F and F ′ . For each i ≥ 1, write F (X, Y ) in the form P F (X, Y ) = b>0 cab X a (Y − m1 X − m2 X 2 − · · · − mi X i )b + X hi Pi (X), (4.23) where cab ∈ K, Pi (X) ∈ K[X], Pi (0) 6= 0; this can be done uniquely. In the expansion of the sum, the term cab X a η b is not eliminated by other terms. Since F has a point of multiplicity r at the origin, it follows that a + b ≥ r. Also, hi ≥ r. Using the form (4.23) for F, the algebraic transform F¯ = v(F¯ (X, Y ′ )) is as follows: F¯ (X, Y ′ ) = F (X, XY ′ ) P = cab X a (XY ′ − m1 X − m2 X 2 − · · · − mi X i )b + X hi Pi (X) P = cab X a+b (Y ′ − m1 − m2 X − · · · − mi X i−1 )b + X hi Pi (X) = 0.

Since, by assumption, the geometric transform F ′ of F has a point of multiplicity r at the origin, so F ′ = v(F ′ ) has F ′ (X, Y ′ ) = F¯ (X, Y ′ )/X r P = cab X a+b−r (Y ′ − m1 − m2 X − · · · − mi X i−1 )b +X hi −r Pi (X) = 0.

Since hi (F) = hi (F ′ ) + r = (hi (F ′′ ) + r) + r, repeating the process i times gives hi (F) = hi (F i ) + ir, from which hi (F) ≥ ir. It follows that hi → ∞ as i → ∞. Substituting Y = m1 X + m2 X 2 + · · · in (4.23) gives P F (X, m1 X + m2 X 2 + · · · ) = cab X a (mi+1 X i+1 + · · · )b + X hi Pi (X). Hence

ordx F (X, m1 X + m2 X 2 + · · · ) ≥ min{i + 1, hi } for each i. Therefore ordx F (X, m1 X + m2 X 2 + · · · ) = ∞; in other words, there is a branch γ = (ξ = t, η = m1 t + m2 t2 + · · · ) centred at O, which is the first part of the theorem. The next proposition completes the proof of Theorem 4.46. 2 P ROPOSITION 4.47 The sequence of quadratic transformations appearing in the above proof does not give rise, for r > 1, to an infinite sequence of points of multiplicity r.

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Proof. Assume at first that the polynomial ∂F/∂Y does not vanish. Then the polar curve G of Y∞ = (0, 0, 1) with respect to F exists. Since F is irreducible, the branch γ = (ξ = t, η = m1 t + m2 t2 + · · · ) cannot be a branch of G by Theorem 4.37. Let G ′ be the polar curve of Y∞ with respect to the geometric transform F ′ of F under the local quadratic transformation. Then G ′ = v(∂F ′ /∂Y ′ ), and      ∂F (X, Y ) ∂F (X, XY ′ ) ∂Y ′ X r−1 = X r−1 ′ ∂Y ∂Y ∂Y ′ Y =XY   ∂ 1 ′ = F (X, XY ) , ∂Y ′ X r whence ∂F ∂F ′ . = X r−1 ∂Y ∂Y ′ 1

With O = (0, 0) the centre of γ and O′ = (0, m1 ) the centre of γ ′ : (ξ ′ = t, η ′ = η/ξ = m1 + m2 t + · · · ), then I(O, G ∩ γ) = ordt = ordt





∂F ∂Y



∂F ′ ∂Y ′

X=t,Y =m1 t+···



X ′ =t,Y ′ =m

+ ordt (X r−1 )X=t 1 +···

= I(O′ , G ′ ∩ γ ′ ) + ordt (tr−1 ) = I(O′ , G ′ ∩ γ ′ ) + r − 1. Therefore, if r > 1, I(O, G ∩ γ) > I(O′ , G ′ ∩ γ ′ ) > 0. Continuing this argument gives an infinite sequence of decreasing positive integers, a contradiction which proves the assertion under the initial hypothesis. If this hypothesis does not hold, then choose another non-tangent line to F to be the Y -axis. Then the polynomial ∂F/∂Y does not vanish any more. Now, the polar curve of F for Y∞ exists. By (4.24), the polar curve of F ′ for Y∞ also exists, and hence this proof still works. 2 Now Theorem 4.36 can be proved. To do this, define P J(P, G ∩ F) = γ I(P, G ∩ γ),

where γ ranges over all branches of F centred at P . Now, check that J(P, G ∩ F) satisfies the seven postulates in Chapter 3.1. By Theorem 4.35, J(P, G ∩ F) is covariant; so it may be assume that P coincides with the origin and that the Y -axis is not tangent to F at P . If F and G have no common component through P , then the branches of F centred at P are distinct from those of G centred at P . Hence J(P, G ∩ F) is a non-negative integer for every branch γ of F. This shows the validity of (I 1).

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If F and G have a common component, say C, through P , then any branch γ of C centred at P is a common branch of F and G. This implies that J(P, G ∩ F) = ∞ showing (I 2). If P is not a point of F, no branch of F is centred at P ; so J(P, G ∩ F) = 0. If P is a point of F and γ is a branch of F centred at P given by the primitive representation (x(t), y(t)), then ordt x(t) > 0, ordt y(t) > 0. Thus G(x(t), y(t)) = G(0, 0) + tH(t). If P is not a point of G, then G(0, 0) 6= 0. Hence J(P, G ∩ F) = ordt G(x(t), y(t)) = 0, and this shows (I 3). Also, (I 4) follows from Theorem 4.36 (i). To show (I 5), the following technical result is required. T HEOREM 4.48 Let the origin P be an r-fold point of F = v(F ) and an s-fold point of G = v(G) such that v(X) is not a tangent at P to F or G. Let F ′ and G ′ be the geometric transforms of F and G under a local quadratic transformation. If P1′ , . . . , Pn′ are the common affine points of F ′ and G ′ on v(X), then Pn J(P, F ∩ G) = rs + i=1 J(Pj′ , F ′ ∩ G ′ ).

Proof. Let γ1 , . . . , γm be the branches of G centred at P , and (xi (t), yi (t)) a primitive representation of γi for i = 1, . . . , m. Let the branches of G ′ centred at affine ′ points of v(X) are γ1′ , . . . , γm , and (x′i (t) = xi (t), yi′ (t) = yi (t)/xi (t)) is a prim′ itive representation of γi , for i = 1, . . . , m. Hence Pm J(P, F ∩ G) = i=1 ordt F (xi (t), yi (t)) Pm = i=1 ordt xi (t)r F ′ (x′i (t), yi′ (t)) Pm Pm = r i=1 ordt xi (t) + i=1 ordt F ′ (x′i (t), yi′ (t)) Pn = rs + i=1 J(Pi′ , F ′ ∩ G ′ ). 2

Now, (I 5) is shown by induction on k = J(P, F ∩ G). Since J(P, F ∩ G) = 0 if and only if P is not a common point of F and G, the assertion is true for k = 0. Now, assume it is true for every integer less than k, and let J(P, F ∩ G) = k. By Theorem 4.48, Pn J(P, F ∩ G) = rs + i=1 J(Pi′ , F ′ ∩ G ′ ). As rs ≥ 1, so J(Pi′ , F ′ ∩ G ′ ) < k. Hence J(Pi′ , F ′ ∩ G ′ ) = J(Pi′ , G ′ ∩ F ′ ), and so J(P, F ∩ G) = J(P, G ∩ F). The validity of (I 6) and (I 7) follows from the next lemma.

L EMMA 4.49 Let γ be a branch of F = v(F ) centred at the point P, and let G = v(G). If Φ = v(G + F H) with H ∈ K[X, Y ], then J(P, G ∩ γ) = J(P, Φ ∩ γ). If Λ is a reducible curve that splits into the components G and Ψ, then I(P, Λ ∩ γ) = I(P, G ∩ γ) + I(P, Ψ ∩ γ).

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Proof. Let (x(t), y(t)) be a primitive representation of γ. Then J(P, Φ ∩ γ) = ordt (G(x(t), y(t)) + F (x(t), y(t))H(x(t), y(t))) = ordt G(x(t), y(t)), whence the first assertion follows. Let Ψ = v(M ) and Λ = v(GM ). Then I(P, Λ ∩ γ) = ordt GM (x(t), y(t)) = ordt G(x(t), y(t))M (x(t), y(t)) = ordt G(x(t), y(t)) + ordt M (x(t), y(t)) = I(P, G ∩ γ) + I(P, Ψ ∩ γ), whence the second assertion follows.

2

The uniqueness of the intersection number, as in Theorem 3.8, gives the following important result. T HEOREM 4.50 Let F and G be curves. Then, for any point P, I(P, F ∩ G) = J(P, F ∩ G). It is possible to show directly that J(P, F ∩ G) = J(P, G ∩ F) when G is a line. To do this, fix the coordinate system in such a way that P = (0, 0), G = v(Y ) and that no tangents to F at P is v(X). By Theorem 4.36, if F = v(F ), then J(P, F ∩ v(Y )) = ordx F (x, 0). So, the following result is required: J(P, v(Y ) ∩ F) = ordx F (x, 0).

(4.24)

To prove this, suppose P to be an r-fold point of F with r > 1, as the assertion is true for r = 1 by Theorem 4.36. Then Qr F (X, Y ) = i=1 (Y − mi X) + G(X, Y ),

with ord G > r. Let γi , for i = 1, . . . , k, be the branches of F centred at P and let (xi (t), yi (t)) be a primitive representation of γi . Under a local quadratic transformation, γi is transformed into a branch γi′ of the geometric quadratic transform F ′ of F. With γi′ centred at Pi′ = (0, mi ), let (x′i (t), yi′ (t)) be a primitive representation of γi′ . Since the algebraic quadratic transform of G = v(Y ) is the conic v(XY ), from (4.21), Pk Pk Pk J(P, v(Y ) ∩ F) = i=1 ordt y(t) = i=1 ordt x′i (t) + i=1 ordt yi′ (t). By (4.22),

J(P, v(Y ) ∩ F) = r +

Pk

i=1

ordt yi′ (t).

In fact, as yi′ (t) = yi (t)/xi (t), the branches giving a positive contribution to the sum are those centred at P . If their number is s, relabel the indices i so that the branches of F ′ centred at P have indices 1, . . . , s. Then Ps J(P, v(Y ) ∩ F) = r + i=1 ordt yi′ (t) = r + J(P, v(Y ) ∩ F ′ ).

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Now, if a least two of Pi′ = (0, mi ) are distinct, then each point Pi′ = (0, mi ) is an ri′ -fold point of F ′ with ri′ < r, and induction can be applied on r. Hence, Pk ′ ′ i=1 ordt yi (t)r + ordx F (x, 0).

On the other hand, r = ordx xr . Thus

r + ordx F (x, 0) = ordx (xr F ′ (x, 0)) = ordx F (x, 0). Even if there is only one point Pi′ , this can be done when ri′ < r. Otherwise, there is only one Pi′ , say P ′ , and ri′ = r. In this case, take P ′ to the origin by a translation, and apply the local quadratic transformation. Perhaps r is reduced. If not, repeat the procedure. As in the proof of Theorem 4.46, this procedure ends in a finite number of steps. This completes the proof.

4.5 NOETHER’S THEOREM Let F = v(F (X, Y )) and G = v(G(X, Y )) be two curves with no common components. By B´ezout’s Theorem 3.14, F ∩ G = {P1 , . . . , Ps }. Given a polynomial H(X, Y ), it is proposed to find conditions for it to belong to the ideal generated by F and G; that is, H = AF + BG for some polynomials A and B. A necessary condition is that H vanishes at the points P1 , . . . , Ps . This condition is also sufficient if the multiplicity of F and G at each point Pi is equal to 1. The result is not proved directly, but follows from other results. D EFINITION 4.51 For a point P = (x0 , y0 ), the local ring OP at the point P is the ring of all rational functions defined at P ; that is, OP = {A/d | A, d ∈ K[X, Y ], d(P ) 6= 0}. D EFINITION 4.52 Given any point P of the plane, let qP = {H ∈ K[X, Y ] | H = (AF + BG)/d; d(P ) 6= 0; d, A, B ∈ K[X, Y ]}. P ROPOSITION 4.53 The set qP is an ideal. Proof. It must be shown that (i) if H1 and H2 are in qP , then (H1 + H2 ) ∈ qP ; (ii) if H ∈ qP and D ∈ K[X, Y ], then HD ∈ qP . Let H1 = (A1 F + B1 G)/d1 , H2 = (A2 F + B2 G)/d2 with d1 (P ), d2 (P ) 6= 0. Then (A1 F + B1 G) (A2 F + B2 G) CF + BG H1 + H 2 = + = d1 d2 h with C = d2 A1 + d1 A2 , D = d2 B1 + d1 B2 and 0 6= h(P ) = d1 (P )d2 (P ). Hence H1 + H2 ∈ qP . Also,   (DA)F + (DB)G AF + BG = , DH = D d d

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and so DH ∈ qP .

93 2

The following lemma is a consequence of Hilbert’s Nullstellensatz; see Exercise 7 in Chapter 2. L EMMA 4.54 Let {di (X, Y )} be a set of polynomials with P no common zeros. Then the ideal generated by the di (X, Y ) is (1); that is, 1 = Ai di for some polynomials Ai . D EFINITION 4.55 The polynomial H satisfies Noether’s conditions for F, G and P if H ∈ qP . T HEOREM 4.56 If H satisfies Noether’s conditions for F, G and for every point P, then H ∈ (F, G). Proof. For every point P , there is a polynomial dP such that dP (P ) 6= 0 and dP · H ∈ (F, G). From the previous Plemma, it follows that the ideal generated by the dP is equal to (1), whence 1 = AP dP . Since (F, G) is anP ideal and dP · H ∈ (F, G), so also AP · (dP H) ∈ (F, G) and hence H = 1H = ( AP dP )H ∈ (F, G). 2 T HEOREM 4.57 The ideal qP is (1) except for P = P1 , . . . , Ps .

Proof. Let P 6= Pi for i = 1, . . . , s. Then one of the polynomials F, G, say F, is not zero at P ; that is, F (P ) 6= 0. Since 1 = F/F , so 1 ∈ qP , where A = 1 and B = 0 in (AF + BG)/F. But, if 1 ∈ qP , then qP = (1). Now, let P = Pi and suppose that 1 ∈ qP = qPi . By the definition of qPi , 1 = (AF + BG)/d with d(Pi ) 6= 0. From this, d = AF + BG. However, as F and G pass through Pi , 2 so d(Pi ) = 0, a contradiction. So, qPi 6= (1). C OROLLARY 4.58 (Lasker–Noether) The ideal generated by F and G can be decomposed as follows: (F, G) = qP1 ∩ · · · ∩ qPs . Proof. If H ∈ (F, G) then H = AF + BG = (AF + BG)/1. TsTake d = 1; then d(Pi ) = 1 6= 0 for all Pi . So H ∈ qPi for each i; that is, H ∈ i=1 qPi . Hence (F, G) ⊂ qP1 ∩ · · · ∩ qPs .

Conversely, if H ∈ qP1 ∩ · · · ∩ qPs , then there exists a polynomial d such that d(Pi ) 6= 0 for each i = 1, . . . , s and H = (AF + BG)/d. So H satisfies Noether’s conditions by Theorem 4.56, giving that H ∈ (F, G); that is, qP1 ∩ · · · ∩ qPs ⊂ (F, G). T From the two inclusions, it follows that (F, G) = si=1 qPi

2

C OROLLARY 4.59 If P1 = (0, 0), then all powers of sufficiently high degree of X, Y and the ideal (X, Y ) are in qP1 .

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Proof. Write F and G in descending powers of Y , and calculate the resultant with respect to Y , that is, taking X as parameter. This gives a polynomial in X, namely RY (F, G) = AF + BG = X ρ (c0 + c1 X + · · · + cm X m ), with ci ∈ K, c0 6= 0 and A, B ∈ K[X, Y ]. Since P1 = (0, 0) belongs to both F and G, so ρ > 0. Then X ρ = (AF + BG)/(c0 + c1 X + · · · + cm X m ) ∈ qP1 ; so also all powers greater than ρ belong to qP1 . If, instead of considering X as unknown and Y as parameter, it is the other way round, then there exists σ > 0 such that Y σ ∈ qP1 . Let τ = ρ + σ − 1. It is shown that (X, Y )τ ⊂ qP1 . Now, (X, Y )τ = (X, Y ) · . . . · (X, Y ); if u ∈ (X, Y )τ then u = u1 · . . . · uτ with ui = Ai X + Bi Y ∈ (X, Y ) for i = 1, . . . , τ . So u = (A1 X + B1 Y ) · . . . · (Aτ X + Bτ Y ) = Cτ X τ + X τ −1 Y Cτ −1 + · · · + C1 XY τ −1 + C0 Y τ . It remains to see that X i Y j ∈ qP1 . Then, since qP1 is an ideal, it follows that u ∈ qP1 and so (X, Y )τ ⊂ qP1 . Since, in the powers of the product X i Y j with i + j = τ , either i ≥ ρ or j ≥ σ, since, if i ≤ (ρ − 1) and j ≤ (σ − 1), then τ = i + j ≤ ρ + σ − 2; but this is impossible by the definition of τ . Hence X i Y j ∈ qP1 . In fact, if i ≥ ρ, then it has 2 been shown above that X i ∈ qP1 . Since qP1 is an ideal, so X i Y j ∈ qP1 . T HEOREM 4.60 Let P1 = (0, 0). Then H is in the ideal qP1 if and only if there exist A, B ∈ K[[X, Y ]] such that H = AF + BG. Proof. Let H ∈ qP1 . Then H = (A1 F + B1 G)/d with A1 , B1 , d ∈ K[X, Y ] and d = d0,0 + d1,0 X + d0,1 Y + · · · , with d0,0 6= 0 because d(P1 ) 6= 0. Hence d is a unit in K[[X, Y ]] and also A = A1 d−1 , B = B1 d−1 are the required elements in K[[X, Y ]]. Conversely, suppose that H = AF + BG with A, B ∈ K[[X, Y ]]. Now, write A = Ar + A′r and B = Br + Br′ , where Ar and Br are the sums of the terms of degree less than or equal to r in A and B. So H = Ar F + Br G + A′r F + Br′ G = Ar F + Br G + Tr+1 , where Tr+1 contains only terms of degree greater than r. Since H and Ar F + Br G are polynomials, their difference Tr+1 is also a polynomial. By Corollary 4.59, Tr+1 ∈ qP1 for r sufficiently large. Further, Ar F + Br G ∈ qP1 , by taking d = 1, 2 and so H ∈ qP1 . Consider the ring K[[X]][Y ], and let G, H, f be three of its elements, where Q f = ri=1 (Y − mi X − · · · ) with mi 6= mj for i 6= j.

T HEOREM 4.61 If, for i = 1, . . . , r, ordX H(X, mi X + · · · ) ≥ ordX G(X, mi X + · · · ) + r − 1, then H = Af + BG with A, B ∈ K[[X]][Y ]. Also, B can be chosen so that ordX B ≥ r − 1.

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95

Proof. The theorem is proved by induction on r. For r = 1, ordX H(X, mi X + · · · ) ≥ ordX G(X, mi X + · · · ). Then H(X, m1 X + · · · ) = B(X)G(X, m1 X + · · · ); so Y = m1 X +· · · is a root of the polynomial H(X, Y )−B(X)G(X, Y ), whence H(X, Y ) − B(X)G(X, Y ) = A(X, Y )(Y − m1 X − · · · ). So the case r = 1 has been proved. Suppose the theorem is true in the following case: Qr f1 = i=2 (Y − mi X − · · · ), G1 = XG. By the inductive hypothesis, H = A1 f1 + B1 XG with ordX B1 ≥ r − 2. Since mi 6= mj for i 6= j, ordX A1 (X, m1 X + · · · ) + r − 1 = ordX (H(X, m1 X + · · · ) − B1 (X, m1 X + · · · )XG(X, m1 X + · · · )) ≥ ordX G(X, m1 X + · · · ) + r − 1. Here, use has been made of the following two inequalities: ordX H(X, m1 X + · · · ) ≥ ordX G(X, m1 X + · · · ) + r − 1; ordX (B1 (X, m1 X + · · · )XG(X, m1 X + · · · )) ≥ ordX G(X, m1 X + · · · ) + r − 1. It has therefore been shown that ordX A1 (X, m1 X + · · · ) ≥ ordX G(X, m1 X + · · · ); so A1 (X, Y ) = A2 (X, Y )(Y − m1 X − · · · ) + B2 (X, Y )G(X, Y ). Substituting the expression for A1 in H = A1 f1 + B1 XG gives that H(X, Y ) = [A3 (Y − m1 X − · · · ) + B3 G(X, Y )]f1 + B1 XG(X, Y )

= A4 f + [A5 f1 + B1 X]G(X, Y ) = Af + BG, where A3 , A4 , A5 , B1 , B3 ∈ K[X, Y ] and ordX B = ordX B1 + ordX X ≥ r − 2 + 1 = r − 1.

2

Let F = v(F (X, Y )) be a plane curve for which O = (0, 0) is an ordinary singular point with multiplicity r. Suppose that v(X) is not a tangent to F at O. Then Qr F (X, Y ) = i=1 (Y − mi X) + · · · , with mi 6= mj for i 6= j. From the previous chapter, the curve F has r branches at O that are of the following type: x = t, y = mi t + · · · . Then, by the definition of a branch, the point (X, mi X + · · · ) is a root of F (X, Y ); that is, F (X, mi X + · · · ) = 0. Hence F (X, Y ) is divisible by (Y − mi X − · · · ) in K[[X]][Y ] and, if mi 6= mj for i 6= j, then (Y −mi X −· · · ) and (Y −mj X −· · · ) are not associated. This proves the following result.

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C OROLLARY 4.62 The polynomial Qr F (X, Y ) = i=1 (Y − mi X − · · · ) · g(X, Y ),

with g ∈ K[[X]][Y ] and g(0, 0) 6= 0, where ordX F = r.

C OROLLARY 4.63 Let F = v(F (X, Y )) be a plane curve for which P1 is an ordinary singular point with multiplicity r, and let γ1 , . . . , γr be the branches of F centred at P1 . If G = v(G(X, Y )) and H = v(H(X, Y )) are two curves such that I(P1 , H ∩ γj ) ≥ I(P1 , G ∩ γj ) + r − 1 for j = 1, . . . , r, then H ∈ qP1 . Proof. By the previous corollary, Qr F (X, Y ) = i=1 (Y − mi X − · · · )g(X, Y ) Q with g(0, 0) 6= 0. With ri=1 (Y − mi X − · · · ) = f , then F = f g. Since g is a unit, f = g −1 F. Let γj = (x, y), with x y

= t, = mi t + · · · ,

be a branch of F. Then I(P1 , G ∩ γj ) = ordt G(t, mj t + · · · ) = ordX G(X, mj X + · · · ); analogously, I(P1 , H ∩ γj ) = ordt H(t, mj t + · · · ) = ordX H(X, mj X + · · · ). Since ordX H ≥ ordX G + r − 1 by Theorem 4.61, H = Af + BG Since f = g H ∈ qP1 .

−1

with A, B ∈ K[[X, Y ]].

F , it follows that H = (Ag −1 )F + BG and, by Theorem 4.60, 2

Let {γ} be the set of all branches of an irreducible curve F. P D EFINITION 4.64 A formal sum nγ γ, where each nγ is an integer and nγ 6= 0 only for a finite number of γ, is a divisor of F. The set of divisors is an abelian group under addition: P P P nγ γ + mγ γ = (nγ + mγ )γ.

Its identity is the zero divisor, that is, the divisor for which every nγ is zero. P D EFINITION 4.65 (i) A divisor nγ γ is effective if nγ ≥ 0 for each branch P γ; write nγ γ ≻ 0. (ii) A non-effective divisor is virtual.

(iii) Given two divisors D and D′ , define D ≻ D′ if D − D′ ≻ 0.

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Let F be a plane curve with only ordinary singularities P1 , . . . , Ps having respective multiplicities r1 , . . . , rs . Also, let γi,j , for j = 1, . . . , ri , be the branches of F centred at Pi . Then the divisor P D = i,j (ri − 1)γi,j is the double-point divisor of F. For two curves F and G, denote by G · F the divisor P G · F = γ I(P, G ∩ γ)γ,

where γ varies over all branches of F. Then G · F represents the divisor cut out on F by G. To formulate Noether’s Theorem, let F = v(F (X, Y )) be a plane curve with only ordinary singularities and suppose that G = v(G(X, Y )), H = v(H(X, Y )) are two other curves, where F and G have no common components. T HEOREM 4.66 (Noether) If H · F ≻ D + G · F, then H = AF + BG, with A, B ∈ K[X, Y ]. Proof. Let γ be a branch of F with centre a point P of multiplicity r ≥ 1 for F. By hypothesis, I(P, H ∩ γ) ≥ I(P, G ∩ γ) + r − 1. By Corollary 4.63, H ∈ qP and so H satisfies Noether’s conditions. Hence, by Theorem 4.56, H belongs to the ideal (F, G). 2 Now, the projective formulation of this theorem is given. Let F = v(F ∗ (X0 , X1 , X2 ), G = v(G∗ (X0 , X1 , X2 )), H = v(H ∗ (X0 , X1 , X2 )) be the corresponding projective curves. T HEOREM 4.67 (Noether) If H · F ≻ D + G · F, then H ∗ = A∗ F ∗ + B ∗ G∗ ,

with A∗ , B ∗ homogeneous polynomials in K[X0 , X1 , X2 ]. Proof. Applying a projectivity if necessary, assume that G and F do not meet on v(X0 ) and that v(X0 ) is a component of neither G nor F; therefore G∗ (0, X1 , X2 ) 6= 0,

F ∗ (0, X1 , X2 ) 6= 0.

Consider the affine polynomials of the curves: F (X, Y ) = f0,0 + f1,0 X + f0,1 Y + · · · + fn,0 X n + f0,n Y n , deg F = n; G(X, Y ) = g0,0 + g1,0 X + f0,1 Y + · · · + gm,0 X m + g0,m Y m , deg G = m;

H(X, Y ) = h0,0 + h1,0 X + h0,1 Y + · · · + hs,0 X s + h0,s Y s ,

deg H = s.

By the preceding theorem, H ∗ (1, X, Y ) = A(X, Y )F ∗ (1, X, Y ) + B(X, Y )G∗ (1, X, Y ),

(4.25)

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with deg A = n0 , deg B = m0 . Putting X = X1 /X0 , Y = X2 /X0 in (4.25) gives 1 ∗ 1 H (X0 , X1 , X2 ) = n0 +n A∗ (X0 , X1 , X2 )F ∗ (X0 , X1 , X2 ) s X0 X0 1 + m0 +m B ∗ (X0 , X1 , X2 )G∗ (X0 , X1 , X2 ) X0 with A∗ and B ∗ homogeneous. Put t = max{s, n0 + n, m0 + m}; then t−(n0 +n)

X0t−s H ∗ = X0

t−(m0 +m)

A∗ F ∗ + X0

B ∗ G∗ ,

which can be re-written as follows: X0ρ H ∗ (X0 , X1 , X2 ) = A∗1 (X0 , X1 , X2 )F ∗ (X0 , X1 , X2 ) +B1∗ (X0 , X1 , X2 )G∗ (X0 , X1 , X2 ) A∗1

(4.26)

B1∗

homogeneous. Suppose that ρ > 0, and put X0 = 0: ∗ A1 (0, X1 , X2 )F ∗ (0, X1 , X2 ) + B1∗ (0, X1 , X2 )G∗ (0, X1 , X2 ) = 0. Noting that F ∗ (0, X1 , X2 ) is a homogeneous polynomial in the indeterminates

with

and

X1 , X2 , it may be factorised as F ∗ (0, X1 , X2 ) = with (0, αi , βi ) ∈ F. Analogously,

G∗ (0, X1 , X2 ) =

Qn

i=1

Qm

j=1

(αi X1 − βi X2 ) (γj X1 − δj X2 )

with (0, γj , Gj ) ∈ G. Since the curves F and G have no common component, so points on v(X0 ), namely of type (0, βi , αi ), are in v(B1∗ ). Hence the terms of the product Qn i=1 (αi X1 − βi X2 ) divide B1∗ , whence

B1∗ (0, X1 , X2 ) ∗ G (0, X1 , X2 ) F ∗ (0, X1 , X2 ) = A∗2 (X1 , X2 )G∗ (0, X1 , X2 ).

A∗1 (0, X1 , X2 ) = − Therefore

A∗1 (X0 , X1 , X2 ) = A∗2 (X1 , X2 )G∗ (X0 , X1 , X2 ) + X0 A∗3 (X0 , X1 , X2 ) with A∗2 , A∗3 homogeneous. Substituting this expression for A∗1 in (4.26) gives X0ρ H ∗ (X0 , X1 , X2 ) = X0 A∗3 (X0 , X1 , X2 )F ∗ (X0 , X1 , X2 ) +B2∗ (X0 , X1 , X2 )G∗ (X0 , X1 , X2 ). Now put X0 = 0; hence B2∗ (0, X1 , X2 ) = 0 and so B2∗ (X0 , X1 , X2 ) = X0 B3∗ (X0 , X1 , X2 ). Eliminating X0 gives X0ρ−1 H ∗ (X0 , X1 , X2 ) = A∗3 (X0 , X1 , X2 )F ∗ (X0 , X1 , X2 ) +B3∗ (X0 , X1 , X2 )G∗ (X0 , X1 , X2 ). Repeating this procedure reduces ρ to 0, which proves the theorem.

2

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4.6 ANALYTIC BRANCHES In this section, another definition of the intersection multiplicity of two algebraic curves is given, and it is shown that the new definition coincides with the previous one. For a curve F = v(F ) with F (X, Y ) ∈ K[X, Y ], the idea of a branch was introduced. Now, this idea is extended to the case that F (X, Y ) is a formal power series; that is, F (X, Y ) ∈ K[[X, Y ]]. D EFINITION 4.68 An analytic branch centred at O is a branch centred at O in the sense of Definition 4.29. The branch belongs to F (X, Y ) ∈ K[[X, Y ]] if, for any representation (x(t), y(t)), then F (x(t), y(t)) = 0. It is shown below that there is a unique branch that belongs to a given irreducible power series other than 1 and that, given F1 , F2 ∈ K[[X, Y ]], both irreducible and different from 1, if a branch belongs to both of them, then they are associates. So, it is possible to introduce the idea of analytic branch as an irreducible formal power series different from 1 or, more precisely, as an equivalence class of irreducible power series other than 1 with respect to the equivalence relation: F1 ∼ F2 if F1 = EF2 with E a unit. Given F (X, Y ) ∈ K[[X, Y ]] with F = Fr + Fr+1 + · · · , perform a transformation of the type X = aX ′ + bY ′ , Y = cX ′ + dY ′ , with ad − bc 6= 0, in such a way that Fr is not divisible by X. Now, since Fr is not divisible by X, then F (0, Y ) = ar Y r + ar+1 Y r+1 + · · · ,

with ar 6= 0.

The Weierstrass Preparation Theorem 4.10 applied to F (X, Y ) determines a polynomial in Y with coefficients in K[[X]] which is an associate of F (X, Y ); that is, there exists a unit E(X, Y ) ∈ K[[X, Y ]] such that F (X, Y ) = E(X, Y )(Y r + A1 (X)Y r−1 + · · · + Ar (X)),

with Ai (X) ∈ K[[X]] and Ai (0) = 0 for i = 1, . . . , r. This allows the substitution, for each formal power series F (X, Y ), of its associated polynomial in K[[X]][Y ]. D EFINITION 4.69 (i) An analytic curve is an element F (X, Y ) ∈ K[[X, Y ]] that is not a unit and that has no repeated factors. (ii) If F (X, Y ) is a formal power series with F 6= 0, then F is an analytic cycle and F (X, Y ) = 0 is its equation. D EFINITION 4.70 An analytic cycle F is centred at O = (0, 0) if F (0, 0) = 0. More generally, F (X, Y ) is centred at (0, m) if F (0, m) = 0. D EFINITION 4.71 The centre (0, 0) is r-ple for F (X, Y ) if and is simple if r = 1.

F = Fr + Fr+1 + · · · , Fr 6= 0,

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The factors that appear in Fr represent the tangents of F (X, Y ) at its centre. R EMARK 4.72 Both F and EF, where E is a unit, refer to the same analytic curve. T HEOREM 4.73 Let F (X, Y ) ∈ K[[X, Y ]] with ord F = 1. Then there is one and only one branch (x(t), y(t)) such that F (x(t), y(t)) = 0. Proof. The proof is similar to that of Theorem 4.32, considering a formal power series instead of a polynomial. The existence of the branch is obtained as in Theorem 4.6 and, if F = Y − c1 X − · · · , then the branch is (t, c1 t + · · · ). 2 D EFINITION 4.74 If (x(t), y(t)) is a primitive representation of the branch γ and G is an analytic cycle with equation G(X, Y ) = 0, then put I(P, G ∩ γ) = ordt G(x(t), y(t)).

The idea of multiplicity of a branch at its centre is defined as in the case of a branch of a plane curve. T HEOREM 4.75 Two distinct irreducible analytic curves F = v(F (X, Y )) and G = v(G(X, Y )) cannot have a branch in common. Proof. The transformation X = aX ′ + bY ′ ,

Y = cX ′ + dY ′ ,

with ad − bc 6= 0, can make F and G regular in Y and substitute the corresponding special polynomials F1 and G1 obtained from the Weierstrass Preparation Theorem 4.10. The special polynomials are irreducible in K[[X]][Y ] and also in K((X))[Y ]. By hypothesis, F and G are not associates in K[[X, Y ]]; so F1 and G1 are also associates neither in K[[X]][Y ] nor in K((X))[Y ]. Therefore their greatest common divisor in K((X))[Y ] is 1; so 1 = aF1 + bG1 with a, b ∈ K((X))[Y ]. Note that a(Y ) = an (X)Y n + · · · + ai (X)Y i + · · · + a0 (X)

with ai (X) ∈ K((X)); that is,

ai (X) =

si (X) = X ki Ei (X), ti (X)

where Ei (X) is a unit for i = 0, . . . , n. Also, b(Y ) = bm (X)Y m + · · · + bj (X)Y j + · · · + b0 (X),

with bj (X) ∈ K((X)); that is,

bj (X) =

rj (X) = X hj Hj (X), uj (X)

with Hj (X) a unit for j = 0, . . . , m. It may happen that some ki and hj are negative; write ρ = max{|ki |, |hj |} for i = 0, . . . , n and j = 0, . . . , m. Hence, multiplying by X ρ gives X ρ = AF1 + BG1 ,

(4.27)

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with A, B ∈ K[[X]][Y ]. Suppose that there is a branch (x(t), y(t)) common to the two analytic curves. Substituting in (4.27) gives that xρ = 0, whence x = 0. Hence F1 is special; that is, F1 = Y r + c1 (X)Y r−1 + · · · + cr (X) and (x(t), y(t)) is a branch of F1 , as it is an associate of F . Therefore 0 = F1 (0, y(t)) = y(t), whence y = 0. This contradicts the definition of a branch; so they cannot have branches in common. 2 C OROLLARY 4.76 Let (ξ(t), η(t)) be a branch with ξ(t) = 0; then it is a branch of the line v(X) and of no other irreducible analytic curve. Proof. The branch considered satisfies the equation X = 0 and no other irreducible analytic curve can have such a branch. 2 T HEOREM 4.77 Let F be an analytic cycle centred at O = (0, 0) not containing v(X) as a component. Then F has a unique special equation F = 0, where F is reducible in K[[X]][Y ] if and only if it is reducible in K[[X, Y ]]. Proof. Suppose that F is regular in Y ; otherwise, it can be made so by an affine transformation. Then, by the Weierstrass Preparation Theorem 4.10, there exist unique G and H such that F = GH with H a special polynomial which is an associate of F. By Theorem 4.15 the second part follows. 2 D EFINITION 4.78 An analytic cycle along v(X) is a finite number of irreducible analytic cycles centred on v(X) together with their associated multiplicity. The notation is m1 F1 + · · · + ms Fs . If F ∈ K[[X]][Y ] is monic and irreducible, Hensel’s Lemma 4.9 shows that the polynomial F (0, Y ) has only one root λ and F defines an irreducible analytic cycle centred at (0, λ) and different from v(X). If λ = 0, then F is special; in general, F is special relative to Y1 = Y − λ. Conversely, every irreducible analytic cycle centred on v(X), but not this line, is represented by a unique monic irreducible element F ∈ K[[X]][Y ]. If F1 , . . . , Fs are such cycles with monic irreducible series F1 , . . . , Fs , then by definition F1m1 · · · Fsms is the monic series m1 F1 + · · · + ms Fs . Conversely every monic element F ∈ K[[X]][Y ] has a unique factorisation into monic irreducible factors: F = F1m1 · · · · · Fsms . It defines the cycle m1 F1 + · · · + ms Fs . Also, in the theory of analytic branches, it is useful to consider the local quadratic transformation, X ′ = X,

Y ′ = Y /X.

Let F = v(F (X, Y )) be an analytic cycle centred at O = (0, 0), and define the transform of F only in the case that v(X) is not tangent to F. Defining F ′ as the special series of F, it may be assumed directly that F is special. D EFINITION 4.79 The curve given by F ′ = v(F (X, XY ′ )/X r ), where (0, 0) is r-ple for F, is the transform F ′ of F. Theorems 4.42 to 4.45 that connect F and F ′ in the context of branches continue to hold for analytic curves F centred at (0, 0) and that do not have the line v(X)

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as a tangent. The proofs are the same, apart from taking the special series F (X, Y ) of F. Corresponding to Theorem 4.46 is the following result. T HEOREM 4.80 Let F = v(F (X, Y )) be an analytic curve centred at O = (0, 0). Then there is at least one and at most a finite number of branches (x(t), y(t)) that satisfy F. Proof. The proof is similar to that of Theorem 4.46 with some small changes. Let F = Fr (X, Y ) + Fr+1 (X, Y ) + · · · ; then, by a transformation X ′ = aX + bY,

Y ′ = cX + dY,

it can be ensured that Fr is not divisible by X. Applying Weierstrass Preparation Theorem 4.10 again gives the series Y r + a1 (X)Y r−1 + · · · + ar (X), which is the special series of F associated to F (X, Y ). Suppose further that F is irreducible; then the transform F ′ of F having polynomial F ′ (X, Y ′ ) = F (X, XY ′ )/X r

is still irreducible. By Hensel’s Lemma 4.9, F ′ (0, Y ′ ) has only one root λ. By a translation Y ′′ = Y ′ − λ, it may be assumed that λ = 0; then r

F (X, Y ′ ) = Y ′ + b1 (X)Y ′

r−1

+ ··· .

If r does not decrease, then F ′ is still special and the axis v(X) is not tangent to F ′ . The difference from Theorem 4.46 is the need to make transformations apart from translations on the analytic curves before applying the local quadratic transformations. 2 It remains to prove, as in Theorem 4.46, that r decreases. This then gives the following result. C OROLLARY 4.81 The sequence of quadratic transformations and translations that occur in the preceding proof cannot produce a sequence of r-ple points for r > 1. Proof. Let F be irreducible with F (X, Y ) = 0 as its special equation, and assume that F is not tangent to the line v(X). As in the proof of Theorem 4.47, if the sequence of quadratic transformations does not produce a reduction in r, then F (X, Y ) = (Y − m1 X − m2 X 2 − · · · )G(X, Y ). Since F is irreducible and monic, it follows that F (X, Y ) = Y − m1 X − m2 X 2 − · · · , whence r = 1. T HEOREM 4.82 An irreducible analytic cycle F has only one branch.

2

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Proof. The proof is by induction on r. Let O = (0, 0) be r-ple for F. If r = 1, then, by Theorem 4.73, there is one and only one branch of F. Suppose therefore that r > 1 and ensure by a suitable transformation that v(X) is not tangent to F. Consider the special equation F (X, Y ) = 0 and apply the local quadratic transformation. If there is a reduction in r, the result is obtained by the inductive hypothesis. If r does not reduce, first make a translation and then another quadratic transformation; by Corollary 4.81, r reduces and the result follows. 2 Let F (X, Y ) = Y e + a1 (X)Y e−1 + · · · + ae (X) ∈ K[[X]][Y ] be monic and irreducible. By Hensel’s Lemma 4.9, F (0, Y ) has only one root λ, which may be taken as λ = 0. Let x = ctd + · · · , y = y(t), c 6= 0,

be a branch of v(F (X, Y )). It is shown that d = e.

L EMMA 4.83 If (x(t), y(t)) is a primitive branch representation, then the field K(x(t), y(t)) contains an element of order 1. Proof. The proof is given in Theorem 4.21.

2

D EFINITION 4.84 An element of order 1 is a uniformising parameter. L EMMA 4.85 Let x = ctd + · · · , c 6= 0. Then

K[[x]][t] = K[[t]] and [K((x))(t) : K((x))] = d.

Proof. Let f ∈ K[[x]][t]. Then f is a polynomial in t with coefficients in K[[x]]; that is, f = ah (x)th + ah−1 (x)th−1 + · · · + a0 (x),

with ai (x) = a0,i + a1,i x + a2,i x2 + · · · for i = 0, . . . , h. Substituting in ai (x) the value of x, in f only powers of t are obtained. So f ∈ K[[t]]; this shows that K[[x]][t] ⊂ K[[t]]. Conversely, consider 1, t, . . . , td−1 , x, tx, . . . , td−1 x, x2 , tx2 , . . . ; they have degrees 0, 1, 2, . . . in t and constitute a basis for K[[t]]. Take an element h ∈ K[[t]]; it can be expressed as a linear combination of the basis elements: h = h0 + h1 t + h2 t2 + · · · + hd−1 td−1 + hd x + · · · .

So h can be written as a polynomial in t of degree d − 1 with coefficients in K[[x]]. In fact, with

h = h′0 + h′1 t + · · · + h′d−1 td−1 , h′0 = h0 + hd x + h2d x2 + · · · + hid xi + · · · , h′1 = h1 + hd+1 x + h2d+1 x2 + · · · + hid+1 xi + · · · , .. .. . . h′j = hj + hd+j x + h2d+j x2 + · · · + hid+j xi + · · · .

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Hence h ∈ K[[x]][t] and so K[[x]][t] = K[[t]]. For the second part of the theorem, consider an element F ∈ K[[X, T ]] with F = X − cT d − · · · and c 6= 0. Note that F is irreducible with subdegree 1. Since F (0, T ) = −cT d + · · · with c 6= 0, the Weierstrass Preparation Theorem 4.10 shows the existence of a unit E(X, T ) such that E(X, T )F (X, T ) = T d + a1 (X)T d−1 + · · · .

The polynomial associated to F is irreducible in K[[X]][T ]. As in the proof of Theorem 4.15, it is also irreducible in K((X))[T ]. By the choice of F , the point (x(t), t) with F (x(t), t) = 0 also satisfies the equation T d + a1 (X)T d−1 + · · · = 0.

Hence t is a root of the polynomial

T d + a1 (x(t))T d−1 + · · · ,

which is a polynomial of degree d with coefficients in K((x(t))). So, it has been shown that [K((x(t)))(t) : K((x(t)))] = d. 2 C OROLLARY 4.86 If (x = ctd + · · · , y = etf + · · · ) is a primitive branch representation centred at (0, 0) and x 6= 0, then y is algebraic over K((x)). Proof. As in Lemma 4.83, choose an element τ = at + · · · , a 6= 0 of order 1 in K(x, y). Then x = c¯τ d +· · · , y = e¯τ f +· · · and, further, τ = a(x, y)/b(x, y) with a(X, Y ), b(X, Y ) ∈ K[X, Y ]. If τ ∈ K(x), then, a fortiori, τ ∈ K((x)), whence y ∈ K(x). If, instead, τ 6∈ K(x), write τ b(x, y) = a(x, y) and then re-order it in powers of y, giving a1 (x, τ )y m + a2 (x, τ )y m−1 + · · · = 0,

with m ≥ 1 and ai (x, τ ) ∈ K(x, τ ). Therefore y is algebraic over K((x))(τ ). By Lemma 4.85, the result follows. 2 T HEOREM 4.87 Given a monic, irreducible polynomial F = Y e + a1 (X)Y e−1 + · · · ∈ K[[X]][Y ],

if (x = ctd + · · · , y = y(t)), with c 6= 0, is a branch representation of F, then d = e. Proof. By Lemma 4.85, y(t) ∈ K[[t]] = K[[x(t)]][t]

and [K((x(t)))(t) : K((x(t)))] = d.

It follows that y is a root of a polynomial F of degree ℓ ≤ d with coefficients in K((x)). Then F is a multiple of Y e + a1 (X)Y e−1 + · · · . This shows that e ≤ ℓ ≤ d. To see that also d ≤ e, choose an element τ of order 1 in K(x, y). Then τ ∈ K((x))(y) and [K((x))(y) : K((x))] = e, from which it follows that τ is a root of a polynomial of degree at most e. But, then d ≤ e, as required. 2 T HEOREM 4.88 With p = 0, consider a monic, irreducible polynomial F = Y d + a1 (X)Y d−1 + · · · ∈ K[[X]][Y ]

and one of its branches γ = (x(t), y(t)). Then F (x(t), Y ) splits completely into linear factors in K((t)).

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Proof. Let (x = ctd + · · · , y = y(t)) with c 6= 0 be the branch representation γ. By applying Theorem 4.27 it is possible to transform such a pair into an equivalent branch representation (x = td , y = η(t)). Hence, it may be assumed that x = td . Let P (4.28) y= cij tij with cij 6= 0. Write only the terms that have non-zero coefficients. Then, with e = gcd(d, . . . , ij , . . .), e = 1. For, if e > 1, then (x(t), y(t)) can be re-written in terms of te , but this contradicts the primitive character of (x(t), y(t)). Before continuing with the proof, some algebraic arguments need to be made. Suppose that K = C; then the d-th roots of unity are the d complex numbers ζd,0 , . . . , ζd,d−1 , where ζd,k = cos(2kπ/d) + i sin(2kπ/d), 0 ≤ k < d. A d-th root of unity ζ is primitive if ζ k 6= 1 for 1 ≤ k ≤ d − 1 and ζ d = 1; then, cos(2π/d) + i sin(2π/d) is primitive in C. Similarly, every algebraically closed field of characteristic 0 has a primitive d-th root of 1. Let ζ be a primitive d-th root of 1 and consider the d substitutions t 7→ ζ i t, d

d

i = 0, . . . , d − 1.

(4.29)

Such a substitution sends t to t and y(t) to yi (t) for i = 0, . . . , d − 1. Since such substitutions are K-automorphisms of K[[t]], the yi (t) are roots of F (td , Y ). Hence, (x(t), y(t)) is a branch of F; that is, F (x(t), y(t)) = 0, whence y(t) is a root of F (x(t), Y ) = 0. By (4.29), y(t) 7→ yi (t) by an automorphism; if y(t) is a root, so also is yi (t). If it is shown that the yi are distinct, then they are the d roots of F (td , Y ) and the proof is complete. The expression for yλ is P (4.30) yλ = cij ζ λij tij ,

where the cij are the same as in (4.28). If yλ = yµ , then ζ λij = ζ µij , whence ζ (λ−µ)ij = 1 and (λ − µ)ij ≡ 0 (mod d). Put e = gcd(. . . , ij , . . .); then (λ − µ)e ≡ 0

(mod d).

Since gcd(d, e) = 1, it follows that λ−µ ≡ 0 (mod d), so λ = µ, as 0 ≤ λ ≤ d−1 and 0 ≤ µ ≤ d − 1. Since the yi are distinct, the result is proved. 2 There is a more general result valid in any characteristic. T HEOREM 4.89 Let F = Y d + a1 (X)Y d−1 + · · · ∈ K[[X]][Y ]

be a monic polynomial. Then there exists x(v) ∈ K[[v]] such that F (x(v), Y ) splits completely into linear factors in K[[v]][Y ].

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Proof. If (x(t), y(t)) is a branch representation of F, then y(t) is a root of the polynomial F (x(t), Y ) ∈ K[[t]][Y ]. Therefore F (x(t), Y ) = (Y − y(t))F1 (t, Y ), with F1 (t, Y ) = Y d−1 + b1 (t)Y d−2 + · · · ∈ K[[t]][Y ]. If d > 1, there is a branch representation (t(t1 ), y1 (t1 )) of F1 = v(F1 ) and the argument can be repeated to obtain that F1 (t1 , Y ) = (Y − t(t1 ))(Y d−2 + c1 (t1 )Y d−3 + · · · ) ∈ K[[t1 ]][Y ]. After this procedure has been repeated d − 1 times, F (x(t), Y ) = (Y − y(t))(Y − y1 (t1 )) · (Y − yd−1 (td−1 )). Put v = td−1 , and express td−2 in terms of v; successively, do this for each ti with i = d − 3, . . . , 2, 1, whence the result follows. 2 T HEOREM 4.90 Given a monic, irreducible polynomial F = Y d + a1 (X)Y d−1 + · · · ∈ K[[X]][Y ], let F be an analytic cycle with special equation F = 0 centred at O = (0, 0) and let γ = (x(t), y(t)) be one of its branches. If G = v(G) is another analytic cycle also centred at O, then I(P, G ∩ γ) = ordX RY (F (X, Y ), G(X, Y )); that is, ordt G(x(t), y(t)) = ordX RY (F (X, Y ), G(X, Y )). Proof. First, a proof is given in the case of characteristic zero; the proof in the general case is similar. Recall that RY (F, G) is a polynomial in X. By a preceding theorem, Q F (x(t), Y ) = di=1 (Y − yi (t))

and there are K-automorphisms of K[[t]] that send x to x and y to yi . Then ordt G(x(t), yi (t)) = ordt G(x(t), y(t)). From Section 4.1,

whence

Qd Qd RY ( i=1 (Y − yi ), G) = i=1 G(x(t), yi (t)), ordt RY (F (x, Y ), G(x, Y )) = d ordt G(x, y).

Recall that ordt x = d; so it follows that ordt RY (F (x, Y ), G(x, Y )) = d ordx RY (F (x, Y ), G(x, Y )), and the theorem is proved.

2

In the general case, write F (x(u), Y ) =

Qd

i=1

(Y − y1 (u)),

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where (x(u), y(u)) and (x(u), yi (u)) are not-necessarily-primitive representations. By a substitution of the type ti = ci uρi + · · · , the pair (x(u), yi (u)) is transformed to a primitive representation (x′i (ti ), yi′ (ti )). By Theorem 4.87, ordti x′i (ti ) = d; so ordu x(u) = ρi d. Thus all the redundancies ρi are equal: write ρi = ρ. Now, (x′i (ti ), yi′ (ti )) and (x′j (tj ), yj′ (tj )) are primitive branch representations of F. The substitution ti 7→ tj transforms (x′i (ti ), yi′ (ti )) to (x′j (tj ), yj′ (tj )), which still represents a branch of F. Since such a substitution is an automorphism, then ordti G(x′i (ti ), yi′ (ti )) = ordtj G(x′j (tj ), yj′ (tj )), whence ordu G(x(u), yi (u)) = ordu G(x(u), yj (u)). The remainder of the proof proceeds as before. Now, an alternative proof can be given for the following: J(P, G ∩ F) = I(P, G ∩ F). C OROLLARY 4.91 Let F and G be plane curves with no common components, and let P be a point of the plane. Then the intersection multiplicities defined by the resultant coincide with the intersection multiplicities defined using the branches of F centred at P ; that is, I(P, G ∩ F) = J(P, G ∩ F). Proof. If P does not belong to both curves, then, from the definitions, both intersection multiplicities are zero. Therefore suppose that the point belongs to both curves and that it is the origin; suppose also that (0, 1, 0) does not belong to both curves and that the curves meet v(X) only in P = (0, 0) = (0, 0, 1). Let F = v(F (X, Y )) and G = (G(X, Y )); if F does not pass through (0, 1, 0), then F (X, Y ) is monic in Y. Let F = F1 F2 · · · Fs be the complete factorisation of F in K[[X]][Y ] with Fi monic and irreducible for each i. Let F1 , . . . , Fk the polynomials centred at (0, 0),. The polynomial Fi defines the branch γi of F centred at the origin. Note that RY (Fi , G) = Pi (X) is a unit in K[[X]] for i > k, since G(0, Y ) and Fi (0, Y ) could have a root in common only if the two curves met v(X) at a point other than the origin. Then I(P, G ∩ F) = I(P, F ∩ G) = ordX RY (F, G) Q Q = ordX RY ( si=1 Fi , G) = ordX si=1 RY (Fi , G) Pk Ps = i=1 ordX RY (Fi , G) = i=1 ordX RY (Fi , G) Pk = i=1 I(P, G ∩ F) = J(P, G ∩ F).

4.7 EXERCISES 1. For p > 0, show that the branch representation x(t) = u + tp + tp+1 , y(t) = v + tp + tp+2 has order p, and that it is not equivalent to a branch representation of type x(t) = u + tp , y(t) = v + η(t), with ordt η(t) ≥ p.

2

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2. If R denotes the set of all branch representations in special affine coordinates, for each K-monomorphism σ of K[[t]], define the mapping σ ⋆ : R → R as follows: σ ⋆ : (x(t), y(t)) 7→ (σ(x(t), σ(y(t)).

Let T ⊂ R be the union of the images σ ⋆ (T ) as σ varies in the set of Kmonomorphisms of K[[t]] that are not automorphisms. Show that a branch representation (x(t), y(t)) in special affine coordinates is primitive if and only if it is in R\T . 3. Over K = C, the irreducible plane curve F = v((X 2 + Y 2 )2 − X(5X 2 + 4Y 2 ) + 4X 2 ) has a cusp at the origin. After the linear substitution X ′ = X + Y, Y ′ = Y, apply the quadratic transformation (3.34) and show that the geometric transform of F is F ′ = v((X 2 + 1)2 + Y (X − 1)(5X 2 + 4Y 2 ) + 4Y 2 (Y − X)2 ).

Also, show that F ′ has a node at the infinite point (0, 2, −1). Deduce from this that F has two linear branches centred at the origin. 4. For u ∈ K((t)) with ordt u = m ≥ 1, the field K((u)) can be viewed as a subfield of K((t)). Show that [K((t)) : K((u))] = m. 5. Let p = 3. Show that the irreducible plane curve F = v(X 3 Y 3 + 2X 2 Y 4 + X 4 Y + X 3 Y 2 + 2XY 4 + 2X 4 +2X 2 Y 2 + 2XY 3 + 2X 3 + X 2 Y + XY + Y 2 ) has five singular points, namely, O = (1, 0, 0), E = (1, 1, 1),

X∞ = (0, 1, 0), P = (1, 2, 1),

Y∞ = (0, 0, 1),

each the centre of exactly two branches. 6. Give a direct proof of Noether’s Theorem 4.66 in its simplest case. If the curves F = v(F ) and G = v(G) intersect in mn distinct points, where m = deg F, n = deg G, show that any curve H = v(H) of degree r which passes through all common points of F and G has H = AF + BG, where A and B are polynomials of degrees r − m and r − n. 7. Show that the result in Exercise 6 holds true if each common point P of F and G is a simple point of F and I(P, H ∩ F) ≥ I(P, G ∩ F). 8. Suppose that (a) F = v(F ) and G = v(G) are plane curves of degree m and n;

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(b) at every common point P the tangents to F are distinct from the tangents to G;

(c) H = v(H) is a plane curve of degree r which has an (h + k − 1)-fold point at P whenever P is h-fold on F and k-fold on G.

Then H = AF + BG, where A and B are polynomials of degrees r − m and r − n such that the curves A = v(A) and B = v(B) have points of multiplicity at least k − 1 and h − 1 at P . 9. Let the curves F = v(F ) and G = v(G) intersect in mn distinct points, where m = deg F, n = deg G. Show that, if nr of these points lie on a plane curve of degree r, then the remaining (m − r)n lie on a plane curve of degree m − r. Deduce Pascal’s Theorem, Chapter 3, Exercise 2, from this result. 10. (Cayley–Bacharach Theorem) Let ∆ be the set of mn points of intersection of two plane curves of degrees m and n. Show that that every curve of degree m + n − 3 containing all but one point of ∆ must contain ∆. 4.8 NOTES Chapter 4 is based on Seidenberg [400, Chapters 11,12,15]. The theory of branches in positive characteristic is treated in Campillo [64], and includes Puiseux and Hamburger–Noether expansions, characteristic exponents, and Newton coefficients; see also [201]. For generalisations of the Cayley–Bacharach Theorem, see [403, Chapter 5] and [115].

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Chapter Five The function field of a curve A generalisation of the idea of a branch is a generic point of an irreducible plane curve F = v(F ). The coordinates of a generic point are in some proper extension of K, but still are assumed to satisfy the polynomial F . As in Section 4.2, branch representations are points in this sense. The coordinates of a generic point of F together with K generate a field Σ of transcendence degree 1 over K. Distinct generic points of F give rise to the same field Σ, the function field K(F) associated to F, determined up to a Kisomorphism. It may be noted that no generic point of F has a special property that the other generic points do not have. This justifies the adjective ‘generic’ as the term for a generic point in the classical literature used to indicate a point without any particular property. The idea of branches of F can be transferred to K(F) via the concept of a place in such a way that birationally equivalent branches correspond to the same place. The aim of this chapter is to show how function fields equipped with their places are used in the study of the birational properties of an irreducible plane curve. Almost all results are similar to those for curves over the complex numbers. However, the presence of inseparable variables in K(F) when K has positive characteristic requires the concept of a Hasse derivative, which is an efficient technical tool but must be carefully used. This becomes apparent in Section 5.11, where the behaviour of dual curves in positive characteristic is considered.

5.1 GENERIC POINTS An (affine) point now means any point P = (x, y) with coordinates x, y belonging to some extension of the field K. As in Section 4.2, branch representations are points in this sense. A point is constant if both its coordinates x, y belong to K; in the other case, the point is variable. D EFINITION 5.1 A point P = (x, y) is generic on the curve F = v(F (X, Y )) if, for every G(X, Y ) in K[X, Y ] with G(x, y) = 0, it follows that G(a, b) = 0 for all points Q = (a, b) of F. In other words, the point P = (x, y) is generic if G(x, y) = 0 implies that G ≡ 0 (mod F ). R EMARK 5.2 It follows from this definition that two distinct irreducible curves cannot have a common generic point. T HEOREM 5.3 A reducible curve has no generic points.

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Proof. Let C be a curve with two components F = v(F ) and G = v(G), with F 6⊂ G, G 6⊂ F. Then C = v(F G). So, if P = (a, b) is a point, constant or variable, such that F (a, b)G(a, b) = 0; then F (a, b) = 0 or G(a, b) = 0. Suppose that C has a generic point P = (x, y); then F (x, y) = 0 or G(x, y) = 0. Suppose the former holds; that is, F (x, y) = 0. Then F is zero at all points of C and so also at all points of G. So G must be a component of F, a contradiction. 2 T HEOREM 5.4 A variable point P = (x, y) of an irreducible curve F is a generic point. Proof. Let F = v(F (X, Y )) and suppose that G(x, y) = 0 but G(X, Y ) 6≡ 0 (mod F (X, Y )), with P = (x, y) a variable point of F. Taking F to be irreducible, it follows from B´ezout’s Theorem that the common points of the curves F and G = v(G(X, Y )) are finite in number and so belong to the plane AG(2, K). In other words, each of the common points of F and G is constant. In particular, since P = (x, y) is a variable point of F, it cannot belong to the curve G, and so G(x, y) 6= 0, as required. 2 From now on, the term ‘point’ of an irreducible curve F only indicates any of its constant points, while ‘generic point’ of F means any of its variable points. The above discussion extends to the projective plane. Let F = v(F ) with homogeneous F = F (X0 , X1 , X2 ) ∈ K[X0 , X1 , X2 ]. A generic point of F is defined to be a point P = (x0 , x1 , x2 ) with F (x0 , x1 , x2 ) = 0, whose coordinates x0 , x1 , x2 come from an extension field of K such that, if G is a homogeneous polynomial in K[X0 , X1 , X2 ] that vanishes at P , then G vanishes at every point of F. This amounts to saying that G is divisible by F . As in the affine case, a curve has a generic point if and only if it is irreducible. Using homogeneous polynomials, the function field of F can be described as the subfield in K(x0 , x1 , x2 ) consisting of all elements ξ = u(x0 , x1 , x2 )/v(x0 , x1 , x2 ), where u(X0 , X1 , X2 ), v(X0 , X1 , X2 ) ∈ K[X0 , X1 , X2 ] are homogeneous of the same degree and v(x0 , x1 , x2 ) 6= 0. In fact, if x0 6= 0, this subfield is K-isomorphic to K(1, x1 /x0 , x2 /x0 ) which is just Σ = K(x, y). If all coordinates x0 , x1 , x2 are constant, then P is a constant point; otherwise, it is a variable point. As before, a point of an irreducible curve is generic if and only if it is variable. R EMARK 5.5 With the notion of a universal domain Ω over K, all points are in the plane over Ω. Since Ω is an algebraically closed field, a curve F over K can be considered as a curve over Ω, meaning that the coefficients in the equation of F are in Ω. All theorems previously proved remain true as stated, although they may have a different content because of the change of meaning for point. D EFINITION 5.6 The generic points P = (x, y) and P1 = (x1 , y1 ) of an irreducible curve F are K-isomorphic if there exists an isomorphism σ : K[x, y] → K[x1 , y1 ] such that σ(x) = x1 and σ(y) = y1 .

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T HEOREM 5.7 Any two generic points of an irreducible curve are K-isomorphic. Proof. Let P = (x, y) and P1 = (x1 , y1 ) be generic points of the irreducible curve F. Now, let σ : K[x, y] → K[x1 , y1 ] be a map such that σ(G(x, y)) = G(x1 , y1 ) for every G(X, Y ) ∈ K[X, Y ]. It must be shown that G(x, y) = H(x, y) implies G(x1 , y1 ) = H(x1 , y1 ) for every H(X, Y ) ∈ K[X, Y ]. In other words, if G(x, y) − H(x, y) = 0, then also G(x1 , y1 ) − H(x1 , y1 ) = 0. But this follows immediately from the notion of generic point. Therefore the map σ is well-defined. Both the following relations hold: σ(G1 (x, y) + G2 (x, y)) = σ(G1 (x, y)) + σ(G2 (x, y)); σ(G1 (x, y)G2 (x, y)) = σ(G1 (x, y))σ(G2 (x, y)). This shows that σ is a homomorphism from K[x, y] to K[x1 , y1 ]. Further, the kernel of σ is zero since G(x1 , y1 ) = 0 implies that G(x, y) = 0, since, by hypothesis, P = (x1 , y1 ) is a generic point of F. 2 C OROLLARY 5.8 An irreducible curve F has infinitely many generic points. Proof. To prove the assertion, it is enough to show that F has infinitely many points P = (x, y) such that x, y ∈ K((t)). Given a constant point Q of F, let (x(t), y(t)) be any representation of a branch centred at Q. Then P = (x(t), y(t)) is a variable point of F. By Theorem 5.4, P = (x(t), y(t)) is a generic point of F. For any K-automorphism σ of K[[t]], the point P ′ = (x′ (t), y ′ (t)) with x′ (t) = σ(x(t)) and y ′ (t) = σ(y(t)) is also a variable point of F. By Theorem 4.4, each τ ∈ K[[t]] defines a K-automorphism of K[[t]]. Since K[[t]] has infinitely many such Kautomorphisms, each of them defines a generic point of F. 2 D EFINITION 5.9 In the case that P = (x, y) is a generic point of an irreducible curve F = v(F (X, Y )), then Σ = K(x, y) is the field of rational functions or, more briefly, the function field of F. Note that, by Theorem 5.7, this field is uniquely determined up to K-isomorphism. Also, its transcendence degree over K is 1. In other words, the following theorem holds. T HEOREM 5.10 For any two elements ξ, η ∈ Σ, there exists an irreducible polynomial g(X, Y ) ∈ K[X, Y ] such that g(ξ, η) = 0; that is, there is a polynomial relation between any two elements in Σ. An immediate corollary is the following result. C OROLLARY 5.11 Every point P = (ξ, η) of AG(2, Σ) not in AG(2, K) is a generic point of an irreducible curve F. 5.2 RATIONAL TRANSFORMATIONS In classical literature on projective geometry, a rational transformation is a mapping g(X, Y ) f (X, Y ) , Y′ = , X′ = h(X, Y ) h(X, Y )

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with f, g, h ∈ K[X, Y ]. Since h(X, Y ) could vanish at some points, these formulas may not allow the images of such points to be calculated. Further, if the domain of the transformation is to be non-empty and be contained in a given algebraic curve F = v(F (X, Y )), then it is necessary that h(X, Y ) 6≡ 0 (mod F (X, Y )). In this case, h(a, b) is zero only at a finite number of points of F. Then, for any P = (a, b) ∈ F and h(a, b) 6= 0, the following transformation can be defined:   f (a, b) g(a, b) T(a, b) = . , h(a, b) h(a, b) In particular, T is defined at every generic point of F and transforms such a generic point P = (x, y) into a point P ′ = (x′ , y ′ ) of AG(2, Σ). Now the proper definition can be given. D EFINITION 5.12 A rational transformation T of an algebraic curve F is given by a generic point P = (x, y) of F and a point P ′ = (x′ , y ′ ), x′ , y ′ ∈ K(x, y). Then, g(x, y) f (x, y) ′ , y = , x′ = h(x, y) h(x, y) with f, g, h ∈ K[X, Y ] and h(x, y) not vanishing identically on F. From this, for each P = (a, b) ∈ F such that h(a, b) 6= 0,   f (a, b) g(a, b) . T(a, b) = , h(a, b) h(a, b) R EMARK 5.13 The expressions for x′ , y ′ as rational functions of x, y are not uniquely determined; but T(a, b) does not depend on the particular functions chosen. Proof. Suppose that x′ =

f1 (x, y) ′ g1 (x, y) , y = h1 (x, y) h1 (x, y)

and T′ (a, b) =



f1 (a, b) g1 (a, b) , h1 (a, b) h1 (a, b)



,

with h1 (a, b) 6= 0. Then it must be shown that T′ (a, b) = T(a, b); that is, if f (x, y) f1 (x, y) = , h(x, y) h1 (x, y)

then f1 (a, b) f (a, b) = . h(a, b) h1 (a, b) Thus, the mapping is well-defined and does not depend on the representative functions. From the hypothesis that f1 (x, y) f (x, y) = , h(x, y) h1 (x, y)

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it follows that f (x, y)h1 (x, y) − f1 (x, y)h(x, y) = 0. So, the polynomial G(X, Y ) = f (X, Y )h1 (X, Y ) − f1 (X, Y )h(X, Y )

vanishes at the generic point P = (x, y) of F. Hence, G(a, b) = 0 for every point Q = (a, b) ∈ F, whence f (a, b)h1 (a, b) − f1 (a, b)h(a, b) = 0.

Since h(a, b)h1 (a, b) 6= 0, it may be concluded that

f (a, b)/h(a, b) = f1 (a, b)/h1 (a, b).

2 ′

′

The mapping T is defined at P = (a, b) if, for some expression in h(x , y ), the value h(a, b) 6= 0. If there exists no such expression, then T is not defined at P = (a, b). E XAMPLE 5.14 Let C = v(Y − X 2 ) be a conic and P = (x, y) one of its generic points. Let T be defined by the generic point P ′ = (x′ , y ′ ) where x′ = x, y ′ = y/x. Then T((0, 0)) = (0, 0) is defined as, although 0/0 is not defined, y/x = x. In fact, x′ , y ′ can be written in the form x′ = y, y ′ = x, whence T((0, 0)) = O. On the other hand, consider the transformation T determined by x′ = y, y ′ = x/y. The expression x/y cannot be rewritten in the form g(x, y)/h(x, y) with h(0, 0) 6= 0. In fact, since h(x, y)x − g(x, y)y = 0, so h(X, Y )X − g(X, Y )Y = d(X, Y )(Y − X 2 )

with d(X, Y ) ∈ K[X, Y ], which is impossible, as the terms of least degree in h(X, Y )X − g(X, Y )Y contain a term cX with c ∈ K, c 6= 0, which cannot be the case in the right-hand side. In terms of homogeneous coordinates, a rational transformation T of F is defined by a generic point P = (x0 , x1 , x2 ) of F and a point P ′ = (x′0 , x′1 , x′2 ) such that K(P ) ⊃ K(P ′ ). Then x′0 : x′1 : x′2 = f0 (x0 , x1 , x2 ) : f1 (x0 , x1 , x2 ) : f2 (x0 , x1 , x2 ),

with f0 , f1 , f2 homogeneous polynomials of the same degree. If Q = (a0 , a1 , a2 ) is a point of F, and f0 , f1 , f2 do not vanish at Q, then T(Q) is defined to be (f0 (Q), f1 (Q), f2 (Q)). The triple (f0 , f1 , f2 ) is not unique. But, if f0 (x0 , x1 , x2 ) : f1 (x0 , x1 , x2 ) : f2 (x0 , x1 , x2 ) = g0 (x0 , x1 , x2 ) : g1 (x0 , x1 , x2 ) : g2 (x0 , x1 , x2 ), where the gi are also homogeneous of the same degree, and if g0 , g1 , g2 do not all vanish at Q, then (f0 (Q), f1 (Q), f2 (Q)) and (g0 (Q), g1 (Q), g2 (Q)) define the same point Q′ = T(Q). In this way, T is defined at all but a finite number of points of F.

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E XAMPLE 5.15 In Example 5.14 in which C = v(Y − X 2 ) and T is given by the equations x′ = y, y ′ = x/y, the mapping is not defined at the point P = (0, 0). In PG(2, K), however, (1, x′ , y ′ ) = (y, y 2 , x) = (x2 , yx2 , x) = (x, yx, 1) = (f0 (1, x, y), f1 (1, x, y), f2 (1, x, y)), where f0 = X0 X1 , f1 = X1 X2 , f2 = X02 . Then, with P = (1, 0, 0), T(P ) = P ′ = (0, 0, 1). In other words, T is not defined at P viewed as a point of AG(2, K), but it is defined at P when that point is regarded as a point in PG(2, K). For an example of a rational transformation in the projective plane not everywhere defined on a given curve, consider the curve C ′ = v(Y 2 − X 2 − X 3 ) and the transformation T given by x′ = x, y ′ = y/x. It can be shown that T is not defined at P = (1, 0, 0). Returning to the affine case, two cases occur for x′ , y ′ ∈ K(x, y). (i) If x′ , y ′ ∈ K, then all points of F are transformed into the same point, namely P ′ = (x′ , y ′ ). This is the trivial case. (ii) If at least one of x′ , y ′ is not in K, then the field extension K(x′ , y ′ )/K has still transcendence degree 1. Hence P ′ = (x′ , y ′ ) is a generic point of an irreducible curve F ′ , the image curve of F under T. This is the non-trivial case, and it is usually tacitly assumed that this is the case. These definitions give the following two results. T HEOREM 5.16 All transformed points of F under T are on the image curve. T HEOREM 5.17 A rational transformation S of F is always defined on generic points of F and the image of a generic point of F is a generic point of the image curve F ′ . As a special case, take a branch representation of F as a generic point. T HEOREM 5.18 The image of a branch representation of a curve F under a rational transformation is a branch representation of the image curve F ′ . The image of a primitive branch representation need not be primitive. Still, equivalent branch representations yield equivalent branch representations, so that a given branch γ of F gives rise, under T, to a definite branch γ ′ of F ′ . The branch γ ′ is the image of γ under T. Let T be given by a generic point P ′ = (x′ , y ′ ) of F, and define a map T∗ as follows: if P = (a, b) is a point of F and γ is a branch of F centred at P (a, b), set T∗ (P ) = P ′ where P ′ is the centre of the image γ ′ of γ under T. Here T∗ need not be a univalent map, as F may have several branches centred at P . Thus T∗ (P )

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should be defined to be the set of all centres of the branches of F ′ arising from the branches of F centred at P . It will also not be univalent, in general. The centre of the transformed branch γ ′ of γ need not be at finite distance. However, from the discussion on the projective plane, T∗ will be defined at every point of F, since there is at least one branch of F centred at any point of F. Let P = (a0 , a1 , a2 ) be a point of F and assume that T is defined at P . Then 1 : x′ : y ′ = f0 (1, x, y) : f1 (1, x, y) : f2 (1, x, y),

where f0 , f1 , f2 are homogeneous of the same degree and fi (a0 , a1 , a2 ) 6= 0 for some i. Let x0 (t) = a0 + ty0 (t), x1 (t) = a1 + ty1 (t), x2 (t) = a2 + ty2 (t) be a special representation of a branch of F. Put t = 0 in fi (x0 (t), x1 (t), x2 (t)); this shows that γ ′ is centred at the point P ′ = (f0 (a0 , a1 , a2 ), f1 (a0 , a1 , a2 ), f2 (a0 , a1 , a2 )). In other words, T∗ extends T, and it has the advantage of being defined everywhere on F. Nevertheless, T cannot be defined at a point where T∗ is multivalued. D EFINITION 5.19 A rational transformation S : F → F ′ , given by the generic points P = (x, y) ∈ F and P ′ = (x′ , y ′ ) ∈ F ′ , is birational if K(x′ , y ′ ) = K(x, y).

In this case, the generic point P ′ = (x′ , y ′ ) of F ′ together with the pair of elements ¯ : F ′ → F. (x, y) yield a rational transformation S ¯ T HEOREM 5.20 If a birational transformation S is defined at the point P and S ¯ is defined at the point S(P ), then S(S(P )) = P. Proof. Let P = (a, b) and let r, s, t ∈ K[X, Y ] such that S : x′ =

r(x, y) ′ s(x, y) , y = , t(x, y) t(x, y)

with t(a, b) 6= 0. For P ′ = (a′ , b′ ) = S(P ), let r′ , s′ , t′ ∈ K[X, Y ] be such that ′ ′ ′ ′ ′ ′ ¯ : x = r (x , y ) , y = s (x , y ) , S ′ ′ ′ ′ t (x , y ) t (x′ , y ′ )

with t′ (a′ , b′ ) 6= 0. Now,

r′ (r(x, y)/t(x, y), s(x, y)/t(x, y)) , t′ (r(x, y)/t(x, y), s(x, y)/t(x, y)) s′ (r(x, y)/t(x, y), s(x, y)/t(x, y)) y= ′ . t (r(x, y)/t(x, y), s(x, y)/t(x, y))

x=

There exists an integer h ≥ 1 such that      r(X, Y ) s(X, Y ) r(X, Y ) s(X, Y ) − r′ , , w(X, Y ) = (t(X, Y ))h Xt′ t(X, Y ) t(X, Y ) t(X, Y ) t(X, Y )

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is a polynomial with coefficients in K. It follows simply that w(x, y) = 0, and so the point P = (x, y) is generic point of F. Hence w(a, b) = 0 for each point Q = (a, b) ∈ F. Since t(a, b) 6= 0, t′ (a′ , b′ ) 6= 0, a=

r′ (r(a, b)/t(a, b), s(a, b)/t(a, b)) t′ (r(a, b)/t(a, b), s(a, b)/t(a, b))

b=

s′ (r(a, b)/t(a, b), s(a, b)/t(a, b)) . t′ (r(a, b)/t(a, b), s(a, b)/t(a, b))

and, similarly,

Therefore a=

s′ (a′ , b′ ) r′ (a′ , b′ ) , b = , t′ (a′ , b′ ) t′ (a′ , b′ )

¯ ′ , b′ ) = (a, b). However, (a′ , b′ ) = S(P ) and (a, b) = P , from which it and S(a ¯ follows that S(S(P )) = P, as required. 2 ¯ is the inverse of S and so is written S ¯ = S−1 . The rational transformation S Since S is defined at every generic point P of F and since S(P ) is generic point of F ′ , the following corollary is immediate. C OROLLARY 5.21 A birational transformation S of a curve F defines a bijective correspondence between the generic points of F and those of the image curve F ′ . D EFINITION 5.22 Let Σ be the function field of a curve; equivalently, let K have an extension Σ = K(x, y) of transcendence degree 1 over K. A model of Σ is given by a point P = (ξ, η) such that Σ = K(ξ, η) and by the curve F having P as generic point; in symbols, write (F; (ξ, η)). It should be noted that two models (F; (x, y)) and (F ′ ; (x′ , y ′ )) are distinct if (x, y) 6= (x′ , y ′ ). It may, however, happen that F = F ′ but (x, y) 6= (x′ , y ′ ). So, a given curve F may give rise to more than one model, consistent with the fact that a curve can have non-trivial automorphisms, that is, can be birational images of themselves under a non-identical birational transformation. Automorphisms are investigated in Chapter 11. Here two examples illustrate this situation. E XAMPLE 5.23 Let C and P = (x, y) be as in Example 5.14. If S is the map x′ y′

= =

x + c, 2cx + y + c2

with c ∈ K, then S−1 is as follows: x y

Hence

= x′ − c, = y ′ − 2cx′ + c2 . 2

y − x2 = (y ′ − 2cx′ + c2 ) − (x′ − c)2 = y ′ − x′ .

This shows that S transforms C into itself, as y − x2 = 0 implies that y ′ − x′ 2 = 0. Therefore (C; (x, y)) is transformed into the model (C; (x′ , y ′ )) by S.

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E XAMPLE 5.24 Let q0 = 2e and q = 2q02 . The DLS curve is the irreducible plane curve F = v(X 2q0 (X q + X) + Y q + Y ). It has only one singular point, namely Y∞ , which is a 2q0 -fold point. For a generic point P = (x, y), let K(F) = K(x, y) be the function field of F. Now, put h = xy + x2q0 +2 + y 2q0 . A straightforward computation shows that the rational transformation y x S : x′ = , y ′ = h h leaves D invariant. In particular, the associated map T on F is defined on all points of F apart from Y∞ and O. Further, the branch γ centred at O is transformed to a branch centred at Y∞ . Also, S has order two as it coincides with its inverse. Therefore Y∞ is the centre of only one branch, say γ ′ , which is the image of γ under T. Since (x(t), y(t)), with x(t) = t,

y(t) = t2q0 + tq+2q0 + t2qq0 + tq(q+2q0 ) + . . . ,

is a primitive representation of γ, so (ξ(t), η(t)), with ξ(t) =

1 + tq + t2q0 (q−1) + . . . , t(1 + t + t2q−2q0 −1 + . . .)

η(t) =

1 , t2q0 (1 + t + t2q−2q0 −1 + . . .)

is a primitive representation of γ ′ . If (F; (ξ, η)) and (F ′ ; (ξ ′ , η ′ )) are two models of Σ, then Σ = K(ξ, η) and Σ = K(ξ ′ , η ′ ). Hence K(ξ, η) = K(ξ ′ , η ′ ) and so, by Definition 5.19, the points P = (ξ, η) and P ′ = (ξ ′ , η ′ ) determine a birational transformation from F to F ′ . T HEOREM 5.25 Any two models of a given field Σ are birationally equivalent. A property of the field Σ is a birationally invariant property of any one of its models. Now, consider the irreducible curves F = v(F (X, Y )) and G = v(G(X, Y )), and take generic points P = (x, y) of F and Q = (ξ, η) of G. Assume that their function fields Σ = K(x, y) and Σ′ = K(ξ, η) are isomorphic over K. Let ϕ : Σ → Σ′ be a K-isomorphism, and put x′ = ϕ(ξ), y ′ = ϕ(η). Then Σ′ = K(x′ , y ′ ) and ϕF (x, y) = 0 implies F (x′ , y ′ ) = 0. Hence (F; (x′ , y ′ )) is a model of Σ′ , in particular a birationally equivalent model to the model (G; (ξ, η)). This discussion motivates the following definition. D EFINITION 5.26 Two irreducible plane curves are birationally isomorphic or birationally equivalent if their function fields are isomorphic over K.

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5.3 PLACES Let (F, (x, y)) be a model of a field Σ of transcendence degree 1. If (ξ(t), η(t)) is a not-necessarily-primitive branch representation of F, then, by Theorem 5.7, there is a K-monomorphism σ of Σ into K((t)) such that σ(x) = ξ(t), σ(y) = η(t). For an automorphism ρ of K[[t]], extended to K((t)) in a natural way, put σ ′ = ρ◦σ. Then σ ′ is also a K-monomorphism of Σ into K((t)) such that the branch representations (σ ′ (x) = ξ ′ (t), σ ′ (y) = η ′ (t)) and (ξ, η) are equivalent. In a diagram, K(ξ(t), η(t))  * σ    ρ Σ = K(x, y) H HH σ ′ = ρ ◦ σ HH ? j H K(ξ ′ (τ ), η ′ (τ )). This leads to the following fundamental concepts. D EFINITION 5.27 sentation.

(i) A K-monomorphism σ : Σ → K((t)) is a place repre-

(ii) A place representation σ is primitive if (σ(x), σ(y)) for a pair, and hence for any pair, of generators x, y of Σ is a primitive branch representation of F. (iii) Two place representations σ and σ ′ are equivalent if there is a K-automorphism ρ of K((t)) such that σ ′ = ρ ◦ σ. (iv) A place is an equivalence class of primitive place representations. (v) The set of all places of Σ is denoted by P(Σ). A place is merely the field-theoretic or birational counterpart of the geometric concept of a branch. The following two theorems explain this point. T HEOREM 5.28 The places of Σ and the branches of any model of Σ are in oneto-one correspondence, in a natural way. Proof. Let (F; (x, y)) be a model of Σ. It has been shown how a primitive branch representation (ξ(t), η(t)) leads to a primitive place representation σ in a natural way, and also how to go back to a primitive branch representation (ξ(t), η(t)) from a primitive place representation σ by σ(ξ) = ξ(t), σ(η) = η(t). In this way, a branch representation and a place representation give rise to one other. Also, equivalent branch representations correspond to equivalent place representations; so branches and places are in one-to-one correspondence. 2 T HEOREM 5.29 Let (F; (x, y)) and (F; (x′ , y ′ )) be two models of Σ, and let S be the birational transformation sending P = (x, y) into P ′ = (x′ , y ′ ). Then S

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induces a one-to-one correspondence between the branches of F and those of F ′ , two branches corresponding if and only if they correspond to the same place of Σ. Proof. Let the place P, that is, one of its primitive representations, send (x, y) into (x(t), y(t)) and (x′ , y ′ ) into (x′ (t), y ′ (t)). In a diagram, P(x, y) (x(t), y(t)) pp pp pp τ pp p? ? P- ′ ′ ′ (x , y ) (x (t), y ′ (t)). There exist r(X, Y ), s(X, Y ) ∈ K[X, Y ] such that x′ = r(x, y)/s(x, y). Applying P gives x′ (t) = r(x(t), y(t))/s(x(t), y(t)), and similarly for y ′ . This shows that S(P (x(t), y(t))) = (x′ (t), y ′ (t)) justifying the fourth arrow in the diagram, and proving one part of the theorem. A similar argument gives the converse. 2

5.4 ZEROS AND POLES Let F = v(F (X, Y )) be an irreducible curve with a generic point P = (x, y). For a place P of the field Σ = K(x, y), let τ : Σ → K((t)) be a primitive representation of P. If ξ is an element of Σ, then ordt τ (ξ) does not depend on the choice of τ . To show this, take another primitive representation τ ′ : Σ → K((t)) of P. Then τ ′ = ρ ◦ τ with an automorphism ρ of K[[t]] naturally extended to K((t)). Hence ordt τ ′ (ξ) = ordt ρ(τ (ξ)) = ordt τ (xi ) by Theorem 4.5. This leads to the following concept. D EFINITION 5.30 Let τ be any primitive representation of a place P of Σ. (i) The order of ξ at P is ordP ξ = ordt τ (ξ). (ii) If ordP ξ = 1, then ξ is a uniformising element or local parameter at P. Let γ be the branch of F associated to the place P. The centre of P is the centre of γ. Since, by Theorem 5.28, there is a bijection between the branches of F and the places of Σ, and by Theorem 4.46 there is a only a finite number of branches centred at the same point, there is a only a finite number of places of Σ with the same centre. Put ordγ ξ = ordP ξ. In particular, ord γ x = ordt x(t), ord γ y = ordt y(t) which can be thought of as alternative notation for ordt x(t) and ordt y(t). From part (iii) of Theorem 4.21 the following result appears. L EMMA 5.31 The function field Σ has a uniformising element at every place. D EFINITION 5.32 (i) The place P is a zero of ξ if ordP ξ > 0; if P is a zero of ξ, it is a zero of multiplicity ordP ξ.

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(ii) The place P is a pole of ξ if ordP ξ < 0; if P is a pole of ξ, it is a pole of multiplicity −ordP ξ. (iii) If the place P is not a pole of ξ, then ξ is regular at P. Henceforth, ζ denotes a function of Σ, that is, an element of Σ\K. T HEOREM 5.33 The function ζ has only finitely many zeros. The number of zeros counted with their multiplicity is [Σ : K(ζ)]. Proof. Since ζ is transcendental over K, then Σ is a finite algebraic extension of K(ζ) and [Σ : K(ζ)] is an integer. If L is a finite algebraic extension of a field M , then L is a simple extension of M . By the Theorem of Primitive Element, there exists η ∈ Σ such that Σ = K(ζ, η). Write ζ = x, η = y; in other words, consider ζ as the first coordinate of a generic point of a model F of Σ. Let F (X, Y ) = c0 (X)Y m + c1 (X)Y m−1 + · · · + cm (X) be an irreducible polynomial with coefficients in K of which (x, y) is a zero. Then c0 (X)m−1 F (X, Y ) = (c0 (X)Y )m + c1 (X)(c0 (X)Y )m−1 + · · · + cm (X)c0 (X)m−1 = Z m + c1 (X)Z m−1 + · · · ,

having put c0 (X)Y = Z. For this polynomial in K[X, Z], the pair (x, c0 (x)y) is a zero. Since c0 (x)y is a generator of Σ over K(x), then c0 (x)y may be substituted for y; in other words, it may be supposed that c0 (X) = 1. Then [Σ : K(x)] = m. Let (F; (x, y)) be a model of Σ = K(x, y) with (x, y) as a generic point. Then the branches γ of F such that ordγ x > 0 are to be counted each with its order. These branches have centres on the Y -axis, but a priori possibly at Y∞ . This can happen but not when c0 (X) = 1. To check it, assume γ is a branch centred at Y∞ . Note that if (x(t), y(t)) is a primitive branch representation of γ, then ordγ y = ord y(t) < 0. So ordγ (y m + c1 (x)y m−1 + · · · ) = ordγ y m < 0, which contradicts that

y m + c1 (x)y m−1 + · · · = 0,

ordγ 0 = ∞.

Let γ1 , γ2 , . . . , γs be the branches for which ordγ x > 0; they are precisely the branches centred at points (0, b, 1) on the y-axis. Consider now such a point Pj on F. Let γj1 , . . . , γjk be the branches of F centred at Pj . Then I(Pj , v(X) ∩ F) = ordγj1 x + · · · + ordγjk x.

Summing for all such points Pj , it follows that P P I(Pj , v(X) ∩ F) = ordt x(t);

the right-hand side counts the zeros of x with appropriate multiplicity. To finish the proof, it suffices to note that, if Pj = (0, yj , 1) and yj is an mj -ple root of m m−1 the + · · · , then I(Pj , v(X) ∩ F) = mj ; hence P polynomial Y + c0 (0)Y I(P , v(X) ∩ F) = m. 2 j j

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T HEOREM 5.34 The function ζ has finitely many poles. The number of poles each counted with its multiplicity is [Σ : K(ζ)]. Proof. Poles of ζ are zeros of 1/ζ,. Hence, there are only a finite number of them, and this number, counting with multiplicity, is [Σ : K(1/ζ)] = [Σ : K(ζ)]. 2 C OROLLARY 5.35 The number of zeros of ζ equals the number of its poles. Equivalently, P P∈P(Σ) ordP ζ = 0,

where the summation extends over all places P.

E XAMPLE 5.36 Suppose that p = 5, and let K = F5 . Take F = v(f (X, Y ) with f (X, Y ) = Y 5 + Y − X 3 ; then K(F) = Σ = K(x, y) with y 5 + y − x3 . The irreducible plane curve F has a unique singular point, namely the double point X∞ = (0, 1, 0). It is the centre of a unique branch. Therefore there is a bijection between branches and points of F; so branches and points may be identified. To illustrate Corollary 5.35, let g(x) = x2 (1 − x12 ) and compute its zeros and poles. The roots ξ of the polynomial g(X) = X 2 (1 − X 12 ) are 0, counted twice, together with the 12 non-zero square elements of F25 , the quadratic extension of F5 in K. If w is a primitive element of F25 , then wi = w2i with i = 1, . . . , 12 are the non-zero square elements. Therefore the zeros of g(x) are the places associated with the points P = (a, b) such that b5 + b = a3 and either a = 0, counted twice, or a is a non-zero square element of F5 . The former points are five, say P1 , . . . , P5 , while the latter ones are 60, say Q1 , . . . , Q60 . These 65 points together with X∞ are those points of F which lie in PG(2, 25). If Pi and Qi are the corresponding places, then the zero divisor of g(x) is P60 P5 div(x2 (1 − x12 ))0 = 2 i=1 Pi + i=1 Qi . Also, g(x) has a unique pole, namely the place P∞ arising from X∞ . Since (x(t) = t−5 + · · · , y(t) = t−3 + · · · ) is a primitive representation of the unique branch of F centred at X∞ , the pole divisor of g(x) is div(x2 (1 − x12 ))∞ = 70P∞ .

5.5 SEPARABILITY AND INSEPARABILITY Any field of transcendence degree 1 over K can be obtained by a transcendental extension K(x)/K followed by an algebraic extension Σ/K(x). In positive characteristic p, the latter extension Σ/K(x) can be split into a separable step Σs /K(x) and a purely inseparable step Σ/Σs ; see Section A.1. Separable extensions have similar properties to algebraic extensions in characteristic zero. Purely inseparable extensions are related to certain subfields of Σ containing K, namely, the subfields Σq = {ξ q | ξ ∈ Σ}, where q denotes a power of p. T HEOREM 5.37 For every q, the extension Σ/Σq is purely inseparable of degree q. Conversely, if K ⊂ Σ′ ⊂ Σ and Σ/Σ′ is purely inseparable of degree q, then Σ′ = Σq .

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Proof. Since K is perfect, it follows that Σq is a field that contains K. The extension Σ/Σq is inseparable, as ξ q ∈ Σq for every ξ ∈ Σ. To calculate the degree of Σ/Σq , choose a local parameter x ∈ Σ at some place P of Σ. By the Theorem of the Primitive Element, Theorem A.1, there exists y ∈ Σ such that Σ = K(x, y). Also, Σ = K(x, y q ). In fact, Σ = K(x, y q )(y)/K(x, y q ) is a purely inseparable extension since y satisfies the equation X q − y q = 0 over K(x, y q ). On the other hand, K(x) ⊂ K(x, y q ) ⊂ Σ, and therefore Σ/K(x, y q ) is also a separable extension. This is only possible when Σ = K(x, y q ). Now, Σq = K(xq , y q ) and Σ = K(x, y q ) imply that Σ = Σq (x). Since x is a root of the polynomial X q −xq with coefficients in Σq , it follows that [Σ : Σq ] ≤ q. To prove the reverse inequality, let aq0 X m + · · · + aqm be a minimal polynomial of the extension Σ = Σq (x)/Σq . It is enough to show that m ≥ q. Assume that ordP aqi xi is minimal; that is, ordP aqi xi ≤ ordP aqj xj for every j = 0, . . . , m, Then aq0 xm + · · · + aqm = 0 yields that ordP aqi xi = ordP aqk xk for some k, with 0 ≤ k ≤ m and i 6= k. Since ordP x = 1, it follows that i ≡ k (mod q). Therefore m ≥ q. For the converse, assume that the extension Σ/Σ′ is purely inseparable of degree q. Then ξ q ∈ Σ′ for every ξ ∈ Σ. Hence Σq ⊂ Σ′ ⊂ Σ. Since the degree of Σ/Σq is q, so it follows that Σq = Σ′ . 2 Let Σ′ be a subfield of Σ which contains K and has transcendence degree 1 over K. The extension Σ/Σ′ is finite. Also, there is a unique intermediate field Σ′′ , with Σ′ ⊂ Σ′′ ⊂ Σ such that Σ′′ /Σ′ is separable and Σ/Σ′′ is purely inseparable. Then, [Σ : Σ′ ] = [Σ : Σ′′ ][Σ′′ : Σ′ ], where [Σ′′ : Σ′ ] is the separable degree and [Σ : Σ′′ ] is the inseparable degree of the extension [Σ : Σ′ ]. An element ξ ∈ Σ is a separable or inseparable variable of Σ according as the finite extension Σ/K(ξ) is separable or inseparable. A separable variable ζ of Σ is characterised in the following manner. For any element ξ ∈ Σ\K, the irreducible polynomial f (X, Y ) ∈ K[X, Y ] linking ξ and ζ, that is f (ξ, ζ) = 0, has the property that the polynomial f (X, ζ) ∈ K(ζ)[X] is separable; that is, its derivative df (X, ζ)/dX is not identically zero. Note that df (X, ζ)/dX = ∂f (X, Y )/∂X with Y = ζ. A useful criterion for separability in Σ is given in the following lemma. L EMMA 5.38 An element ζ ∈ Σ is a separable variable if and only if ζ 6∈ Σp , that is, ζ is not a p-th power of an element in Σ. Proof. If ζ is a separable variable, K(ζ) 6⊂ Σp , because Σ/Σp is purely inseparable of degree greater than 1. Conversely, if ζ ∈ Σ\K is not separable, there is an intermediate field Σ′ with K(ζ) ⊂ Σ′ ⊂ Σ such that Σ/Σ′ is purely inseparable of degree p. By Theorem 5.37, Σ′ = Σp , whence it follows that ζ ∈ Σp . 2 5.6 FROBENIUS RATIONAL TRANSFORMATIONS In this section, K has positive characteristic q denotes a power of p. Also, P p, and if g(X, Y ) ∈ K[X, Y ] and g(X, Y ) = cij X i Y j , then g q (X, Y ) denotes the

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P

polynomial cqij X i Y j . The previous notation is maintained: F = v(f (X, Y )) is an irreducible curve; P = (x, y) is a generic point of F; the field Σ = K(x, y) is the associated function field; ω is a rational transformation of Σ; the subfield ω(Σ) = {ω(ξ) | ξ ∈ Σ} is the image of Σ under ω. D EFINITION 5.39 The rational transformation ωq : x′ = xq ,

y′ = yq

of Σ is the q-th Frobenius transformation. P Let f (X, Y ) = aij X i Y j . Note that f q (X, Y ) is also irreducible. Also, the image curve of F under ωq is F ′ = v(f q (X, Y )). In fact, as x′ = xq , y ′ = y q , P q ′i ′j P so f q (x′ , y ′ ) = aij x y = ( aij xi y j )q = 0. In particular, P ′ = (x′ , y ′ ) is a generic point of F ′ , and the subfield Σ′ = K(x′ , y ′ ) of Σ can be viewed as the function field associated with F ′ . T HEOREM 5.40 The mapping ωq is not birational. However, Σ ∼ = Σ′ . Proof. If ωq were birational, then both x and y, and hence every element in Σ, would be a q-th power; but this is impossible. Now, it is shown that the mapping ρ : Σ −→ Σ′ , U q (x′ , y ′ ) U (x, y) 7−→ q ′ ′ , V (x, y) V (x , y ) with U (X, Y ), V (X, Y ) ∈ K[X, Y ], V (x, y) 6= 0, is well-defined and does not depend on the representative functions. To do this, choose U1 , V1 ∈ K[X, Y ] so that U1 (x, y) U (x, y) = . V1 (x, y) V (x, y) Then U (x, y)V1 (x, y) − U1 (x, y)V (x, y) = 0. As P = (x, y) is a generic point of F, there exists h(X, Y ) ∈ K[X, Y ] such that f (X, Y )h(X, Y ) = U (X, Y )V1 (X, Y ) − U1 (X, Y )V (X, Y ). It follows that f q (X, Y )hq (X, Y ) = U q (X, Y )V1q (X, Y ) − U1q (X, Y )V q (X, Y ).

As f q (x,′ y ′ ) = 0, this shows that

U (x′ , y ′ )V1 (x′ , y ′ ) − U1 (x′ , y ′ )V (x′ , y ′ ) = 0, whence U q (x′ , y ′ ) U1q (x′ , y ′ ) . q ′ ′ = V1 (x , y ) V q (x′ , y ′ ) Since the mapping ρ induces an automorphism of K, so ρ is a homomorphism ′ from . Actually, ρ isPinjective, as ξ q = 0 implies ξ = 0, and surjective as P Σ′ ito ′Σ j bij x y is the image of aij xi y j when aqij = bij for every i and j. Therefore ρ is an isomorphism. 2

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D EFINITION 5.41 A rational transformation ω of Σ is either separable or inseparable according as the field extension Σ/ω(Σ) is separable or inseparable. T HEOREM 5.42 (i) Every inseparable rational transformation ω of Σ is the product of a separable rational transformation by a q-th Frobenius transformation. (ii) The inseparability degree of ω is equal to q. Proof. Let ω be given by the generic points P = (x, y) of F and P ′ = (x′ , y ′ ) of F ′ . If either x or y is not a p-th power, Σ/ω(Σ) is separable. Otherwise, let q denote the highest power of p such that x′ = ξ q and y ′ = η q . By Theorem 5.37, ω(Σ) is purely inseparable and its inseparability degree is equal to q. Further, the rational transformation ωs given by the generic point (ξ, η) is separable. Hence ω is the product of ωs and the q-th Frobenius transformation. 2

5.7 DERIVATIONS AND DIFFERENTIALS D EFINITION 5.43 A derivation D in a field L is a mapping D : L → L given by a 7→ a′ = D(a) such that (i) (a + b)′ = a′ + b′ ; (ii) (ab)′ = a′ b + ab′ , for all a, b ∈ L. L EMMA 5.44

(i) 0′ = 0;

(ii) 1′ = 0; (iii) (a/b)′ = (a′ b − ab′ )/b2 for all a, b ∈ L with b 6= 0.

Proof. (i) Here, 0′ = (0 + 0)′ = 0′ + 0′ , whence 0′ = 0′ − 0′ = 0. (ii) In this case, 1′ = (1 · 1)′ = 1′ · 1 + 1 · 1′ = 1′ + 1′ , whence 1′ = 1′ − 1′ = 0. (iii) Finally, a′ = (b · a/b)′ = b′ · a/b + b · (a/b)′ . So (a/b)′ = (a′ − b′ · a/b)/b = a′ /b − b′ · a/b2 = (a′ b − ab′ )/b2 .

2

D EFINITION 5.45 Given a subfield M of L, a derivation D of L is said to be over M if D(c) = 0 for each c ∈ M. Let D be a derivation of L over M and let θ, φ ∈ L; then

D(θi φj ) = iθi−1 φj Dθ + jθi φj−1 Dφ.

So D(F (θ, φ)) =

∂F ∂F Dθ + Dφ, ∂θ ∂φ

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where F ∈ M [X, Y ]. Here, ∂F/∂θ means ∂F/∂X evaluated for X = θ, Y = φ. Consider now the particular case in which L = K((t)). In K((t)), a derivation is given by the following rule: dP i P ci t = ici ti−1 , dt with ci ∈ K. Take dt as an indeterminate and consider the field K((t))(dt). If θ ∈ K((t)), define the differential of θ in the following way:   dθ dt. dθ = dt Let φ ∈ K((t))\K and, when p > 0, suppose also that dφ/dt 6= 0; that is, φ 6∈ K((tp )). Define, in the field K((t))(dt), dθ dφ as dθ divided by dφ. From the rule that dθ2 dθ1 d (θ1 θ2 ) = θ1 + θ2 , dt dt dt multiplication by dt gives d(θ1 θ2 ) = θ1 dθ2 + θ2 dθ1 and similarly d(θ1 + θ2 ) = dθ1 + dθ2 . Dividing by dφ gives dθ2 dθ1 d(θ1 θ2 ) = θ1 + θ2 , dφ dφ dφ d(θ1 + θ2 ) dθ1 dθ2 = + . dφ dφ dφ Therefore dθ dφ is a derivation of K((t)), which coincides with d/dt for φ = t. Such a derivation is denoted by d/dφ. As ∂F dθ ∂F dφ d F (θ, φ) = + , dt ∂θ dt ∂φ dt multiplying by dt gives   d ∂F dθ ∂F dφ F (θ, φ) dt = dt + dt, dt ∂θ dt ∂φ dt and hence ∂F ∂F dθ + dφ, dF (θ, φ) = ∂θ ∂φ where F is a rational function f /g with f, g ∈ K[X, Y ] and g(θ, φ) 6= 0. To deal with Σ, a characterisation for separability in terms of derivatives is needed. θ 7→

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T HEOREM 5.46 The element ζ ∈ Σ is a separable variable if and only if dζ¯ 6= 0 for each place P of Σ. Proof. By the Theorem of the Primitive Element, there exists ξ ∈ Σ such that Σ = K(ξ, ζ). Note that for any place P, either dξ or dζ does not vanish. In fact, for a primitive place representation τ of P, the equality dτ (ξ) = 0 occurs if and only if τ (ξ) ∈ K((t)) has no term ci ti with i 6≡ 0 (mod p). If dτ (ζ) = 0 also happens then neither τ (ζ) ∈ K((t)) has a term ci ti with i 6≡ 0 (mod p). Then (τ (ξ), τ (ζ)) is an imprimitive representation of the branch of the model (F; (ξ, ζ)). But this would mean that τ is an imprimitive place representation. Let U (X, Y ) ∈ K[X, Y ] be an irreducible polynomial for which U (ξ, ζ) = 0, and F = v(U ) the corresponding irreducible curve having P = (ξ, ζ) as a generic ¯ ζ) ¯ = 0 holds; hence, by derivation, point. For any place P of Σ, the equation U (ξ, ¯ ξ¯ + (∂U/∂ ζ)d ¯ ζ¯ = 0. (∂U/∂ ξ)d Assume the existence of a place P ∈ P(Σ) such that dζ¯ = 0. Then ζ¯ = τ (ζ) for a primitive representation τ of P, and so ¯ ξ¯ = 0. (∂U/∂ ξ)d Since dξ¯ 6= 0, it follows that

∂U/∂ ξ¯ = 0.

Therefore, either ∂U (X, Y )/∂X = 0 and ζ is inseparable or ∂U (X, Y )∂X 6= 0 and P = (ξ, ζ) is also a generic point of the curve v(∂U (X, Y )∂X). The latter case cannot actually occur, as deg ∂U (X, Y )∂X < deg U (X, Y ), while U (X, Y ) is irreducible. Conversely, suppose ζ is inseparable; then there exist ξ in Σ and U (X, Y ) in K[X, Y ] such that U (ξ, ζ) = 0, Therefore, for each place P of Σ, ¯ ζ) ¯ = 0, U (ξ,

∂U (X, ζ)/∂X = 0. ¯ ζ)/∂ ¯ ξ¯ = 0. ∂U (ξ,

It follows that also ¯ ζ)/∂ ¯ ζ)d ¯ ζ¯ = 0. (∂U (ξ, Here, ¯ ζ)/∂ ¯ ζ¯ 6= 0 ∂U (ξ, by the irreducibility of U (X, Y ). So dζ¯ = 0, as required.

2

Given a place P of Σ, let τ : Σ → K((t)) be a primitive representation of P. Having fixed a separable variable ζ of Σ, let ξ be any element of Σ. Then there exists a polynomial f (X, Y ) ∈ K[X, Y ] such that f (ξ, ζ) = 0. it follows that ¯ ζ) ¯ = 0, where, with the usual conventions, ξ(t) ¯ = τ (ξ) and ζ(t) ¯ f (ξ, = τ (ζ). ¯ ¯ Therefore df (ξ, ζ) = 0, and so ∂f ¯ ∂f ¯ dξ + ¯ dζ = 0. ∂ ξ¯ ∂ζ

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¯ ζ); ¯ Note that ∂f /∂ ξ¯ 6= 0. If this were not true, then ∂f /∂X = 0 is satisfied by (ξ, but this implies that the polynomial ∂f /∂X is identically zero, simply from the fact that such a polynomial, if not zero, would have degree less than that of f (X, Y ), contradicting the irreducibility of f (X, Y ). Since ∂f /∂X = 0, so ∂f /∂ξ = 0; but this is impossible with ζ a separable variable of Σ. By the previous theorem, dζ¯ 6= 0. Therefore it makes sense to write dξ¯ ∂f /∂ ζ¯ =− ∈ K((t)), (5.1) ¯ dζ ∂f /∂ ξ¯ and to make the following definition. D EFINITION 5.47 dξ ∂f /∂ζ =− ∈ Σ. dζ ∂f /∂ξ

(5.2)

In a diagram, ξ τ

d/dζ - dξ/dζ

τ ? ?d/dζ - dξ/dζ ξ

¯ ζ¯ ∈ K((t)) and the The mapping ξ 7→ dξ/dζ is a derivation of Σ, since ξ¯ 7→ dξ/d mapping τ : Σ → K((t)) is a K-monomorphism. Now, the idea of the differential of a separable variable of Σ is introduced in a similar way to the analogous idea for K((t)). D EFINITION 5.48 (i) For a separable variable ζ of Σ, a differential is any element xdζ, with x ∈ Σ, of the field Σ(dζ), a transcendental extension of Σ by the symbol dζ. (ii) The differential dξ of ξ ∈ Σ is defined by the expression dξ =

dξ dζ. dζ

(5.3)

(iii) The differentials with respect to the fixed element ζ form a vector space of dimension 1 over Σ, the vector space of exact differentials. It should be noted that the differential does not depend on the choice of ζ in the following sense. If ζ1 is another separable variable of Σ, then dζ1 /dζ 6= 0, and if the new indeterminate dζ1 is identified with (dζ1 /dζ)dζ, then dξ as defined from dζ1 can be considered as the same as dξ defined from dζ. If this is done, then dξ dξ dζ1 , dζ = dζ dζ1 by the following theorem.

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T HEOREM 5.49 If ζ, ζ1 ∈ Σ are both separable, then, for each ξ ∈ Σ, dξ dζ1 dξ · = . dζ1 dζ dζ

129

(5.4)

Proof. Let P be a place of Σ. If σ is a primitive representation of P, then       dξ¯ dξ¯ dζ¯1 dξ dζ1 dξ dζ1 =σ = ¯ · ¯ = ¯. ·σ · σ dζ1 dζ dζ1 dζ dζ1 dζ dζ Since σ is a K-monomorphism from Σ into K((t)), the result is proved.

2

The above definition of differential is functional but awkward in that it singles out a separable variable ζ for a special role. An alternative introduction to the concept of the vector space of exact differentials which avoids such an inconvenience is possible. To do this, define for every ζ ∈ Σ, including inseparable variables and elements of K, a symbol dζ and consider the infinite-dimensional vector space over Σ whose elements are the formal finite sums α1 dζ1 + α2 dζ2 + · · · ,

with ζ1 , ζ2 , . . . ∈ Σ and α1 , α2 , . . . ∈ K; then these symbols satisfy the equations ∂F ∂F dζ1 + dζ2 + . . . = 0, ∂ζ1 ∂ζ2 where F (X1 , X2 , . . .) is a rational function over K satisfied by ζ1 , ζ2 , . . .. This one-dimensional vector space over Σ is the desired vector space of exact differentials. Familiar results on derivatives, such as the first assertion in the following theorem, hold true in zero characteristic. Extensions to positive characteristic are also possible but inseparability needs to be considered carefully. T HEOREM 5.50

(i) If p = 0, then dξ = 0 is only for ξ ∈ K.

(ii) If p > 0, then ξ is a separable variable of Σ if and only if dξ 6= 0. Proof. Let f (X, Y ) ∈ K[X, Y ] be irreducible and such that f (ξ, ζ) = 0. If f (X, Y ) is written in the usual canonical form f (X, Y ) = a00 + a10 X + a01 Y + · · · + aij X i Y j + . . . ,

then, with g(X, Y ) = ∂f (X, Y )/∂Y ,

g(X, Y ) = a01 + a11 X + 2a02 Y + · · · + jaij X i Y j−1 + · · · .

Suppose that dξ/dζ = 0. From (5.2) it follows that (ξ, ζ) satisfies the equation g(X, Y ) = 0. Since P = (ξ, ζ) is a generic point of the curve v(f (X, Y )), that can happen only when g(X, Y ) is the zero polynomial. Then, in zero characteristic, f (X, Y ) = f (X, 0) = a00 + a10 X + . . . + an0 X n , whence f (ξ, 0) = 0, which implies that ξ ∈ K. Instead, in positive characteristic p, P f (X, Y ) = aik X i Y pk .

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Since ∂f (ξ, Y )/∂Y = 0, so ξ is inseparable. Finally, as dx = 0 for any inseparable variable in Σ, so (ii) holds. 2 By (5.2), dξ/dζ is an element in Σ. Hence, it makes sense to consider the second derivative of ξ, that is, d2 ξ d(dξ/dζ) = , 2 dζ dζ and iteratively the higher derivatives of ξ: d(di ξ/dζ i ) di+1 ξ = . i+1 dζ dζ Hence n   dn (ξη) X n dj ξ dn−j η = · , j dζ j dζ n−j dζ n j=0

(5.5)

which is essentially a consequence of the product rule: dξ dη d(ξη) = η+ ξ. dζ dζ dζ

5.8 THE GENUS OF A CURVE Given a place P of Σ and a primitive representation τ, if ξ is a separable variable ¯ the order of dξ at P is defined as follows. of Σ and τ (ξ) = ξ, D EFINITION 5.51 ordP dξ = ordt

¯ dξ(t) . dt

(5.6)

This concept is well defined by Theorem 5.46 and does not depend on the choice of the representative τ . D EFINITION 5.52 Let ξ be a separable variable of Σ. (i) A place P with ordP dξ > 0 is a zero of dξ; if P is a zero, then it is a zero of order ordP dξ. (ii) A place P with ordP dξ < 0 is a pole of dξ; if P is a pole, then it is a pole of order −ordP dξ. L EMMA 5.53 If x ∈ Σ is separable, then ordP dx = 0 except for a finite number of places; that is, dx has only a finite number of zeros and poles.

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Proof. Let y be an element of Σ such that Σ = K(x, y). Consider the irreducible curve F = v(F (X, Y )) for which P = (x, y) is a generic point. Now, it is shown that the affine simple points of F with tangent v(X) are finite in number. In fact, such a point P = (a, b) of F also belongs to the polar curve F ′ of F at the point Q = (0, 1, 0). Note that, since x is a separable variable of Σ, so ∂F (X, Y )/∂Y is not identically zero; hence the polar curve F ′ is v(∂F (X, Y )/∂Y ). Then the number in question is at most (deg F )(deg F − 1) by B´ezout’s Theorem. It remains to calculate ordP dx at the places P of Σ centred at affine simple points P with tangent distinct from v(X). As before, this includes all places except perhaps for a finite number. If P = (a, b) is such a point, then x = a + ct + · · · , where c 6= 0; otherwise, F either has v(X) as tangent at P or P is a singular point. It follows that dx/dt = c, whence ordP dx = 0. This proves the result. 2 P Lemma 5.53 shows that the sum P∈P(Σ) ordP dζ is meaningful, since all but a finite number of terms are zero. A fundamental result that enables the genus of a curve to be defined is the following. T HEOREM 5.54 For every two separable variable ξ, η of Σ, P P P∈P(Σ) ordP dη, P∈P(Σ) ordP dξ =

where the summation is over all places of Σ.

Proof. Since dξ = (dξ/dζ)dζ, so P P P P∈P(Σ) ordP dζ. P∈P(Σ) ordP (dξ/dζ) + P∈P(Σ) ordP dξ = P Since dξ/dζ ∈ Σ, it follows from Corollary 5.35 that P∈P(Σ) ordP (dξ/dζ) = 0, P P and so P∈P(Σ) ordP dξ = P∈P(Σ) ordP dζ, as required. 2 D EFINITION 5.55 Let g be the integer for which P P∈P(Σ) ordP dζ = 2g − 2.

Then g, necessarily non-negative, is the genus of Σ.

It is shown in the next section that the genus of Σ is less than or equal to the virtual genus of each curve F which is a model of Σ; equality holds if and only if F has only ordinary singularities. The following theorem provides a formula for ordP for a place centred at a simple point. T HEOREM 5.56 Let P be a simple point of an irreducible curve F = v(F (X, Y )) such that the vertical line ℓ through P is the tangent to F at P, and let P be the unique place of F centred at P . Given a generic point (x, y) of F such that x is separable, then   ∂F (x, y) ordP dx = ordP . ∂y

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Proof. Without loss of generality, take P to be the origin. Then F (X, Y ) = X + Φ2 (X, Y ) + · · · + Φn (X, Y ).

Suppose that Y r , with r ≥ 2, is the term of least degree not containing X. Write F (X, Y ) = X + cY r + · · · ,

where all other terms are divisible by X 2 , XY or Y r+1 . Then I(P, F ∩ v(X)) = r and a primitive representation of the branch of F centred at P is x(t) = −ctr + · · · , y(t) = t.

Hence dx(t)/dt = −crtr−1 + · · · ; that is, ordP dx = r − 1. Also,

∂F (X, Y ) = cr Y r−1 + · · · , ∂Y where all other terms are divisible by X or Y r . Hence, ordP (∂F (x, y)/∂y) = r − 1,

as required. This proof fails when p > 0 and r = np with n a positive integer. To prove the result in this case, take the derivative of F (x(t), y(t)) = 0. Then     ∂F (x(t), y(t)) ∂F (x(t), y(t)) x′ (t) + y ′ (t) = 0. ∂X ∂Y Hence ∂F (x(t), y(t)) ∂F (x(t), y(t)) ′ x (t) = − . ∂X ∂Y Since ordt y ′ (t) = 0, so ordt (∂F (x(t), y(t))/∂Y ) = ordt (∂F (x(t), y(t))/∂X) + ordt x′ (t). To obtain the result, it is necessary to show that ordt (∂F (x(t), y(t))/∂X) = 0. Suppose, therefore, that ordt(∂F (x(t), y(t))/∂X) > 0 and consider the polar curve F ′ = v(∂F (X, Y )/∂X) at (1, 0, 0). Hence ∂F (x(0), y(0))/∂X = 0, from which it follows that F ′ passes through P . So the tangent to F at P passes through (1, 0, 0). This gives a contradiction since the tangent to F at P is v(X). Therefore ordt (∂F (x(t), y(t))/∂X) = 0, as required.

2

T HEOREM 5.57 If P1 , . . . , Pk are the singular points of an irreducible curve F, each of them ordinary, then the virtual genus of F is equal to g; that is, Pk g = 12 (d − 1)(d − 2) − i=1 12 mi (mi − 1) where d is the degree of F and mi is the multiplicity of the point Pi .

Proof. Choose a reference system (X0 , X1 , X2 ) in the plane so that v(X0 ) is the line at infinity and so that the following conditions are satisfied:

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(i) the points at infinity of F are d distinct, simple points, none of which is Y∞ = (0, 0, 1); (ii) no tangent to F at a singular point is the Y -axis.

P Let P = (x, y) be a generic point of F. Now, calculate P ordP dx that, by definition, equals 2g − 2. Given these assumptions, the following properties are demonstrated: (a) dx has poles only at places centred on the line at infinity and ordP dx = −2 for each of these places; (b) dx has zeros only at places centred at simple points of F with vertical tangent and ordP dx = ordP ∂F (X, Y )/∂Y ; (c) if P is a point of multiplicity m of F and F ′ = v(∂F (X, Y )/∂Y ), then I(P, F ∩ F ′ ) = m(m − 1). To show (a), a primitive representation of a branch γ of F centred at the simple point P = (0, 1, a), with a 6= 0, must be obtained. Interchanging X0 and X1 , the point P becomes the affine point (1, 0, a) and the branch γ is centred at (0, a) with tangent v(X2 − aX0 − bX1 ), with b ∈ K; in non-homogeneous coordinates, this becomes v(Y − bX − a), which, it should be noted, is not v(X) for (0, a). A primitive representation of the branch γ has the form x(t) = t, y(t) = a + ctr + · · ·

with r ≥ 1; in homogeneous coordinates, x0 (t) = 1, x1 (t) = t,

x2 (t) = a + ctr + · · · .

It follows that a representation of γ centred at (0, 1, a) has the form x0 (t) = t, x1 (t) = 1, x2 (t) = a + ctr + · · · ; this, in non-homogeneous coordinates, becomes x(t) = t−1 , y(t) = (a + ctr + · · · )/t.

Hence, dx(t)/dt = −1/t2 and ordP dx = −2. To prove (b), consider a simple affine point P = (a, b) of F. If the line v(X − a) through P is not the tangent to F at P, a branch representation centred at P is given by x(t) = a + t, y(t) = b + ctr + · · · ,

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whence ordP dx = 0. If, instead, the tangent to F at P is v(X − a), a branch representation is given by x(t) = a + dtr + · · · , y(t) = b + t,

with ≥ 2. By Theorem 5.56, ordP dx = ordP



∂F (x(t), y(t)) ∂Y



.

This means that ordP dx = I(P, F ∩ F ′ ), where F ′ = v(∂F /∂Y ). To finish the proof, it remains to show (c); that is, it is necessary to calculate I(P, F ∩ F ′ ), where P is an m-fold singular point of F. Note that F ′ has an (m − 1)-fold singularity at P . Now it is shown that the tangents at P for F are all distinct from those for G, and hence that I(P, F ∩ F ′ ) = m(m − 1). Without loss of generality, let P = (0, 0). Then F (X, Y ) = Φm (X, Y ) + · · · + Φd (X, Y ), with Φm (X, Y ) = am,0 X m + · · · + ai,m−i X i Y m−i + · · · + a0,m Y m . Note that a0,m 6= 0, as otherwise v(X) is a tangent. Let t1 , . . . , tm be the roots of the polynomial Φm (Z) = Φm (1, Z) = am,0 + · · · + ai,m−i Z m−i + · · · + a0,m Z m . Since P is an ordinary singular point of F, the roots of Φm (Z) are distinct; so Φm (Z) and its derivative Φ′m (Z) have no common roots. To determine the tangents to F ′ , let ∂F (X, Y ) = Ψi (X, Y ) + · · · + Ψd−1 (X, Y ), ∂Y where ∂Φm (X, Y ) ; ∂Y also ∂Φm (X, Y )/∂Y cannot be identically zero, as otherwise Φm (Z) would be the p-th power of a polynomial and its roots would have multiplicity p. Hence, the tangents to F ′ at P are given by the roots of Ψi (Z) = Ψi (1, Z), except possibly for tangents equal tov(X). The curves F and F ′ have a tangent at P in common if and only if Φm (Z) and Ψi (Z) have a root in common; but that cannot happen since Ψi (Z) = ∂Φm (Z)/∂Z and Φm (Z) has no multiple roots. This proves (c). Now, if γ is a branch centred at a singular point of F, it must be linear since P is an ordinary singularity. Hence a primitive representation of γ has the form x(t) = a + t, y(t) = b + ctr + · · · . It follows that ordγ dx = 0. Ψi (X, Y ) =

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As a consequence, P

P∈P(Σ)

ordP dx =

P

I(P, F ∩ F ′ ) − 2d,

where the second summation is over all simple points of F with tangent v(X − a). Applying B´ezout’s Theorem, the intersection of F and F ′ can be determined. With d = deg F and d − 1 = deg F ′ ,   k X X ∂F (x, y) ordP mi (mi − 1) + d(d − 1) = , ∂y i=1 P∈P(Σ)

where the second summation is over all places centred at affine non-singular points of F with vertical tangent. From (b), Pk P d(d − 1) = i=1 mi (mi − 1) + P∈P(Σ) ordP dx, and so

X

P∈P(Σ)

ordP



∂F (x, y) ∂y



= d(d − 1) −

k X i=1

mi (mi − 1).

Therefore

Finally,

P

P∈P(Σ)

ordP dx = d(d − 1) −

d(d − 3) − whence

Pk

i=1

Pk

i=1

mi (mi − 1) − 2d.

mi (mi − 1) = 2g − 2,

g = 21 (d − 1)(d − 2) −

1 2

Pk

i=1

mi (mi − 1).

2

It has been shown that the genus of an irreducible algebraic curve with only ordinary singularities coincides with its virtual genus. This leads to the following definition. D EFINITION 5.58 The genus of an irreducible algebraic curve is the genus of its function field Σ, whence it is the genus of each model (F; (x, y)) of Σ, and so is the virtual genus of a plane model with only ordinary singular points. However, the genus of an irreducible curve with some non-ordinary singularities is in general smaller than its virtual genus; more precisely, P g ≤ 21 (n − 1)(n − 2) − P 21 r(r − 1),

where n = deg F, where P ranges over the singular points of F, and where r is the multiplicity of P . This follows from Theorem 3.27 and (i) of Lemma 3.28. The following example illustrates this case.

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E XAMPLE 5.59 Let p 6= 2 and, for a positive integer h ≥ 4, let Qh F (X, Y ) = Y 2 − f (X), with f (X) = i=1 (X − αi ),

where αi 6= αj for 1 ≤ i < j ≤ h. Let F be the irreducible curve v(F (X, Y )) and let P = (x, y) be a generic point of F. First, all the affine points of F are simple, since all the roots of f (X) are distinct, and Y∞ is a singular point of multiplicity h − 2. So, the virtual genus gv of F is equal to h − 2. P To determine the genus g of F, the sum P∈P(Σ) ordP dx is calculated. For this, the simple points of F with vertical tangent and the unique point at infinity must be considered. Among the simple points of F the only ones with vertical tangent are the points Qi = (αi , 0) for i = 1, . . . , h. If P = (x0 , y0 ) is a simple point of F, its tangent is v(G) with G = FX (x0 , y0 ) (X − x0 ) + FY (x0 , y0 ) (Y − y0 ); this is vertical when FY (x0 , y0 ) = 0, that is, if and only if y0 = 0. A primitive branch representation centred at Qi is x(t) = αi + ct2 + · · · , c 6= 0, y(t) = t. Then, dx(t)/dt = 2ct + . . . and so ordt (dx(t)/dt) = 1. Hence, if P is the place corresponding to Qi , then ordP dx = 1. Actually, F has only one point at infinity, which is Y∞ = (0, 0, 1). Now, it is shown that are one or two branches centred at Y∞ , according as h is odd or even. The corresponding places are poles whose order is 3 in the former case, and 2 for both places in the latter case. Take Y∞ to the origin O = (0, 0) by interchanging the X-axis and the line at infinity. Then the transformed curve is F ′ , given by Qh Y h−2 − X h + f (X, Y ), with f (X, Y ) = X h − i=1 (X − αi Y ), which shows that X∞ is an (h − 2)-ple point of F with only the one tangent, given by ℓ∞ . A primitive representation (x′ (t), y ′ (t)) of a branch γ ′ of F ′ centred at P = (0, 0) has the form x′ (t) y ′ (t)

= ti + · · · , = a tj + · · ·

(5.7)

with i < j. From the equation y(t)h−2 − x(t)h + y(t)h = 0, it follows that (h − 2)j = hi, whence one of the following occurs: (i) i = h − 2, j = h; (ii) h is even and i = 21 (h − 2), j = 12 h. The two cases are investigated separately. In case (i), F ′ has only one branch centred at P = (0, 0). In fact, I(P, v(X) ∩ F ′ ) = h − 2,

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while ordt x′ (t) = h − 2 for every branch γ ′ . By Theorem 4.50, P h−2= ordt x′ (t),

where the summation is for all branches γ ′ of F ′ centred at P (0, 0), whence the assertion follows. Now, (5.7) becomes x′ (t) = y ′ (t) =

th−2 + · · · , a th + · · · .

(5.8)

t−2 E1 (t), t−h E2 (t)),

(5.9)

Going back to F, the corresponding branch γ of F centred at Y∞ has a primitive representation x(t) = y(t) =

with invertible E1 (t), E2 (t) ∈ K[[t]]. So ordt dx(t)/dt = −3, since p 6= 2. Hence, ordP∞ dx = −3, where P∞ is the place coming from γ. This implies that 2g − 2 = h − 3, whence g = 12 (h − 1) and h is odd. In case (ii), h is even. Now, Theorem 4.50 together with x′ (t) = 21 (h − 2) shows that F ′ has exactly two branches centred at O = (0, 0), each having a primitive representation x′ (t) y ′ (t)

= t(h−2)/2 + · · · , = a th/2 + · · · .

(5.10)

Each of the two corresponding branches of F centred at Y∞ has primitive representation x(t) = t−1 F1 (t)), y(t) = t−h/2 F2 (t),

(5.11) (i)

with invertible F1 (t), F2 (t) ∈ K[[t]]. It follows that ordt dx(t)/dt = −2. If P∞ with i = 1, 2 are the corresponding places, ordP (i) dx = −2. Therefore ∞ P 2g − 2 = P ordP dx = h − 4, and so the genus of F is g = 21 (h − 2) for h even.

T HEOREM 5.60 Let m be the smallest non-gap of Σ at a place P ∈ PK(X ). Suppose that ξ ∈ K(X ) has a pole of order m at P but is regular elsewhere. If F is a subfield of Σ containing K(ξ) such that [F : K(ξ)] > 1, then F has positive genus. Proof. Suppose F is rational, and let F = K(η). Then ξ = f (η)/g(η), where f (X), g(X) ∈ K[X] and the roots of f (X) are distinct from those of g(X). Assume first that g(X) is constant. Then f (X) has positive degree k; in fact, k > 1 as [F : K(ξ)] > 1. Since any pole of η is also a pole of ξ, the hypothesis implies that η has a unique pole, that is, div(η)∞ = −nP. Now, m = nk; but this contradicts the definition of ξ. Assume now that g(X) is divisible by X −a. Then P is a zero of η −a. Actually, P is the unique zero of η − a, because P is the unique pole of ξ. Thus (η − a)−1 has a pole at P but is regular elsewhere; let n = div ((η − a)−1 )∞ . The fact that P

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is the unique pole of ξ also implies that g(X) = c(X − a)r with r ≥ 1. As P is a pole of order n = mr of (η − a)−l , the definition of m implies that r = 1. If f (X) is constant, then ξ = c(η − a)−1 with c ∈ K, c 6= 0, which shows that F = K(ξ), a contradiction. If f (X) has degree k > 0, then every pole of η is a pole of order k of ξ. Again, by the definition of ξ, this can happen only when k = 1. But then c(η−b)(η−a)−1 = ξ with a, b, c ∈ K, c 6= 0, whence K(ξ) = F , a contradiction. 2 T HEOREM 5.61 Suppose that Σ = K(x, y). If Σ has genus g, then g ≤ ([Σ : K(x)] − 1)([Σ : K(y)] − 1). Proof. Put n = [Σ : K(x)] and m = [Σ : K(y)]. Let F = v(f (X, Y )) be the irreducible plane curve with P = (x, y) as a generic point. Let d denote the degree of F. If Y∞ ∈ F, choose a non-tangent line ℓ = v(X − a) to F at Y∞ and transform X to X − a, and similarly for X∞ . Then F = v(g(X, Y )) with g(X, Y ) = f (X + a, Y + b), and P = (ξ, η) with ξ = x − a, η = y − b is a generic point of F. Note that K(ξ) = K(x), K(η) = K(y). Also, no place arising from a branch of F centred at Y∞ is a zero of ξ. By Theorem 5.33, Y∞ is a (d − n)–fold point of F. Similarly, X∞ is a (d − m)–fold point of F. Hence, the virtual genus g ∗ ≤ 21 (d2 − 3d + 2 − (d − n)(d − n − 1) − (d − m)(d − m − 1)).

Since g ≤ g ∗ , it suffices to check that 1 2 2 (d

− 3d + 2 − (d − n)(d − n − 1) − (d − m)(d − m − 1)) ≤ (n − 1)(m − 1).

A straightforward calculation also shows that equality occurs in two cases, namely, for d = n + m and d = n + m − 1. Since g ∗ = g only happens when F has only ordinary singularities, it follows that g = (n − 1)(m − 1) if and only if each of the following occurs: (i) either d = n + m or d = n + m − 1; (ii) F has at most two singular points, namely X∞ and Y∞ ; (iii) F has only ordinary singularities.

2

5.9 RESIDUES OF DIFFERENTIAL FORMS P If y ∈ K((t)) with y = ak tk , then a−1 , that is, the coefficient of t−1 is the residue of y with respect to t, denoted by Rest y. Note that the residue is a function, K((t)) → K, which is K-linear. Also, Rest y = 0 when y ∈ K[[t]]. L EMMA 5.62 If x, y ∈ K((t)), and if ordt (u) = 1, that is, u is a local parameter of K((t)), then     dx dx Resu y = Rest y . du dt

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Proof. To simplify the computations, note that it suffices to prove the lemma in the particular case that x = t and y = tn with n any integer. Also, the lemma remains true when t is replaced by at with a 6= 0. Hence it may be assumed that t = u + a2 u 2 + · · · ,

dt/du = 1 + 2a2 u + · · · .

The following must be shown:    dt 1 Resu tn = 0 du

for n = −1, otherwise.

For n ≥ 0, the assertion follows, as tn dt/du has no term in negative powers of t. If n = −1, then 1 + 2a2 u + · · · 1 1 dt = = + ··· , t du u + a2 u 2 + · · · u and the residue is equal to 1. When n < −1, the case p = 0 is considered first. Here,      d 1 n dt n+1 Resu t = Resu , t du du n + 1

which is zero for n 6= −1. When p ≥ 0, put m = −n for a fixed n < −1. Then there is an integer N and a polynomial F (X1 , . . . , XN ) with integer coefficients such that 1 + 2a2 u + · · · F (a2 , a3 , . . . , aN +1 ) 1 dt = m = + ··· . tm du u (1 + a3 u + · · · ) u

Actually, F (X1 , . . . , XN ) is the same for any field K. Hence the truth of the assertion in positive characteristic is a formal consequence of the result in zero characteristic, because in that case the assertion holds, showing that F (X1 , . . . , XN ) must be the zero polynomial. This completes the proof. 2 Note that, by Lemma 5.62, Rest (ydx), that is, the residue of the differential form ydx is meaningful as it is independent of the choice of t. Next, the case that ordt u = m > 1 is investigated. Up to a multiplication by a constant, u = tm + b1 tm+1 + b2 tm+2 + · · · = tm (1 + b1 t + b2 t2 + · · · ).

If K((u)) is viewed as subfield of K((t)), then [K((t)) : K((u))] = m; see Exercise 4 in Chapter 4. So, the trace function T from K((t)) to K((u)) is defined. L EMMA 5.63 Let u ∈ K((t)) be a non-zero element of order m ≥ 1 such that m 6≡ 0 (mod p). If y ∈ K((t)), then   du (5.12) Rest y dt = Resu (T(y)du). dt Proof. By Exercise 4 in Chapter 4, the powers 1, t, . . . , tm−1 are linearly independent over K((u)). Hence they form a basis for K((t)) over K((u)), and the trace of an element y of K((t)) can be computed from the matrix (aij ) representing y with respect to this basis, as in (III) of Section A.3.

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Multiplying u by a non-zero constant does not alter the validity of the lemma. Hence, it may assumed that u = tm + b1 tm+1 + · · · = tm (1 + b1 t + b2 t2 + · · · ).

with b1 , b2 , . . . ∈ K. Hence, there exist f0 (u), f1 (u), . . . , fm−1 (u) ∈ K((u)) such that tm = f0 (u) + f1 (u)t + · · · + fm−1 (u)tm−1 .

These elements can be found recursively and the coefficients of fi (u) are polynomials in b1 , b2 , . . . with integer coefficients; that is, P (i) fi (u) = Fα (b)uα , (i)

where each Fv (b) is a polynomial with integer coefficients involving finitely many bj . Therefore the matrix representing an arbitrary element

in K((t)) is of type,

g0 (u) + g1 (u)t + · · · + gm−1 (u)tm−1   

···

G0,1 (u) .. .

··· Gm−1,1 (u) · · ·

G0,m−1 (u) .. . Gm−1,m−1 (u)



 ,

with Gν,µ (u) ∈ K((u)), where the coefficients of Gν,µ (u) are polynomials with integer coefficients in the bj and in the coefficients of the gi (u). This means that (5.12), if it is true, is a formal identity. The proof of (5.12) may therefore be carried out in zero characteristic; the case of positive characteristic is then a corollary, since if two expression only involving integers are equal then they are also equal (mod p). So, take p = 0. From Theorem 4.27, u = v m where v = t + c2 t2 + · · · . By Lemma 5.62, to prove (5.12) it suffices to show that   du Resv y dv = Resu (T(y)du). (5.13) dv Now, (5.13) holds if ordt y is large enough, in which case both sides are equal to zero. Actually, it suffices to consider the case y = v j with j ∈ Z. This simplification is possible by linearity since y can be written as a sum of a finite number of terms aj v j and an element whose order is as high as required. Write j = ms + r with 0 ≤ r < m. Then v j = us v r and T(v j ) = us T(v r ). Also  0 for r 6= 0, r T(v ) = m for r = 0,

whence

j

T(v ) = Thus



mus 0

Resu (T(v j )du) =



for j = ms, otherwise. m 0

for j = −m, otherwise.

(5.14)

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For y = v j , the left-hand side in (5.13) is Resv (v j mv m−1 dv), which is equal to the right-hand side in (5.14). Note that, if p | m, both sides of the equation are zero. This completes the proof. 2 As before, let Σ denote a field of transcendence degree 1 over K. In the same way that the derivation d/dt of K((t)) gives rise to a derivation in Σ, it is possible to define a residue on differential forms. To do this, choose a differential form ydx and a place P of Σ. If σ : Σ → K((t)) is a primitive representation of P, put z¯ = σ(z) for z ∈ Σ. Then Rest (¯ y d¯ x) is the residue of the differential form y¯d¯ x at P. As already noted after the proof of Lemma 5.62, this is independent of the choice of σ. D EFINITION 5.64 Let P be a place of Σ. Then the residue of the differential form ydx at P is ResP (ydx) = Rest (¯ y d¯ x),

where x ¯ = σ(x) and y¯ = σ(y) for a primitive representation σ of P. The next lemma, which is an essential ingredient in the proof of the main result on residue of differential forms, requires some preliminary observations. Suppose y ∈ Σ is a primitive element satisfying an irreducible monic polynomial g(Y ) ∈ K[x][Y ]. The idea is to view g(Y ) as a polynomial over K((x)). This can be done as follows. Let K((˜ x)) be a field of power series. If g(Y ) = Y n + a1 (x)Y n−1 + · · · + an (x),

with a1 (X), . . . , an (X) ∈ K[X], put

g˜(Y ) = Y n + a1 (˜ x)Y n−1 + · · · + an (˜ x).

Since the subfield K(˜ x) of K((˜ x)) is naturally K-isomorphic to K(x), the rational field K(x) may be assumed to be embedded in K((˜ x)). This makes it possible to replace x ˜ by x. With this convention, g˜(Y ) = g(Y ) and g(Y ) ∈ K[[x]][Y ]. Since K[[x]][Y ] is a unique factorisation domain, g(Y ) splits into monic and irreducible factors over K[[x]]: g(Y ) = g1 (Y ) · · · gr (Y ).

(5.15)

Put dj = deg gj (Y ) for j = 1, . . . , r. The factors are distinct, x being a separable variable of Σ. Write gj (Y ) = Y dj + α1 (x)Y dj −1 + · · · , gj (X, Y ) = Y dj + α1 (X)Y dj −1 + · · · ,

with α1 (X), . . . ∈ K[[X]]. By Hensel’s Lemma 4.9, gj (0, Y ) = 0 has only one solution, say bj . Then, hj (X, Y ) = gj (X, Y + bj ) ∈ K[[X]][Y ],

with hj (0, 0) = 0 and hj (X, Y ) is still monic and irreducible. By Theorem 4.89 there is a finite extension K((x))(vj )/K((x)) such that hj (x(vj ), Y ) splits completely into monic linear factors in K[[vj ]][Y ]. Each such factor Y − f (vj ) defines

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a branch representation (x = x(vj ), y = f (vj )) of the irreducible analytic cycle hj (X, Y ) = 0. From the proof of Theorem 4.89, one of these branch representations is primitive. By Theorem 4.87, it is of the form d

(x = cvj j + · · · , y = f (vj )),

with c 6= 0, and f (X) ∈ K[[X]]. In particular, K((x))(vj )/K((x)) is a finite extension of degree dj . Then (x = x(vj ), yj = −bj + f (vj ))

defines the unique branch of the analytic cycle gj (X, Y ) = 0. The same primitive branch representation (x = x(vj ), yj = −bj + f (vj ))

defines a branch γ of the irreducible plane curve F = v(g(X, Y )). Note that γ is a centred at the point Pj = (0, −bj ). Also, by Theorem 4.82, all these dj factors give rise to the same branch of gj (X, Y ) and hence the same branch γ of F. If Pj is the associated place of Σ, the subfield K((vj )) of the algebraic closure K((x)) of K((x)) is denoted by ΣPj . Note that, if σi is the primitive representation arising from this primitive representation of γ, that is, σi (x) = x, σi (y) = yj , then σi is a K-monomorphism from Σ into ΣPj . Now, each of the dj elements −bj + f (vj ) is a root of g(Y ) Also, two such elements arising from distinct values of j are different, by Theorem 4.75. Therefore every root of g(Y ) is one of these elements for some j with 1 ≤ j ≤ r. Finally, every place P of F over P ′ is obtained in this way, where P ′ is the place of K(x) arising from the origin viewed as a point of the line Γ = v(Y ), a non-singular model (Γ; (x, 0)) of K(x). For each j = 1, . . . , r, let Tj be the trace from ΣPj to K((x)), and T the trace from Σ to K(x). L EMMA 5.65 With this notation, T(y) =

Pr

i=1

Tj (σj (y)).

(5.16)

Proof. If [Σ : K(x)] = n, then T(y) is the coefficient of Y n−1 in the irreducible polynomial g(Y ) as above. The same holds for the local traces in ΣPj . Hence the equation follows from (5.15). 2 R EMARK 5.66 Lemma 5.65 holds true for any y ∈ Σ. In fact, if y is not a primitive element of Σ over K(x), let z be such a primitive element. By Theorem A.1, which is the Theorem of the Primitive Element in its strong form, there exists c ∈ K such that w = y + cz is a primitive element. Note that the formula (5.16) is valid for both cz and w. Since (5.16) is linear on both sides, it follows that (5.16) also holds for y. L EMMA 5.67 Let x be separable variable of Σ. If P ′ is a place of K(x) and P1 , . . . , Pr are the places of Σ lying over P, then Pr (5.17) ResP ′ (T(y)dx) = i=1 ResPi (ydx).

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Proof. Without loss of generality, suppose that P ′ arises from the origin as in the preceding discussion. By definition,   σi (x) ResPi (ydx) = Resvi σi (y) dvi . dvi By Lemma (5.12), the right-hand side is equal to Resx (Ti (σi (y))dx). Taking (5.16) into account, this gives Pr Pr i=1 Ti (σi (y))dx) = Resx (T(y)dx), i=1 ResPi (ydx) = Resx (

whence (5.17) follows.

2

The main result on residues of differential forms in Σ is the following. T HEOREM 5.68 Let Σ be a field of transcendence degree 1 over K. If ydx is a differential form, then P ResP (ydx) = 0. (5.18) Note that the sum is taken over all places P, but it is a finite sum since any differential form has only a finite number of poles, by Lemma 5.53.

Proof. Consider first the rational case; that is, suppose Σ = K(x). There exist h(X), k(X) ∈ K[X] with k(X) 6= 0 such that y = h(x)/k(x). Places of K(x) are in one-to-one correspondence with branches of the line Γ = v(Y ). If P is a place of K(x) corresponding to the branch γ centred at the point P = (c, 0) and having primitive representation (x = c + t, y = 0), then   h(c + t) . ResP (ydx) = Rest k(c + t) Expressing the rational function h(X)/k(X) as partial fractions and replacing x by t + b gives the equation, h(x) X fµ (x) + f (x) (5.19) = k(x) (x − bµ )iµ where f (X), fµ (X) ∈ K[X], and deg fµ (X) < iµ . Write

fµ (X) = cµ,1 X iµ −1 + cµ,2 X iµ −2 + · · · + cµ,iµ .

To calculate ResP (ydx), only the coefficient of x−1 matters. With this notation, the sum of the residues taken P over all P whose corresponding branches are centred at affine points is equal to µ cµ,1 . Suppose that the branch γ arising from P is centred at the infinite point of Γ. Then (x = t−1 , y = 0) is a primitive representation of γ. Since dx/dt = −t−2 , the coefficient of t−1 in (5.19) can be calculated. The residue of −f (1/t)/t2 at t is zero. The other terms in (5.19) can be written as follows: X fµ (t−1 ) −t−2 (t−1 − bµ )iµ X
tiµ cµ,1 (t−1 )iµ −1 + cµ,2 (t−1 )iµ −2 + · · · + cµ,iµ = −t−2 (1 − tbµ )iµ X =− (cµ,1 t−1 + cµ,2 + cµ,3 t + · · · + cµ,iµ tiµ −2 )(1 + tbµ + · · · )iµ .

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This shows that the contribution toP the residue only comes P from the coefficient of the first term, which is precisely − µ cµi . Therefore ResP (ydx) = 0. In the general case, the assertion follows from the rational case together with Lemma 5.67. 2 R EMARK 5.69 In the classical case, K = C, any H irreducible curve has the natural structure of a Riemann surface, and ResP (ω) = P ω/(2πi). Hence Theorem 5.68 for K = C follows from Stokes’ Theorem. E XAMPLE 5.70 For an odd integer n not divisible by p, with p odd, let F (X, Y ) = Y n−2 − X n + Y n . Then Σ = K(x, y), with F (x, y) = 0, has x as a separable variable and ydx as a non-trivial differential form. The place P of Σ corresponds to a branch γ on the model (F, (x, y)) of Σ, where F = v(F ), Assume first that γ is centred at an affine point. Then, for any primitive representation σ, both σ(x) and σ(y) are in K[[t]]. Hence ResP (ydx) = 0. The points of F at infinity are Pi = (ǫi , 1, 0), where ǫ is a primitive n-th root of unity and i = 0, 1, . . . , n − 1. If the centre of γ is P = (0, ǫi , 1), then σ can be chosen such that ǫi + (1/n)ǫi t2 + · · · , t is a primitive representation of γ. Then σ(x) = x ¯=

y¯d¯ x=−

σ(y) = y¯ =

1 , t

ǫi + (1/n)ǫi t2 + · · · . t3

This shows that Rest (¯ y d¯ x) = −(1/n)ǫi . Therefore, apart from the constant factor Pn−1 i −(1/n), the right-hand side in (5.18) is i=0 ǫ , that is, the sum of all n-th root of unity. Since this sum is equal to zero, (5.18) follows. 5.10 HIGHER DERIVATIVES IN POSITIVE CHARACTERISTIC Formula (5.5) in Section 5.7 shows that dn ζ n /dζ n = n!. However, this implies that dn ζ n /dζ n = 0 for any separable variable ζ ∈ Σ when p is less than or equal to n. To avoid all the negative consequences of this unusual behaviour, a modification to higher derivatives is made by
replacing factorials with binomial coefficients. n The binomial coefficient m is usually defined for non-negative integers n and m with n ≥ m. Here, it is convenient to extend this definition to all integers n and all non-negative integers m by putting     n n = n(n − 1) · · · (n − (m − 1))/[m(m − 1) · · · 1], = 1. m 0
n Note that m = 0 when 0 ≤ n < m. Higher derivatives in K((t)) are considered first.

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D EFINITION 5.71 For i = 0, 1, 2, . . ., the i-th Hasse derivative of P k
k−i (i) P D t ( a k tk ) = . i ak t (i)

P

ak tk is

(i)

Note that Dt for i ≥ 1 is over K, that is Dt c = 0 for c ∈ K and i ≥ 1. Also, (i)

Dt f = (1/i!)di f /dti , (0)

for f ∈ K((t)) provided that p = 0, or i < p. In particular, Dt f = f for every (1) f ∈ K((t)), while Dt f coincides with the usual derivative df /dt of f . Further,  1 for i = j, (i) j Dt t = (5.20) 0 for i > j ≥ 0. Now, some fundamental equations for Hasse derivatives are established. L EMMA 5.72 Let f, g ∈ K((t)) and s ∈ K. Then (i)

(i)

(i)

(i)

(i)

Dt (f + g) = Dt f + Dt g, Dt (sf ) = sDt f ; P∞ (i) (i) P∞ Dt ( k=0 ck f k ) = k=0 Dt (ck f k ) for ordt f > 0; P (i) (j) (i−j) Dt (f g) = ij=0 Dt f Dt g;
(i) (j) (i+j) Dt Dt f = i+j f; i Dt P (k1 ) (i) (k ) Dt (f1 · · · fr ) = Dt f1 · · · Dt r fr ,

(5.21) (5.22) (5.23) (5.24) (5.25)

where the last sum is over all r-ples of non-negative integers (k1 , · · · , kr ) with k1 + . . . + kr = i. Proof. Both equations in (5.21) are immediate. As a consequence, for every positive integer m, Pm (i) Pm (i) Dt ( k=0 ck f k ) = k=0 Dt (ck f k ). (i) P∞ If ordt f > 0, then all terms of a given order m in Dt ( k=0 ck f k ) are alP∞ (i) Pm+i (i) ready present in Dt ( k=0 ck f k ), and the same holds for k=0 Dt (ck f k ) and Pm+i (i) k k=0 Dt (ck f ). Thus (5.22) follows from a finite number of repeated applications of (5.21). P v P To show (5.23), let f = cv t , g = dv tv . Then the coefficient of tv in f g
P P P v−i (i) is ck dv−k . Therefore Dt (f g) = ev t with ev = vi ck dv−k . On the

P P (j) (i−j) k w k−j w−(i−j) other hand, from Dt f = c t and D g = d , t j k i−j w t    X X k w (j) (i−j) Dt f Dt g= ck dw tk+w−i . j i − j w k

Hence, the coefficient of tv−i in

Pi

j=0

(j)

(i−j)

Dt f Dt

g is

 i X   X k v−k ck dv−k . j i−j j=0 k

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Since

   i   X k v−k v = , j i − j i j=0

which may be established by comparing the coefficient of X i on both sides of (1 + X)v = (1 + X)k (1 + X)v−k , so (5.23) follows. Now, X v v − j  (i) (j) tv−(i+j) . Dt Dt f = cv j i This, together with the identity,       v v−j i+j v = , j i i i+j proves (5.24). Finally, (5.25) follows from (5.23) by induction on r. 2 For any b ∈ K and non-negative integers i, m, since   b + t if j = 0, (j) 1 if j = 1, Dt (b + t) =  0 if j ≥ 2, so (5.25) implies that   m (i) m Dt (b + t) = (b + t)m−i . i Another fundamental equation is the chain rule.

(5.26)

L EMMA 5.73 Let f, g ∈ K[[t]] and ordt g > 0. Then i h Pi−1 (j) (i) (1) (i) Dt f (g(t)) = (Dt g(t))i DT f (T ) + j=1 gj DT f (T )

T =g(t)

,

(5.27)

where

P (r1 ) (r ) gj = Dt g(t) · · · Dt j g(t), and the sum is over all j-ples of positive integers (r1 , . . . , rj ) with r1 +· · ·+rj = i.

Proof. By (5.22), it is enough to prove (5.27) for f (t) = tn . From (5.25), P (k ) (k ) (i) Dt g(t)n = k1 +···+kn =i Dt 1 g(t) · · · Dt n g(t). In each summand, among the ki some positive integers occur; let (s1 , . . . , sm ) be the m-ple arising from (k1 , . . . , kn ) by omitting the zeros. Then 1 ≤ m ≤ i, s1 + · · · + sm = i, and (s ) (s ) (k ) (k ) Dt 1 g(t) · · · Dt n g(t) = g(t)n−m (Dt 1 g(t) · · · Dt m g(t)).
(s ) (s ) n Note that each term g(t)n−m (Dt 1 g(t) · · · Dt m g(t)) comes from exactly m (k ) (k ) terms Dt 1 g(t) · · · Dt n g(t). Hence   i X X n (s ) (s ) (i) n g(t)n−m Dt 1 g(t) · · · Dt m g(t). (5.28) Dt g(t) = m m=1 s +...+s =i 1

Since

the assertion follows.

m

  n (m) n n−m g(t) = DT T , m T =g(t)

2

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E XAMPLE 5.74 Let f (t) = t3 and g(t) = t2 + 1. If i = 3, then g1 = 0, g2 = 4t (3) and Dt = 6t3 + 12t2 . Another basic relation is given in Lemma 5.75. The proof requires the numbertheoretical result given in Lemma A.6. L EMMA 5.75 Let q = ph . For f ∈ K((t)) with ordt f > 0, and 0 ≤ j, k < q, (q)

(q)

Dt f jq+k = f jq Dt f k + f k (Dt f j )q .

(5.29)

Proof. From (5.23), (q)

Dt f jq+k =

Pq

m=0

(m) jq

Dt

(q−m) k

f Dt

f .

By Lemma A.6, this sum consists of two terms only, namely the first and the last. (q) (1) It remains toP show that Dt f jq = (Dt f j )q . j n Put f = cn t . Then P (1) Dt f j = ncn tn−1 , P (1) nq cqn tnq−q , (Dt f j )q = P q f jq = (f j ) = cq tqn ,  n P q nq nq−q (q) t . cn Dt f jq = q
As nq 2 q ≡ n (mod p), by Lemma A.6, so the result follows. Using similar arguments, the following lemma is established.

L EMMA 5.76 Let i = rpm + s with 1 ≤ r < p and 0 ≤ s < pm . Then, for all f ∈ K((t)),  −1    1 i (pm ) (pm ) (s) (i) · · · Dt f . Dt Dt Dt f = r! s | {z } r times

Proof. By (5.24),

(pm ) Dt

···



(pm ) Dt f m

where, as in Lemma A.6, np pm s < i, so r! 6≡ 0 (mod p) and






  m   m  i rp 2p (i) = ··· Dt f, s pm pm

≡ n (mod p) for all 1 ≤ n ≤ r. Since r < p and
6≡ 0 (mod p). Hence the result follows. 2

i s

Now, Hasse derivatives in Σ are introduced using a similar idea that led from d/dt to d/dζ via any place of Σ; see Section 5.7. This time, however, not all places are suitable, or admissible, as it may happen at some places that the idea does not work. As a matter of fact, computational and some theoretical complications arise when either ordP ζ < 0, or ordP ζ ≥ 0 but ordP (ζ − b) > 1 with σ(ζ) = b + · · · for a primitive representation of σ of P. This obstacle can be overcome by using the argument below.

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By Lemma 5.53, all but finitely many places of Σ are neither zeros nor poles of dζ. Choose one of these infinite places, say P. Let σ be a primitive representation of P, and put σ(ζ) = ζ. Then ζ = b + φ(t) with φ(t) ∈ K[[t]] and ordt φ(t) = 1. Since t 7→ φ(t) is an automorphism of K((t)), it may be assumed that ζ = b + t. In other words, ζ − b is a local parameter at P. For an element ξ ∈ Σ, let f (X, Y ) ∈ K[X, Y ] be an irreducible polynomial such that f (ξ, ζ) = 0. Let F = v(f (X, Y )) be the corresponding irreducible curve. All but finitely many branches of F are centred at simple affine points such that the corresponding places are neither zeros nor poles of dζ. Hence, P may be assumed to come from a branch γ of F with a primitive representation in the form (ξ, ζ) with ξ = a+ψ(t), ζ = b+t, and ψ(t) ∈ K[[t]]. Such a place P is admissible with respect to ζ and ξ. Now, Hasse derivatives are generalised to more than one variable. As pointed out in Remark 1.37,P in positive characteristic the usual partial derivatives of a polynomial f (X, Y ) = am,k X m Y k , that is,     m k ∂ α+β f (X, Y ) X = a α! β! X m−α Y k−β m,k ∂X α ∂X β α β are changed to Hasse partial derivatives,    m k ∂ (α+β) f (X, Y ) X X m−α Y k−β . = a m,k α β ∂X (α) ∂Y (β) It may be noted that, if u and v are indeterminates, then X ∂ (µ+ν) f (X, Y ) uµ v ν . f (X + u, Y + v) = (µ) ∂Y (ν) ∂X µ,ν≥0 L EMMA 5.77 The i-th Hasse derivative of ξ with respect to t can be computed as follows:  i−1 1 ∂ (i) f (ξ, ζ) X ∂ (i−j+1) f (ξ, ζ) (j) (i)  + Dt ξ+ Dt ξ = − (i) (i−j) ∂f (ξ, ζ)/∂ξ ∂ζ ∂ξ∂ζ j=1  (5.30) i X i (i−j+n) X X ∂ f (ξ, ζ) (r1 ) (rn )  Dt ξ · · · Dt ξ , (n) (i−j) ∂ζ n=2 j=n r1 +···+rn =j ∂ξ

where r1 , . . . , rn > 0 and ∂ (i−j+n) f (ξ, ζ) ∂ξ

(n)

∂ζ

(i−j)

=

∂ (n) f (ξ, ζ) ∂ξ

(n)

when j = i. P P k m Proof. Let f (X, Y ) = akm X k Y m . Then akm ξ ζ = 0. From (5.21),
m−j P (j) m (i) k m . From (5.23), akm Dt (ξ ζ ) = 0. By (5.26), Dt ζ = m j ζ Pi (j) k (i−j) m (i) k m ζ Dt (ξ ζ ) = j=0 Dt ξ Dt     i m m m−i k X m−(i−j) (j) k ζ ξ + ζ = Dt ξ . i − j i j=1

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By (5.28), for every j with 1 ≤ j ≤ i, (j) k

Dt ξ = kξ

k−1

(j)

Dt ξ +

j X

X

n=2 r1 +...+rn

Hence X

(i)

+

·

k m

akm Dt (ξ ζ ) =

X

amk

  k k−n (r1 ) (r ) ξ Dt ξ · · · Dt n ξ. n =j

  m (m−i) k ζ ξ i

 i−1  X m m−(i−j) k−1 (j) m k−1 (i) ξ Dt ξ + kζ ξ Dt ξ kζ i − j j=1

i X i X

X

n=2 j=n r1 +...+rn =j



m i−j

   k m−i+j k−n (r1 ) (r ) ζ ξ Dt ξ · · · Dt n ξ  = 0. n

(5.31)

Since ∂f (ξ, ζ)/∂ξ 6= 0, the assertion now follows by dividing the last equation by ∂f (ξ, ζ)/∂ξ. 2 Although the above proof is constructive, for higher values of i the computation of the coefficients of c(i) (X) is usually long, since it depends on the number of partitions of a natural number. For j = n, the condition that r1 + · · · + rn = j implies that r1 = · · · = rn = 1 and the corresponding term is ∂ (i) f (ξ, ζ) (n)

(1)

(i−n)

(Dt ξ)n ;

∂ξ ∂ζ but, for n = 2 and j > n, there are already ⌊j/2⌋ summands with coefficient ∂ (i−j+2) f (ξ, ζ) ∂ξ

(2)

∂ζ

(i−j)

.

For small values of i, (1)

Dt ξ = Dt ξ = −

∂f (ξ, ζ)/∂ζ ; ∂f (ξ, ζ)/∂ξ

(5.32)

(2)

Dt ξ 1 =− ∂f (ξ, ζ)/∂ξ

∂ (2) f (ξ, ζ) ∂ξ

(2)

∂ (2) f (ξ, ζ) ∂ (2) f (ξ, ζ) Dt ξ + (Dt ξ)2 + (2) ∂ξ∂ζ ∂ζ

!

.

(5.33) To check (5.33), the procedure in the proof of Lemma 5.30 is used. It suffices to (2) k m calculate Dt (ξ ζ ), and repeat the final argument:   m m−2 k m (2) k m−1 k (2) k m ζ ξ + mζ Dt ξ Dt (ξ ζ ) = ζ Dt ξ + 2     k m k−2 m m−2 k k−1 m (2) ζ Dt ξ + ζ ξ (Dt ξ)2 + ζ ξ = kξ 2 2 +mkζ

m−1 k−1

ξ

Dt ξ.

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Similarly, the third and fourth derivatives are calculated as follows: (3)

Dt ξ = −

(4)

Dt ξ = − +3 +

∂

(3)

(3)

∂ ∂

f (ξ, ζ)

(3)

∂ξ f (ξ, gz)

(4)

∂ξ

(2)

∂ζ

f (ξ, ζ)

(2)

∂ζ

(2)

∂ (2) f (ξ, ζ)

(2)

∂ (3) f (ξ, ζ)

(Dt ξ)(Dt ξ) +

∂ (2) f (ξ, ζ) (2) 2 (Dt ξ) ∂ξ

(Dt ξ)3 (2) (3) ∂ξ ∂ξ ∂ (3) f (ξ, ζ) ∂ (2) f (ξ, ζ) (2) (Dt ξ) + + (Dt ξ)2 (2) ∂ξ∂ζ ∂ξ ∂ζ ! ∂ (3) f (ξ, ζ) ∂ (3) f (ξ, ζ) + ; (Dt ξ) + (2) (3) ∂ξ∂ζ ∂ζ (5.34) 2

∂ (2) f (ξ, ζ)

1 ∂f (ξ, ζ)/∂ξ

∂ξ +

1 ∂f (ξ, ζ)/∂ξ

∂ξ

(2)

(3)

(2)

(Dt ξ)(Dt ξ)2 + (2)

(Dt ξ)(Dt ξ) +

(Dt ξ)2 +

∂

(4)

∂ (4) f (ξ, ζ)

∂

(4)

∂ξ f (ξ, ζ)

(4)

∂ξ

f (ξ, ζ)

∂ξ∂ζ

(Dt ξ)(Dt ξ) +

(3)

(3)

(Dt ξ)4 +

(Dt ξ)3 +

∂ (2) f (ξ, ζ) (3) (Dt ξ) ∂ξ∂ζ

∂ (3) f (ξ, ζ)

∂ζ

(Dt ξ) +

∂

(4)

f (ξ, ζ)

∂ζ

(4)

∂ξ∂ζ !

(2)

(2)

(Dt ξ)

.

(5.35) As in Lemma 5.77, higher derivatives can be defined using the same approach that produced Dζ from Dt via a place of Σ. D EFINITION 5.78 Let ζ be a separable variable of Σ. For i = 0, 1, 2, . . ., the i-th (i) (0) Hasse derivative Dζ ξ of ξ ∈ Σ is defined iteratively: Dζ ξ = ξ, and  i−1 (i−j+1) (i) X ∂ f (ξ, ζ) (j) 1 (i)  ∂ f (ξ, ζ) + Dt ξ Dζ ξ = − (i−j) ∂f (ξ, ζ)/∂ξ ∂ζ i ∂ξ∂ζ j=1  (5.36) i X i (i−j+n) X X ∂ f (ξ, ζ) (r1 ) (r ) Dt ξ · · · Dt n ξ  . + n ∂ζ (i−j) ∂ξ n=2 j=n r +...+r =j 1

n

Diagrammatically,

(i)

ξ τ

Dζ - D(i) ξ ζ

τ ? ? D(i) t(i) ξ Dt ξ (i)

Some previous remarks also hold true for Dζ . For i ≥ 1, the Hasse derivative (i)

Dζ is over K and, for i < p,

(i)

i! Dζ ξ = di ξ/dζ i .

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(i)

R EMARK 5.79 By definition, σ(Dζ ξ) = Dt σ(ξ) for every admissible place P of Σ, where σ is a primitive representation of P such that σ(ζ) = b + t with b ∈ K. (i) Since σ is a K-monomorphism from Σ onto K((t)), the previous results on Dt (i) extend to Dζ . L EMMA 5.80 All equations but (5.22) in Lemma 5.72 as well as (5.20) and Lemma 5.76 remain true when K((t)) and t are replaced by Σ and ζ. E XAMPLE 5.81 (i) Suppose that p = 2. Let f (X, Y ) = Y + 1 + Y 5 X 23 and Σ = K(x, y) with y + 1 = y 5 x23 . Then x is a separable variable of Σ. By straightforward calculations, ∂f (x, y) ∂f (x, y) 1 = x22 y 5 ; = ; ∂x ∂y y (2) ∂ (2) f (x, y) f (x, y) 21 5 ∂ = x22 y 4 ; = x y ; (2) ∂x∂y ∂x (3) f (x, y) ∂ (3) f (x, y) 20 5 ∂ = x y ; = x21 y 4 ; (3) ∂x ∂x(2) ∂y ∂ (4) f (x, y) ∂ (4) f (x, y) = x11 y 5 ; = x20 y 4 ; (4) ∂x ∂x(3) ∂y ∂ (4) f (x, y) ∂ (4) f (x, y) ∂ (4) f (x, y) = = 0; (2) (2) (3) ∂x ∂y ∂x∂y ∂y (4)

∂ (2) f (x, y) = 0; ∂y (2) ∂ (3) f (x, y) ∂ (3) f (x, y) = = 0; ∂x∂y (2) ∂y (3)

= yx23 .

From (5.32), Dx y =

x22 y 5 = x22 y 6 . 1 + x23 y 4

From (5.33), (5.34) and (5.35), Dx(2) y = x21 y 7 ; Dx(3) y = x20 y 8 ; Dx(4) y = y 6 (y 4 + y 3 + 1)x19 . (ii) In Example 5.36, x is a separable variable of Σ, and Dx x = 1, Dx y = −2x2 , Dx y 2 = x2 y;

Dx(2) x = 1, Dx(2) y = −2x, Dx(2) y 2 = −xy − x4 ; Dx(3) x = 0, Dx(3) y = 1, Dx(3) y 2 = −2x3 + 2y; Dx(4) x = 0, Dx(4) y = 0, Dx(3) y 2 = 0; Dx(5) x = 0, Dx(5) y = 2x10 , Dx(5) y 2 = x − x10 y;

Dx(6) x = 0, Dx(5) y = 0, Dx(6) y 2 = 2x12 + 1.

T HEOREM 5.82 Let ζ and η be separable variables of Σ. Then there are elements (j) d1 , . . . , di−1 in Σ that are polynomials in the indeterminates Dη ζ with integer coefficients for 1 ≤ j ≤ i such that, for any ξ ∈ Σ, Pi−1 (j) (i) (5.37) Dη(i) ξ = (dζ/dη)i Dζ ξ + j=1 dj Dζ ξ.

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Proof. Let P be an admissible place of Σ with respect to the pair (ξ, ζ). To compare Dη and Dζ , admissibility with respect to the pair (ξ, η) also enters into consideration. Since there remain infinitely many places with the required properties, P may be assumed admissible with respect to both pairs (ξ, ζ) and (ξ, η). Let σ be a primitive representation σ of P. Then σ(ζ) = b + t, (j)

σ(ξ) = ξ = a + f with ordt f > 0.

(i)

(j)

Hence, σ(Dζ ξ) = Dt ξ = Dt f , for every j ≥ 0. Also, σ(η) = η = c + h with ordt h = 1. By Theorem 4.4, the substitution t → h is a K-automorphism of K[[t]]. Its inverse µ also operates as a substitution, namely t → g with g ∈ K[[t]]. ¯ = b + g(t). Let Then µ(η) = c + t, µ(ξ) = a + µ(f ) = a + f (g(t))), and µ(ζ) ξ ∗ = a + f (g(t)), η ∗ = c + t, ζ ∗ = b + g(t). Since µσ is another primitive representation of P, for every j ≥ 0, (j)

(j)

(j)

(j)

(µσ)(Dη(j) ξ) = Dt ξ ∗ = Dt f (g(t)); (µσ)(Dη(j) ζ) = Dt ζ ∗ = Dt g(t). Also, (j) (j) ¯ (i) (µσ)(Dζ ξ) = µ(Dt ξ) = DT f (T ))|T =g(t) ,

Since µσ is a monomorphism from Σ into K((t)), Theorem 5.82 follows from Lemma 5.73. 2 Take p > 0. Again, let ζ be a separable variable of Σ. For any positive integer m, put (i)

Σm = {ξ ∈ Σ | Dζ ξ = 0, 1 ≤ i < pm }, (i)

Σ∞ = {ξ ∈ Σ | Dζ ξ = 0, i = 1, 2, . . .}. Then Σm ⊃ K and Σ∞ ⊃ K. L EMMA 5.83 For any positive integer m, (i) Σm is a subfield of Σ containing K; (pm )

(ii) Dζ

is a derivation of Σm ; (pm )

(iii) Σm+1 = {ξ ∈ Σm | Dζ

ξ = 0}.

Proof. (i) This follows from Lemma 5.72. (ii) From (5.23), (pm )

Dζ

(pm )

(ξη) = ηDζ

(pm )

ξ + ξDζ

η

(pm )

for ξ, η ∈ Σm . This, together with (5.21), shows that Dζ (iii) Put T = {ξ ∈ Σm |

m

(p ) Dζ ξ

is a derivation of Σm .

= 0}. If it is shown that T = Σm+1 , then the (i)

lemma follows by induction. By definition, it suffices to show that Dζ ξ = 0 for

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ξ ∈ T and pm < i < pm+1 . Now, write i = rpm + s with 1 ≤ r ≤ p − 1 and 0 ≤ s ≤ pm − 1. From Lemmas 5.76 and 5.80,    r i (s) (rpm ) (i) (s) (pm ) ξ = r! Dζ Dζ ξ = r! Dζ ξ. Dζ Dζ s
From Lemma A.6, si 6≡ 0 (mod p). Hence, for ξ ∈ T ,  −1  r 1 i (i) (s) (pm ) Dζ ξ = Dζ Dζ ξ = 0. r! s 2 L EMMA 5.84 For any positive integer m, m

m

Σm = Σp = {up | u ∈ Σ}. Proof. Let η ∈ Σ. Choose an admissible place P with respect to ζ and η. For a P P m m m primitive representation τ of P, τ (η) = η = ck tk . As τ (η p ) = cpk tkp , it follows that X pm kpm  m (i) pm Dt η = ck tkp −i , (5.38) i

which equals zero for 1 ≤ i < pm . Since τ is a monomorphism from Σ into m (i) m K((t)), this only occurs when Dζ η p = 0. Thus η p ∈ Σm for every η ∈ Σ. To show the converse, note that every ξ ∈ Σ1 is either a constant or an inseparable variable of Σ, by Theorem 5.50. Hence, the assertion for m = 1 follows from Lemma 5.38. Now, the proof is by induction on m. Assume that every element in Σm is a pm -th power of an element in Σ. Then ξ m (i) is in Σm+1 when ξ = η p and Dζ ξ = 0 for pm ≤ i < pm+1 . Again, choose an P k ck t with τ admissible place P with respect to ζ and η, and write τ (η) = η = a primitive representation of P. Arguing as before, (5.38) follows, and this is zero since pm < i < pm+1 is assumed. By the p-adic criterion of Lemma A.6, this implies that every k in the expansion of η is divisible by p. Thus Dt η = 0. By Theorem 5.50 η is an inseparable variable of Σ. Hence, Lemma 5.38 implies that m m+1 η = up for some u ∈ Σ. Therefore ξ = η p = up . 2 L EMMA 5.85 Σ∞ = K. Proof. Let ξ be any non-zero element in ∈ Σ∞ . For a place P of Σ, let k = ordP ξ. From the definition, Σ∞ ⊂ Σm for any positive integer m. Choose m such that m pm > k. By Lemma 5.84, ξ = η p for some η ∈ Σ. Hence ordP ξ = pm ordP η. But this only occurs when ordP ξ = 0 and ordP η = 0. Therefore ordP ξ has no zeros. By Theorem 5.33, ξ ∈ K. 2

T HEOREM 5.86 Let x0 , . . . , xr ∈ Σ. Then x0 , . . . , xr are linearly independent over Σm if and only if there exist integers ǫ0 , . . . , ǫr , with (ǫ )

0 = ǫ 0 < ǫ1 < · · · < ǫ r < pm ,

such that det(Dζ i xj ) 6= 0.

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Proof. Suppose that x0 , . . . , xr ∈ Σ are linearly Prdependent over Σm ; that is, there exist z0 , . . . , zr ∈ Σm not all zero for which j=0 zj xj = 0. By (5.23) and the definition of Σm , Pr (i) j=0 zj D xj = 0.

This proves the ‘if’ part of the theorem. The converse is proved by induction on r. The case r = 0 is trivial. Let x0 , . . . , xr be linearly independent over Σm , and assume that the assertion of the theorem does not hold; that is, the r + 1 vectors, (1)

(2)

(pm −1)

uj = (xj , Dζ xj , Dζ xj , . . . , Dζ

xj ),

are linearly dependent over Σ. Then there exist z0 , . . . , zr ∈ Σ such that Pr (i) m j=0 zj Dζ xj = 0 for 0 ≤ i ≤ p − 1.

(5.39)

Since x0 , . . . , xr−1 are linearly independent over Σm , from the inductive hypothesis it follows, for 0 ≤ j ≤ r − 1, that the above r vectors are linearly independent over Σ. Hence zr 6= 0. Therefore it may be assumed that zr = 1. It suffices to show that zj ∈ Σm for j = 0, . . . , r − 1. In fact, this, together with (5.39) for i = 0, gives a contradiction that completes the proof. (n) To prove that zj ∈ Σm , it is enough to show by Lemma 5.83 that Dζ zj = 0 for 1 ≤ n ≤ pm−1 . This is done by induction on n. For n = 1, from (5.39), Pr Pr−1 (1) (1) Pr (i) (i) (i+1) Dζ xj + j=0 Dζ zj Dζ xj = 0. j=0 zj Dζ xj = (i + 1) j=0 zj D Pr (i) From (5.39), (i + 1) j=0 zj Dζ xj = 0 for i < pm − 1. This holds true for i = pm − 1 since in this case i + 1 ≡ 0 (mod p). Hence Pr−1 (1) (i) j=0 Dζ zj Dζ xj = 0 for 0 ≤ i ≤ pm − 1. Since the above r vectors for 0 ≤ j ≤ r − 1 are linearly (1) independent over Σ, the conclusion is that Dζ zj = 0 for j = 0, . . . , r − 1. (s)

Assume that Dζ zj = 0 for 1 ≤ s < n ≤ pm−1 and for 0 ≤ j ≤ r − 1. From (5.39) by derivation,   r r−1 X i+n X (n) (i) (i+n) Dζ zj Dζ zj = 0. z j Dζ xj + i j=0 j=0 (n)

As in the case n = 1, to prove that Dζ zj = 0, it suffices to show that   r i+n X (i+n) z j Dζ xj = 0 for 0 ≤ i ≤ pm − 1. i j=0 For i + n < pm , this follows from (5.39). For i + n ≥ pm , pm ≤ i + n < pm + pm−1 ,

pm > i ≥ pm − n ≥ (p − 1)pm−1 .

Hence the coefficient of pm−1 in
the p-adic expansion of i + n is zero and that of i is p − 1. From Lemma A.6, i+n ≡ 0 (mod p), and this completes the proof. 2 i The following result is a consequence of Theorem 5.86.

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155

T HEOREM 5.87 Let x0 , . . . , xr ∈ Σ. Then x0 , . . . , xr are linearly independent over Σ∞ if and only if there exist integers ǫ0 , . . . , ǫr , with 0 = ǫ0 < ǫ1 < . . . < ǫr , (ǫ ) such that det(Dζ i xj ) 6= 0.

5.11 THE DUAL AND BIDUAL OF A CURVE Let F be an irreducible plane curve of degree n > 1. To each non-singular point of F there corresponds a unique tangent line to F at the point. In the dual plane, tangent lines determine points and, at least intuitively, the point set obtained is a curve. A rigorous proof is the main purpose of the present section, in which some unexpected behaviour of the dual curves in positive characteristic is also shown. D EFINITION 5.88 The dual curve of F is an irreducible plane curve F ′ containing all but finitely many points P ′ = (b0 , b1 , b2 ) such that the line b0 X0 +b1 X1 +b2 X2 in PG(2, K) is tangent to F. Assume that F = v(F ) for a homogeneous F ∈ K[X0 , X1 , X2 ] of degree n. With the notation of (1.5), Euler’s formula gives that nF = X0 F0 + X1 F1 + X2 F2 . Since F is irreducible, one of the partial derivatives, say F0 , is not vanishing; see the proof of Theorem 1.24. Then, with coordinates x = (x0 , x1 , x2 ) for the generic point P , x0 F0 (x) + x1 F1 (x) + x2 F2 (x) = 0.

(5.40)

One of the coordinates of P , say x0 , is distinct from 0. Without loss of generality, ξ = x1 /x0 is a separable variable of the function field Σ of F. D EFINITION 5.89 The rational transformation ωF : Σ → Σ,

(x′0 , x′1 , x′2 ) = (F0 (x), F1 (x), F2 (x))

(5.41)

is the rational Gauss map associated to F. First, it is shown that the rational Gauss map is non-trivial. Without loss of generality, suppose that F0 (x) 6= 0. In the trivial case, both polynomials F1 − sF0 and F2 − tF0 are divisible by F . Since they have both degree less than n, the trivial case only occurs when F1 = sF0 and F2 = tF0 . Then, Euler’s formula gives that nF = X0 F0 + X1 F1 + X2 F2 = (X0 + sX1 + tX2 )F0 . Since F is irreducible and F0 6= 0, this only happens for n = 1 and F0 is constant, contradicting the hypothesis n > 1. So, the transform of F under the rational Gauss map is an irreducible curve F ′ which is actually the dual curve of F. Now, some important properties of the dual curve are established. Choose any separable variable ζ of Σ, and derive F (x) = 0. Then dx1 dx2 dx0 F0 (x) + F1 (x) + F2 (x) = 0. dζ dζ dζ

(5.42)

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So the triple (F0 (x), F1 (x), F2 (x)) provides a non-trivial solution of the following system of homogeneous linear equations in the unknowns u0 , u1 , u2 : x0 u0 + x1 u1 + x2 u2 = 0, dx0 dx1 dx2 u0 + u1 + u2 = 0. dζ dζ dζ

(5.43) (5.44)

There is a unique non-trivial solution, up to a non-zero factor, since the matrix of the system has rank 2. To show this, assume that x0 = 1. Then dx0 /dζ = 0, and the rank is 1 if and only if dx1 /dζ = 0 and dx2 /dζ = 0. But x1 /x0 is a separable variable, and thus dx1 /dζ 6= 0 by Theorem 5.50(ii). Hence, the rank is equal to 2. Taking derivatives and subtracting, a further equation from (5.40) and (5.42) is obtained: x0

dF1 (x) dF2 (x) dF0 (x) + x1 + x2 = 0. dζ dζ dζ

(5.45)

Now consider the dual curve F ′′ of F ′ , that is, the bidual of F. In the real and complex planes, F ′′ coincides with F. But this is not true for every curve in positive characteristic, and this behaviour is now investigated. Put yi = Fi (x) for i = 0, 1, 2, and write y = (y0 , y1 , y2 ). If F ′ = v(F ′ ) for an irreducible homogeneous polynomial F ′ with coefficients in K, then F ′ (y) = 0; that is, Q = (y0 , y1 , y2 ) is a generic point of F ′ . Let Σ′ = K(F ′ ); then Σ′ is a subfield of Σ, and the rational Gauss map associated to F ′ is ω F ′ : Σ′ → Σ ′ ,

(y0′ , y1′ , y2′ ) = (F0′ (y), F1′ (y), F2′ (y)),

Fi′

′

(5.46)

′

where is the partial derivative of F with respect to Xi . If F is not a line, that is, F is not a strange curve, then F ′ is transformed into the curve F ′′ , and R = (F0′ (y), F1′ (y), F2′ (y)) is a generic point of F ′′ . Since F ′′ arises from F ′ in the same way as F ′ from F, the triple (F0′ (y), F1′ (y), F2′ (y)) is a solution of the system of homogeneous linear equations in the unknowns v0 , v1 , v2 : y0 v0 + y1 v1 + y2 v2 = 0, dy0 dy1 dy2 v0 + v1 + v2 = 0. dζ dζ dζ

(5.47) (5.48)

Another solution is the triple (x0 , x1 , x2 ). This follows from (5.45) and (5.40). In this system, two cases occur according to whether the matrix     y0 y1 y2 F0 (x) F1 (x) F2 (x)     (5.49)  dy0 dy1 dy2  =  F0 (x) F1 (x) F2 (x)  dζ

dζ

dζ

dζ

dζ

dζ

has maximal rank or not. If the rank is 2, then there is a unique solution up to a non-zero factor, and F and F ′′ share a generic point. Since both F and F ′′ are irreducible, this is only possible when they coincide. Therefore ωF ′ ωF is a birational transformation leaving F and also F ′ invariant. In terms of separability, see Theorems 5.40 and 5.42, this implies that both Gauss maps ωF and ωF ′ , and hence also their product, are separable; that is, F is a reflexive curve.

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Now, the possibility that the matrix in (5.49) has rank 1 is examined. For this purpose, let x0 = 1 and ζ = x1 . Note that F2 (x) 6= 0, since x1 is a separable variable. If the rank is 1, then F1 (x)

dF2 (x) dF1 (x) = F2 (x) . dx1 dx1

(5.50)

Also, from (5.42), F1 (x) +

dx2 F2 (x) = 0. dx1

(5.51)

By taking derivatives, dF1 (x) d2 x2 dx2 dF2 (x) = − 2 F2 (x) − . dx1 dx1 dx1 dx1 This, together with (5.50) and (5.51), gives d2 x2 = 0. dx21

(5.52)

Note that, if p = 2, then (5.52) holds trivially. For p 6= 2, (5.52) implies that all non-singular points of F are inflexions. To show this, derive (5.51). In terms of F ,  2 2 dx2 dx2 ∂ 2 F (x) ∂ F (x) d2 x2 ∂F (x) ∂ 2 F (x) + 2 + + = 0. ∂x21 dx1 ∂x1 ∂x2 dx1 ∂x22 dx21 ∂x2 Writing f (X, Y ) = F (1, X, Y ), this, together with (5.51) and (5.52), yields 2 2 2 2   ∂f ∂f ∂ 2 f ∂ f ∂ f ∂f ∂f + −2 = 0. 2 2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 Since f (x1 , x2 ) = 0, by Theorem 1.36 the assertion holds. Another consequence of (5.52) for p > 2 is that the Gauss map ωF is inseparable; that is, F is a nonreflexive curve. In fact, both F0 (x)/F2 (x) and F1 (x)/F2 (x) are in Σp . This depends on the above equations: from (5.52), d(F1 (x)/F2 (x)) = 0, dx1 and this, together with (5.40), implies that d(F0 (x)/F2 (x)) = 0. dx1 Let pm be the largest power of p for which m

F0 (x)/F2 (x) = z0p ,

m

F1 (x)/F2 (x) = z1p , m

m

for some z0 , z1 ∈ Σ. Then (5.40) becomes z0p + z1p x1 + x2 = 0. Therefore m

m

m

H(X, Y )f (X, Y ) = Z0 (X, Y )p + Z1 (X, Y )p X + Z2 (X, Y )p Y,

(5.53)

with Z0 , Z1 , Z2 , H ∈ K[X, Y ]. In summary, the following theorem is established. T HEOREM 5.90 Let p 6= 2.

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(i) A sufficient condition for an irreducible plane curve C to coincide with its bidual curve is that C is reflexive. (ii) The curve F = v(f (X, Y )) is non-reflexive if and only if f (X, Y ) satisfies (5.53) with pm the inseparability degree of the Gauss map of C. (iii) F is non-reflexive if and only if it is the locus of its inflexion and singular points. For p = 2, the situation is different. T HEOREM 5.91 Let p = 2. (i) Every irreducible plane curve F = v(f (X, Y ) is non-reflexive, and f (X, Y ) satisfies (5.53) with pm the inseparability degree of the Gauss map of F. (ii) If pm > 2, then F is the locus of its inflexion and singular points. Proof. The hypothesis P p =i 2 jimplies that d(F1 (x)F2 (x))/dx1 = 0. To show this, let F (1, X, Y ) = aij X Y , and, for every non-negative integer n, define χ(n) to be 0 or 1 according as n is even or odd. Then P P F1 = χ(i)aij X i−1 Y j , F2 = χ(j)aij X i Y j−1 .

Therefore F1 (x) P dF1 (x) dx2 P χ(i)χ(j)xi−1 y j−1 , = χ(i)χ(j)xi−1 y j−1 = dx1 dx1 F2 (x) dF2 (x) P = χ(i)χ(j)xi−1 y j−1 , dx1 whence F1 (x)F2 (x) dF1 (x) dF2 (x) = F2 (x) + F1 (x) = 0. dx1 dx1 dx1 Similarly, d(F0 (x)F1 (x))/dx1 = 0. From Theorem 5.50 and Lemma 5.38, F1 (x)F2 (x) is in Σ2 ; that is, it is a square in Σ. Since F2 (x) 6= 0, this gives that F1 (x)/F2 (x) ∈ Σ2 . Similarly, F1 (x)F0 (x) is in Σ2 , and hence F0 (x)/F2 (x) = (F0 (x)F1 (x))/(F1 (x)F2 (x)) ∈ Σ2 . This means that d(F0 (x)F1 (x))/dx1 = 0, and hence the matrix (5.49) has rank 1. In particular, the rational Gauss map of F is inseparable. By (5.40), this shows that (5.53) holds for p = 2, where 2m is the inseparability degree of the rational Gauss map. Assertion (ii) follows from (1.9) and (5.50). 2 R EMARK 5.92 For pm = 2, Theorem 5.90 (iii) does not hold true. A counterexample is the Hermitian curve H2 = v(X03 + X13 + X23 ) which is non-reflexive, but has only nine inflexion points. A necessary and sufficient condition for a nonreflexive curve F to have only finitely many inflexion points is that the matrix " # F0 (x) F1 (x) F2 (x) F0 (x) Dx(2) 1

F1 (x) Dx(2) 1

has rank 2. This is shown in Section 7.7.

F0 (x) Dx(2) 1

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R EMARK 5.93 It is possible for a non-reflexive curve to coincide with its dual curve, up to a Frobenius collineation. An example of this is the Hermitian curve Hq = v(X0q+1 + X1q+1 + X2q+1 ) for any power q of p. In fact, the Gauss map is ωHq : Hq → Hq ,

(x0 , x1 , x2 ) 7→ (xq0 , xq1 , xq2 ),

and hence is the product of the identity rational transformation and the q-th Frobenius collineation. R EMARK 5.94 The dual curve of a strange curve is a line, and hence strange curves have no bidual curve. For the dual of a non-singular plane curve the following results hold. T HEOREM 5.95 (i) If F is a non-singular plane curve of degree n, and if its dual curve F ′ has degree n′ , then qn′ = n(n− 1), where q is the inseparable degree of the Gauss map. (ii) If F and G are two distinct non-singular plane curves, then their duals F ′ and G ′ coincide if and only if p = 2 and they are conics with the same nucleus.

5.12 EXERCISES 1. Let Σ = K(x, y). It follows from Lemma 5.38 that either x or y is a separable variable of Σ. Give another proof here by using Theorem 1.24. 2. With Σ a field of transcendence degree 1, give a new proof for B´ezout’s Theorem using Theorem 5.35. 3. Show that a uniformising element at any place is a separable variable of Σ. 4. Show that, if p = 0, then dξ/dζ = c if and only if ξ = cξ + d, with c, d ∈ K. This result extends to positive characteristic provided that ξ is separable. 5. Prove that the image of a rational curve under a rational transformation is rational. 6. Prove that an irreducible plane curve has genus zero if and only if it is rational. 7. Let γ be a branch of F with order sequence (0, j1 , j2 ). For p > 0, assume that p ≥ 3 but that p does not divide either j1 or j2 − j1 . Show that the corresponding branch γ ′ of the dual curve F ′ of F has order j2 − j1 . 8. Prove the following generalisation of Theorem 5.61. Let Σ1 and Σ2 be two subfields of Σ both of transcendence degree 1. For i = 1, 2, put ni = [Σ : Σi ] and denote by gi the genus of Σi . If Σ = Σ1 Σ2 is the compositum of Σ1 and Σ2 , that is, the smallest subfield of Σ containing both Σ1 and Σ2 , then g ≤ n1 g1 + n2 g2 + (n1 − 1)(n2 − 1).

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9. Let ζ be a separable variable of Σ. Take x, y ∈ Σ and let G = G(x, y) with G(X, Y ) ∈ K[X, Y ]. Show that the higher derivatives of G may be computed from those of x and y using the following formula: (i)

Dζ (G) X ∂ (r1 +r2 +···+s1 +s2 ) G(x, y) r1 + r2 + · · · = × r1 , r2 , . . . ∂x(r1 +r2 +··· ) ∂y (s1 +s2 +··· )   s1 + s2 + · · · (1) (1) (2) (1) (Dζ x)r1 (Dζ x)r2 · · · (Dζ x)s1 (Dζ x)s2 · · · , s1 , s2 , . . . where the summation is for all non-negative integers r1 , r2 , . . . , s1 , s2 , . . . such that r1 + 2r1 + · · · + s1 + 2s2 + · · · = i. 5.13 NOTES The first four sections and Sections 5.7, 5.8 are based on [400]. Introductory works dealing with function fields of transcendence degree 1 over an algebraically closed field are [385], [387], [77], [102], [114], [84], [489]. Section 5.9 is based on [291], and has applications in coding theory; see [178]. Another proof of Theorem 5.68 is found in [196]. Hasse derivatives were introduced in [198]. Lemmas 5.72 and 5.73 come from [451]. The other lemmas and theorems in Section 5.10 are due to Garcia and Voloch; see [161]. Theorem 5.87 is also found in [386]. For Theorem 5.82, see [197] and [176]. The curve in Example 5.81 is birationally equivalent to that in Section 12.5. The exceptional behaviour of duals and biduals of irreducible plane curves in positive characteristic was noted by Wallace [499]. For a detailed investigation of this topic, see Hefez [200], in which some generalisations of Theorems 5.90 and 5.91 are also found. See also [254], [271], [272], [357], [229], [230], [232], [231]. For a proof of Theorem 5.95, see Homma [235]. A generalisation of the concept of dual curve to higher-dimensional projective spaces is treated in Section 7.9. Exercise 8 comes from [427]; see also [428, Theorem 10.3]. For Exercise 9, see [197].

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Chapter Six Linear series and the Riemann–Roch Theorem In the investigation of the birational invariants of an irreducible plane curve, a central role is played by divisors, which are finite formal sums of places, and by linear series, which are certain sets of effective divisors. This chapter develops the theory of linear series; this includes the Riemann–Roch Theorem 6.61, the Weierstrass Gap Theorem 6.89 and the study of divisor class groups, which are also called Picard groups.

6.1 DIVISORS AND LINEAR SERIES As in the previous chapter, the field Σ = K(x, y) is associated to an irreducible algebraic curve F equipped with a generic point P = (x, y). From an algebraic point of view, Σ is a function field of transcendence degree 1 whose places are in a one-to-one correspondence with branches of F. A natural generalisation of a place is a divisor. D EFINITION 6.1

(i) A divisor of Σ is a formal sum of places: P D = P∈P(Σ) nP P,

with nP integers such that nP = 0 except for a finite number of places. (ii) The coefficient nP is the multiplicity or weight of P in D. (iii) The support of D, denoted by Supp D, is the set of all places appearing with non-zero coefficients in D. L EMMA 6.2 The set of all divisors of Σ is an abelian group with respect to the addition, the divisor group Div(Σ) of Σ. P P Proof. The P sum D + E of two of divisors D = nP P and E = mP P is the divisor (nP + mP )P. The identity element (0) is the zero divisor which has no non-zero weight. 2 D EFINITION 6.3 (i) If nP ≥ 0 for every place P, the divisor D = effective or positive.

P nP P is

(ii) A non-effective divisor is virtual. P P (iii) Given two divisors D = nP P and E = mP P, the notation D ≻ E is used when D − E is effective.

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The most important divisors arise directly from elements of Σ, due to Theorem 5.33. For ξ ∈ Σ\{0}, these are the following: P (i) div(ξ) = P∈P(Σ) nP P with nP = ordP ξ is the principal divisor of ξ; P (ii) div(ξ)0 = P∈P(Σ) nP P with nP = ordP ξ and nP ≥ 0 is the divisor of zeros of ξ; P (iii) div(ξ)∞ = P∈P(Σ) (−nP )P with nP = ordP ξ and nP ≤ 0 is the divisor of poles of ξ. Note that div(ξ) = div(ξ)0 − div(ξ)∞ . T HEOREM 6.4 Let ξ ∈ Σ\{0}. Then div(ξ) = (0) if and only if ξ ∈ K\{0}. Proof. If ξ ∈ K\{0}, then ξ has neither zeros nor poles. Assume x 6∈ K. From the proof of Theorem 5.33, Σ has a model (F; (ξ, η)) such that some branch of F is centred at an affine point on the X-axis. The corresponding place of Σ is a zero of div(ξ). Hence div(ξ)0 6= (0). Since zeros of ζ ∈ Σ\{0} are poles of its inverse ζ −1 , so div(ξ)∞ 6= (0) also holds. Hence div(ξ) 6= (0). 2 L EMMA 6.5 The elements in Div(Σ) which are principal divisors constitute a subgroup Prin(Σ). Proof. If f, g ∈ Σ\{0}, then div(f g) = div(f ) + div(g).

2

D EFINITION 6.6 (i) The quotient group Pic(Σ) = Div(Σ)/Prin(Σ) is the divisor class group of Σ. (ii) For a divisor D of Σ, the corresponding element in the factor group Pic(Σ) is denoted by [D], the divisor class of D. D EFINITION 6.7 Two divisors A and B of Σ are equivalent, written A ≡ B, if [A] = [B], that is if there exists ξ ∈ Σ\{0} for which A − B = div(ξ). L EMMA 6.8

(i) The relation of Definition 6.7 is an equivalence relation.

(ii) For divisors A, B, C, D of Σ, (a) if A ≡ B, then A + C ≡ B + C;

(b) if A ≡ B, then −A ≡ −B;

(c) if A ≡ B, C ≡ D, then A + C ≡ B + D.

Proof. (ii) (a) Since A ≡ B, there exists ξ ∈ Σ\{0} for which A − B = div(ξ). Since A − B = (A + C) − (B + C), the assertion follows. (ii) (b) If A ≡ B, then A − B = div(ξ) with ξ ∈ Σ\{0}, whence it follows that −A − (−B) = div(ξ −1 ). (ii) (c) If C ≡ D also holds, that is, C − D = div(η) with η ∈ Σ\{0}, then A + C − (B + D) = A − B + (C − D) = div(ξ) + div(η) = div(ξη). 2 P P D EFINITION 6.9 The degree of a divisor D = P∈P(Σ) nP P is deg D = nP .

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Note that ϕ : A 7→ deg A is an additive map. In other words, ϕ is a homomorphism from the divisor group to the additive group of integers. T HEOREM 6.10 If A ≡ B, then deg A = deg B. Proof. Let A − B = div(ξ) with ∈ Σ\{0}. If ξ ∈ K, then A = B. Otherwise, as div(ξ) = 0 by Corollary 5.35, deg(A − B) = 0, whence the assertion follows. 2 Theorem 6.10 shows that deg[A] is well-defined. Hence the map ϕ : [A] 7→ deg[A] is a homomorphism from the divisor class group to the additive group of integers. D EFINITION 6.11 The subgroup Div0 (Σ) of divisors of degree zero of Div(Σ) consists of all divisors of Σ of degree zero. Under the natural homomorphism Div(Σ) 7→ Pic(Σ), the subgroup Div0 (Σ) corresponds to the subgroup Pic0 (Σ) of divisor classes of degree zero; that is, Pic0 (Σ) = {[A] ∈ Pic(Σ) | deg A = 0}. Certain subsets of divisors, not necessarily subgroups, play an important role in the study of Σ. The most relevant are the linear series. Let x0 , x1 , . . . , xr be elements of Σ, not all zero, and let B be a fixed divisor of Σ. An ordered r-ple c = (c0 , c1 , . . . , cr ) of elements in K is permissible when c0 x0 + c1 x1 + · · · + cr xr 6= 0. For such an r-ple c, consider the divisor Ac = div(c0 x0 + c1 x1 + · · · + cr xr ) + B.

(6.1)

D EFINITION 6.12 The set L of divisors Ac for all permissible r-ples c is a virtual linear series. Note that there may be a finite number of places occurring in every divisor of L. Such places are the fixed places of L. A sum of fixed places is a fixed divisor of L. Generally, it is supposed that x0 , x1 , . . . , xr are linearly independent over K. Then, the condition of permissibility on c is that c 6= 0 = (0, 0, . . . , 0). The next two results follow readily. T HEOREM 6.13 All divisors of a virtual linear series L are equivalent to B and hence are equivalent to each other. D EFINITION 6.14 The common degree of the divisors of a virtual linear series is the order or degree of the virtual linear series. T HEOREM 6.15 For every ζ 6= 0,

Ac = div(c0 x0 ζ + c1 x1 ζ + · · · + cr xr ζ) − div(ζ) + B.

C OROLLARY 6.16 If B ≡ B ′ , then divisors in L can be represented in the form div(c0 x0 + c1 x1 + · · · + cr xr ) + B ′

for some x0 , x1 , . . . , xr of Σ linearly independent over K.

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Proof. If B ≡ B ′ , then B−B ′ = div(ζ) for an element ζ ∈ Σ\{0}. Then Theorem 6.15 applies. 2 T HEOREM 6.17 Let y0 , . . . , ys ∈ Σ be linearly independent elements over K such that the divisors A′d = div(d0 y0 +d1 y1 +· · ·+ds ys )+B ′ , with a fixed divisor B ′ , are exactly the divisors of L arising from x0 , x1 , . . . , xr , linearly independent elements over K. Then r = s and, for some non-zero element ζ ∈ Σ, P yi = ζ cij xj , with det(cij ) 6= 0.

Proof. Since div(yi ) + B ′ ∈ L, there exists ci such that div(yi ) + B ′ = Aci . Thus Aci ≡ B ′ , whence B ≡ B ′ . Therefore B − B ′ = div(ζ) for a non-zero ζ ∈ Σ. Hence P P div(yi ) = div( rj=0 cij xj ) + B − B ′ = div( rj=0 cij xj ) + div(ζ) Pr Pr = div( j=0 (cij xj )ζ) = div( j=0 cij xj ζ). P Thus div( rj=0 (cij xj )/yi ) = 0. By Theorem 6.4, there exists k ∈ K\{0} such Pr that yi = k j=0 cij xj ζ. Absorbing the constant k into the coefficients cij , this Pr reads yj = j=0 cij xj ζ. As a consequence, s ≤ r. Similarly, r ≤ s, whence r = s follows. 2 C OROLLARY 6.18 The integer r depends only on the series L and not on its representations. D EFINITION 6.19 The integer r is the dimension of the virtual linear series L. It should be noted that the virtual linear series L is a projective space over K whose points are the divisors of L. The dimension of L is equal to the dimension of the projective space. Hence PG(r, K) can be regarded as a representative of L. If d = kc, with k ∈ K\{0}, then Ad = Ac . Conversely, if Pr Pr div( i=0 ci xi ) + B = div( i=0 di xi ) + B, Pr Pr then div( i=0 ci xi ) = div( i=0 di xi ), whence, by Theorem 6.4, Pr Pr i=0 ci xi = k i=0 di xi ,

with k ∈ K\{0}. Assuming the elements xi to be linearly independent over K, this only occurs when d = kc. Thus the set of divisors Ac of the linear series L is in one-to-one correspondence with the points of the r-dimensional projective space PG(r, K) over K, the divisor Ac corresponding to the point with coordinates (c0 , c1 , . . . , cr ). A change of representation (6.1) of divisors in L changes c by a projectivity of PG(r, K). To check this, use the same notation as in the proof of Theorem 6.17. If A′d = Ac , that is, P P div( ri=0 di yi ) + B = div( ri=0 ci xi ) + B ′ , P P P then div( ri=0 di yi ) = div( ri=0 ci yi ). Since yi = ζ rj=0 cij xj , so Pr Pr Pr div( i=0 j=0 di cij xj ) = div( i=0 ci xi ).

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By Corollary 5.35, for a non-zero k ∈ K, Pr k cj = i=0 cij di .

Thus c = (c0 , . . . , cr ) and d = (d0 , . . . , dr ) are related in the way stated. Put C = (cij ) and let t C be the transpose of the matrix C; then the relation can also be −1 written as c = k −1 t Cd and d = k t C c. D EFINITION 6.20 A subseries of a virtual linear series L is the set of all divisors corresponding to a projective subspace of PG(r, K), the projective space representative of L. The following result shows that PG(r, K) is a good representative of L. T HEOREM 6.21 Every subseries of a virtual linear series is a virtual linear series. Proof. Let Πs be an s-dimensional subspace of PG(r, K) whose points are in oneto-one correspondence with the divisors of a linear series L. Given s + 1 linearly independent points in Πs , say, c0 = (c00 , . . . , c0r ), . . . , cs = (cs0 , . . . , csr ), the subseries corresponding to Πs consists of the divisors Ps Ps div(( i=0 λi ci0 )x0 + · · · + ( i=0 λi cir )xr ) + B

(6.2)

= div(λ0 (c00 x0 + · · · + c0r xr ) + · · · + λs (cs0 x0 + · · · + csr xr )) + B.

Put yi = ci0 x0 + P· · · + cir xr for i = 0, 1 . . . , s. Then the divisor (6.2) can be rewritten as div( si=0 yi ) + B, that is, as the divisor of a linear series L′ . A straightforward calculation shows that y0 , . . . , ys are linearly independent if and only if the matrix C = (cij ) has maximum rank s + 1. This condition is satisfied here, since c0 , . . . , cs have been chosen to represent linearly independent points in PG(r, K). 2 Let L be Par virtual linear series consisting of all divisors (6.1). If there exists c such that i=0 ci xi = 1, then L is normalised.

T HEOREM 6.22 A virtual linear series L is normalised if and only if its fixed divisor B belongs to L. P Proof. If L is normalised, take the (r + 1)-tuple c so that ri=0 ci xi = 1; then Ac = div(1) + B = B, showing that B ∈ L. Conversely, if B ∈ L, then P there exists c 6= 0 such that B = AcP . For such an r (r + 1)-ple c, the divisor div( ri=0 ci xP i ) is equal to 0. Therefore i=0 ci xi has r no zeros or poles. By Corollary 5.35, i=0 ci xi is equal to a non-zero constant P k ∈ K. Putting di = ci /k, this sum can be rewritten in the form ri=0 di xi = 1, as required. 2 Actually, it may be supposed that L is given in normalised form. In fact, as in Theorem 6.17, in (6.1) the elements xi may be replaced Pr by xi ζ with an arbitrary ζ ∈ Σ\{0} and B by a suitable B ′ . Now, put ζ = i=0 ci xi , and yi = xi /ζi , bearing in mind thatP ζ 6= 0 as x0 , . . . , xr are Prlinearly independent over K. This r gives that Ac = div( i=0 ci yi ) + B ′ , with i=0 ci yi = 1.

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T HEOREM 6.23 If L and L′ are two virtual linear series and an element of L is equivalent to one of L′ , then L and L′ are both contained in a virtual linear series. Proof. Let P L = {Ac | Ac = div( ri=0 ci xi ) + B, c 6= 0}, Ps L′ = {A′d | A′d = div( j=0 dj yj ) + B ′ , d 6= 0}.

In L′ , the divisor B ′ may be replaced by any divisor equivalent to it, in particular B. Hence B = B ′ may assumed. Now, the virtual linear series Pr Ps L0 = div( i=0 ci xi + j=0 dj yj ) + B

contains both L and L′ .

2

In this proof, x0 , . . . , xr , y0 , . . . , ys are not necessarily linearly independent. Hence dim L0 ≤ 1 + dim L + dim L′ . T HEOREM 6.24 An arbitrary divisor C can be added to or subtracted from the divisors of a virtual linear series to obtain another virtual linear series of the same dimension. T HEOREM 6.25 The multiplicities with which a place P appears in the divisors of a virtual linear series L are bounded above and below. Proof. The divisor B in L does not affect the question of boundedness; so assume that B = (0). Let P be a place of Σ. For i = 0, P. r. . , r, let −νi = ordP xi . Since ordP (x + y) ≥ min{ordP x, ordP y}, so ordP ( i=0 ci xi ) ≥ nP , where  0 if P is not a pole of any xi , nP = min{−νi } if P is a pole of some xi , which gives a lower bound. This also shows that Pr −nQ ≥ −ordQ ( i=0 ci xi )

for each place Q of Σ. Hence P Pr −nQ Q ≻ div( i=0 ci xi )∞ ,

with the summation over all places Q of Σ. Since Pr Pr Pr div( i=0 ci xi ) = div( i=0 ci xi )0 − div( i=0 ci xi )∞ , so

Hence

P

Pr Pr −nQ Q + div( i=0 ci xi ) ≻ div( i=0 ci xi )0 . P

P P ordQ ( ri=0 ci xi ) ≥ mQ , Pr Pr where mQ is theP zero number i=0 ci xi )0 . On i=0 ci xi ; that is, mQ = ordQ ( Pof r the other hand, ord (div c x ) = 0, by Corollary 5.35. This means that Q i i i=0 P P P P nQ ≥ mQ , whence nQ ≥ mP . Since nQ depends only on L but is independent Pr of the particular choice of the coefficients c0 , . . . , cr , it follows that 2 ordP ( i=0 ci xi ) is bounded above by a number only depending on L. −nQ +

P

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T HEOREM 6.26 Let P be a place of Σ and ν the smallest multiplicity with which P occurs in the divisors of L. Then the divisors of L in which P occurs with multiplicity at least ν + 1 make up a subseries of L of dimension dim L − 1. Proof. Let (6.1) be a divisor of L. Choose B ∈ L in which P occurs in B with multiplicity ν; that is, nP (B) = ν. Then P is not a pole of any xi , for otherwise P would occur with multiplicity less than ν in some div(x) + B. In fact, nP (div(xi ) + B) = ordP (xi ) + nP (B) = ordP (xi ) + ν, and, if ordP (xi ) < 0 were true, then nP (div(xi ) + B) < ν would also be true, contradicting the minimality of ν. Also, P is not a zero for every xi . If this were not true, then P P nP (div( ri=0 ci xi ) + B) = ordP ( ri=0 ci xi ) + ν > ν, since

Pr ordP ( i=0 ci xi ) ≥ min {ordP (xi )} ≥ ν

and ordP (xi ) > 0 for 0 ≤ i ≤ r. Let σ : Σ → K((t)) be a primitive representation of the place P. Then σ(xi ) = ai0 + ai1 t + · · · .

Since ordP (xi ) = 0 for some i, the coefficient ai0 does not vanish for every i. Therefore Ac = div(c0 x0 + · · · + cr xr ) + B, div(c0 (a00 + a01 t + · · · ) + · · · + cr (ar0 + ar1 t + · · · )) + B = div((c0 a00 + · · · + cr ar0 ) + tM1 + · · · ) + B,

whence nP (Ac ) = nP (div(c0 a00 + · · · + cr ar0 ) + tL1 + · · · ) + nP (B) = nP (div(c0 a00 + · · · + cr ar0 ) + tL1 + · · · ) + ν.

Now, nP (Ac ) ≥ ν + 1 if and only if

nP (div(c0 a00 + · · · + cr ar0 ) + tL1 + · · · ) ≥ 1,

whence c0 a00 + · · · + cr ar0 = 0. This means that, in the projective space PG(r, K) representing L, the hyperplane H = v(a00 X0 + . . . , a0r Xr ) passes through the point A = (c0 , . . . , cr ). Hence the divisors of L in which P appears with multiplicity at least ν + 1 correspond to those points A lying in H, and hence they constitute a virtual linear series of dimension dim L − 1. 2 C OROLLARY 6.27 The divisors of L in which a given place P appears with multiplicity at least µ constitute a subseries of L. Proof. If µ ≤ ν, then L itself is the subseries. If µ = ν + 1, then the corollary is just Theorem 6.26. If µ = ν + 2, apply Theorem 6.26 to the subseries consisting of all divisors in which P occurs with multiplicity at least ν + 1; continue this process. 2

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R EMARK 6.28 With the notation of Theorem 6.13, if ν < µ, the subseries is empty. Also, the divisors of L in which P appears with multiplicity at least ν + 2 is a subseries L′ of L whose dimension is at least dim L − 2. In fact, L may happen not to contain divisors in which P appears with multiplicity ν + 1. If this is the case, Theorem 6.13 does not apply to L′ and ν + 1. C OROLLARY 6.29 If P1 , . . . , Ps is a set of places of Σ and µ1 , . . . , µs are integers, then the divisors of L in which each Pi appears with multiplicity at least µi constitute a subseries of L. T HEOREM 6.30 The effective divisors of L constitute a subseries. Proof. There are only finitely many places of Σ in which the divisors (6.1) can have negative degree, namely, the places occurring in B and the poles of the elements xi . The theorem now follows from the previous corollary when P1 , . . . , Ps are defined to be these places and µi = 0 for i = 1, . . . , s. 2 D EFINITION 6.31 A linear series is a virtual linear series consisting of effective divisors; it is denoted by gnr , where n is its order and r is its dimension. T HEOREM 6.32 In a linear series gnr , the order and dimension satisfy r ≤ n. Proof. If r = 0, then r ≤ n. Let r > 0, and take a place P that does not occur in every divisor of gnr . Then the divisors of gnr in which P occurs constitute a r−1 subseries gnr−1 . Subtracting P from each of these divisors, a subseries gn−1 is obtained. Applying induction on r shows that r − 1 ≤ n − 1, whence the assertion follows. 2 s s D EFINITION 6.33 Let gm and gnr be two linear series of Σ. Then gm is contained r s r in gn if every divisor D ∈ gm is contained in some divisor E ∈ gn ; that is, E ≻ D.

Here, s gm = {div(c0 x0 + · · · + cm xm ) + B | c = (c0 , . . . , cm ) ∈ PG(m, K)}

is contained in gnr if and only if there exist xs+1 , . . . , xr ∈ Σ, and F ≻ B such that gnr = {div(c0 x0 + · · · + cr xr ) + F | c = (c0 , . . . , cr ) ∈ PG(r, K)}. ′

T HEOREM 6.34 If gnr and gnr ′ are linear series of Σ such that a divisor of one of them is equivalent to a divisor of the other, then n = n′ and both are contained in a linear series. Proof. Theorem 6.25 together with Theorem 6.13 yield n = n′ . From Theorem ′ 6.23, gnr and gnr are both contained in a virtual linear series L. Let L′ be the set of all effective divisors in L. By Theorem 6.30, L′ is a linear series, and both gnr and ′ gnr ′ are subseries of L′ . 2 T HEOREM 6.35 The effective divisors equivalent to a divisor C constitute a linear series.

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Proof. It may be that no effective divisor is equivalent to C, for instance when ordP C < 0. If this is the case then such effective divisors constitute the empty linear series. Otherwise there is at least one effective divisor B equivalent to C, and C may be replaced by B since the divisors equivalent to B are the same as those equivalent to C. Let ordP B = n. The virtual linear series L0 = div(x0 )+B consists of effective divisors, more precisely, L0 = gn0 . Every divisor E equivalent to B can be written as E = div(f ) + B for some f ∈ Σ. Let Γ the set of such elements f ∈ Σ with E ranging over the set of all effective divisors equivalent to B. Note that Γ is a subspace of Σ regarded as a vector space over K. To show this, take a place P of Σ and two elements f1 , f2 ∈ Γ together with two constants c1 , c2 ∈ K. Since ordP (c1 f1 + c2 f2 ) ≥ min{ordP f1 , ordP f2 }, it follows that div(c1 f1 + c2 f2 ) ≻

P

P∈ΣP nP P,

with nP defined to be min{ordP f1 , ordP f2 }. Since f1 , f2 ∈ Γ implies nP ≥ 0, the assertion follows. To show that Γ has dimension at most n, suppose on the contrary that Γ contains a K-independent set {x0 , . . . , xr } with r > n. Then L = {Ac | c 6= 0} is a linear series with dimension r > n; but this contradicts Theorem 6.32. 2 D EFINITION 6.36 The linear series consisting of all effective divisors equivalent to a divisor C is a complete linear series and is denoted by |C|. E XAMPLE 6.37 (i) Given an irreducible non-singular cubic C and a fixed point P0 ∈ C, the lines through P0 cut out a complete g21 on C. (ii) Given P1 ∈ / C, the lines through P1 cut out a g31 on C contained in the com2 plete g3 cut out by all lines of the plane. L EMMA 6.38 Let |A|, |B| be two complete linear series, with A and B not necessarily effective. If A1 ≡ A and B1 ≡ B, then |A1 + B1 | = |A + B|,

|A1 − B1 | = |A − B|.

Proof. By Lemma 6.8 (iii), A1 + B1 ≡ A + B, whence |A1 + B1 | = |A + B|. The same holds when ‘+’ is replaced by ‘−’. 2 D EFINITION 6.39 (i) The sum of two complete linear series |A| and |B| consists of all effective divisors equivalent to A + B, is denoted by ||A| + |B||. (ii) The difference of |A| and |B| is the linear series of all effective divisors equivalent to A − B. T HEOREM 6.40 Let A and B be two effective divisors. Then ||A| − |B|| consists of all divisors D − B, where D ∈ |A| and D ≻ B.

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Proof. Suppose C ∈ ||A| − |B|| = |A − B|. Then C ≡ A − B. Hence C + B ≡ A, whence C + B ∈ |A|. Therefore C + B = A1 with A1 ∈ |A|, which yields C = A1 − B. Conversely, if C = A1 − B with A1 ∈ |A| and A1 ≥ B, then C ∈ |A1 − B| = ||A1 | − |B|| = ||A| − |B||.

2

In Section 5.7 the concept of a differential of the form dx for a separating variable x ∈ Σ is defined and investigated. In the context of divisors, associate with dx the divisor of the differential dx defined by P div(dx) = ordP dx P. Every differential is of the form ξdx with ξ ∈ Σ. Hence (ξdx) = div(ξ) + div(dx), which shows that all divisors ξdx belong to the same equivalence class of divisors, the canonical class. A canonical divisor has degree 2g − 2, by Definition 5.55. D EFINITION 6.41 The complete linear series |dx| is the canonical series. The canonical series has order 2g − 2. In the next section, an explicit description of the canonical series is given which demonstrates that its dimension is g − 1. 6.2 LINEAR SYSTEMS OF CURVES The aim of this section is to describe the construction of a linear series from a linear system of curves by means of intersection divisors. Let Σ be a function field of transcendence degree 1 and (F; (x, y)) a model of Σ, where F is an irreducible curve and P = (x, y) is a generic point of F. A divisor of Σ arises from every curve G that contains F as a component. Such a divisor is the intersection divisor cut out on F by Φ and defined to be P F ·G = I(P, G ∩ γ)P,

where the summation is over all places of Σ, and γ is the branch of F corresponding to P. T HEOREM 6.42 Suppose that (a) F = v(F (X0 , X1 , X2 )) is an irreducible curve with generic point (1, x, y); (b) G = v(G(X0 , X1 , X2 ) is any curve of degree m; (c) F = 6 ℓ∞ = v(X0 ), the line at infinity; (d) F does not divide G.

Then div G(1, x, y) = G · F − m(ℓ∞ · F).

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Proof. For a place P of Σ, let (x0 (t), x1 (t), x2 (t)) be a primitive representation in special coordinates of the branch of F corresponding to P. Note that ℓ∞ 6= F implies x0 (t) 6= 0 . As usual, let x(t) = x1 (t)/x0 (t) and y(t) = x2 (t)/x0 (t). Then ordP G(x0 (t), x1 (t), x2 (t)) = ordP x0 (t)m + ordP G(1, x, y). Multiplying by P and summing over all places P, it follows that P P P ordP G(x0 (t), x1 (t), x2 (t)) P = m ordP x0 (t) P + ordP G(1, x, y) P. P By (4.34), the left-hand side is D · F , while ordP x0 (t)P = ℓ∞ · F. Therefore G · F = m(ℓ∞ · F) + div G(1, x, y),

whence the assertion follows.

2

C OROLLARY 6.43 The curve G cuts out on F the divisor G · F equivalent to m(l∞ · F). C OROLLARY 6.44 If G1 = v(G1 (X0 , X1 , X2 )) is another curve of degree m that does not contain F as a component, then G1 and G cut out equivalent divisors on F. D EFINITION 6.45 Let Φ0 = v(ϕ0 (X0 , X1 , X2 )), . . . , Φr = v(ϕr (X0 , X1 , X2 )) be curves of the same degree m, with ϕ0 , . . . , ϕr linearly independent over K. Pr (i) The set of curves F = v( i=0 ci ϕi ), with c = (c0 , . . . , cr ) ranging over all non-trivial (r + 1)-ples of elements of K, is a linear system of dimension r, and is the linear system generated by Φ0 , . . . , Φr ; (ii) when r = 1, the system is a pencil; (iii) when r = 2, the system is a net. There is a one-to-one correspondence between the curves of a linear system of dimension r and the points of PG(r, K). It is worth noting that, if the polynomials are written in their inhomogeneous form, that is, ϕ′ (X, Y ) = ϕ(1, X, Y ), then some of them could have degree less than m. T HEOREM 6.46 The curves in a linear system of degree m that do not contain F as a component cut out on F the divisors of a linear series. The converse also holds, up to a fixed divisor. Proof. For the converse, let L be a linear series consisting of all divisors (6.1). Since Σ = K(x, y) with P = (x, y) a generic point of F, there are polynomials ϕi (X, Y ), d(X, Y ) ∈ K[X, Y ] such that xi = ϕi (x, y)/d(x, y) for i = 0, . . . , r. Hence Pr Pr i=0 ci ϕi (x, y)/d(x, y). i=0 ci xi = Also,

Pr Ac = div ( i=0 ci ϕi (x, y)/d(x, y)) + B Pr = div ( i=0 ci ϕi (x, y)) − div(d(x, y)) + B.

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If m is the maximum degree of the polynomials ϕi (X, Y ), define the homogeneous polynomial ϕi (X0 , X1 , X2 ) to be X0m ϕ(X1 /X0 , X2 /X0 ), for i = 0, . . . , r. Let Fi = v(ϕ(X0 , X1 , X2 )) be the corresponding curve, and let D = v(D) with D(X0 , X1 , X2 ) a homogeneous polynomial such that D(1, X, Y ) = d(X, Y ). Now, the linear Pr system Γ generated by the curves Fi is well defined. Let Φc be the curve v( i=0 ci ϕi (X0 , X1 , X2 )). From Theorem 6.42, Pr div ( i=0 ci ϕ(x, y)) = Φc · F − m(ℓ∞ · F). Therefore

r X ci ϕi (x, y) div d(x, y) i=0

!

+ B = Φc · F − m(ℓ∞ · F) − div d(x, y) + B.

Since C = −m(ℓ∞ · F) − (d(x, y)) + B does not depend on c, the linear system Γ cuts out on F the linear series consisting of the divisors (6.1), up to the fixed divisor C. The direct part of Theorem 6.46 can be proved by reversing these steps. 2 R EMARK 6.47 Theorem 6.46 establishes a close relationship between linear series and linear systems. However, the dimension dim Γ of a linear system Γ differs from the dimension of the linear series L cut out by Γ on F. In fact, dim Γ ≥ dim L and equality only holds when no two distinct curves in Γ cut out the same divisor in L. To eliminate this difference, linear dependency modulo F is needed. D EFINITION 6.48 If F = v(F (X0 , X1 , X2 )) and F (X0 , X1 , X2 ) divides φ(X0 , X1 , X2 ) − ψ(X0 , X1 , X2 ), then φ ≡ ψ (mod F ).

Pr Given a linear system v( i=0 ci ϕi (X0 , X1 , X2 )), if one of the ϕi , say ϕ0 , is linearly dependent that is, the Pron the others modulo F , then ϕ0 can be eliminated;P r linear system v( i=1 ci ϕi ) cuts out the same linear series on F as v( i=0 ci ϕi ) does. So, the ϕi may be supposed linearly independent modulo F . Note that, if m < deg F , then linear independence means linear independence modulo F . To obtain deeper results on linear series and especially on the canonical series, a model (F; (x, y)) of Σ is chosen such that the irreducible curve F has only ordinary singularities. Let P1 , . . . , Pk denote the singular points of F and let ri be the multiplicity of Pi for i = 1, . . . , k. Let {γij | j = 1, . . . , ri } be the set of all branches of F centred at Pi . D EFINITION 6.49 (i) The double-point divisor of F, more precisely of Σ, with respect to the model (F; (x, y)), is Pk Pri D = i=1 j=1 (ri − 1)Pij , where Pij is the place of Σ corresponding to γij .

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(ii) A curve G is an adjoint of F if G · F ≻ D. In this case, G · F = D + E with E an effective divisor, and E is the divisor cut out by G on F, up to D. Also, G passes through E ′ for any effective divisor E ′ such that E ≻ E ′ . T HEOREM 6.50 A curve G is an adjoint of F if and only if G has at least an (r − 1)-fold point at every r-fold point of F. Proof. Let P be an r-fold point of F. If P is an (r − 1)-fold point of G and γ is a branch of F centred at P , then I(P, G ∩ γ) ≥ r − 1. Suppose now that G has only an s-fold point, with s ≤ r − 2, at a point of multiplicity r of F. Since P is ordinary singularity, G has at most s tangents at P . Since s < r, some tangent to F at P , say ℓ, is not tangent to G at P . Let γ be the branch whose tangent is ℓ. Since γ is linear, I(P, G ∩ γ) = s. As s ≤ r − 2, so G is not an adjoint. 2 T HEOREM 6.51 The adjoints of a given degree cut out a complete linear series on the curve F. Proof. Let Φ be an adjoint of degree m. Then Φ · F = D + E, with E ≻ (0). If E ′ is another effective divisor with E ′ ≡ E, the existence of an adjoint Φ′ of F for which Φ′ · F = D + E ′ needs to be established. First it is shown that E ≡ E ′ implies the existence of two curves Ψ and Ψ′ such that Ψ · F = E + F,

Ψ′ · F = E ′ + F.

Since E = E ′ + div(u) for an element u = a(x, y)/b(x, y) with a(X, Y ), b(X, Y ) both in K[X, Y ], so E + div b(x, y) = E ′ + div a(x, y). Applying Lemma 6.42 to the curves C1 = v(a(X, Y )) and C2 = v(b(X, Y )) gives div a(x, y) div b(x, y)

= C1 · F − (deg C1 )(ℓ∞ · F), = C2 · F − (deg C2 )(ℓ∞ · F).

(6.3)

Hence E + C2 · F − (deg C2 )(ℓ∞ · F) = E ′ + C1 · F − (deg C1 )(ℓ∞ · F). Now, suppose that the reference system is chosen with the following two properties: (i) ℓ∞ ∩ F is disjoint from both C1 ∩ F and C2 ∩ F; (ii) if P appears either in E or in E ′ , then the centre of P is not a point of ℓ∞ . Then (6.3) implies that deg C1 = deg C2 . Therefore

E + C2 · F = E ′ + C1 · F,

whence C1 · F − E = C2 · F − E ′ . Putting F = C1 · F − E, the assertion follows. Note that F need not to be effective even if this happens in some cases. For instance, if E and E ′ do not have places in common. Nevertheless, a slight change

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is enough to ensure that F is effective. In fact, write F = F0 − F∞ with effective divisors F0 and F∞ . If F∞ = n0 P0 +. . .+nt Pt , choose a line ℓi through the centre of Pi and define the totally reducible curve C whose components are the lines ℓi for i = 1, . . . t, each counted ni times. Put U = C · F. Then U is an effective divisor such that U ≻ F∞ . Then also U + F ≻ (0). Replacing Ψ by CΨ and Ψ′ by CΨ′ ensures that (CΨ) · F = C · F + Ψ · F = U + E + F = U + F, (CΨ′ ) · F = C · F + Ψ′ · F = U + E ′ + F = E ′ + U + F.

This shows that F ≻ (0) can be assumed. Then Put

(ΦΨ′ ) · F = D + E + E ′ + F ≻ D + Ψ · F. Φ = v(ϕ),

Ψ = v(ψ)),

Ψ′ = v(ψ ′ ).

By Theorem 4.66, ϕψ ′ = AF + ϕ′ ψ, where A and ϕ′ are homogeneous polynomials, and ϕ′ 6≡ 0 (mod F ). Thus D + E + E ′ + F = Φ · F + Ψ′ · F

= (ΦΨ′ ) · F = (AF + ϕ′ ψ) · F = (ϕ′ ψ) · F = Φ′ · F + Ψ · F

Hence, Φ′ · F = D + E ′ .

= Φ′ · F + E + F. 2

T HEOREM 6.52 Every complete linear series is cut out, up to a fixed divisor, by the adjoints of some degree m. Proof. Let |E| be a complete linear series. Apart from the trivial case when |E| is empty, it may be assumed that E is effective. Choose an adjoint passing through E. Such adjoints certainly exist; for example, take lines passing through the centres of the branches corresponding to the places occurring in E, as well as, for every r-fold point of F, some r − 1 lines passing through it. The totally reducible curve whose components are these lines is an adjoint of F passing through E. Any adjoint Φ of F through E satisfies Φ · F = D + E + E ′ with E ′ ≻ (0). Let m denote the degree of Φ, By Theorem 6.51, the adjoints of degree m cut out the complete linear series |E + E ′ | on F. The divisors of |E + E ′ | containing E ′ are cut out by the adjoints of degree m which pass through E ′ . By Theorem 6.40, the linear series ||E + E ′ | − |E ′ || consists of all divisors H − E ′ , where H is cut out by the adjoints of degree m passing through E ′ . To complete the proof it is enough to note that |E| = ||E + E ′ | − |E ′ ||. 2 T HEOREM 6.53 An irreducible plane curve F has genus 0 if and only if F is birationally equivalent to a line.

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Proof. From Theorem 5.57, lines have genus 0. For the converse, F = v(f (X, Y )) is assumed to have only ordinary singularities. Also, deg f (X, Y ) ≥ 2 may be supposed. From the proof of Theorem 3.24, the complete linear series cut out on F by the adjoint curves of F of degree n − 1 has dimension Pk r ≥ 21 (n − 1)(n + 2) − 21 i=1 ri (ri − 1) − (2n − 3) = 1.

Suppose r > 1. Then there is an adjoint curve G of degree 1 containing two non-singular points of F. By B´ezout’s Theorem applied to F and G, it turns out that they must have a common component, since Pk i=1 ri (ri − 1) + (2n − 3) + 2 > n(n − 1).

This is impossible since F is irreducible and G has lower degree than F. Therefore r = 1. So, adjoint curves of F degree n − 1, with one exception, can be expressed in the form Gλ = v(gλ (X, Y )), with λ ∈ K and gλ (X, Y ) = g1 (X, Y ) + λg2 (X, Y ), where G1 = v(g1 (X, Y ) and G2 = v(g2 (X, Y )) are two fixed, distinct adjoints of degree n − 1. Choose a coordinate system so that f (X, Y ) = uY n + · · · , g1 (X, Y ) = v1 Y n−1 + · · · , g2 (X, Y ) = v2 Y n−1 + · · · , with u, v1 , v2 6= 0, and so that no common point of F and G1 is at infinity. Let R(X, λ) be the resultant with respect to Y of F and G. Then R(X, λ) = b0 (λ) + b1 (λ)X + · · · + bs (λ)X s ,

where s = n(n − 1). Now, R(X, 0) is the resultant of F and G1 , and it has degree n(n − 1). Hence bs (0) 6= 0, and so bs (λ0 ) = 0 for at most a finite set of values of λ0 . Dismissing this finite set, F and Gλ have no common point at infinity. The polynomial R(X, λ) ∈ K[X] has a root of ai of multiplicity at least ri (ri − 1) at each ri -fold point Pi = (ai , bi ), and has a simple root ci at each of the 2n − 3 chosen simple points Qi = (ci , di ) of F. Since Pk i=1 ri (ri − 1) + (2n − 3) = n(n − 1) − 1, all but one of the roots of R(X, λ) = 0 have been accounted for. The remaining root is, therefore, ϕ(λ) =

Pk bs−1 (λ) Pk − i=1 ri (ri − 1)ai − i=1 ci . bs (λ)

Note that ϕ(λ) is a rational function. Similarly, from the resultant with respect to X, another rational function ψ(λ) is obtained. For each value λ which has not been ruled out, the point Q = (ϕ(λ), ψ(λ)) is on F. If Q = (x0 , y0 ) is a point of F not on G2 , there is a unique λ0 such that Q ∈ Gλ0 , namely λ0 = −g1 (x0 , y0 )/g2 (x0 , y0 ). Hence x0 is a root of the polynomial R(λ0 , X) ∈ K[X], and so x0 = ϕ(λ0 ). Similarly, y0 = ψ(λ0 ). There is only a finite set of points of F which are on G2 . Let Σ be the function field of the line ℓ of equation Y = 0. Then Σ = K(ξ).

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Let ω be the rational transformation x′ = ϕ(ξ),

y ′ = ψ(ξ)

of Σ. Then the image curve F ′ of ℓ by ω is an irreducible plane curve which has infinitely many common points with F. By B´ezout’s theorem, they coincide. Hence F is birationally equivalent to the line ℓ. 2 D EFINITION 6.54 For a curve F of degree n > 3, the adjoints of degree n − 3 are the canonical or special adjoints of F. T HEOREM 6.55 If F has degree n > 3 and genus g, there are g linearly independent canonical adjoints of F. Proof. The curves of degree n − 3 form a linear system of dimension 12 n(n − 3). If F is non-singular, all these curves are adjoints and the result follows since g = 21 (n − 1)(n − 2) = 21 n(n − 3) + 1.

Suppose P is an r-fold point of F. If G is an adjoint, P is an (r − 1)-fold point of G by Theorem 6.50. To find a canonical adjoint G of F, take any polynomial G(X, Y ) ∈ K[X, Y ] of degree n − 3. Arguing as in the proof of Theorem 3.24, a curve G = v(G(X, Y )), with G(X, Y ) = a00 + a10 X + . . . + a0,n−3 Y n−3 ,

is an adjoint of F if and only if (a00 , a10 , . . . , a0,n−3 ) is a solution of a certain system of linear equations. The number of the equations is the sum of the number Pk of singular points of F, each counted 1 + 2 + · · · + (r − 1) = i=1 21 ri (ri − 1) times, where P1 , . . . , Pk are the singular points of F and ri is the multiplicity of Pi . These equations can be interpreted as hyperplane equations in PG(N, K) with P N = 21 n(n−3). In this way, 21 r(r−1) is the total number of hyperplanes arising from singular points of F. The P intersection of these hyperplanes is a subspace ΠM of dimension M ≥ N − 21 r(r − 1). To M + 1 linearly independent points in ΠM there correspond the same number of linearly independent canonical adjoints. Now the assertion follows from the fact that M ≥ g −1, a consequence of Theorem 5.57. 2 T HEOREM 6.56 If F has degree n > 3, then the canonical series |(dx)| is cut out by the canonical adjoints. Proof. With a change of coordinates, take F as in the proof of Theorem 5.57. Then the polar curve F ′ = v(∂(F (X, Y )/∂Y ) of F at Y∞ exists and, if P = (x, y) denotes a generic point of F, then x is a separable variable of Σ. From the proof of Theorem 5.57, F ′ · F = D + div(dx)0 ,

div(dx)∞ = 2(ℓ∞ · F).

If E is a canonical adjoint, then (E + 2ℓ∞ ) · F ≡ F ′ · F, and so it follows that E · F ≡ D + div(dx). Therefore every canonical adjoint cuts out a divisor equivalent to div(dx) on F. By Theorem 6.51, the result is established provided that there exists at least one canonical divisor. However, this follows from Theorem 6.55 for g ≥ 1. For g = 0, the canonical series is empty, and the result holds. 2

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R EMARK 6.57 The above approach to the canonical series does not work when the condition that F has only ordinary singularities is dropped. However, in some cases, the canonical linear series may have a nice explicit description in terms of linear systems of curves, as the following example shows. E XAMPLE 6.58 Assume h to be odd in Example 5.59, and consider the linear system ∆ consisting of all plane curves of degree 12 (n − 3) for which Y∞ is an 1 2 (n − 1)-fold point. In other words, ∆ consists of all completely reducible plane curves split into 21 (n − 1) lines through Y∞ . Since Y∞ is an (n − 2)-fold point of F, ∆ cuts out on F a linear series L for which Y∞ is a fixed place of multiplicity 1 1 2 (n − 2)(n − 3). It then follows that L has dimension 2 (n − 3). Discarding 1 d 2 (n − 2)(n − 3)Y∞ from each divisor of L, the resulting linear series is a g2d with 1 d = 2 (n − 3) = g − 1; by Theorem 6.72 (i), this is the canonical series. 6.3 SPECIAL AND NON-SPECIAL LINEAR SERIES Now, the Riemann–Roch Theorem can be proved; it is the main result concerning complete linear series. Again, a model (F; (x, y)) of Σ is used with the property that F is an irreducible algebraic curve with only ordinary singularities. The degree of F is denoted by m, and W stands for a canonical divisor. If m ≤ 3, take a triangle meeting F in 3m distinct points. Then the image curve of F under a quadratic transformation associated to the triangle is birationally equivalent to F, and it has degree 3m, which is larger than 3 for m > 1. If m = 1, repeating this process, an irreducible curve of degree greater than 3 is obtained. Thus, it may be assumed that m > 3. D EFINITION 6.59 For a complete linear series gnr , (i) the index of speciality i is the number of linearly independent canonical divisors through a given divisor; (ii) the series is special if i > 0 and non-special if i = 0. T HEOREM 6.60 (Riemann) If gnr is a complete linear series on the curve F of genus g, then r ≥ n − g. T HEOREM 6.61 (Riemann–Roch) If gnr is a complete linear series |D| on the curve F of genus g, with D an effective divisor and W a canonical divisor, then (i) r = n − g + i; (ii) r = g − 1 and n = 2g − 2 when D = W ; (iii) r = n − g when n > 2g − 2. E XAMPLE 6.62 Let F be as in Example 5.59. Put Pi = (ai , 0) for i = 1, . . . , h, and denote Pn by Pi the place associated to the unique branch of F centred at Pi . Let D = i=1 ǫi Pi with ǫi = 0 or ǫi = 1. By Example 6.58, every canonical divisor

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is cut out on F by a completely reducible curve split into 21 (h − 1) lines through Y∞ minus the fixed divisor 21 (n − 2)(n − 3)P∞ , where P∞ is the place centred at Y∞ . From this, the index of speciality i of D is calculated: ( 1 for deg D ≤ 21 (h − 1), 2 (h − 1) − deg D i= (6.4) 0 for deg D > 12 (h − 1). The Riemann–Roch Theorem shows that dim |D| = 0 for deg D ≤ 21 (h − 1). In the proof of Theorem 3.24 it is shown that the maximum P number of linearly independent adjoints of degree m − 2 of F is 21 m(m − 1) − 12 ri (ri − 1) + e with e ≥ 0. These cut out divisors of degree P m(m − 2) − ri (ri − 1) = 2g − 2 + m

on F. Thus the adjoints of degree m − 2 cut P out out a complete linear series g−2+m+e g2g−2+m . A priori, e may be positive, as the ri (ri − 1) conditions imposed on a curve of degree m − 2 to be adjoint may be linearly independent. A posteriori, by the Riemann–Roch Theorem, e = 0. For the present it may only be asserted that e ≥ 0. However, the assertion that e = 0 can be formulated as a consequence of the Riemann–Roch Theorem. C OROLLARY 6.63 The maximum P number of linearly independent adjoints of degree m − 2 of F is 21 m(m − 1) − 21 ri (ri − 1).

Similarly, the maximum number P of linearly independent adjoints of degree m − 3 of F is 21 (m − 1)(m − 2) − 12 ri (ri − 1) + e with e ≥ 0. Here, again, e = 0 follows from the Riemann–Roch Theorem. To prove Theorem 6.61, a particular case is first examined. L EMMA 6.64 If a complete linear series gnr has order n = g + 1, then r ≥ 1.

Proof. Since m > 2, so 2g + m − 2 ≥ g + 1. The adjoints of degree m − 2 cut out g−2+m+e . Given a divisor C ∈ gnr , then it is first shown that there on F a series g2g−2+m exists D0 ∈ L with D0 ≻ C. By Theorem 6.40, the divisors H − C, with H ∈ L ′ and H ≻ C, is a gnr ′ with r′ = g − 2 + m + e − (g + 1) + e′ = m − 3 + e + e′ , n′ = 2g − 2 + m − (g + 1) = g − 3 + m,

with e′ ≥ 0. Since r′ ≥ 0, so F exists in L with F ≻ C. Consider now the divisors in L which contain F − C; they form a series of dimension, s = (g − 2 + m + e) − (2g − 2 + m − (g + 1)) + e′′ = 1 + e + e′′ ≥ 1.

This series contains C and has dimension at least one.

C OROLLARY 6.65 If a complete linear series gnr has n > g + 1, then r ≥ 1.

2

Proof. Let C +E ∈ gnr , with C, E effective divisors and deg C = g+1. By Lemma 6.64, dim |C| ≥ 1; hence there exists an effective divisor C ′ such that C ′ ≡ C but C 6= C ′ . So C ′ + E ≡ C + E but C + E 6= C ′ + E. Thus the dimension r ≥ 1. 2

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T HEOREM 6.66 (Noether’s Reduction Theorem) Given a place P centred at the simple point P of F, if there exists a special adjoint through the divisor C but not through C + P, then P is a fixed place of |C + P|. Proof. The original proof of Noether which is reproduced here does not work in positive characteristic when P is a terrible point. To deal with terrible points, the same argument as after Definition 3.30 does work; a quadratic transformation is applied to give a curve, the geometric transform of F, whose singular points are ordinary, for which the corresponding point Q of P is no longer terrible. Therefore P may be assumed not to be terrible. Then there is a line ℓ through the centre P of the branch corresponding to P that meets F in m distinct points, viewed as the centres of the branches corresponding to the places P = P1 , . . . , Pm . Then (Fm−3 ℓ) · F = D + C + E + P1 + · · · + Pm . The adjoint curves Fm−2 cut out on F the complete linear series |C + E + P1 + · · · + Pm |. As above, the adjoint curves Fm−2 through E + P2 + · · · + Pm form a linear subsystem Γ′ which cuts out on F the complete series |C + P|; that is, |C + P| = ||C + E + P1 + · · · + Pm | − |E + P2 + · · · + Pm || . The curves of Γ′ meet ℓ in m − 1 points, and therefore contain ℓ as a component. It follows that each curve of Γ′ passes through the centre of P. Since P is not in E + P1 + · · · + Pm , so P is a fixed place of |C + P|. 2 R EMARK 6.67 In the theorem, the case C ≻ P is allowed, since places are counted with multiplicity. Proof of Theorem 6.61. The proof works by induction on i. Suppose first that i = 0; in this case, the proof is by induction on r. The first case is r = 0, and it must be shown that n = g. From Lemma 6.64 and Corollary g−1+e 6.65 it follows that n ≤ g. As above, the canonical curves cut out a g2g−2 on F; therefore through any g − 1 places there is at least one canonical divisor. Since i = 0, so n ≥ g. Therefore n = g and the theorem is proved for i = r = 0. Suppose, still with i = 0, that r > 0. Given a place P centred at a simple point that is not a fixed place for gnr , Theorem 6.13 shows that the divisors of gnr , with P r−1 removed, form a gn−1 . Note that the index of speciality for the latter series is zero. In fact, if it were otherwise, there would exist a special adjoint through a divisor C r−1 of gn−1 , which cannot pass through P since i = 0. By Theorem 6.66, P would be a fixed place of gnr , a contradiction. By induction, r − 1 = n − 1 − g, whence r = n − g. Finally, suppose that i > 0, and choose a divisor C in gnr and a canonical curve ′ C through C. Let P be a simple point of F not lying on C ′ ; then the place Q with centre Q is fixed for |C + P|, by Theorem 6.66. Therefore this series |C + P| is a r gn+1 , and its index of speciality is one less than that of |C|. Hence r = n + 1 − g + (i − 1) = n − g + i.



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6.4 REFORMULATION OF THE RIEMANN–ROCH THEOREM Theorem 6.61 can be written in the following form: dim |C| = deg C − g + 1 + dim |W − C|;

(6.5)

here, C is an effective divisor and W a canonical divisor. This has been established in the case that |C| contains at least one effective divisor; that is, r ≥ 0. Now, it is shown that (6.5) is still valid when |C| contains no effective divisor; that is, r = −1. If dim |W − C| ≥ 0, then, with W − C for C, equation (6.5) takes the form dim |W − C| = deg(W − C) − g + 1 + dim |W − (W − C)|

(6.6)

= 2g − 2 − deg C − g + 1 + dim |C|,

since deg W = 2g − 2. But this implies (6.5); so considering C or W − C gives equivalent results. If dim |W − C| = −1, then W − C contains no effective divisors. Since |C| also contains no effective divisors, it must be shown that deg C = g − 1 or, equivalently, that deg(W − C) = g − 1. Suppose that deg C ≥ g − 1 and let C = A − B with both A and B effective divisors. Then dim |C + B| ≥ deg(C + B) − g

= deg C + deg B − g ≥ deg B.

So deg B conditions can be imposed on the divisors of |C + B|; therefore there exists A′ ∈ |C + B| such that A′ ≻ B. But then A′ − B ≡ C, and A′ − B is effective, contradicting that dim |C| = −1. Now, suppose that deg C < g − 1. Then deg(W − C) > g − 1, and the above argument holds, with W − C in place of C. Thus, (6.5) is valid for any divisor C on F. The Riemann–Roch Theorem can be reformulated in terms of functions on F. P D EFINITION 6.68 For any divisor C = nP P, effective or not, L(C) = {f ∈ Σ | ordP (f ) ≥ −nP for all P ∈ P(Σ)} = {f ∈ Σ | div(f ) + C ≻ 0} ∪ {0};

this is the Riemann–Roch space of C. L EMMA 6.69 The set L(C) is a vector space over K with the following properties: (i) if C ≻ D, then L(D) is a subspace of the vector space L(C); (ii) L(0) = K, and L(C) = {0} when deg C < 0; (iii) dim L(C) is finite, and dim L(C) ≤ deg C + 1 when deg C ≥ 0; (iv) if C ≡ D, then dim L(C) = dim L(D).

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Proof. (i) Let C = D + E with E ≻ 0. If f ∈ L(D), then div(f ) + D ≻ 0, whence div(f ) + D + E ≻ 0. Thus L(C) ⊃ L(D) and (i) follows. (ii) If f ∈ Σ is a function, then div(f ) = div(f )0 − div(f )∞ with div(f )∞ 6= 0. Therefore div(f ) 6≻ 0, whence L(0) = K. When deg C < 0, then there is no function f such that div(f ) + C ≻ 0. Hence L(C) = {0}. (iii) A more general assertion is proved. Let A, B be divisors of Σ. If B ≻ A, then dim L(B)/L(A) ≤ deg B − deg A. It is enough to prove it for B = A + P, as the general case follows from the special case by induction. It may be assumed that L(B) P 6= L(A). Choose an element ξ in L(B) that does not belong to L(A). If A = nQ Q, then ordP ξ = nP + 1 = m, and hence ordP ξ −1 = −m. Now, ordP ξ −1 η ≥ 0 for any element η ∈ L(B). Given a primitive representation τ of P, let τ (ξ −1 ) = ct−m + · · · , c 6= 0, τ (η) = dtm + · · · .

Let V1 (K) be one-dimensional vector space. Then the mapping ϕ : L(B) → V1 (K), η 7→ cd is K-linear, and its kernel is L(A). Thus, dim L(B)/L(A) ≤ 1. It follows that dim L(B)/L(A) = 1. (iv) Let D = C + div(g). The mapping ϕ : L(C) → L(D), f 7→ f g

is K-linear, and its kernel is the zero divisor 0. So ϕ is an isomorphism.

2

T HEOREM 6.70 For any divisor C on the curve F, dim L(C) = deg C − g + 1 + dim L(W − C).

(6.7)

Proof. Let rP = dim L(C) − 1 and suppose that f0 , f1 , . . . , fP r is a basis for L(C). Let AC = { ci fi | ci ∈ K}; then AC is a linear series, and ci fi ∈ L(C) if and only if the divisor div(f ) + C is effective; that is, div(f ) + C ∈ AC . Hence |C| is a linear series and has dimension r. Now, Theorem 6.5 gives the required result. 2 E XAMPLE 6.71 Consider F = X03 + X13 + X23 defined over K = F4 , with F4 = {0, 1, ω, ω ¯ = ω 2 = ω + 1}. The curve F = v(F ) is non-singular, and its points are identified with the places of its function field Σ = K(x, y) with x3 + y 3 + 1 = 0. There are nine points (1, x, y) = (x0 , x1 , x2 ) in PG(2, 4) which belong to F; namely, P0 = (0, 1, 1), P1 = (0, 1, ω), P2 = (0, 1, ω ¯ ),

P3 = (1, 0, 1), P4 = (1, 0, ω), P5 = (1, 0, ω ¯ ),

P6 = (1, 1, 0), P7 = (1, ω, 0), P8 = (1, ω ¯ , 0).

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Let f1 = 1/(x + y) = x0 /(x1 + x2 ). Then div f1 = P1 + P2 − 2P0 . Note that 1 x1 + x2 1 = = 2 x2 = u x20 , f1 x0 x1 + x1 x2 + x22 0

where u = 1/(x21 + x1 x2 + x2 ) has no pole at P0 , and x0 is a uniformising parameter; so f1 has a pole of order 2 at P0 . Similarly, f2 = x/(x + y) = x1 /(x1 + x2 ) has a pole of order 3 at P0 , since x1 + x2 1 1 = = x3 . f2 x1 x1 (x21 + x1 x2 + x22 ) 0 Here, div f2 = P3 + P4 + P5 − 3P0 . Let C = 3P0 ; then f1 and f2 are both in L(C). Since g = 1, deg C = 3, L(W − C) = {0}, so dim L(C) = 3. Therefore a basis for L(C) is {1, f1 , f2 }. 6.5 SOME CONSEQUENCES OF THE RIEMANN–ROCH THEOREM T HEOREM 6.72 (i) The canonical series is the only series on a curve of genus g that has order 2g − 2 and dimension g − 1. (ii) The canonical series has no fixed place. (iii) A divisor of a complete special gnr puts n − r conditions on a canonical divisor to contain it. g−1 Proof. (i) A g2g−2 with dimension r = g − 1 and r > n − g is special; so it is totally contained in the canonical series, and hence coincides with it, having the same dimension. (ii) If the canonical series had a fixed point P , the residual series would be a g−1 g2g−3 . The addition of another place Q to the divisors of this series would give a g−1 g2g−2 different from the canonical series. (iii) If i is the index of speciality of the gnr , then i = g − (n − r), whence i − 1 = g − 1 − (n − r). 2

E XAMPLE 6.73 Let C 6 = v(F ) be a sextic curve with one singular point P0 that is an ordinary quadruple point; for example, if char K 6= 2, let F = XY (X − Y )(X + Y )Z 2 + X 6 + Y 6 .

Then a special adjoint is a cubic curve with a triple point at P0 , and is therefore a set S of three lines through P0 . The canonical series therefore has order 6, since each line of S meets C 6 in two points apart from P0 , and dimension 3, since the pencil of lines through P0 has dimension 1. Since C 6 has genus g = 21 (6 − 1)(6 − 2) − 21 4(4 − 1) = 4,

this is consistent with the canonical series being a g63 . Now, the converse of Theorem 6.66 is established.

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T HEOREM 6.74 Given a linear series gnr on F, if there exists a place P such that the complete linear series gnr + P = {D + P | D ∈ gnr } has P as a fixed place, then the gnr is complete and special, and there exists a canonical divisor containing a divisor of gnr but not containing P.

Proof. If gnr + P has the fixed place P, since between the order and dimension of this series there is the inequality r ≥ (n + 1) − g, so r > n − g; hence the series gnr is special. Further, gnr is complete, since otherwise gnr + P would not be complete. Now, if each canonical divisor containing a divisor of gnr also contained P, then gnr + P would also be special, with the same index of speciality as gnr ; that is, a divisor of gnr and one of gnr + P impose the same number of conditions on a canonical divisor. This contradicts Theorem 6.72. 2 C OROLLARY 6.75 The complete series gnr + P has P as a fixed place if and only if the series gnr is special and there exists a canonical divisor containing a divisor of gnr but not containing P. In other words, for any effective divisor A and any place P, one and only one of the following equations holds: (i) dim |A + P| = dim |A|; (ii) i(|A + P|) = i(|A|). ′

D EFINITION 6.76 The sum of two linear series gnr , gnr ′ is ′

′

gnr + gnr ′ = {D + D′ | D ∈ gnr , D′ ∈ gnr ′ }. C OROLLARY 6.77 A complete series gnr is non-special if n > 2g − 2 or r > g − 1, whereas it is special if n < g. ′

T HEOREM 6.78 (Reciprocity Theorem of Brill and Noether). If gnr and gnr ′ are ′ residual with respect to the canonical series |W |, that is, |gnr + gnr ′ | = |W |, then n − 2r = n′ − 2r′ . ′

Proof. The speciality indices i and i′ of gnr and gnr ′ satisfy i = r′ +1 and i′ = r +1. Thus r = n − g + i = n − g + r′ + 1 and, similarly, r′ = n′ − g + r + 1; subtracting the two equations gives the result. 2 T HEOREM 6.79 (Clifford’s Theorem) If a linear series gnr is special, then n ≥ 2r.

Proof. It may be supposed that the series gnr is complete. If H is the residual of a divisor G of gnr with respect to a canonical divisor, then G imposes r conditions on the canonical divisors through H, while it imposes n − r conditions on an arbitrary canonical divisor containing it, by Theorem 6.72 (iii). As it cannot happen that independent conditions in the general case become dependent for a subseries, so n − r ≥ r, whence n ≥ 2r. Equality can only occur when the series is complete. An alternative view is as follows. Let Φ be the canonical linear system of F. If the gnr is complete, then it is cut out by a subsystem Φ′ of dimension r of Φ. More precisely, for a given divisor G of gnr and a canonical adjoint G through G, let H

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be the residual of G in the canonical divisor C cut out by G; that is, C = G + H. Then Φ′ consists of all special adjoints through H. Now, consider the subsystem Φ′′ of all the canonical adjoints through G. If i − 1 is the dimension of Φ′′ , then r = n − g + i by the Riemann–Roch Theorem. Also, Φ′ and Φ′′ have G in common but not a pencil of curves, as otherwise F would be a component of a curve in such a pencil. Hence the system containing both Φ′ and Φ′′ has dimension at most r + i − 1. Since dim C = g − 1, so r + i − 1 ≤ g − 1 = n − r + i − 1, whence n ≥ 2r. 2 E XAMPLE 6.80 A non-singular quartic curve C 4 has genus 3. The lines of the plane are the special adjoints and cut out the canonical series g42 . The lines through a point Q off C 4 cut out a special g41 . A complement to Theorems 6.32 and 6.79 is the following result. T HEOREM 6.81 (i) A necessary and sufficient condition for a curve to be rational is that it possesses at least one linear series gnn . (ii) The complete linear series of a rational curve are gnn for n ∈ N.

Proof. A series gnn is complete as its dimension is the largest possible for its order; see Theorem 6.32. Suppose the curve is rational; that is, it is the line at infinity ℓ∞ = v(X0 ) up to a birational transformation. The linear system of all plane curves of degree n cuts out on ℓ∞ a linear series of degree n. Since the n + 1 monomials X i Y n−i for i = 0, . . . , n are linearly independent over K, the linear series has dimension n, and hence it is a gnn . Since there is a curve of degree n cutting out any preassigned divisor of degree n, so gnn consists of all divisors of ℓ∞ of degree n. Thus, everything apart from the sufficiency condition is proved. Suppose now that the curve possesses a series gnn , with n > 0. By Theorem 6.32, n gn is complete. Choose a simple point P on the curve such that the corresponding place P is not a fixed place of gnn . The divisors C − P, with C ∈ gnn and C n−1 containing P, constitute a linear subseries gn−1 . By repeating this argument, the 1 existence of a g1 follows. Let g11 = {div(c0 + c1 x) + B | c0 , c1 ∈ K, (c0 , c1 ) 6= (0, 0)}.

By Theorem 7.36, g11 is not composed of an involution. Thus, [Σ : K(x)] = 1, whence Σ = K(x). 2

6.6 THE WEIERSTRASS GAP THEOREM In Section 5.4, it is shown that every function z ∈ Σ has a finite number of poles, giving rise to the effective divisor of poles div(z)∞ . The converse of this theorem is generally false, as not every effective divisor is the pole divisor of a function in Σ. This is explained in the following theorem. T HEOREM 6.82 An effective divisor A is the divisor of poles div(z)∞ of some function of Σ if and only if |A| has no fixed place.

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185

Proof. If A is the divisor of poles of z, then A ≡ div(z)∞ ; that is, |A| contains div(z)0 . Since A and div(z)0 do not share any place, so |A| has no fixed place. Conversely, let |A| have no fixed places. It is shown first that B ≡ A for some Pk divisor B such that A and B have no common place. Write A = i=1 nPi Pi . By assumption, for every j with 1 ≤ j ≤ k, there is a divisor Aj ≡ A such that Aj does not contain Pj . If Fj = v(ϕj (X0 , X1 , X2 )) cuts out Aj , then the Pk curves of the linear system v( i=1 λj ϕj ) cut out divisors in |A|. Such a divisor contains Pj if and only if the k-ple (λ1 , . . . , λk ) is a solution of a certain system of linear equations. Choosing coefficients λj that satisfy none of the equations in that system, a divisor B of the desired type is obtained. Now, the argument in the first part can be reversed. 2 As a corollary, the following theorem is obtained. T HEOREM 6.83 Every effective divisor is the divisor of poles of some function of Σ if and only if Σ is a rational function field. R EMARK 6.84 Theorem 6.83 remains true if the condition is only required for divisors consisting of one place. For the rest of this section, the genus g of Σ is assumed to be positive. T HEOREM 6.85 Let As = P1 + · · · + Ps be an effective, non-special divisor with not-necessarily-distinct places Pi , and let A0 = 0, An = P1 + · · · + Pn , for 1 ≤ n ≤ s. Then (i) there exist at least g values of n for which An is not the divisor of poles of any function in Σ; (ii) the indices of the Pi can be relabelled such that this occurs precisely for g values of n. Proof. Put in = i(|An |); in particular, i0 = g, is = 0. By Theorem 6.75, in = in−1 in = in−1 − 1

if dim |An−1 | < dim |An |, if dim |An−1 | = dim |An |.

This implies the existence of exactly g values of n for which the relationship dim |An−1 | = dim |An | holds, that is, for which Pn is a fixed place in |An |. From Theorem 6.82, there are k values of n, with k ≥ g, for which An is not the divisor of poles of a function in Σ. To show (ii), merely relabel the indices of the Pi as follows. Let n1 be the largest values of n for which |An | has a fixed place. Relabel the places of An1 so that Pn1 is such a fixed place. Proceeding similarly for the new labelling, let n2 be the largest value of n less than n1 for which |An | has a fixed place, and relabel the places of An2 so that Pn2 is fixed. By continuing this procedure, an arrangement of the places Pi is eventually obtained so that whenever An has a fixed place, Pn is fixed. From the preceding argument it then follows that k = g. 2 Now, the case P1 = P2 = · · · = Ps is considered.

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D EFINITION 6.86 For any place P, a non-negative integer n is a pole number of P if there is a function z such that div(z)∞ = nP; otherwise, n is a gap number of P. L EMMA 6.87 The set of pole numbers of P is a sub-semigroup H(P) of the additive semigroup of the non-negative integers. Proof. If div(z1 )∞ = n1 P and div(z2 )∞ = n2 P, then div(z1 z2 )∞ = (n1 + n2 )P. 2 D EFINITION 6.88 The set H(P) is the Weierstrass semigroup at P. The integer n is a pole number of P if and only if dim(|nP|) > dim |(n − 1)P|. By Theorem 6.85, this occurs if and only if i(nP) = i((n − 1)P). In particular, every n > 2g − 1 is a pole number. On the other hand, 1 is a gap number. Suppose, on the contrary, that 1 is a pole number of P. Then H(P) is the whole semigroup of non-negative integers showing that there are no gaps at all, contradicting Theorem 6.85. Therefore, for P1 = · · · = Pn , Theorem 6.85 reads as follows. T HEOREM 6.89 (Weierstrass Gap Theorem) For any place P, there exist exactly g gap numbers, n1 < n2 < · · · < ng , with n1 = 1 and ng ≤ 2g − 1. It is shown in Section 7.6, see Corollary 7.57, that all but finitely many places have the same sequence of gap numbers. These are the ordinary places of Σ. If the gap sequence of Σ at an ordinary point is (1, 2, . . . , g) then Σ has classical gap sequence at a general point. The non-ordinary places are the Weierstrass points, and the i-th gap of an ordinary point is greater than or equal to the i-th gap of any Weierstrass point. Also, if g ≥ 2, then Σ contains at least one Weierstrass point. Two illustrative examples are now given. E XAMPLE 6.90 Let F = v(f (X, Y )) be the irreducible plane curve in Example 5.59. Assume that h is odd. Then F has genus g = 12 (h − 1). It is now shown that the gap numbers of K(F) at ordinary places are 1, 2, . . . , g, and that F has exactly 2g + 2 Weierstrass points where the gap numbers are the odd integers less than or equal to 2g − 1. With same notation employed in Example 5.59,  −2Qi + 2P∞ for h odd, −1 div(x − ai ) = (2) (1) −2Qi + P∞ + P∞ for h even.

This shows that 2 is a pole number at Qi . Since g 6= 0, it follows that the semigroup H(Qi ) consists of all even numbers. Therefore the odd integers less than or equal to 2g − 1 are gap numbers. This holds true for P∞ . These 2g + 2 places are the only Weierstrass points of F. To show the latter assertion, it suffices to prove that 1, 2, . . . , g are gap numbers at any place P corresponding to a branch of F centred at a point P = (a, b) with

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b 6= 0. This occurs if and only if i(nP) = i((n − 1)P) − 1 for n = 1, . . . , g. To do this, the canonical series of K(F) is first determined. Let Q = (x, y) be a generic point of F. The condition p ∤ h = deg f (X, Y ) ensures that x is a separating variable. From the calculations carried out in Example 5.59, div(dx) = Q1 + · · · + Qh − 3P∞ , div y = Q1 + · · · + Qh − hP∞ .

Therefore (y −1 dx) = (h − 3)P∞ . This shows that the canonical series of K(F) is |(h − 3)P∞ |. Also, div (xi ) = i(R1 + R2 ) − 2iP∞ ,

where R1 and R2 are the places corresponding to the branches centred at the points of F lying at finite distance on the Y -axis. The case R1 = R2 is possible, and this occurs when f (0) = 0, that is, the Y -axis meets F at finite distance only in the origin. Hence, if i = 0, 1, . . . , (h − 3)/2, then div (xi ) + (h − 3)P∞ is a canonical divisor. Since 1, x, . . . , x(h−3)/2 are linearly independent over K, it follows in terms of g that canonical series of K(F) consists of all divisors div (c0 + c1 x1 + · · · + cg−1 xg−1 ) + (2g − 2)P∞ , as c = (c0 , . . . , cg−1 ) ranges over the points of PG(g − 1, K). For n = 1, . . . , g − 1, this implies that a canonical divisor W contains nP if and only if W = div (c0 + c1 x1 + · · · + cg−1 xg−1 ) + (2g − 2)P∞

such that the polynomial c0 + c1 X + · · · + cg−1 X g−1 has an n-fold root at a. Therefore i(nP) = i((n − 1)P) − 1, for n = 1, . . . , g. E XAMPLE 6.91 Take p = 3, and let H3 = v(Y 3 + Y − X 4 ), the Hermitian curve. Since H3 has genus 3, it is the canonical curve of its function field K(H3 ). Hence, the canonical divisors are cut out on H3 by lines. Let P be the place arising from the point P = (a, b) of H3 . Assume at first that both a, b are in the prime field F3 of K. There are 27 such points. If ℓ is the tangent to H3 at P , then I(P, ℓ ∩ H3 ) = 4. Thus i(P) = 2,

i(2P) = i(3P) = i(4P) = 1,

i(5P) = 0.

This shows that the gap numbers of K(H3 ) at P are 1, 2, 5. The same holds for the place P∞ centred at the infinite point of the Y -axis. Now, assume that either a or b does not belong to F3 . Then I(P, ℓ ∩ H3 ) = 3, and hence i(P) = 2,

i(2P) = i(3P) = 1,

i(4P) = 0.

Therefore the gap numbers of K(H3 ) at P are 1, 2, 4. To sum up, the gap sequence at ordinary places is (1, 2, 4), while there are 28 Weierstrass points with gap sequence (1, 2, 5). Let P be any place of Σ, and let m denote the smallest pole number of P. Choose an element x ∈ Σ such that div(x)∞ = mP . Every integer n in the Weierstrass

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semigroup H(P ) can be written in the form n = am + i where a and i are nonnegative integers such that a > 0 and i < m. Since there are finitely many gaps at P, all the values i = 0, 1, . . . , m − 1 arise. Let ai m + i be the least pole number congruent to i modulo m. Choose elements z0 , z1 , . . . , zm−1 ∈ Σ such that div(zi )∞ = (ai m + i)P. Note that z0 may be taken to be x. It is shown that every y ∈ Σ which has a pole at P but is regular elsewhere can be written in the form y = A(x) + B(x)z1 + · · · + C(x)zm−1 ,

(6.8)

with A(X), B(X), . . . , C(X) ∈ K[X]. Let n be the pole number arising from y; that is, div(y)∞ = nP . If n ≡ i (mod m), write n = am + i. Since n cannot be less then ai m + i, there is a constant c so that z ′ = y − cxa−ai zi is either zero or div(z ′ )∞ = n′ P with n′ < n. Proceeding then similarly with z ′ , an equation (6.8) is obtained. Applying this equation to the first m − 1 powers of y gives the following: B1 (x)z1 + · · · + C1 (x)zm−1 B2 (x)z1 + · · · + C2 (x)zm−1 .. .

= = .. .

y − A1 (x), y 2 − A2 (x), .. .

Bm−1 (x)z1 + · · · + Cm−1 (x)zm−1

=

y m−1 − Am−1 (x).

(6.9)

Note that (z1 , . . . , zm−1 ) can be viewed as the solution of a system of linear equations over K(x). If these equations are linearly independent, that is, the (Bi (x), . . . , Ci (x)) are m − 1 linear independent vectors over the field K(x), then Cramer’s rule gives the result zi = Qi,1 (x)(y − A1 (x)) + Qi,2 (x)(y 2 − A2 (x)) + · · · (6.10) +Qi,m−1 (x)(y m−1 − Am−1 (x)),

with Qi,1 (X), Qi,2 (X), . . . , Qi,m−1 (X) ∈ K(X). Otherwise,

P (x) + P1 (x)y + · · · + Pm−1 (x)y m−1 = 0,

(6.11)

with P (X), P1 (X), . . . , Pm−1 (X) ∈ K[X]. Since

ordP (Pk (x)y k ) = m deg Pk (X) + kn

from (6.11), so m deg Pk (X) + kn = m deg(Pk′ (X)) + k ′ n, for some k, k ′ with 1 ≤ k < k ′ ≤ m − 1. This implies that

deg Pk (X) − deg Pk′ (X) n . = m k − k′ Now k and k ′ are both less than m. Therefore m and n have a common divisor. Now take n prime to m. Then the equations in (6.9) must be linearly independent, and (6.10) holds. This together with (6.8) applied to y m gives an irreducible equation of the form U0 (x)y m + U1 (x)y m−1 + · · · + Um−1 (x)y + Um (x) = 0,

(6.12)

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with Ui (X) ∈ K[X], and U0 (X) 6= 0. More precisely, U0 (X) is a non-zero constant polynomial as both x and y have a unique pole P. Further, if i < j, then m deg Ui (X) + (m − i)n = ordP (Ui (x)y m−i )

= ordP (Uj (x)y m−j ) = m deg Uj (X) + (m − j)n

only for j ≡ i (mod m). As this congruence is only valid for i = 0, j = m, it follows that deg Um (X) = n and that deg Ui (X) < in/m,

for i = 1, . . . , m − 1.

(6.13)

Since gcd(m, n) = 1 implies that Σ = K(x, y), the following result is obtained. T HEOREM 6.92 (Weierstrass Normal Form) Let F an irreducible curve of genus g ≥ 1. For a place P of K(F), let m be the first non-gap at P and let n be the least non-gap which is prime to m. Then (i) F = v(f (X, Y )) with

f (X, Y ) = Y m + U1 (X)Y m−1 + · · · + Um−1 (X)Y + Um (X), (6.14)

where Ui (X) ∈ K[X], deg Um (X) = n and (6.13) holds;

(ii) P = (x, y) is a generic point of F = v(f (X, Y )) for div(x)∞ = mP and div(y)∞ = nP; (iii) the branch of F associated to P is the unique branch of F with centre at Y∞ . A consequence is the following result. P ROPOSITION 6.93 Let F = v(f (X, Y )) be given in Weierstrass normal form (6.14). Then the genus g has the following properties: (i) g ≤ 21 (n − 1)(m − 1); (ii) if no point of F other than Y∞ is singular, then g = 21 (n − 1)(m − 1). Proof. (i) The idea is to apply the Riemann–Roch Theorem 6.61 to the complete linear series |(mn)P|. For each ordered pair (i, j) of non-negative integers i, j for which im + jn < mn, the divisor div(xi y j ) + (mn)P belongs to |(mn)P|. This also holds for div(xn ) + (mn)P. As in the discussion on (6.9), the hypothesis gcd(m, n) = 1 implies that 1, y, . . . , y m−1 are linearly independent over K(x). Therefore xn and the functions xi y j form a linearly independent set in the vector space L(P). So P dim |(mn)P| ≥ n + m − 1 + m−1 j=1 ⌊n(m − j)/m⌋ = n + m − 1 + 21 (m − 1)(n − 1). 1 2 nm

1 2

(6.15)

= deg(mn)P0 . This and Clifford’s Theorem In particular, dim |(mn)P| > 6.79 imply that |(mn)P| is not special. Then, from (6.15) and Theorem 6.61, 2 g ≤ 21 (n − 1)(m − 1).

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(ii) Both m and n cannot be divisible by p. Suppose first that m 6≡ 0 (mod p). Then x is a separable variable of K(F) and g(X, Y ) = ∂f /∂Y = mY m−1 + · · · + (m − i)Ui (X)Y m−i + · · · + Um−1 (X).

A primitive representation of the branch of F associated to P is x(t) = t−m ,

y(t) = t−n + · · · .

From (6.13), ordt (m − 1)y(t)m−1 = −(m − 1)n

< − ⌊in/m⌋ m − (m − i − 1)n ≤ ordt (m − i)Ui (x(t))y(t)m−i−1 .

Hence ordt g(x(t), y(t)) = −n(m − 1). From this and Corollary 5.35, P Q ordQ g(x, y) = n(m − 1),

where the summation is over all places of K(F) other than P. On the other hand, since the branch of F associated to such a place Q is centred at a non-singular point of F, from Theorem 5.56, ordQ g(x, y) = ordQ dx.

Also, ordP dx = −(m + 1).

Now, by Definition 5.55, 2g − 2 = n(m − 1) − (m + 1), whence (ii) follows. If n 6≡ 0 (mod p), a similar argument can be used, this time with g(X, Y ) = ∂f /∂X = nX n−1 + . . .

and dy.

6.7 THE STRUCTURE OF THE DIVISOR CLASS GROUP In Section 6.1, basic facts on divisors of a function field Σ of transcendence degree 1 are discussed. Now, a fundamental result on the divisor class group Pic(Σ) is stated. As the group Pic(Σ) is written additively, that is, [A + B] = [A] + [B] for [A], [B] ∈ Pic(Σ), the meaning of m · [A] is just [mA], for any positive integer m and divisor A of Σ. So, the set {[A] | m · [A] = 0, [A] ∈ Pic(Σ)} is a subgroup Pic(Σ, m) of Pic(Σ). Therefore Pic0 (Σ, m) = Pic(Σ, m) ∩ Pic0 (Σ)

is a subgroup of Pic0 (Σ), which plays an important role in the study of the structure of the divisor class group, as its order only depends on g. T HEOREM 6.94 For any positive integer m not divisible by p, ∼ (Zm )2g , Pic0 (Σ, m) = where g is the genus of Σ and (Zm )2g is the direct product of 2g copies of the additive group of integers modulo m.

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Theorem 6.94 is a special case of a more general result on abelian varieties. The idea is to prove the theorem first for K = C and then use reduction theory to extend it from C to any K. No more detail is given here. A proof of Theorem 6.94, valid in the case that K = Fq , is given in Section 9.7. The following example illustrates how to determine Pic0 (Σ, 2) in a particular case. E XAMPLE 6.95 Let F and Σ be as in Example 5.59, and suppose that h is odd. With the notation of Example 6.62, let D1 be the set of all divisors D for which ǫ1 = 0. Then deg D < h − 1. For two distinct divisors D1 and D2 in D1 , write E1 − E2 = D1 − D2 with E1 and E2 effective divisors. Note that both E1 and E2 are in D1 . Thus, deg E1 = deg E2 < 12 (h − 1).

Then dim |E1 | = 0 by (6.4), and hence D1 6≡ D2 . Now, for any D ∈ D1 , let F = mP1 − D with m = deg D; then [F ] 6= [0]. Since 2P1 ≡ Pi , as shown in Example 6.62, so 2mP1 ≡ 2D; that is, [2F ] = [0]. Since divisors D in D1 can be chosen in 2h−1 different ways, the total number of inequivalent divisors F amounts to 2h−1 . Since g = 21 (h − 1), Theorem 6.94 shows that such divisors F constitute a full representative system of cosets in the divisor class group Pic(Σ, 2). The hypothesis p ∤ m cannot be omitted in Theorem 6.94, even though the group Pic0 (Σ, m) is finite. In particular, Pic0 (Σ, p) ∼ = (Zp )γ for a non-negative integer γ, the p-rank of Σ. This important particular case appears in Theorem 11.62. Here, the main result is stated, which can be viewed as describing the essence of Theorem 6.94. T HEOREM 6.96 The p-rank γ is less than or equal to the genus g. A proof of Theorem 6.96 valid in the case that K = Fq is given in Section 9.7. D EFINITION 6.97 A plane non-singular cubic curve C is supersingular if the prank of its function field K(C) is equal to 0. An alternative definition of the p-rank is the Hasse–Witt invariant and relies on the Hasse–Witt matrix of Σ. For this purpose, assume that p 6= 0 and choose a separable variable ζ of Σ. From Theorem 5.37, every differential ω of Σ can be written as follows: ω = (up0 + up1 ζ + · · · + up−1 ζ p−1 )dζ, with u0 , u1 , . . . , up−1 ∈ Σ. The Cartier operator C is defined by the relation, C(ω) = up−1 dζ. Some fundamental properties of the Cartier operator are the following: (i) C(ω) is independent of the choice of ζ; (ii) C(y1p ω1 + y2p ω2 ) = y1 C(ω1 ) + y2 C(ω2 ) for every y1 , y2 ∈ Σ; (iii) if ω is holomorphic, that is, div ω is effective, then C(ω) is also holomorphic;

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(iv) C(ω) is independent of the choice of ζ. Choose x0 , x1 , . . . , xg−1 ∈ Σ such that the canonical series |dζ| consists of all divisors Pg−1 Ac = div( i=0 ci xi ) + div (dζ).

Let ωi = xi dζ. Then the representation matrix (hij ) over K of C is the Hasse–Witt matrix, or Cartier–Manin matrix, with respect to ζ, x0 , . . . , xg−1 such that (C(ω0 ), C(ω1 ), . . . , C(ωg−1 ))T = (hij ) (ω0 , ω1 , . . . , ωg−1 )T . The rank of the matrix g−1

M = (hij )(hpij ) · · · (hij p

)

is the Hasse–Witt invariant and is independent of ζ, x0 , . . . , xg−1 . A deep result which implies Theorem 6.96 is stated in the theorem below and is followed by two illustrative examples. T HEOREM 6.98 The p-rank is equal to the Hasse–Witt invariant. E XAMPLE 6.99 For p = 2 and b ∈ K, let F = v(Y 4 + b2 Y 2 + Y + X 5 ) be the non-singular plane curve belonging to the family of curves investigated in Section 12.1. Then F has genus g = 6 and its function field Σ = K(x, y), with y 4 + b2 y 2 + y = x5 , contains a separable variable x. From Lemma 12.1 (iii), div (dx) = 10P∞ , where P∞ is the unique infinite point of F. The canonical linear series of Σ consists of all divisors Ac = div (c0 + c1 x + c2 x2 + c3 y + c4 y 2 + c5 xy) + 10P∞ , with c = (c0 , . . . , c5 ) ranging over the points of PG(5, K). Let ω0 = dx, ω1 = xdx, ω2 = x2 dx, ω3 = ydx, ω4 = y 2 dx, ω5 = xydx. Then (C(ω0 ), C(ω1 ), C(ω2 ), C(ω3 ), C(ω4 ), C(ω5 )) = (0, 1, 0, x2 , 0, y 2 + by)dx, whence the Hasse–Witt matrix (hij ) is equal to  0 0 0 0 0  1 0 0 0 0   0 0 0 0 0   0 0 1 0 0   0 0 0 0 0 0 0 0 b 1 r−1

0 0 0 0 0 0



   .   

2 Further, the matrix, (hij )(h2ij ) · · · (hij ), has rank 3, 1, or 0 according as r = 1, 2, or r > 2. This shows that the Hasse–Witt invariant is equal to zero.

E XAMPLE 6.100 For p ∤ n, let Σ = K(Dn ) = K(x, y), with xn + y n + a = 0, be the function field of the Fermat curve Dn = v(X n + Y n + a), which has genus

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g = 21 (n − 1)(n − 2), where n 6= 3, and a ∈ K\{0}. For any pair (k, i) with integers 0 ≤ k ≤ m − 3 and 0 ≤ i ≤ k, let xki = xi y k−(i+m)+1 .

Then the canonical series |dx| consists of all divisors P Ac = div( k,i cki xki ) + div (dx).

Let ωki = xki dx. To calculate C(ωki ), two number-theoretic results are required.

L EMMA 6.101 For any pair (k, i) with integers satisfying 0 ≤ k ≤ m − 3 and 0 ≤ i ≤ k, there exists exactly one pair (u, v) of integers such that

pu + nv = (p − 1)(n − 1) + k − i, 0 ≤ u ≤ n − 2 and 0 ≤ v ≤ p − 1. (6.16)

For these pairs (k, i) and (u, v), there exists exactly one pair (j, s) of integers such that nj + i = p(s + 1) − 1, 0 ≤ j ≤ p − 1 and 0 ≤ s ≤ n − 2.

(6.17)

Further, if j ≤ v then u + s ≤ n − 3 and if j > v then u + s ≥ n − 1. From this, for any such pair (k, i), let (u, v) be the solution of (6.16) and (j, s) the solution of (6.17). Then (
v−j (−1)j vj (a1/p ) ωu+s,s if j ≤ v, C(ωk,i ) = 0 if j > v. Therefore the Hasse–Witt matrix (hij ) can be calculated. This gives the following results. P ROPOSITION 6.102

(i) The Hasse–Witt matrix (hij ) of Σ has

(a) maximum rank g if and only if p ≡ 1 (mod n);

(b) minimum rank 0 if and only if p ≡ −1 (mod n). (ii) If p = 2 and n > 4, then 1 ≤ rank (hij ) ≤ g − 1. Another equivalent definition of the p-rank of Σ follows from Theorem 11.64. In the rest of this section, the special case when Σ = K(F) for a non-singular plane cubic curve F is worked out. The places of Σ = K(F) and points of F are identified, and P0 denotes a given point of F. P ROPOSITION 6.103 Every zero divisor class of K(F) contains a unique divisor P − P0 with P ∈ F. Proof. If P − P0 ≡ Q − P0 with P, Q ∈ F, then P ≡ Q. Since K(F) has genus 1, from Theorem 6.81 (i) P = Q, and the uniqueness is established. To show the existence, first the following assertion is shown. For any effective divisor D, there exists P ∈ F and m ∈ N0 such that D ≡ P + mP0 .

(6.18)

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If deg D = 1, then (6.18) holds with m = 0. If deg D > 1, then there exists an effective divisor D′ together with a point R ∈ F such that D = D′ + R with deg D = deg D′ + 1. By induction, (6.18) may be assumed true for D′ . Then D′ ≡ P + m′ P0 and hence D′ ≡ R + P + m′ P0 . If there is a point Q ∈ F for which R + P ≡ Q + P0 ,

(6.19)

then (6.18) follows. If R 6= P , the line ℓ through R and P is either a tangent, or meets F in a third point. In both cases, there exists S ∈ F such that the intersection divisor ℓ · F is equal to P + R + S. If P0 6= S, let r be the line through P0 and S. If Q ∈ F is the point such that r · F = Q + P0 + S, then R + P + S ≡ Q + P0 + S, whence (6.18) follows. This argument still works for R = P and S = P0 , when the lines ℓ and r are the tangents to F at R and at S. In the remaining case, deg D = 0, that is, D = D1 − D2 with two effective divisors of the same degree m. From (6.18) applied to D1 and to D2 , D1 ≡ P1 + mP0 ,

D2 ≡ P2 + mP0 .

Therefore D ≡ P1 − P2 , and the assertion follows when P2 = P0 . For the case P0 6= P2 , the above geometric argument may be used. Let s be the line through P0 and P1 , or the tangent at P0 when P0 = P1 . Then there is a point U ∈ F such that s · F = U + P0 + P1 . If u is the line through U and P2 , which is the tangent at U when U = P2 , there exists a point P ∈ F such that u · F = U + P + P2 . Then U + P + P2 ≡ U + P0 + P1 whence P1 − P2 ≡ P − P0 . Thus, D ≡ P − P0 . 2 The group Pic0 (K(F)) has a nice geometric representation. To show this, the following result is needed. T HEOREM 6.104 The points of a non-singular plane cubic curve F = v(F ) can be equipped with an operation to form an abelian group. Proof. For any point T ∈ F, let ℓT denote the tangent at T to F. For P, Q ∈ F, let ℓ be the line through P and Q; take ℓ = ℓP when P = Q. Let A = φ(P, Q) be the point of F such that ℓ · F = P + Q + A. Now, define P ⊕ Q = φ(P0 , A); see Figure 6.1. Then, under this operation, F is an abelian group GF with identity P0 . First, the operation is well defined and abelian. Next, P0 is the identity, since if for a point P the point P ′ is defined by φ(P0 , P ) = P ′ , then φ(P0 , P ′ ) = P ; that is, P ⊕ P0 = P . To show that P has an inverse, let P0′ be point where ℓP0 meets F again; that is, φ(P0 , P0 ) = P0′ . Then φ(P0′ , P ) = −P ; see Figure 6.2. This incidentally means that, when P0′ = P0 , then −P = P ′ .

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P0 r

rP ⊕ Q rA Qr

Pr

Figure 6.1 Abelian group law on an elliptic curve

P0 r

Pr

−P r

′

r P0

Figure 6.2 Inverse of an element

It remains to show that the associative law holds. For P, Q, R ∈ F, define A, B, C, D, P ⊕ Q, Q ⊕ R as follows: φ(P, Q) = A, φ(A, P0 ) = P ⊕ Q, φ(P ⊕ Q, R) = B, φ(Q, R) = C, φ(C, P0 ) = Q ⊕ R, φ(P, Q ⊕ R) = D.

The vertical and horizontal lines of Figure 6.3 make up two cubic curves with eight common points on F. By the Theorem of the Nine Associated Points, see Exercise 4 in Section 9.9, they have a ninth point in common, namely B = D; hence 2 (P ⊕ Q) ⊕ R = φ(P0 , B) = φ(P0 , D) = P ⊕ (Q ⊕ R).

C OROLLARY 6.105 (i) P, Q, R are collinear if and only if P ⊕ Q ⊕ R = P0′ , where ℓP0 .F = 2P0 + P0′ .

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rP0

rP ⊕ Q

rA

rD

rP

rQ ⊕ R r B rC

rR

rQ

Figure 6.3 The associative law

(ii) If P0 is an inflexion, then P, Q, R are collinear if and only if P ⊕Q⊕R = P0 . C OROLLARY 6.106 group.

(i) Any inflexion in F other than P0 has order 3 in the

(ii) If P0 is an inflexion, then the points of order 2 are the points of contact of the tangents through P0 other than the inflexional tangent. For P ⊕ P ⊕ . . . ⊕ P (m times), write [m]P . T HEOREM 6.107 If GF be the group in Theorem 6.104, then GF ∼ = Pic0 (K(F)). Proof. By Proposition 6.103, every zero class is uniquely represented by [P − P0 ] with P ∈ F. In particular, there is a one-to-one correspondence between points of F and zero divisor classes of K(F)). To show that the map GF → Pic0 (K(F)) which sends P ∈ F to the zero divisor class [P − P0 ] is an isomorphism, it must be shown that P ⊕ Q = R implies that [P − P0 ] + [Q − P0 ] = [R − P0 ]. Let ℓ be the line through P and Q (the tangent at P when P = Q). There exists a point S ∈ F such that ℓ · F = P + Q + S. Let r be the line through S and P0 (the tangent at S when S = P0 ). Then R = P ⊕ Q is the point for which r · F = P0 + R + S. Therefore P − P0 + Q − Q0 = P + Q + S − S − 2P0 ≡ S + R + P0 − S − 2P0 = R − P0 , showing that [P − P0 ] + [Q − P0 ] = [R − P0 ].

2

6.8 EXERCISES 1. (Bertini’s Theorem) Assume that K has characteristic zero. Prove that, if any curve in a pencil has a multiple point, this must be a base point. Show that Bertini’s Theorem is not necessarily true in positive characteristic by considering the counter-example of the pencil consisting of all curves Fλ = v(X p + Y 2 − 2λY ).

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2. For any two effective divisors A, B, dim |A| + dim |B| ≤ dim |A + B|. 3. Let P be a place of Σ. Show that the following conditions are equivalent: (a) P is a Weierstrass point;

(b) gP is a special divisor;

(c) mP is a special divisor for some m ≥ g. 4. Show that the curve in Example 6.91 is the only irreducible plane curve of genus 3 with non-classical gap sequence. 5. Let p = 3. For function fields of genus 3 show that ordinary places have the classical gap sequence (1, 2, 3, 4). 6. For q odd, let F = v(X (q+1)/2 + Y (q+1)/2 + 1) be the Fermat curve of degree 12 (q + 1). Show that the Weierstrass semigroup of F at each inflexion point of F is h 21 (q − 1), 12 (q + 1)i. 7. For q ≡ 3 (mod 4), let F = v(Y q + Y − X (q+1)/4 ). Show that the Weierstrass semigroup of F at the place P∞ arising from the unique branch centred at Y∞ is hq, 41 (q + 1)i. 8. Let F be an irreducible curve of genus g ≥ 0. For a place P of K(F), let m be the first non-gap at P and n another non-gap at P. Choose x, y ∈ Σ such that div(x)∞ = mP and div(y)∞ = nP. Show that there exists an irreducible polynomial g(X, Y ) ∈ K[X, Y ] with the following properties: (a) g(x, y) = 0; (b) g(X, Y ) = Y ℓ + U1 (X)Y ℓ−1 + . . . + Uℓ−1 (X)Y + Uℓ (X), where Ui (X) ∈ K[X]; (c) if n/m = n′ /m′ with gcd(n′ , m′ ) = 1, then m′ | ℓ | m;

(d) deg Uℓ (X) = (nℓ)/m;

(e) deg Ui (X) ≤ ⌊(ni)/m⌋ for 1 ≤ i ≤ ℓ − 1;

(f) the subfield K(x, y) of K(F) is a separable extension of K(x);

(g) if m does not divide n, then g > 0; (h) if gcd(m, n) = 1, then K(F) = K(x, y). 9. Let H3 be the Hermitian curve, as in Example 6.91. After showing the properties below, find all the 22g = 64 divisor classes that represent an involution in Pic0 (K(H3 )), where K = F9 . This gives an explicit presentation of Pic0 (K(H3 ), 2) as in Example 6.95. (a) The set H3 (F9 ) of points of H3 lying in PG(2, 9) has size 28.

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(b) For any three non-collinear points P0 , P1 , P2 ∈ H3 (F9 ), there is a unique irreducible conic C2 such that the intersection divisor C2 · H3 has the following property: C2 · H3 ≻ 2P0 + 2P1 + 2P2 .

(6.20)

(c) If (6.20) holds, then C2 is a 4-tangent irreducible conic to H3 ; that is, C2 · H3 = 2P0 + 2P1 + 2P2 + 2P3 with a fourth point P4 ∈ H3 (F9 ) distinct from P1 , P2 , P3 . (d) For a fixed point P0 ∈ H3 (F9 ), the total number of such 4-tangent irreducible conics to H3 is 36. (e) For each of these thirty-six 4-tangent irreducible conics C2 to H3 , the divisor class [P1 + P2 + P3 − 3P0 ] is an involution in Pic0 (K(H3 )) such that no two of them lie in the same divisor class. (f) For a fixed point P0 ∈ H3 (F9 ), each of the twenty-seven divisor classes [2P − 2P0 ] with P ∈ H3 (F9 ) is an involution in Pic0 (K(H3 )).

(g) Divisor classes arising from distinct points P ∈ H3 (F9 ) are distinct.

(h) No two of the 63 = 36+27 divisor classes in Pic0 (K(H3 ), 2) coincide.

6.9 NOTES The idea of a linear series arose in classical accounts of the subject, such as [122], as a set of effective divisors cut out on a plane algebraic curve by a linear system of plane algebraic curves. Later, it assumed a more general form, so that linear equivalence of non-effective divisors was also considered, as in Section 6.1 based on Seidenberg’s book [400]. It should be noted however that other authors, notably Lefschetz [301] and Fulton [135], only allowed effective divisors in a linear series in order to ensure the equivalence between linear series and Riemann–Roch spaces. This has now become prevalent in the literature. In the present book, the name virtual linear series is adopted for what Seiedenberg called linear series, while linear series is reserved for linear series of effective divisors in Seidenberg’s terminology. Weierstrass points in positive characteristic were introduced by Schmidt [387]; see also Laksov [288] and Neeman [346]. The proof of Theorem 6.92 is adapted from Baker’s proof in [25, Chapter 5]; see also [208], [260], [261] and [502]. For a different proof of Proposition 6.93 (i), see [252]. For direct proofs of Theorem 6.94, see [338] and [130]. For a survey on p-ranks of a curves, see [58]. Fundamental papers on the Hasse–Witt matrix and invariant are Hasse and Witt [199], Manin [321], and St¨ohr-Voloch [433]. Examples 6.99 and 6.100 come from [431] and [275]; see also [329], [276], [351], [177]. For Example 6.91 and Exercises 4, 5, see [277]. Exercise 8 can be viewed as a generalisation of Theorem 6.92; see [383].

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Chapter Seven Algebraic curves in higher-dimensional spaces Algebraic curves in higher-dimensional spaces are usually introduced in the more general context of arbitrary algebraic varieties. However, an equivalent ad hoc definition which fits in with the plan of this book is also possible. Essentially, an irreducible algebraic curve of PG(n, K) is a set of points which can be put in a birational correspondence with the points of an irreducible algebraic plane curve. The idea of deriving the concept of an algebraic curve in a higher-dimensional space directly from that of an algebraic plane curve has at least two advantages. On the one hand, it allows the development of a theory of algebraic curves depending only on the basic concepts that have been investigated so far. On the other hand, it makes it possible to go directly to important results on linear series and rational transformations.

7.1 BASIC DEFINITIONS AND PROPERTIES A point in the n-dimensional projective space PG(n, K) is a homogeneous (n+1)tuple (a0 , a1 , . . . , an ) with ai ∈ K. As before, define a branch representation to be a point in PG(n, K(t)) but not in PG(n, K). All the basic definitions from Chapter 5 extend to projective spaces. For example, if x0 (t) = a0 + b0 ti0 + · · · , .. .. . .

b0 6= 0,

xn (t) = an + bn tin + · · · ,

bn 6= 0,

is a special representation of a branch, then its centre is the point A = (a0 , . . . , an ), and its order is min{i0 , . . . , in }. Let Q = (x0 , . . . , xn ) be a point in an n-dimensional projective space whose coordinates xk belong to an extension field of K. Then, xi 6= 0 for some 0 ≤ i ≤ n. The field K(x0 /xi , . . . , xn /xi ) is the field of the point Q = (x0 , . . . , xn ) and denoted by K(x0 , . . . , xn ) or K(Q). Here, K(x0 /xj , . . . , xn /xj ) = K(x0 /xi , . . . , xn /xi ) provided that also xj 6= 0. In fact, xk /xj = (xk /xi )(xi /xj ) holds for every k = 0, . . . , n. Hence K(Q) is independent of the choice of the non-zero coordinate xi of P . As before, Σ = K(x, y) stands for a finite extension of K of transcendence degree 1. So, x is transcendental over K and y is related to x by an irreducible

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polynomial equation over K: F (x, y) = 0 with F (X, Y ) ∈ K[X, Y ]. As usual, consider Σ as the function field K(F) of an irreducible plane curve F = v(F ) arising from its generic point P = (x, y). With the terminology of Chapter 5, (F; (x, y)) is a birational model of Σ. From now on, assume that K(F) = K(Q). In particular, x0 , . . . , xn ∈ K(F) and, if xi 6= 0, then some xj /xi 6∈ K. An immediate consequence of the hypothesis is that some coordinate xj /xi must be a separable variable in K(F), otherwise both x and y would be inseparable which is impossible; see Exercise 1 in Chapter 5. Let γ be a branch of F and P the corresponding place of K(F). If τ : K(F) → K((t)) is a primitive representation of P, then the corresponding branch in PG(n, K) with respect to the point Q = (x0 , . . . , xn ) is defined, namely the branch with primitive representation (τ (x0 ), . . . , τ (xn )). The centres of all such branches form a point set in PG(n, K) which gives the idea of a curve. The aim here is to develop the theory of space curves based on this idea. D EFINITION 7.1 An irreducible algebraic curve Γ in PG(n, K) consists of all points which are centres of branches corresponding to the places of K(F) with respect to a point Q = (x0 , . . . , xn ). These branches are the branches of Γ, and (Γ, (x0 , . . . , xn )) = (Γ, Q) is a model of K(F) in PG(n, K). The xj are the coordinate functions of Γ. An advantage of Definition 7.1 is that it is a straightforward extension of that of an irreducible plane curve. Several results, especially the following theorems, are higher-dimensional versions of the analogous results for plane curves. T HEOREM 7.2 Each point of an irreducible curve Γ is the centre of finitely many branches of Γ. Proof. Let A = (a0 , . . . , an ) be a point of Γ, and γ a branch centred at A. Choose a special representation (x0 (t), . . . , xn (t)) of γ. Suppose that ai 6= 0. Then xi 6= 0. Set yk = xk /xi for k = 0, . . . , n. One of them, say yj , is not a constant. Then yj − aj 6∈ K. Hence P is a zero of yj − aj . By Theorem 5.33, any function in K(F) has only finitely many zeros. Hence there are only finitely many branches centred at A. 2 As a finite number of points can be the centres for only a finite number of branches, Theorem 7.2 has the following corollary. T HEOREM 7.3 An irreducible curve Γ has infinitely many points. D EFINITION 7.4 (i) A hypersurface Γ in PG(n, K) is the set of all points satisfying a polynomial equation F (X0 , . . . , Xn ) = 0, where F ∈ K[X0 , . . . , Xn ] is a homogeneous non-constant polynomial; write Γ = v(F ).

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(ii) The degree of the hypersurface Γ is deg F . (iii) A hyperplane H is a hypersurface of degree 1; that is, H = v(a0 X0 + . . . + an Xn ). Most of the material in Chapter 6.2 extends from curves to hypersurfaces. In particular, two hypersurfaces Φ1 = v(F1 ) and Φ2 = v(F2 ) have the same set of points if and only if the polynomials F1 and F2 have the same irreducible factors. If γ is a branch with centre A and a primitive representation (x0 (t), . . . , xn (t)) in special coordinates, and ∆ is a hypersurface v(G), then the intersection multiplicity is I(A, ∆ ∩ γ) = ordt G(x0 (t), . . . , xn (t)). The branch γ is linear if I(A, H ∩ γ) = 1 for some hyperplane H. T HEOREM 7.5 Let Γ be the irreducible curve of PG(n, K) given by the point Q = (x0 , . . . , xn ). (i) The hypersurface ∆ = v(G) contains Γ if and only if G(x0 , . . . , xn ) = 0. (ii) If ∆ does not contain Γ, then they have only finitely many common points. Proof. (i) If G(x0 , . . . , xn ) = 0, then it is immediate that ∆ contains Γ. For the converse, suppose G(x0 , . . . , xn ) 6= 0 and show that Γ is not contained in ∆. By relabelling the indices, assume that x0 6= 0 and put yi = xi /x0 . Then G(1, y1 , . . . , yn ) 6= 0, and hence G(1, y1 , . . . , yn ) has a finite number of zeros. If A = (1, a1 , . . . , an ) is a point of Γ and (1, y1 (t) = a1 + · · · , . . . , yn (t) = an + · · · ) is a primitive representation of a branch γ of Γ centred at A, then the corresponding place P is a zero of G(1, y1 , . . . , yn ). Hence ∆ can contain only a finite number of affine points of Γ. On the other hand, Γ has infinitely many affine points, as in the proof of Theorem 7.3. Hence Γ cannot be contained in ∆. (ii) It is only necessary to show that the hyperplane v(X0 ) meets Γ in a finite number of points. But this follows since x0 has only a finite number of zeros. 2 T HEOREM 7.6 Let γ be a branch of Γ centred at A, and assume that the hypersurface ∆ does not contain Γ. Then I(A, ∆ ∩ γ) is finite; that is, G(x0 (t), . . . , xn (t)) 6= 0, where (x0 (t), . . . , xn (t)) is a primitive representation of γ and ∆ = v(G). Proof. If σ is a primitive representation of the place associated to γ, then σ maps K(Q) isomorphically onto K(x0 (t), . . . , xn (t)) with xi mapped to xi (t). Thus the xi (t) satisfy the same polynomial relations over K as do the xi . It follows that G(x0 , . . . , xn ) 6= 0 implies G(x0 (t), . . . , xn (t)) 6= 0. 2

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This theorem, together with Theorem 7.2, provides the extension of the concept of intersection divisor: if ∆ does not contain Γ, then P ∆ · Γ = γ I(A, ∆ ∩ γ) P, where P is the place associated to γ and γ ranges over all branches of Γ. Some slight changes in the wording and proof of Theorem 6.42 are sufficient to obtain the following result.

T HEOREM 7.7 Let the curve Γ be given by the point P = (x0 , . . . , xn ), and let ∆ = v(G(X0 , . . . , Xn )) be a hypersurface of degree m. If ∆ does not contain Γ, and H∞ is the hyperplane v(X0 ), then div (G(x0 , . . . , xn )) = ∆ · Γ − m(H∞ · Γ). As in Section 6.2, deg(∆ · Γ) stands for the degree of the intersection divisor of ∆ and Γ. C OROLLARY 7.8 For any hyperplane not containing Γ, deg(∆ · Γ) = deg ∆ · deg(H∞ · Γ),

deg(H · Γ) = deg(H∞ · Γ).

The positive integer deg(H · Γ) is the degree of Γ. Then the first equation in Corollary 7.8 can be viewed as the extension of B´ezout’s Theorem 3.14 to space curves. C OROLLARY 7.9 (B´ezout) If a hypersurface ∆ does not contain Γ, then deg(∆ · Γ) = deg ∆ · deg Γ. Also, Theorem 6.43 and Corollaries 6.44 and 6.46 extend to space curves with similar proofs. T HEOREM 7.10 Let ∆ be a hypersurface of degree m not containing Γ. Then ∆ cuts out on Γ a divisor equivalent to m(H∞ · Γ). T HEOREM 7.11 If ∆1 is another surface of degree m not containing Γ, then ∆1 ·Γ is equivalent to ∆ · Γ. T HEOREM 7.12 Every linear system of hypersurfaces of a given degree cuts out a linear series on Γ. Conversely, every linear series of K(F), except for a fixed divisor, is cut out on K(F) by a linear system of hypersurfaces of the same degree. Let ∆0 = v(ϕ0 ), . . . , ∆s = v(ϕn ) be hypersurfaces of degree m, where ϕ0 , . . . , ϕs ∈ K[X0 , . . . , Xn ].

Ps It is possible that some hypersurface in the linear system v( i=0 ci ϕi ) contains Γ even though none of the ∆i does. In the case of plane curves, there was the concept of linear independence modulo F , where Γ = v(F ); see Remark 6.47. Here, a similar concept is required.

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D EFINITION 7.13 The hypersurfaces ∆0 , . . .P , ∆s are linearly dependent modulo s Γ if some hypersurface in the linear system i=0 ci ϕi vanishes over Γ; equivalently, if Ps i=0 ci ϕi (x0 , . . . , xn ) = 0, where Γ is given by the point P = (x0 , . . . , xn ). If this happens, one or more of the ∆i can be eliminated and in this way the linear system becomes linearly independent modulo Γ. D EFINITION 7.14 A singular point of an curve Γ is the centre of either at least two branches or just one non-linear branch of Γ. A non-singular point is also called simple. T HEOREM 7.15 Every irreducible curve has only finitely many singular points. Proof. The notation of the proof of Theorem 7.2 is used. There are finitely many branches centred on H∞ , and these are put aside. Let γ be one of the branches centred at A = (1, a1 , . . . , an ). One of the xi , say x1 , is separable. Let x1 = a1 + ctv + · · · , c 6= 0. If v > 1, then the place corresponding to γ is a zero of dx1 . By Lemma 5.53, this only happens for a finite number of places. Hence all but finitely many branches of Γ are linear. Eliminating the non-linear branches, it may be supposed that the branches centred at A are linear. Since K(F) = K(Q), g(x1 , . . . , xn ) f (x1 , . . . , xn ) , y= . x= d(x1 , . . . , xn ) d(x1 , . . . , xn ) with d, f, g ∈ K[X1 , . . . , Xn ]. Since d(x1 , . . . , xn ) 6= 0, the hypersurface v(d) does not contain Γ. By Theorem 7.5, this hypersurface meets Γ in only a finite number of points. All branches centred at these points are eliminated. Any branch γ centred at A = (1, a1 , . . . , an ) goes into a branch of the plane curve C centred at the point   f (a1 , . . . , an ) g(a1 , . . . , an ) ′ . , A = d(a1 , . . . , an ) d(a1 , . . . , an ) If A′ is a singular point of C, eliminate the branch γ. In this way, finitely many branches are eliminated. Then there remains at most one branch of C centred at A′ , and hence with at most one branch of Γ centred at A. 2

7.2 RATIONAL TRANSFORMATIONS Rational transformations are defined similarly to Section 5.2 when considering points Q = (x0 , x1 , . . . , xn ), Q′ = (x′0 , x′1 . . . , x′n′ ) satisfying K(Q′ ) ⊂ K(Q). Up to relabelling indices, neither x0 nor x′0 is zero; so it is possible to assume that x0 = x′0 = 1. Let Γ and Γ′ denote the irreducible curves in PG(n, K) associated with Q and Q′ .

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D EFINITION 7.16 A rational transformation ω : K(Q) → K(Q′ ) is given by x′i = fi (x1 , . . . , xn )/d(x1 , . . . , xn ) for i = 1, . . . n′ ,

where fi , d ∈ K[X1 , . . . , Xn ], and the hypersurface H = v(d) does not contain Γ. Basic facts on rational transformations stated in Section 5.2 for n = 2 hold true in any dimension n ≥ 3. The image of a branch γ of Γ under ω is a branch of Γ′ . The image A′ of a point A ∈ Γ exists and A′ ∈ Γ′ provided that A is an affine point and that A does not lie in the hypersurface H = v(d). Actually, both conditions can be dropped when A is the centre of only one branch, in particular when A is a non-singular point of Γ. In fact, computing the coordinates of the image A′ of such a point A ∈ Γ is possible using the fact that the coordinate functions are determined up to a non-zero factor. Now, it is shown how to carry out the calculation after replacing (1, . . . , x′n′ ) by (z, . . . , zx′n′ ) for a suitably chosen z ∈ K(Q). Putting mP = min{ordP f1 /d, . . . , ordP fn /d}, take an index k for which mP = ordP fk /d. In the definition of ω, replace (1, f1 /d, . . . , fn /d) by (1/fk , f1 /fk , . . . , fn /fk ). Note that x′k = 1. Now, all fj′ (A) = fj′ (a0 , . . . , an ) exist. Therefore A′ = φ(A) = (f0′ (A), . . . , fn′ ′ (A)) provided that A is the centre of a unique branch of Γ. In particular, if Γ is an irreducible non-singular curve, then the rational transformation ω defines the morphism φ : Γ −→ Γ′ . If K(Q) = K(Q′ ), then Γ is birationally equivalent or birationally isomorphic to the curve Γ′ given by Q′ , and branches correspond to each other when derived from the same place of K(Q). A non-singular point of Γ and a non-singular point of Γ′ correspond if they are the centres of corresponding branches. This definition extends to singular points provided that they are the centre of only one branch. If both Γ and Γ′ are non-singular curves, the morphism φ is an isomorphism. Since every field of transcendence degree 1 over K has a pair of generators, the following result is obtained. T HEOREM 7.17 Any irreducible curve is birationally equivalent to a plane curve. E XAMPLE 7.18 The Klein quartic is the non-singular plane curve F = v(X + Y 3 + X 3 Y ) of genus 3. If P = (x, y) is a generic point of F, then K(F) = K(x, y), with x + y 3 + x3 y = 0,

is the function field of F. Put

x0 = x, x1 = x2 y, x2 = y 2 , x3 = −3xy.

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Let Γ be the irreducible curve in PG(3, K) given by P = (x0 , x1 , x2 , x3 ). The associated rational transformation ω defines a morphism φ : F → Γ such that, if A = (a, b), then A′ = φ(A) = (a, a2 b, b2 , −3ab), provided that A is not a vertex of the triangle of reference in PG(2, K). So, as in Section 7.2, the image of such a vertex is also uniquely determined. Assume that p = 5. First, A = (1, 0, 0) is considered. To carry out the computation, choose a primitive representation σ of the unique place of K(F) whose corresponding branch of F centred at A. If σ(x) = x(t), σ(y) = y(t), then x(t) = a1 ti + a2 tj + · · · , y(t) = t.

a1 a2 6= 0,

From f (a1 ti + a2 tj + · · · , t) = 0,

x(t) = −t3 + t10 + · · · ,

y(t) = t.

Let σ(xi ) = xi (t) for i = 0, 1, 2, 3. Then x0 (t) = −t3 + t10 + · · · , x2 (t) = t2 ,

x1 (t) = (−t3 + t10 + · · · )2 t, x3 (t) = −3(−t3 + t10 + · · · )t.

So eP = −2 and φ(A) = B2 with B2 = (0, 0, 1, 0). The place P corresponds to the branch of Γ which is centred at B2 and has the following primitive branch representation: y0 (t) = −t + t8 + · · · , y2 (t) = 1, Similar calculations show that φ(X∞ ) = B0 φ(Y∞ ) = B1

y1 (t) = t5 − 2t12 + · · · , y3 (t) = 3t2 − 3t9 + · · · .

where B0 = (1, 0, 0, 0), where B1 = (0, 1, 0, 0).

The intersection divisor of the plane H = v(X0 ) with Γ is H · Γ = 5P1 + P2 , where the places P1 and P2 correspond to the branches of C centred at the points Y∞ and O. It is opportune to investigate rational transformations that are not birational. In the non-trivial case, the transcendence degree of K(Q′ ) is 1; in the trivial case, K(Q′ ) = K, and Γ is transformed into a point. Unless specifically mentioned, only the non-trivial case is considered. If P is a place of K(Q), given by a monomorphism τ : K(Q) → K((t)), then τ induces a monomorphism τ ′ : K(Q′ ) → K((t)), giving a place P ′ of K(Q′ ); then P lies over P ′ . In terms of curves, if Q, Q′ define the curves Γ, Γ′ , then a branch of Γ gives rise to a branch of Γ′ . The point A of Γ, which is the centre of the first branch, is taken to the point A′ of Γ′ , which is the centre of the second. As before, A′ is well defined if just one branch is centred at A. This is the case when A is a non-singular point, and hence for all but finitely many points by Theorem 7.15. From a geometric point of view, Γ covers Γ′ , as the images of the non-singular points of Γ cover all but finitely many points of Γ′ .

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To pursue the discussion on rational transformations and morphisms, a purely field-theoretic question is first considered. Let P ′ be a place of K(Q′ ) and P a place of K(Q) over P ′ . Let P be given by τ and let τ induce τ ′ . Then, even if τ is primitive, which is assumed, τ ′ need not be. L EMMA 7.19 The redundancy of τ ′ is independent of the choice of τ . Proof. As usual, Σ = K(x, y), x(t) = τ (x), y(t) = τ (y), and f ∈ K[X, Y ] is an irreducible polynomial for which f (x, y) = 0. Choose ξ ∈ Σ, and write ξ = u(x, y)/v(x, y) with u, v ∈ K[X, Y ], and f ∤ v. Then ordP ξ = ordt u(x(t), y(t))/v(x(t), y(t)). ′

′

′

Let Σ = K(x , y ), and f ′ (x′ , y ′ ) = 0 for an irreducible f ′ ∈ K[X, Y ]. The pair (x′ (t), y ′ (t)) with x′ (t) = τ ′ (x′ ), y ′ (t) = τ ′ (y ′ ) can be viewed as a representation of the branch corresponding to P ′ . If τ ′ has redundancy ν ≥ 1, then there exists s(t) ∈ K[[t]] such that s(t) = cν tν + cν+1 tν+1 + · · · , cν 6= 0

and x′ (t) = x ¯(s(t)), y ′ (t) = y¯(s(t)) for a primitive representation (¯ x(s), y¯(s)) of ′ P in K[[s]]. Suppose that ξ ∈ Σ′ . Let ξ = u′ (x′ , y ′ )/v(x′ , y ′ ) with u′ , v ′ ∈ K[X, Y ], and ′ f ∤ v ′ . Then τ (ξ) = u(τ (x′ ), τ (y ′ ))/v(τ (x′ ), τ (y ′ )). Now, since u(τ (x), τ (y))/v(τ (x), τ (y)) = u′ (τ ′ (x′ ), τ ′ (y ′ ))/v ′ (τ ′ (x′ ), τ ′ (y ′ )), it follows that ordP ξ = ν · ordP ′ ξ, which proves the assertion.

2

From Lemma 7.19, the notation eP = ordP ξ/ordP ′ ξ is meaningful. Hence, the proof of Lemma 7.19 gives the following formula: ordP ξ = eP · ordP ′ ξ.

(7.1)

Now, eP is the ramification index of P. Also, if eP > 1, say that ω ramifies at P. L EMMA 7.20 Over every place P ′ of Σ′ , there lies at least one and at most a finite number of places of Σ. Proof. Consider the linear series |mP ′ |. By the Riemann–Roch Theorem 6.61, dim |mP ′ | ≥ m − g ′ ,

r where g ′ denotes the genus of Σ′ . In other words, |mP ′ | = gm . Then r ≥ 1 for sufficiently large m. So there exists an effective divisor B ≡ mP ′ such that B 6= mP ′ . If B ≻ P ′ , cancel P ′ as often as possible; in the end, B 6≻ P ′ . Hence there exists an element x ∈ Σ′ such that div(x) = mP ′ − B. Now, check that P ′ is a zero of x. Since B 6≻ P ′ , so div(x)0 = mP and div(x)∞ = B. It follows that ordP ′ x = m. Actually, it has been shown that x

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has P ′ but no other place as a zero. By (7.1), the places P of Σ lying over P ′ are also zeros of x. As the number of zeros of x is equal to [Σ : K(x)], this number gives not only an upper bound for the number of places of Σ lying over P ′ , but also ensures the existence of at least one place P of Σ at which x vanishes. Such a place P lies over a place Q′ of Σ′ . By (7.1) Q′ is a zero of x; but then Q′ = P ′ . This completes the proof. 2 Let P1 , . . . , Pj be the places of Σ lying over P ′ , and let eP1 , . . . , ePj be their ramification indices, so that ordPi x = ePi · ordPi′ x. Then P ′ corresponds to D(P ′ ) = eP1 P1 + · · · + ePj Pj .

Also, [Σ : K(x)] = whence

Pj

i=1

ordPi x =

Pj

i=1

Pj

i=1

ePi ordPi ′ x =

(7.2)

P

j i=1

 ePi · [Σ′ : K(x)],

ePi = [Σ : Σ′ ] .

(7.3)

As shown in Lemma 7.28, eP1 = . . . = ePj = 1 in general and, if this occurs, then each of the [Σ : Σ′ ] places over P ′ is an unramified place. On the other hand, P1 ramifies completely when j = 1 or, equivalently, when eP1 = [Σ : Σ′ ]. An interpretation of (7.3) is given in the following theorem. T HEOREM 7.21 To each place P ′ of Σ′ there correspond [Σ : Σ′ ] places of Σ, counted with multiplicity. Geometrically speaking, Γ → Γ′ is a [Σ : Σ′ ]-fold covering. In a rational transformation, multiplicity is counted in a similar way. Hence the following criterion for a rational transformation to be birational is obtained. C OROLLARY 7.22 A rational transformation from Γ to Γ′ is birational if and only if to each branch of Γ′ there corresponds just one branch of Γ; equivalently, if and only if to some branch of Γ′ there corresponds just one branch of Γ. T HEOREM 7.23 Let Γ be given by P = (1, x1 , . . . , xn ). Then, for some pair of elements y1 = a01 + a11 x1 + · · · + an1 xn , y2 = a02 + a12 x1 + · · · + an2 xn ,

with aij ∈ K, the curve Γ′ having Q = (1, y1 , y2 ) as a generic point is birationally equivalent to Γ. Proof. Let P be a place of K(Q) centred at a non-singular point A of Γ. To avoid poles of the coordinate functions xi , suppose that A is an affine point; that is, A does not lie in the hyperplane v(X0 ). Then, for some aj1 ∈ K, the hyperplane H1 = v(a01 + a11 X1 + · · · + an1 Xn )

passes through A and ordP (a01 + a11 x1 + · · · + an1 xn ) = 1 at the place P centred at A. Take y1 = a01 + a11 x1 + · · · + an1 xn ;

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this element y1 has only a finite number of zeros. Choose a hyperplane H2 = v(a02 + a12 X1 + · · · + an2 Xn ) through A but not through the other zeros of y1 . Take ′

y2 = a02 + a12 x1 + · · · + an2 xn ,

and let Σ = K(y1 , y2 ). Since Σ′ ⊂ Σ, there is a place P ′ lying under P. Then, P ′ is a common zero of y1 and P is the only place lying over P ′ . Also, the multiplicity is 1. This follows from ordP y1 = 1 together with (7.1). Corollary 7.22 completes the proof. 2 R EMARK 7.24 A geometric interpretation of Theorem 7.23 can be given in terms of projection. With H1 and H2 as above and (aij ), j = 1, . . . n + 1, i = 0, . . . , n, a matrix such that det(aij ) 6= 0, apply to PG(n, K) the projectivity associated to the matrix (aij ). This can be done so that y1 = x1 , y2 = x2 in Theorem 7.23. The mapping π : AG(n, K) −→ AG(2, K), (a1 , a2 , . . . , an ) 7→ (a1 , a2 )

(7.4)

is a projection. Thus the rational transformation given in Theorem 7.23 determined by (P, Q) is induced by a projection. Using projective coordinates, π can be viewed as induced by the transformation (a0 , a1 , . . . , an ) 7→ (a0 , a1 , a2 ), which is defined at points outside the subspace Πn−3 = v(X0 , X1 , X2 ), and which in fact is the projection from Πn−3 as vertex to the plane Π2 = v(X3 , . . . , Xn ). The join of (a0 , a1 , . . . , an ) and Πn−3 is a subspace Πn−2 intersecting Π2 in the point (a0 , a1 , a2 , 0, . . . , 0) or, omitting the last coordinates as superfluous, in (a0 , a1 , a2 ). It is to be noted that Γ may intersect the vertex Πn−3 and, if so, the transform, or transforms, of the points of intersection are defined via the branches centred at those points. In summary, every irreducible curve can be transformed by means of a projection into a birationally equivalent plane curve.

7.3 HURWITZ’S THEOREM This section establishes a basic theorem of Hurwitz concerning the genera of subfields Σ′ of a function field Σ of transcendence degree 1 over K. For a finite separable extension Σ/Σ′ of degree n, Hurwitz’s Theorem states the numerical relation 2g − 2 = n(2g ′ − 2) + d,

where g and g ′ are the genera of Σ and Σ′ , and d is a non-negative integer; it also gives an interpretation for d together with a method for computing it. By Hurwitz’s Theorem applied to Σ = K(Γ) and Σ′ = K(Γ′ ), certain properties of Γ′ can be derived from those of Γ; in many cases the degree of the covering Γ → Γ′ can be computed.

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If Σ/Σ′ is a Galois extension, then the corresponding covering Γ → Γ′ is a Galois covering. In terms of automorphism groups of K(Γ), the covering Γ → Γ′ is Galois if and only if K(Γ′ ) is the set of all fixed elements of an automorphism group of K(Γ). A detailed treatment of Hurwitz’s Theorem for Galois covers is given in Chapter 11. A key lemma is a formula similar to (7.1). L EMMA 7.25 If ξ ∈ Σ′ is a separable variable of Σ,

ds (7.5) + ordP ′ dξ. dt Proof. From the proof of (7.1), ξ(t) = ξ ′ (s(t)), whence dξ/dt = ds/dt · dξ ′ /ds by derivation. Note that ds/dt 6= 0, since otherwise dξ/dt = 0 contradicting the separability of ξ. This gives the result. 2 ordP dξ = ordt

By (7.5), P ds P = ordP dξ − ordP ′ dξ, dt where the summation is over the places of S. From (7.1), P′ P ordP ′ dξ, ordP ′ dξ = [Σ : Σ′ ] · P

ordt

′ the latter summation being over P the places of Σ . Since dξ has a finite Pnumber of zeros and poles, it follows that ordt (ds/dt) is a finite sum. Also, ordt (ds/dt) is independent of the choice of ξ in Σ′ . This leads to the following definition.

D EFINITION P 7.26 The different of a finite separable extension Σ/Σ′ is the divisor D(Σ/Σ′ ) = dP P with dP = ordt (ds/dt), where the summation is over all places P of Σ. T HEOREM 7.27 (Hurwitz) Let Σ be a finite separable extension of Σ′ , and let g and g ′ be the genera of Σ and Σ′ . Then 2g − 2 = [Σ : Σ′ ] · (2g ′ − 2) + deg D(Σ/Σ′ ). (7.6) P P′ Proof. By Definition 5.55, ordP dξ = 2g − 2 and ordP ′ dξ ′ = 2g ′ − 2. Now, the result follows from Lemma 7.25. 2 If eP is prime to p, then (7.5) becomes ordP dξ = (eP − 1) + ordP ′ dξ. Otherwise, it can only be asserted that ordP dξ ≥ eP + ordP ′ dξ. In any case, ordP dξ ≥ eP − 1 + ordP ′ dξ. Hence P

(eP − 1) ≤

P

ordt (ds/dt) = deg D(Σ/Σ′ ),

(7.7)

where the summation is for all places of Σ at which ω ramifies. In particular, P (eP − 1) is a finite sum. This gives the following.

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L EMMA 7.28 If Σ be a finite separable extension of Σ′ , then ω : Σ → Σ′ ramifies at a finite number of places. Let Σ/Σ1 and Σ1 /Σ2 be finite separable extensions. Then Σ/Σ2 is also a finite separable extension, and it is straightforward to prove the equation dP (Σ/Σ2 ) = dP (Σ/Σ1 ) + [Σ : Σ1 ] dP ′ (Σ1 /Σ2 ),

(7.8)

′

where P is any place of Σ and P is the place of Σ1 lying under P. The following example shows how this machinery works. E XAMPLE 7.29 Let Hq = v(Y + X q + XY q ) be the Hermitian curve. For a generic point P = (x, y) of Hq , let Σ = K(x, y) the function field of Hq . Since dx 6= 0, so x is a separable variable of Σ by Theorem 5.50. Now, fix a divisor m of q 2 − q + 1, and consider the rational transformation ω : ξ = xm , η = y m .

Let Γ′ be the curve with generic point Q = (ξ, η), and let Σ′ = K(ξ, η). Since p ∤ m, so ξ is a separable variable of Σ, by Lemma 5.38. The different D(Σ/Σ′ ) is calculated explicitly. Take the point P1 = (1, 0, 0) of Γ. There is a unique branch γ of Γ centred at P1 . Since the tangent to Γ at P1 is the X-axis, there is a primitive branch representation of γ of type (x = t, y = y(t)), with y(t) ∈ K[[t]]. Note that f (t, y(t)) = y(t) + tq + ty(t)q = 0

shows that y(t) = −(tq + tq

2

+1

+ · · · + aj tq+αj (q

2

−q+1)

+ · · · ).

Let P be the place of Σ corresponding to the branch γ, given by the primitive place representation τ : Σ → K((t)). Let τ ′ : Σ′ → K((t)) be the monomorphism induced by τ on Σ′ , and let P ′ be the place of Σ′ arising from τ ′ . From the definition of ω, the place P is the only place of Σ lying over P ′ . Now define τ ′ (ξ) = ξ(t), τ ′ (η) = η(t). Then (ξ(t), η(t)) is a representation of the branch γ ′ of Γ′ corresponding to P ′ . From the definition of ω, ξ(t) = η(t) =

tm ,

h im 2 (−1)m (tm )q 1 + (tm )(q −q+1)/m + · · · .

Therefore it is a reducible representation with ramification index m. Replacement of tm by s provides a primitive representation of γ ′ : ξ(s) η(s)

= =

s, 2 (−1)m sq [1 + s(q −q+1)/m + · · · ].

This shows that ordP ′ η = q. Since ordP η = mq, and no other place of Σ lies over P ′ , so eP = m; that is, [Σ : Σ′ ] = m. Also, as s(t) = tm yields ds/dt = mtm−1 , the place P contributes (m − 1)P to the different divisor D(Σ/Σ′ ). Similar arguments can be used in doing computations for the other two fundamental points P2 = (0, 1, 0) and P3 = (0, 0, 1). The results for P1 hold true for

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both P2 and P3 , which can be shown directly. The change of the coordinate system by (X0 , X1 , X2 ) → (X1 , X2 , X0 ) leaves Γ invariant and takes P1 to P2 and P2 to P3 , Therefore the contribution of the three fundamental points to D(Σ/Σ′ ) is (m − 1)P1 + (m − 1)P2 + (m − 1)P3 . Each of the remaining points P of Γ is affine: P = (a, b) with a, b 6= 0. The tangent to Γ at P is not the horizontal line v(Y − a). Hence, a primitive representation of the unique branch γ of Γ centred at P is of type x = a + t+ ··· ,

y = b + t.

Therefore ξ(t) = am + · · · ,

η = (b + t)m .

Now, choose η ′ ∈ K[[s]] and s(t) ∈ K[[t]] such that η(t) = η ′ (s(t)). Then m · η(t)/dt = m(b + t)m−1 = m · η ′ /dt · ds(t)/dt,

whence ordt ds/dt = 0. Therefore eP = 1 for the place P corresponding to γ. This shows that P does not contribute to D(Σ/Σ′ ). Summing up, the different divisor D(Σ/Σ′ ) of the rational transformation ω is equal to (m − 1)P1 + (m − 1)P2 + (m − 1)P3 ,

and its degree is equal to 3(m − 1). By Hurwitz’s Theorem 7.6, (2g − 2) = m(2g ′ − 2) + 3(m − 1),

where g = 21 (q 2 − q) is the genus of Γ. Hence the genus g ′ of Γ′ is equal to   2 q −q+1 g ′ = 12 −1 . m 7.4 LINEAR SERIES COMPOSED OF AN INVOLUTION Let Γ′ be a curve given by the point Q = (y0 , . . . , yr ) with yi ∈ Σ and K(Q) ⊂ Σ. There are two basic ideas related to Q: the rational transformation τ : (1, x, y) 7→ (y0 , . . . , yr ),

and the virtual linear series L consisting of all divisors, P Ac = div ( ci yi ) + B, c = (c0 , . . . , cr ) ∈ PG(r, K),

(7.9)

where B is a fixed divisor. The aim now is to give a geometric representation of L on Γ′ . The simplest case occurs when τ is birational and, in this case, the divisors of L are the hyperplane sections on Γ′ . The curve Γ′ lies in PG(r, K). However, if y0 , . . . , yr are linearly dependent over K, that is, if c0 y0 + · · · + cr yr = 0 with some ci ∈ K not all zero, the hyperplane H = v(c0 Y0 + · · · + cr Yr ) contains Γ′ by Theorem 7.5. Conversely, if Γ′ lies in H, then the same theorem yields that c0 y0 + · · · + cr yr = 0. The significance of the usual assumption when discussing virtual linear series, that y0 , . . . , yr are linearly independent, is now clear: this property holds if and only if Γ′ is not contained in a proper subspace of PG(r, K).

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If a fixed divisor C is added to the divisors of L, a new virtual linear series is obtained, but both Γ′ and the rational transformation τ remain the same. Hence it is assumed that the virtual linear series L in question is a linear series and has no fixed place. Also, it is helpful to make the following purely field-theoretic definition. Let P ′ be a place of Σ′ and let D(P ′ ) = eP1 P1 + · · · + ePs Ps

be the divisor as in (7.2). The set of all divisors D(P ′ ) obtained as P ′ ranges over the places of Σ′ is an involution of Σ of order [Σ : Σ′ ] = eP1 + · · · + ePs .

Also, write ν = [Σ : Σ′ ]; then γν indicates an involution of order ν. Also, by abuse ′ ′ of notation, Pwrite τ (P)P= P when P lies over P , and extend this to divisors by writing τ ( µi Pi ) = µi τ (Pi ). In other words, τ is a homomorphism from the divisor group of Σ into the divisor group of Σ′ . T HEOREM 7.30 For x ∈ Σ′ , let div(x)0 and div(x)′0 denote the divisor of zeros of x for Σ and Σ′ . Then τ (div(x)0 ) = [Σ : Σ′ ] · div(x)′0 . Proof. Let div(x)′0 = µ1 P1′ + · · · + µs Ps′ , P with Pi′ = 6 Pj′ , for i 6= j. If j ePij Pij is the divisor corresponding to Pi′ , then P τ ( j ePij Pij ) = [Σ : Σ′ ] · Pi′ .

Also, ordPij x = ePij ordPj′ x = ePij µj . Hence, P div(x)0 = i,j µi (ordPij x)Pij ,

and

P P τ (div(x)0 ) = τ ( ij µi (ordPij x) Pij ) = ij µi (ordPij x)τ (Pij )  P  P P P ′ ′ = i µi j ordPij x Pi = i j ePij µi Pi P = [Σ : Σ′ ] i µi Pi′ = [Σ : Σ′ ] div(x)′0 .

2

C OROLLARY 7.31 Let div(x)′0 = P1′ + · · · + Pt′ . Then div(x)0 is the sum of the divisors corresponding to the Pi′ . Hence, it is made up of the divisors of an involution. P ROPOSITION 7.32 The divisors of a linear series L are made up of divisors of an involution. Proof. If the representation (7.9) of L is not normalised, change it to a normalised one; see Theorem 6.22. This involves replacing yi by a multiple zyi , z ∈ Σ, and hence it does not change Γ′ . By Theorem 6.22, B is then a divisor of L.

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Let P1 , . . . , Ps be the places occurring inPB. The condition that the divisor Ac ∈ L contains Pi is a linear condition λij cj = 0 on the cj ; see Theorem 6.26. This condition is not trivial; thatPis, some λij 6= 0, since Pi is not a fixed divisorP of L. Thus the polynomial λij Xj is not identically zero, and hence H = v( λij Xj ) is a hyperplane of PG(r, K). Since finitely many hyperplanes cannot P all points of PG(r, K), there are points off the hypersurface Qs cover ∆ = v( i=1 ( λij Xj )). Choose such a point P = (c0 , . . . , cm ). Then the corresponding divisor AcPcontains none of the Pi . Thus, Ac is the set of zeros and B r is the set of poles of i=0 ci yi . Hence, B is the set of zeros of an element in Σ. By Corollary 7.31, B is made up of elements of an involution. 2 Thus, the linear series L, or the rational transformation τ , defines an associated involution, γ(L) or γ(τ ). If γ(L) = γν with ν > 1, then {Ac } is composed of the involution γν ; otherwise {Ac } is simple. Then, τ is birational if and only if the corresponding series is simple. In fact, Σ = Σ′ if and only if ν = 1. Applying τ to (7.9) gives P τ (Ac ) = ν · div( ci yi ) + τ (B). Each place in τ (B) occurs as many times as a multiple of ν, and the same holds for τ (Ac ). Dividing by ν gives P ν −1 τ (Ac ) = div( ci yi )′ + ν −1 τ (B), ν = [Σ : Σ′ ]. (7.10) The divisors on the right-hand side cut out a linear series L′ on Γ′ . If P1′ + · · · + Pt′ is a divisor in L′ , then the sum of the divisors corresponding to the Pi′ is an Ac ∈ L. Actually, each Ac ∈ L is obtained in this way. The linear series L′ has no fixed place. To prove this, suppose otherwise and that ′ P is a fixed point. Let P1 , . . . , Ps be the places over P ′ . As seen before, there is a divisor Ac containing none of the Pi and such that τ (Ac ) does not contain P ′ , a contradiction. Also, L′ is effective. The hyperplanes of PG(r, K) cut out on Γ′ the linear series P div( ci yi )′ + D. As this has no fixed place andP is effective, the multiplicity λ with which any place P enters D is −minc {(ordP ( ci yi )}. Since (1/ν)τ (B) can be described in the same way, this also shows that D = (1/ν)τ (B). Hence, (7.10) is the linear series cut out on Γ′ by hyperplanes. From this the following result is obtained. T HEOREM 7.33 Let Γ′ be a curve arising from a rational transformation τ of Σ. If τ is birational, then the corresponding linear series L is cut out on Γ′ by hyperplanes. R EMARK 7.34 Definition 6.33 can be interpreted geometrically. With a slight change in the wording in Remark 7.24, the projection πs from PG(r, K) to its s-dimensional subspace Πs = v(Xs+1 , . . . , Xr ), for 2 ≤ s < r, is defined, with the vertex of πs being Πr−(s+1) = v(X0 , . . . , Xs ). s Let gm and gnr be two simple linear series of Σ such that m ≤ n, s ≤ r − 1. s Suppose that gm is contained in gnr . Then s gm = {div (c0 x0 + · · · + cs xs ) + B | c = (c0 , . . . , cs ) ∈ PG(s, K)}, gnr = {div (c0 x0 + · · · + cr xr ) + F | c = (c0 , . . . , cr ) ∈ PG(r, K)}

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with F ≻ B. Let Q = (x0 , . . . , xs ) and Q′ = (x0 , . . . , xr ). If Γ and ∆ are the irreducible curves determined by K(Q) and K(Q′ ), then ∆ is the projection of Γ with vertex Πr−(s+1) . The converse also holds. An irreducible curve Γ is normal if it cannot be obtained in this way by a projection of another curve. Equivalently, if Γ is determined by K(Q) with Q = (x0 , . . . , xr ) then the associated simple, fixedplace-free linear series {div (c0 x0 + · · · + cr xr ) + B | c = (c0 , . . . , cr ) ∈ PG(r, K)} is complete. E XAMPLE 7.35 Let Σ = K(x), and Q = (1, x, . . . , xr ). The algebraic curve Γ of PG(r, K) determined by Q is normal, and it is the normal rational curve of PG(r, K). Another fundamental result is the following. T HEOREM 7.36 Let gnr be a linear series without fixed points but composed of an involution. If P is a place of Σ, then every divisor of gnr containing P also contains at least another place Q, where Q depends only on P, and conversely. The case P = Q can occur when a divisor of gnr contains P with multiplicity at least 2.

Proof. Let gnr be composed of the involution γν with ν > 1. Take any divisor P + P2 + · · · + Ps of γν . Then every divisor containing P contains each of the remaining places P2 , . . . , Pν . This proves the direct part. For the converse, assume that gnr is simple, so that it defines a birational transformation, and hence gnr is cut out on Γ′ by hyperplanes. Let P be any place centred at a simple point of Γ′ . If Q another place of Σ, then there is a divisor in gnr containing P but Q. In fact, if the centre Q of Q is distinct from the centre P of P, then there is a hyperplane through P but not through Q, and hence the divisor cut out has the required property. For P = Q, such a divisor arises from any hyperplane containing P but not 2P, or geometrically, from any non-tangent hyperplane to Γ′ at P . 2 A characterisation for a point of a space curve to be non-singular is given in the following theorem. T HEOREM 7.37 Let gnr be the linear series cut out on Γ by hyperplanes. Then a point P of Γ is non-singular if and only if the subseries of gnr containing a place P centred at P, minus P, has no fixed place. Proof. Let γ1 , . . . , γk be the branches of Γ centred at P , and P1 , . . . , Pk the corresponding places of Σ. For i = 1, . . . , k, put nPi = ordP γi . Then the divisor Pk E = i=1 nPi Pi is the fixed divisor of the linear subseries L′ of gnr cut out on Γ by the hyperplanes through P . Also, deg E ≥ 1 and equality holds if and only if P is a non-singular point of Γ. On the other hand, deg E = 1 occurs if and only if L minus P1 has no fixed divisor. 2 D EFINITION 7.38 A singular point of Γ is an s-fold point if deg E = s.

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R EMARK 7.39 A point P of Γ is an s-fold point if and only if the fixed divisor of the linear series cut out on Γ by the hyperplanes containing P has degree s. As a consequence of Theorem 7.37, there is the following result. T HEOREM 7.40 Let gnr be a complete linear series of Σ. If n > 2g, then the curve Γ, arising from gnr via the associated rational transformation, is free of singularities; that is, it is a non-singular model of Σ. Proof. By the Riemann–Roch Theorem 6.61, r = n − g since n > 2g > 2g − 2 implies that gnr is not special. Also, gnr has no fixed point; otherwise, a complete r non-special linear series gn−1 would be obtained by subtracting a fixed point. However, then r = n − 1 − g, again by the Riemann–Roch Theorem 6.61, contradicting the previous equation, r = n − g. Now, let P be any point of Γ, and let P be a place centred at P . The subseries of r−1 all divisors of gnr containing P minus P is a linear series gn−1 which has no fixed r−1 point. The latter property can be shown as before. The hypothesis that gn−1 has a r−1 fixed point would imply the existence of a complete non-special linear series gn−2 , contradicting the Riemann–Roch Theorem 6.61. 2 g−1 For g > 1, the most important linear series is the canonical series g2g−2 . It is g−1 possible that g2g−2 may be composed of an involution. However, the following theorem shows that this happens only in a very special situation.

T HEOREM 7.41 Let g > 1. The canonical series is composed of an involution if and only if Σ has a complete g21 . If this is the case, then Σ has a subfield Σ′ such that [Σ : Σ′ ] = 2. Proof. Assume first that Σ admits a complete g21 , and take one of the divisors in g21 , namely C = P1 + P2 . Since the canonical series |W | has no fixed point, its subseries of divisors containing P has dimension 1. If |W | were not composed of an involution, then requiring a W ∈ |W | to contain C would impose two linear conditions. So, this would imply that dim |W − C| = (g − 1) − 2 = g − 3 and hence i(C) = g − 2, contradicting the Riemann–Roch Theorem 6.61 applied to g21 . g−1 Actually, it has been shown that g2g−2 is made up of divisors of g21 . Conversely, let |W | be composed of an involution γν . By Theorem 7.36, the rational transformation associated to the canonical series gives rise to a curve Γ′ in PG(g − 1, K) of degree (2g − 2)/ν such that the hyperplanes cut out on Γ′ a linear g−1 series g(2g−2)/ν . Since r ≤ n for any gnr , this is only possible for ν ≤ 2. As |W | is assumed to be composed, so ν = 2, showing that |W | is composed of an involution γ2 of order 2. It remains to show that γ2 is a linear series, that is, its divisors are the divisors of g−1 a g21 . As ν = 2, the linear series cut out on Γ′ by hyperplanes is gg−1 . Hence Γ′ ′ ′ has genus 0 by Theorem 6.81. Therefore the field Σ of Γ is K(ξ) for some ξ ∈ Σ, and [Σ : K(ξ)] = 2. Let P be a place of Σ which is not a pole of x. For a primitive representation σ of P, σ(ξ) = a + bti + · · · .

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Then P is a zero of ξ − a with multiplicity i. Let P ′ be the place of Σ′ under P. Hence, (7.1) implies that P ′ is a zero of ξ − a regarded as an element in Σ′ . If i = 1, just one more place of Σ , say Q, lies over P ′ . By (7.1), Q is also a zero of ξ − a. Therefore div(ξ − a)0 = P + Q. Now, the case i > 1 is examined. By (7.1), i = eP ordP ′ (ξ − a). Since Γ′ is rational, ordP ′ (x − a) = 1. Thus, i = eP . Since eP ≤ ν = 2, so i = eP = 2. This shows that P is the only place of Σ lying over P ′ . Hence 2P ∈ γ2 , and div(ξ − a)0 = 2P. Thus, it has been shown that γ2 consists of all divisors div(c0 + c1 ξ) + div(ξ)∞ . The completeness of g21 depends on the hypothesis that g > 0.

(7.11) 2

T HEOREM 7.42 If Σ has more than one g21 , then its genus is 0 or 1; that is, Σ is either rational or elliptic. Proof. Assume that Σ admits a g21 . From the proof of Theorem 7.41, the degree [Σ : K(ξ)] = 2 for some ξ, and g21 consists of all divisors (7.11). If Σ has another g21 , then also [Σ : K(η)] = 2 with some η 6∈ K(ξ). Therefore Σ = K(ξ, η). From Theorem 5.61, the assertion g ≤ 1 follows. 2 D EFINITION 7.43 A function field of genus g > 1 that has a complete, necessarily unique, linear series g21 is hyperelliptic. An irreducible curve Γ is hyperelliptic if its function field is hyperelliptic. T HEOREM 7.44 Let K(x) be the unique subfield of a hyperelliptic Σ such that [Σ : K(x)] = 2. If [Σ : K(y)] ≤ g, with g the genus of Σ, then K(y) is contained in K(x). Proof. Suppose that [Σ : K(z)] ≤ g but z 6∈ K(x). Then Σ = K(x, z). From Theorem 5.61, g ≤ ([Σ : K(x)] − 1)([Σ : K(y)] − 1) ≤ g − 1, a contradiction.

2

Curves of genus 2 are examples of hyperelliptic curves. For a more detailed account of hyperelliptic curves, see Section 7.10. For non-hyperelliptic curves, the relevant results are stated in the following section.

7.5 THE CANONICAL CURVE In this section, C denotes a non-hyperelliptic curve of genus g. Since, the canonical g−1 series on C is a simple g2g−2 with no fixed points, the projective image of such a series is a curve Γ of degree 2g−2 in PG(g−1, K), birationally equivalent to C and unique up to projectivities; see Theorem 7.45 (ii) below. It is the canonical curve of genus g. Conversely, if Γ is a curve of degree 2g − 2 in PG(g − 1, K) which is g−1 cut out by the hyperplanes is the birationally equivalent to C, then the series g2g−2 canonical series, and so Γ is the canonical curve.

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T HEOREM 7.45

217

(i) The canonical curve is non-singular.

(ii) Two canonical curves that are birationally equivalent are projectively equivalent. (iii) A divisor D = P1 + · · · + Pn defines a complete special series |D| = gnr if and only if the centres of the n branches corresponding to the places Pi , i = 1, . . . , n, span an (n − r − 1)-dimensional subspace of PG(g − 1, K). (iv) The canonical curve is normal. Proof. (i) Assume on the contrary that M is a singular point of Γ, that is, a (k + 2)fold point with k non-negative. By Theorem 7.37, the hyperplanes through M g−2 cut out on Γ a linear series g2g−4−k without fixed places. If γ1 , . . . , γs are the branches of Γ centred at M and P1 , . . . , Ps are the corresponding places of Σ, the Ps hyperplanes cut out in addition a fixed divisor D = n P, where the degree Ps Ps i=1′ P ′ n = k + 2. Hence a fixed divisor, D = n P, with 0 ≤ n′P ≤ nP i=1 i=1 P Ps P ′ g−2 and i=1 nP = k, can be added to g2g−4−k , so as to produce a special series g g−2 g2g−4 = |D′ + g2g−4−k |.

The dimension of this series is smaller than g − 1, but also at least g − 2, and hence g−2 it is a special series g2g−1 . The residual divisors relative to the canonical series constitute a g2s . By the Reciprocity Theorem 6.78 of Brill and Noether, 2g − 6 = 2(g − 2 − s), whence s = 1, and the series is a g21 . This is ruled out, however, since Γ is supposed not to be hyperelliptic. (ii) This follows from Theorem 6.17 together with Theorem 6.72 (ii). (iii) If G is a special divisor of n points on the canonical curve Γ that defines a complete gnr , the linear system of hyperplanes of PG(g − 1, K) through G has dimension g − 1 − n + r; so G is in a subspace Πn−r−1 . Conversely, if a divisor G of n points is in a subspace Πn−r−1 and not in any lower dimension, the index of speciality of G is g − n + r, and so defines a complete gnr . (iv) This follows from Remark 7.34, since the canonical series is complete. 2

7.6 OSCULATING HYPERPLANES AND RAMIFICATION DIVISORS The idea is to consider for any point P of a non-singular space curve the different intersection multiplicities of hyperplanes with the curve at P . There is only a finite number of these intersection multiplicities, this number being equal to the dimension of the space. There is a unique hyperplane, called the osculating hyperplane, with the maximum intersection multiplicity. The concept of osculating hyperplane will also be defined for singular curves. Let Γ be an irreducible curve of degree n in PG(r, K). The hyperplanes cut out on Γ a simple, fixed-point-free, not-necessarily-complete, linear series L. Note that

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L has degree n and dimension r, and hence it is a gnr . Let Γ be given by the point P = (x0 , . . . , xn ). Without loss of generality, suppose that x0 6= 0; then K(Γ) = K(x1 /x0 , . . . , xr /x0 ) is the function field of Γ. Further, L consists of all divisors Ac = div(c0 x0 + c1 x1 + · · · + cr xr ) + B P with c = (c0 , . . . , cr ) ∈ PG(r, K) and B = eP P, where eP = − min{ordP x0 , . . . , ordP xr }.

Here, ordP B = n. Geometrically speaking, Ac is cut out on Γ by the hyperplane H = v(c0 X0 + · · · + cr Xr ); P that is, Ac is the intersection divisor H · Γ = I(P, H ∩ γ)P, where Pr I(P, H ∩ γ) = ordt ( i=0 ci xi (t)) + nP .

Here xi (t) = τ (xi ), i = 0, . . . , r, is a primitive representation τ of P, and the (r + 1)-tuple (x0 (t), . . . , xr (t)) is a primitive representation of the branch γ of Γ corresponding to P, with P the centre of γ. To develop local properties of Γ, a finite sequence of integers is defined. D EFINITION 7.46 An integer j is a Hermitian P-invariant or an (L, P)-order if there exists a hyperplane H such that I(P, H ∩ γ) = j.

In Example 7.18, if P = (0, 0, 1, 0) then the (L, P)-orders are 0, 1, 2, 5. In the case that L is the canonical series, it follows from the Riemann–Roch Theorem 6.61 that j is a (L, P)-order if and only if j + 1 is a Weierstrass gap; that is, there exists no rational function in Σ which has a pole of order j + 1 at P but is regular at all places different from P; see Section 6.6. For any non-negative integer i, consider the set of all hyperplanes H of PG(r, K) for which I(P, H ∩γ) ≥ i. Such hyperplanes correspond to the points of a subspace Πi in the dual space of PG(r, K). Then PG(r, K) = Π0 ⊃ Π1 ⊃ Π2 ⊃ · · · .

An integer j is an (L, P)-order if and only if Πj 6= Πj+1 , in which case Πj+1 has codimension 1 in Πj . Since deg L = n, so Πi is empty as soon as i > n. The number of (L, P)-orders is exactly r + 1; they are j0 , j1 , . . . , jr in increasing order. Since P is not a base point of L, so j0 = 0. Note also that j1 = 1 if and only if the branch γ is linear. In the case that the linear series L is complete, from the Riemann–Roch Theorem 6.61, dim Πji = n − ji − g,

for ji < n − 2g + 2; in particular, ji = i when i ≤ n − 2g. Consider the intersection Πi of hyperplanes H of PG(r, K), for which I(P, H ∩ γ) ≥ ji+1 .

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Then Π0 is simply P , the centre of the branch γ, and Π1 is the tangent line to the branch γ at P . Also, Πi is the i-th osculating space or the osculating i-space; in particular, Πr−1 is the osculating hyperplane. The flag Π0 ⊂ Π1 ⊂ . . . ⊂ Πr−1 ⊂ PG(r, K)

can be viewed as the algebraic analogue of the Frenet frame in differential geometry. Now, an equation of the osculating hyperplane is given in terms of the coordinate functions xi . The essential tool is a generalisation of the Wronskian determinant whose entries are Hasse higher derivatives of the coordinate functions. Choose a local parameter ζ at P, and divide xk by ζ eP for k = 0, 1, . . . , r. Then (i) ordP Dζ xk ≥ 0; that is, (i)

(i)

τ (Dζ xk ) = c0k + · · · , (i)

with c0k ∈ K. To write the Wronskian determinant, it is better to use simpler and more expressive notation. For this purpose, in the rest of this section, points of Γ are thought of as branch points; that is, the term a point P of Γ is used to indicate (i) the branch of Γ associated to the place P. In this spirit, Dζ xk (P ) is an appropriate (i)

notation for c0k . In particular, the centre of P is the point (x0 (P ), . . . , xr (P )) of Γ. Also, I(P, Γ∩Φ) is used to indicate I(A, γ ∩Φ) for a hypersurface Φ of PG(r, K).

T HEOREM 7.47 The (L, P)-orders can be computed iteratively. If the (L, P)orders j0 , . . . , ji−1 are known, then the successive order ji is the smallest integer such that, in PG(r, K), the i + 1 points (j )

(j )

(j )

Pm = (Dζ m x0 )(P ), (Dζ m x1 )(P ), . . . , (Dζ m xr )(P )), for m = 0, 1, . . . , i, are linearly independent. The i-th osculating space at P is spanned by these points. Proof. Up to a projectivity of PG(r, K), for every i = 0, . . . , r the subspace Li is the intersection of the r − i hyperplanes v(Xi+1 ), . . . , v(Xr ). Then ji+1 = min{ordP (ai+1 xi+1 + · · · + ar xr ) | ai+1 , . . . , ar ∈ K}.

Hence, considering the suffixes in descending order,

jr = ordP xr , jr−1 = ordP xr−1 , . . . , j0 = ordP x0 . Now, define the (r + 1) × (r + 1) matrix A = (ami ), where (jm )

ami = (Dt

xi )(P ), (j)

for 0 ≤ r ≤ n, 0 ≤ i ≤ n. Since, by Lemma 5.80 and (5.20), Dζ ζ j = 1 (k)

but Dζ ζ j = 0 for k > j, so A is lower-triangular with every diagonal element non-zero. Hence Li is spanned by the points (j )

(j )

Pm = (Dζ m x0 (P ), . . . , Dζ m xr (P )), for m = 0, 1, . . . , i. The minimality of the ji holds in an even stronger sense.

2

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C OROLLARY 7.48 If m0 , m1 , . . . , mk are integers with 0 ≤ m0 < · · · < mk (m ) (m ) such that the points Pi = (Dζ i x0 (P ), . . . , Dζ i xr (P )), for i = 0, 1, . . . , k, are linearly independent, then ji ≤ mi for i = 0, . . . , k. (s)

(s)

Proof. Since the points Ps = (Dt x0 (P ), . . . , Dt xr (P )), with s = 0, . . . , ji −1 span a subspace of dimension i, while the i + 1 points with s = m0 , . . . , mi are linearly independent, so ji − 1 < mi ; that is, ji ≤ mi . 2 C OROLLARY 7.49 The osculating hyperplane at P has equation X0 X1 ... Xr (j ) (j ) (j ) 0 0 0 D x0 (P ) Dζ x1 (P ) ... Dζ xr (P ) ζ (j ) (j ) D(j1 ) x0 (P ) Dσ 1 x1 (P ) ... Dζ 1 xr (P ) ζ .. .. .. . . ... . (jr−1 ) (j ) (j ) r−1 r−1 Dζ x0 (P ) Dζ x1 (P ) ... Dζ xr (P )

= 0.

The point P is an osculation point or, more precisely, an L-osculation point if jr > r, that is, if there is a hyperplane intersecting Γ at P with multiplicity greater than r. E XAMPLE 7.50 (i) Let Σ = K(x) be the rational function field. Choose the Xaxis C = v(Y ) as a model (C; (x, 0)) of Σ. Let s ≥ 1 be any integer; then the s s elements 1, x, xp , xp +1 are linearly independent over K. Let Γ be the associated curve of PG(3, K). Note that the divisors Ac + B, with s

and B =

P

s

Ac = div (c0 + c1 x + c2 xp + c3 xp

+1

)

eP P, constitute a simple, fixed-point-free linear series L. Then (j0 , j1 , j2 , j3 ) = (0, 1, ps , ps+1 )

for any P of Γ, and the osculating plane HP has equation X0 X1 X2 X3 s s 1 t tp tp +1 = 0; s 0 1 0 tp 0 0 1 t

that is,

s

HP = v(−tp

+1

s

X0 + tp X1 + t X2 − X3 ).

(ii) Let p = 5. Take F = v(f (X, Y )) as in Examples 5.36 and 5.81 (ii). Let r = 3 and x0 = 1, x1 = x, x2 = y, x3 = y 2 . If Ri = (Dx(i) x0 , Dx(i) x1 , Dx(i) x2 , Dx(i) x3 ), then R0 R2 R4 R6

= (1, x, y, y 2 ), = (0, 0, −2x, xy − x4 ), = (0, 0, 0, 0), = (0, 0, 0, 2x12 + 1).

R1 = (0, 1, −2x2, x2 y), R3 = (0, 0, 1, −2x3 + 2y), R5 = (0, 0, 2x10 , x − x10 y),

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Let A = (1, 0, 0) ∈ F. Then the corresponding point P = (1, 0, 0, 0) is in Γ. The order sequence of Γ at P is (0, 1, 3, 6), and the osculating plane HP to Γ at P has equation X0 X1 X2 X3 1 0 0 0 = 0; 0 1 0 0 0 0 1 0

that is, HP = v(X3 ). Let A = (0, 1, 0) ∈ F. Then the corresponding point P∞ = (0, 0, 0, 1) and a primitive representation of the unique branch of Γ centred at P∞ is as follows: x0 = t6 + · · · , x1 = t + · · · , x2 = t3 + · · · , x3 = 1.

The order sequence of Γ at P∞ is (0, 1, 3, 6). Let

Ri (P∞ ) = (Dx(i) x0 (P∞ ), Dx(i) x1 (P∞ ), Dx(i) x2 (P∞ ), Dx(i) x3 (P∞ )). Then R0 (P∞ ) = (0, 0, 0, 1), R1 (P∞ ) = (0, 1, 0, 0), R2 (P∞ ) = (0, 0, 0, 0), R3 (P∞ ) = (0, 0, 1, 0), R4 (P∞ ) = (0, 0, 0, 0), R5 (P∞ ) = (0, 0, 0, 0), R6 (P∞ ) = (1, 0, 0, 0). The osculating plane HP to Γ at P∞ has equation X0 X1 X2 X3 0 0 0 1 0 1 0 0 0 0 1 0 that is, HP = v(X0 ).

= 0;

To investigate global properties of Γ, the condition on ζ to be a local parameter is weakened to be a separable variable. However, the idea of considering a determinant such as that in Theorem 7.47 is still useful. This depends on the existence of a strictly increasing sequence of non-negative integers ǫ0 , . . . , ǫn , the orders, such that the generalised Wronskian (ǫ )

W = det(Dζ i xj )

(7.12)

with 0 ≤ i, j ≤ r does not vanish. To determine such a sequence the idea is to choose ǫ0 , . . . , ǫr in strictly increasing order such that ǫ0 = 0 and define ǫi inductively so that, if ǫ0 , . . . , ǫi−1 are known, then ǫi is the smallest integer such that the points (ǫk )

Pk = (Dζ

(ǫk )

x0 (P ), . . . , Dζ

xr (P )),

k = 0, . . . , i, are linearly independent in PG(r, Σ). As in Corollary 7.48, the ǫi are minimal in a stronger sense: if m0 , . . . , mi are integers with 0 ≤ m0 < · · · < mi (m ) (m ) such that the points Qk = (Dζ k x0 (P ), . . . , Dζ k xr (P )), k = 0, 1, . . . , i, are linearly independent in PG(r, Σ), then ǫk ≤ mk for k = 0, . . . , i. To show that this inductive procedure works properly, the following lemma is required.

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CHAPTER 7

(i) If yi =

P

aij xj with (aij ) ∈ GL(r + 1, K), then

(ǫ )

(ǫ )

(ǫ )

(ǫ )

det(Dζ i yj ) = det(aij ) det(Dζ i xj ). (ii) If h ∈ Σ, then det(Dζ i (hxj )) = hr+1 det(Dζ i xj ). (iii) If η is another separable variable, then (ǫ )

det(Dη(ǫi ) xj ) = (dζ/dη)ǫ1 +···+ǫr det(Dζ i xj ). Proof. (i) This comes from projective geometry, and does not depend on the minimality of the ǫi . (ii) By Leibniz’ rule, as in Lemma 5.80 and (5.23), Pi (ǫ ) (ǫ ) (s) (ǫ −s) Dζ i (hxj ) = h · Dζ i xj + ǫs=1 (Dζ h) · (Dζ i xj ). In the first row of W , every element has h as a factor. The second row is (ǫ1 )

h · Dζ

(ǫ1 )

(x0 ) + · · · , . . . , h · Dζ

(xk ) + · · · ,

where the ellipses in each entry indicate terms that are the components of a vec(k) (k) tor w which is a linear combination of vectors (Dζ x0 (P ), . . . , Dζ xr (P )) with (ǫ )

(ǫ )

0 ≤ k < ǫ1 . This vector w is a multiple of (Dζ 0 x0 (P ), . . . , Dζ 0 xr P )), by the minimality of ǫ1 . Hence, in the determinant, w can be omitted, and so h is again a factor. Proceeding inductively on the other rows of W gives the result. (iii) The same argument can be employed if the chain rule (5.77) is used in place (ǫ ) of Leibniz’ rule. Here, Dη i xj in W may be replaced by (ǫi )

Dt

xj · (dζ/dη)ǫi .

2

From Lemma 7.51, it follows that ǫ0 , . . . , ǫn depend only on the linear series L, and therefore can be called the L-orders or, simply, the orders of the curve Γ. To show the actual existence of the orders, choose a place P and let ζ be a local parameter of Σ at P, and suppose that eP = 0. By Theorem 7.47, there exist j0 , . . . , jr , the (L, P)-orders, for which (j )

det(Dζ i xj (P )) 6= 0. (j )

This implies that det(Dζ i xj )) 6= 0. Hence there exists the order sequence, that is, (ǫ )

the strictly increasing minimal sequence ǫ0 , . . . , ǫr for which det(Dζ i xj )) 6= 0 and ǫi ≤ ji for i = 0, . . . , r. D EFINITION 7.52 The ramification divisor of L is (ǫ )

R = div (det(Dζ i xj )) + (ǫ0 + · · · + ǫr )div (dζ) + (r + 1)E, P where E = eP P and eP = −min{ordP x0 , . . . , ordP xr }.

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E XAMPLE 7.53 Let p = 5. With the notation in Example 7.50, the order sequence of Γ is (0, 1, 2, 5), and 1 x y y2 2 2 0 1 3x x y 2 12 det(Dx(ǫi ) ) = 4 = 3x (1 − x ). 0 0 3x xy − x 0 0 2x10 x − x10 y The unique branch of Γ centred at P∞ = (0, 0, 0, 1) gives rise to the place P∞ . Then div(dx) = 6P∞ and E = 6P∞ ; see Lemma 12.1. Therefore the ramification divisor is P P R = div(x2 (1 − x12 )) + 72P∞ = 2 5i=1 Pi + 60 i=1 Qi + 2P∞ , where Pi and Qi are as in Example 5.36.

L EMMA 7.54 The divisor R depends only on the linear series L. Proof. From Theorem 6.17, that the divisors in L can also be written as Psuppose r Ad + B where Ad = div( k=0 dk yk ) + B with d = (d0 , . . . , dr ) and P = d in PG(r, K). From Lemma 7.51 (i), (ǫ )

(ǫ )

ordP (div(det(Dζ i xj )) = ordP (div(det(Dζ i yj )),

for every place P of Σ. Given any non-zero element h ∈ Σ, let P ∗ e∗P = −min{ordP (hxi )}, E ∗ = eP P. Then E ∗ + div(h) = E and (ǫ )

div(det(Dζ i (hxj ))) + (ǫ0 + · · · + ǫr )div(dζ) + (r + 1)E ∗ (ǫ ) = (r + 1)div(h) + div(det(Dζ i xj )) +(ǫ0 + · · · + ǫr )div(dζ) + (r + 1)E ∗ (ǫ ) = div(det(Dζ i xj )) + (ǫ0 + · · · + ǫr )div(dζ) + (r + 1)E.

Also, if η is another separable variable in Σ, then (ǫ )

div(det(Dη i xj )) + (ǫ0 + · · · + ǫr )div(dη) + (r + 1)E (ǫ ) = (ǫ0 + · · · + ǫr )(div(dζ/dη) + div(det(Dζ i xj )) +(ǫ0 + · · · + ǫr )(div(dη) + (r + 1)E (ǫ ) = div(det(Dζ i xj )) + (ǫ0 + · · · + ǫr )div[(dη) · (dζ/dη)] + (r + 1)E (ǫ ) = div(det(Dζ i xj )) + (ǫ0 + · · · + ǫr )div(dζ) + (r + 1)E.

This completes the proof.

2

From Corollary 5.35 and the definition of the genus (5.55), deg R = (ǫ1 + · · · + ǫr )(2g − 2) + (r + 1)n. T HEOREM 7.55 If j0 , . . . , jr are the (L, P)-orders, then Pr vP (R) ≥ i=0 (ji − ǫi )

with equality if and only if

det

  ji 6≡ 0 (mod p). ǫk

(7.13)

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Proof. Suppose again that eP = 0. For a local parameter ζ at P, vP (R) = ordP (det(Dtǫk xi )).

Arguing as in Theorem 7.47, it is possible to take yi ∈ Σ such that τ (yi ) = ci tji + · · · for i = 0, . . . , n,

where, as usual, ellipses indicate terms of higher orders. Then    ji ji −ǫr (ǫk ) + ··· t det(Dt τ (yi )) = det ǫr    ji ji = det t + · · · t−ǫ0 −···−ǫn ǫr   ji tj0 +···+jn −ǫ0 −···−ǫn + · · · . = det ǫr P (ǫ ) Thus, ordP (det(Dζ k yi ) ≥ (ji − ǫi ), with equality for   ji 6≡ 0 (mod p). det ǫk

2

By TheoremP7.55, the ramification divisor R is effective, and nP = 0 in the divisor R = nP P if and only if ji = ǫi all i. Hence ǫ0 , ǫ1 , . . . , ǫr are the (L, P)-orders for almost all places P of Σ; so these places are L-ordinary. The other places, finite in number, satisfy the inequality (j0 , . . . , jr ) 6= (ǫ0 , . . . , ǫr ) and are L-Weierstrass points; also, nP is the weight of P, and is denoted by vP (R). Note that, if ζ is a local parameter at P, then x0 x1 ... xr (ǫ1 ) (ǫ1 ) D(ǫ1 ) x Dζ x1 ... Dζ xr 0 ζ . (7.14) vP (R) = .. .. .. . . ... . (ǫr ) (ǫ ) (ǫ ) Dζ x0 Dζ r x1 ... Dζ r xr In this terminology, the number of L-Weierstrass points, each counted according to its weight, is P vP (R) = (ǫ0 + · · · + ǫr )(2g − 2) + (r + 1)n.

R EMARK 7.56 In the particular case that L is the canonical linear series of Σ, the notion of L-Weierstrass point is the same as in Section 6.6. An example for this particular case is worked out in Section 12.5. This has the following important corollary. C OROLLARY 7.57 Every curve which is neither rational, nor elliptic, nor hyperelliptic has finitely many Weierstrass points. Note that this corollary holds true for hyperelliptic curves; see Theorem 7.103.

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D EFINITION 7.58 The linear series L, or equivalently the curve Γ, is classical if the order sequence (ǫ0 , ǫ1 , . . . , ǫr ) is (0, 1, . . . , r). Now, some criteria are given that are useful in deciding whether or not a given linear series is classical. First, a proposition follows that refines the preceding inequality, ǫi ≤ ji . P ROPOSITION 7.59 Let j0 , . . . , jr be the (L, P)-orders. If m0 , . . . , mr are integers such that 0 ≤ m0 < · · · < mr and   ji 6≡ 0 (mod p), det mk then ǫi ≤ mi for each i = 0, . . . , r. Proof. As in the proof of Theorem 7.55,   ji (m ) tj0 −m0 +···+jn −mn + · · · . det(Dt k τ (yi )) = det mk Therefore ǫi ≤ mi for all i by the minimality of the ǫi .

2

C OROLLARY 7.60 Let j0 , . . . , jr be the (L, P)-orders. If Q i>s (ji − js )/(i − s)

is not divisible by the characteristic p of K, then L is classical and Pr vP (R) = i=0 (ji − i). Proof.

det

  ji k

= det(ji k /i! + · · · ) = det(ji k )/(1! 2! · · · n!) =

Q

(ji − js )/(i − s). Pr Hence ǫi = i by Proposition 7.59 and the weight of P in R equals i=0 (ji − i) by Theorem 7.55. 2 i>s

C OROLLARY 7.61 If the Prcharacteristic p of K is 0, or larger than n, then L is classical and vP (R) = i=0 (ji − i).

Proof. The condition in Corollary 7.60 is satisfied if ji 6≡ js (mod p) for all i 6= s; in particular, this holds when jn ≤ n. 2


L EMMA 7.62 If ǫ is an L-order and µ is such that µǫ 6≡ 0 (mod p), then µ is also an L-order. In particular, if ǫ < p, then 0, 1, . . . , ǫ − 1 are also L-orders.
Proof. Since µǫ 6= 0, so 0 ≤ µ ≤ ǫ. It may be supposed that µ > 0. Let k be the largest integer such that ǫk < µ. The matrix

 ǫ0
ǫ ǫk ... ǫ0 ǫ0 ǫ0  . .. ..    . . .   .  


ǫ ǫk   ǫ0 ...  ǫk ǫr  ǫk


ǫ0 ǫk ǫ ... µ µ µ

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ǫ

is triangular with 1, . . . , 1, µ on the main diagonal. Hence the rows are linearly independent over K. Thus ǫk+1 ≤ µ by Proposition 7.59, and so µ = ǫk+1 by the definition of k. 2 L EMMA 7.63 Assume that p ≥ r, and that the L-order ǫi = i for 0, 1, . . . , r − 1. If L is non-classical, then ǫr is a power of p. Proof. From Lemma 7.62, ǫr ≥ p. Assume, on the contrary, that ǫr is not a power
of p. By Lemma A.6, there is an integer k such that p ≤ k < ǫr and ǫkr 6≡ 0 (mod p). Then k is also an L-order. But this contradicts that ǫr−1 < k < ǫr . 2 In the classical case, the L-Weierstrass points are precisely the L-osculating points. In zero characteristic, every curve Γ is classical. T HEOREM 7.64 Let N be the number of Weierstrass points of a function field Σ which is neither rational, nor elliptic, nor hyperelliptic. If either p = 0 or p ≥ 2g − 2, then 2g + 3 ≤ N ≤ (g − 1)g(g + 1).

(7.15)

Proof. By Corollary 7.61, the canonical curve Γ of Σ is classical, and P vP (R) = (g − 1)g(g + 1).

For a point P ∈ Γ, let j0 = 0, j1 = 1, . . . , jg−1 be the orders of Γ at P . By Corollary 7.55, Pg−1 vP (R) = i=0 (ji − i).

Since vP (R) ≥ 1 for any Weierstrass point P, the upper bound follows. An upper bound for ji is ji ≤ 2i.

(7.16)

This comes from Clifford’s Theorem 6.79 applied to the fixed-point-free special linear series cut out on Γ by the hyperplanes H through P with intersection multiplicity I(P, H ∩Γ) at least ji . So, ji −i ≤ i for i = 2, . . . , g−1, and 0 = j1 −1 < 1. Hence vP (R) < 1 + 2 + · · · + g − 1 = 21 g(g − 1). Since the Weierstrass points are the points P ∈ Γ with vP (R) > 0, it follows that P vP (R) < 12 g(g − 1)N . Thus (g − 1)g(g + 1) < 12 g(g − 1)N . Therefore 2g + 3 is a lower bound for N . 2 If L is non-classical, every point is L-osculating. However, it should be noted that non-classical linear series are rare. T HEOREM 7.65 Let ǫ0 < ǫ1 < · · · < ǫr be the L-orders. (i) If ǫr ≥ pm , then there exist z0 , . . . , zr ∈ Σ such that m

m

m

z0p x0 + z1p x1 + · · · + zrp xr = 0.

(7.17)

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(ii) Let ǫr = pm and assume that (7.17) holds. Then the osculating hyperplane at all but finitely many points P is m

m

m

v(z0p (P )X0 + z1p (P )X1 + · · · + zrp (P )Xr ).

(7.18)

(iii) If ǫr−1 < pm , and (7.17) holds, then ǫr = pk with k ≥ m. Equality holds if and only if some zi is a separable variable of Σ. (ǫ )

Proof. Since det(Dζ i xj ) 6= 0, (i) follows from Theorem 5.86 and Lemma 5.84. Choose a point P ∈ Γ or, more precisely, a branch of Γ centred at P such that P is neither a pole of any zk , nor a zero of all the zk . For a primitive representation τ of the place corresponding to P , τ (zk ) = ak0 + ak1 t + · · · , for k = 0, . . . , r, and at least one term au0 is distinct from 0. Now, (7.17) can also be written as follows: m

m

m

m

apk0 x0 + · · · + ar0 zrp xr + (z0 − ak0 )p x0 + · · · + (zr − akr )p xr = 0. Then m

m

ordt (apk0 x0 (t) + · · · + ar0 zrp xr (t)) ≥ pm .

As ǫr = pm , this shows that

m

m

HP = v(apk0 X0 + · · · + apr0 Xr ) is the osculating hyperplane at P . Since, by definition, zk (P ) = ak0 for 0 ≤ k ≤ r, assertion (ii) holds. The previous argument also shows that, if ǫr−1 < pm−1 and (7.17) holds, then ǫr ≥ pm . From Lemma 7.62, ǫr = pk with k ≥ m. If zi is inseparable, then Lemma5.38 provides ui such that zi = upi . Hence, if no zi is separable, then m+1

u0p m+1

m+1

x0 + · · · + upr

xr = 0,

and hence ǫr ≥ p . Conversely, if ǫr = pk with k > m and ǫr−1 < pm , then x0 , . . . , xr are linearly dependent over Σk but independent over Σm . Hence, if (7.17) holds, then each m zip is Σk . By Lemma 5.84, zi = upi with ui ∈ Σ for 0 ≤ i ≤ r. Hence the last assertion of Theorem 7.65 follows from Lemma 5.38. 2 R EMARK 7.66 If ǫr = pm in Theorem 7.65, then the (r + 1)-ple (z0 , . . . , zr ) is determined up to a non-zero factor u ∈ Σ. This follows from (ii) by the uniqueness of the osculating hyperplane. E XAMPLE 7.67 Let p = 5. Take F and Γ as in Example 7.50(ii). Then ǫi = i for i = 0, 1, 2 but ǫ3 = 5. The hypotheses of Theorem 7.65 are satisfied for m = 1. Since (Y 5 + Y − X 3 )(Y 5 + Y + X 3 ) = (Y 2 )5 + 2Y Y 5 + Y 2 − XX 5 ,

so (7.17) holds for z0 = y 2 , z1 = −x, z2 = 2y, z3 = 1.

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7.7 NON-CLASSICAL CURVES AND LINEAR SYSTEMS OF LINES In this section, classical and non-classical curves with respect to lines are considered. Let F = v(f (X, Y )) be an irreducible plane curve. Choose any positive integer s smaller than the degree n of f (X, Y ). The linear system of all plane curves of degree s cuts out on F a gdr that has no fixed place. Here d = ns and r = 21 s(s+3). Theorem 7.65 together with Lemma 7.63 has the following corollary. T HEOREM 7.68 Suppose ns > p ≥ r = 12 s(s + 3). Then F is non-classical with respect to the linear series cut out on F by all curves of a given degree s if and only if there exist z0 (X, Y ), . . . , zr (X, Y ), h(X, Y ) ∈ K[X, Y ] and an integer m ≥ 1 such that m

m

h(X, Y )f (X, Y ) = z0 (X, Y )p + · · · + zr (X, Y )p Y s .

(7.19)

R EMARK 7.69 Rewording Theorem 7.68 in its homogeneous form, (7.19) reads as follows: Pr m h(X0 , X1 , X2 )f (X0 , X1 , X2 ) = j=0 zj (X0 , X1 , X2 )p X0u X1v X2w ;

here, zj (X0 , X1 , X2 ) are homogeneous of the same degree. This shows that interchanging X0 with X1 or X2 produces a similar equation. Therefore, in the local study of F, it is enough to consider branches of F centred at an affine point. For this purpose, the inhomogeneous equation (7.19) is used, as it is more manageable than the homogeneous one. The simplest case occurs when the linear system consists of all lines, that is, the linear series L1 cut out on F = v(f (X, Y )) consists of all divisors Ac = div (c0 + c1 x + c2 y) + B,

c = (c0 , c1 , c2 ) ∈ PG(2, K).

Without loss of generality,Ptake x to be a separable variable. So, ∂f (X, Y )/∂Y does not vanish. Let R = nP P be the ramification divisor. First, the classical case is considered; that is, the order sequence of L is assumed to be (0, 1, 2). Then deg R = 3(2g − 2) + 3n.

(7.20)

Assume p 6= 2. Let P be a a non-singular point of F. Then P is an L-Weierstrass point if and only if it is inflexion point of F. If this is the case, j2 = I(P, ℓ ∩ F) with ℓ the inflexional tangent to F at P , and nP = j2 − 2 where P is the place arising from the unique branch centred at P . Now, let P be a singular point of F, and γ a branch of Γ centred at P . Then γ, namely the branch point P of F, is an L-Weierstrass point if and only if P is non-linear and, when this occurs, nP = j1 + j2 − 3 for the place arising from P . All these results are straightforward consequences of the previous general results, especially Lemmas 7.62 and 7.61. A direct, elementary argument is also possible. Again let p 6= 2. Since ǫ2 = 2 6= p, the Hessian curve of F does not vanish; write its equation in inhomogeneous form  2  2 ∂f 2 ∂f ∂f ∂f 2 ∂f ∂f 2 ∂f − 2 = 0. + ∂ 2 X ∂Y ∂X∂Y ∂X ∂Y ∂ 2 Y ∂X

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Note that d2 y ∂f 2 = dx2 ∂2x



∂f ∂y

2

∂f 2 ∂f ∂f ∂f 2 −2 + 2 ∂x∂y ∂x ∂y ∂ y



∂f ∂x

2

.

Let P be a branch point of F. If (¯ x, y¯) is a primitive representation of P , put x¯′ = d¯ x/dt and x ¯′′ (t) = d2 x ¯/dt2 , and similarly for y¯. Then d2 y¯ (¯ x′ y¯′′ − x ¯′′ y¯′ ) = . 2 ′3 d¯ x x ¯ By Theorem 7.51 (i), x¯′ y¯′′ − x ¯′′ y¯′ is covariant. Therefore  2  2 ! ∂ f¯2 ∂ f¯ ∂ f¯ ∂ f¯2 ∂ f¯ ∂ f¯2 ∂ f¯ −2 + ordt ∂2x ¯ ∂ y¯ ∂x ¯∂y ∂ x ¯ ∂ y¯ ∂ 2 y¯ ∂ x ¯   ∂ f¯ = ordt (¯ x′ y¯′′ − x ¯′′ y¯′ ) + 3 ordt − ordt x ¯′ . ∂ y¯ Since the Hessian has degree 3(n − 2), B´ezout’s Theorem 3.14 gives the equation   ∂ f¯ − ordt x ¯′ . 3n(n − 2) = deg R + 3 ordt ∂ y¯ B´ezout’s Theorem applied to the polar curve at Y∞ shows that X ∂ f¯ = n(n − 1), ordt ∂ y¯ P while ordt x ¯′ = 2g − 2 by definition. Therefore (7.20) holds. In the non-classical case, the order sequence is (0, 1, pm ) with m ≥ 1 for p odd, and m ≥ 2 for p = 2. In particular, F is a non-reflexive curve whose Gauss map has inseparability degree pm ; this is the General Order of Contact Theorem. Theorem 7.65 together with Lemma 7.63 shows that m

m

m

z0p + z1p x + z2p y = 0, where z0 , z1 , z2 ∈ K(F) and at least one of them is separable. By Remark 7.66, the triple (z0 , z1 , z2 ) is determined up to a non-zero factor in K(F). After clearing denominators, m

m

m

h(X, Y )f (X, Y ) = z0 (X, Y )p + z1 (X, Y )p X + z2 (X, Y )p Y,

(7.21)

with z0 , z1 , z2 , h ∈ K[X, Y ]. P ROPOSITION 7.70 The factor h(X, Y ) is not required in (7.21) for non-singular curves; that is, h(X, Y ) = 1 may be assumed. T HEOREM 7.71 If deg F = q, then F is a rational strange curve and, up to a change of the coordinate system, F = v(X2q − (X1q−1 X0 + b2 X1q−2 X02 + · · · + bq−2 X12 X0q−2 + X1 X0q−1 )), with b2 , . . . , bq−2 ∈ K.

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T HEOREM 7.72 If deg F = q + 1, then, up to a change of the coordinate system, F is the Hermitian curve Hq , the rational nodal cubic or a strange curve v(X0 X2q + X0q−1 + a1 X0q X1 + a2 X0q−1 X22 + · · · + aq X0 X1q + X1q+1 )

with suitable a1 , . . . , aq ∈ K. E XAMPLE 7.73 The DLS curve F given in Example 5.24 is a non-classical curve with order sequence (0, 1, 2q0). This follows from Theorem 7.65 for z0 (X, Y ) = X q0 +1 + Y q0 ,

z1 (X, Y ) = X,

z2 (X, Y ) = 1.

7.8 NON-CLASSICAL CURVES AND LINEAR SYSTEMS OF CONICS In this section, classical and non-classical curves with respect to conics are considered. So, the case s = 2 in Theorem 7.68 is examined. The linear system of all 5 conics cuts out on F a simple, fixed-point-free linear series L2 = g2n . This gives rise to the irreducible curve Γ of PG(5, K) of degree 2n which is the image curve of F under the Veronese map, that is, the birational transformation ϕ:

(1, x, y) 7→ (1, x, y, x2 , xy, y 2 ).

Some or all L2 -orders may be calculated from the L1 -orders, where L1 = gn2 is cut out on F by all lines. To do this, two possibilities are considered separately according as L1 is classical or non-classical. Take any non-singular point P ∈ Γ which is not an L1 -Weierstrass point of F, and let P be the corresponding place. If F is non-classical for L1 , then the (L1 , P)-orders are (j0 = 0, j1 = 1, j2 = pm ) with pm > 2. Choose three lines ℓ0 , ℓ1 , ℓ2 such that I(P, ℓi ∩ Γ) = ǫi , for 0 ≤ i ≤ 2. There are six degenerate conics whose components are two of these three lines. They provide six different intersection multiplicities with F at P , namely 0, 1, 2, pm, pm + 1, 2pm . Therefore these integers are the (L2 , P)-orders, and also the L2 -orders due to the general choice of P . From now on, let F be classical with respect to L1 . The above argument shows that 0, 1, 2, 3, 4 are (L2 , P)-orders, and hence L2 -orders. Since the hypotheses of Theorem 7.68 are satisfied for p ≥ 5, so ǫ5 is a power of p. The condition p ≥ 5 cannot be omitted; see Exercise 3. For p ≥ 5, Theorem 7.65 shows that ǫ5 ≥ pm if and only if there exist z0 , . . . , z5 , h ∈ K[X, Y ] such that h(X, Y )f (X, Y ) m m m = z0 (X, Y )p + z1 (X, Y )p X + z2 (X, Y )p Y m m m +z3 (X, Y )p X 2 + z4 (X, Y )p XY + z5 (X, Y )p Y 2 .

(7.22)

Unlike the case r = 2, the non-singularity condition on F is not enough to ensure that h(X, Y ) can be assumed in this equation. An example is given in Exercise 5. Non-classicality of L2 gives rise to several interesting properties of the curve. Here, consider the case that p > 2, the curve F is classical with respect to L1 , and

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the L2 -orders of F are 0, 1, 2, 3, 4, pm. Put F = v(f (X, Y )), and let Q = (x, y) be a generic point of F. Then (7.22) holds, and at least one of the elements zi (x, y) in K(F) is a separable variable. Let P be a place of K(F) with (L2 , P)-orders 0, j1 , j2 . Choose an index j such that ordP zj (x, y) ≤ ordP zi (x, y), and put vi = zi (x, y)/zj (x, y). From (7.22), m

m

m

m

m

m

v0p + v1p x + v2p y + v3p x2 + v4p xy + v5p y 2 = 0.

(7.23)

Let γ be the branch of F corresponding to P. Without loss of generality, assume that γ is centred at an affine point, say at the point P = (a, b). Given a primitive representation (x(t), y(t)) of γ, write the formal power series vi (t) explicitly in the following form: (k)

(1)

(0)

+ µi t + · · · + µi tk + · · · .

vi (t) = µi

(7.24)

So (0)

m

(0)

(0)

m

(0)

m

m

(µ0 )p + (µ1 )p x(t) + (µ2 )p y(t) + (µ3 )p x(t)2 + m (1) m (1) m (0) m (0) m (µ4 )p x(t)y(t) + (µ5 )p y(t)2 + tp [(µ0 )p + (µ1 )p x(t)+ m m (1) pm (1) pm (1) (1) (µ2 ) y(t) + (µ3 ) x(t)2 + (µ4 )p x(t)y(t) + (µ5 )p y(t)2 ] + · · · + m (k) m (k) m (k) m (k) m tkp [(µ0 )p + (µ1 )p x(t) + (µ2 )p y(t) + (µ3 )p x(t)2 + (k) pm (k) pm (µ4 ) x(t)y(t) + (µ5 ) y(t)2 ] + · · · = 0.

For k = 0, 1, . . ., put

m

(k)

(k)

m

(k)

m

sk (X, Y ) = (µ0 )p + (µ1 )p X + (µ2 )p Y m

(k)

(k)

m

(k)

m

+(µ3 )p X 2 + (µ4 )p XY + (µ5 )p Y 2 . Then, with sk (t) = sk (x(t), y(t)), it follows that m

m

s0 (t) + tp s1 (t) + · · · + tkp sk (t) + · · · = 0.

(7.25)

As s0 (x, y) contains a non-zero coefficient, v(s0 (X, Y )) is a conic, possibly re(i) ducible. Putting µ0 = µi , the following result is obtained. L EMMA 7.74

(i) The conic C0 = v(s0 (X, Y )), with

m

m

m

m

m

m

s0 (X, Y ) = µp0 + µp1 X + µp2 Y + µp3 X 2 + µp4 XY + µp5 Y 2 , intersects the branch γ of F with multiplicity at least pm . (ii) This multiplicity is greater than pm if and only if s1 (a, b) = 0, where (1)

m

(1)

m

(1)

m

(1)

m

s1 (a, b) = (µ0 )p + (µ1 )p a + (µ2 )p b + (µ3 )p a2 (1)

m

(1)

m

+(µ4 )p ab + (µ5 )p b2 . (iii) If (a, b) is not a common zero of the six polynomials zi (X, Y ), then m

m

m

C0 = v(z0 (a, b)p + z1 (a, b)p X + z2 (a, b)p Y m

m

m

+z3 (a, b)p X 2 + z4 (a, b)p XY + z5 (a, b)p Y 2 ). (7.26)

In particular, for all but finitely many points P = (a, b) of F, the osculating conic of F at P is C0 .

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R EMARK 7.75 It may happen that a particular branch does not have C0 as osculating conic, as the following example shows. The Fermat curve F = v(F ), with F = X p−1 + Y p−1 + 1,

satisfies (i), since XY (X p−1 + Y p−1 + 1) = Y p X + X p Y + XY. Now, take a point P = (0, c) with cp−1 = −1 of F on the Y -axis. Then P is an inflexion of F; thus the osculating conic of F at P is degenerate and is v((Y −c)2 ), consisting of the horizontal line through P counted twice. On the other hand, the conic C0 is also degenerate but is v(X(Y − c)) consisting of the horizontal and vertical lines through P , since cp X + XY = X(Y − c). Looking back through this section, or alternatively looking forward to Section 8.7, the conic C0 plays a central role in the study of non-classical curves with respect to conics. When the symbol C0 is used it should be understood that C0 denotes the conic with equation (7.26). Introduce a new system of reference taking the centre (a, b) of the branch γ to the origin and the tangent ℓ of γ to the X−axis. ¯ Y¯ ) is given by This change of coordinates from (X, Y ) to (X, ¯ + v12 Y¯ + a, X = v11 X (7.27) ¯ + v22 Y¯ + b, Y = v21 X with v11 v22 − v12 v21 6= 0. Correspondingly, let K(F) = K(¯ x, y¯) such that x = y =

v11 x ¯ + v12 y¯ + a, v21 x ¯ + v22 y¯ + b.

(7.28)

Then ordP x ¯ = j1 , ordP y¯ = j2 , Equation (7.22) is invariant under this transformation. To see this, put ¯ + v22 Y + b) = z¯i (X, ¯ Y¯ ), ¯ + v12 Y¯ + a, v21 X zi (X, Y ) = zi (v11 X ¯ Y¯ ), ¯ + v22 Y¯ + b) = F (X, ¯ + v12 Y¯ + a, v21 X f (X, Y ) = f (v11 X ¯ Y¯ ), ¯ + v22 Y¯ + b) = H(X, ¯ + v12 Y¯ + a, v21 X h(X, Y ) = h(v11 X m

a = cp ,

m

b = dp ,

m

vij = nij p , i, j = 1, 2,

¯ Y¯ ). Then, with and write z¯i = z¯i (X, ¯ Y¯ ) = z¯0 + c¯ Z0 (X, z1 + d¯ z2 + c2 z¯3 + cd¯ z4 + d2 z¯5 , ¯ ¯ Z1 (X, Y ) = n11 z¯1 + n21 z¯2 + 2cn11 z¯3 + (cn11 + dn21 )¯ z4 + 2dn21 z¯5 , ¯ Y¯ ) = n12 z¯1 + n22 z¯2 + 2cn12 z¯3 + (cn22 + dn12 )¯ Z2 (X, z4 + 2dn22 z¯5 , ¯ Y¯ ) = n11 2 z¯3 + n11 n21 z¯4 + n21 2 z¯5 , Z3 (X, ¯ Z4 (X, Y¯ ) = 2n11 n12 z¯3 + (n12 n21 + n11 n22 )¯ z4 + 2n21 n22 z¯5 , ¯ Y¯ ) = n12 2 z¯3 + n12 n22 z¯4 + n22 2 z¯5 , Z5 (X, equation (7.22) becomes ¯ Y¯ )F (X, ¯ Y¯ ) H(X, ¯ + Z2 (X, ¯ Y¯ )pm Y¯ ¯ Y¯ )pm + Z1 (X, ¯ Y¯ )pm X = Z0 (X, ¯ Y¯ )pm X ¯ 2 + Z4 (X, ¯ Y¯ )pm X ¯ Y¯ + Z5 (X, ¯ Y¯ )pm Y¯ 2 . +Z3 (X,

(7.29)

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Since the relationship between Zi and zi is linear and invertible, min{ordP zi (x, y))} = min{ordP Zi (¯ x, y¯)}. Also, if w0 , . . . , w5 ∈ K[X, Y ], put

With this notation,



w0  1 ∆(w0 , . . . , w5 ) =  2 w1 1 2 w2

1 2 w2 1 2 w4

1 2 w1

w3 1 2 w4

w5

(7.30)



 .

ordP ∆(z0 (x, y), . . . , z5 (x, y)) = ordP ∆(Z0 (¯ x, y¯), . . . , Z5 (¯ x, y¯)). (7.31) ¯ Y¯ )/Zj (X, ¯ Y¯ ), and Vi = Wi (¯ Set Wi = Zi (X, x, y¯) where j is chosen such that ordP Zj (¯ x, y¯) = min{ordP Zi (¯ x, y¯)}. ¯ ¯ Then, in the new reference system (X, Y ), the conic m m ¯ + W2 (a, b)pm Y¯ C0 = v(W0 (a, b)p + W1 (a, b)p X m ¯ 2 + W4 (a, b)pm X ¯ Y¯ + W5 (a, b)pm Y¯ 2 ), +W3 (a, b)p X

which shows the covariance of C0 . From (7.29), m

m

m

m

(7.32)

m

m

¯2 + V4p x¯y¯ + V5p y¯2 = 0. ¯ + V2p y¯ + V3p x V0p + V1p x

Let ki = ordP Vi . Then the left-hand side is the sum of six elements from K(F) whose orders are as follows: m

m

¯ = k0 pm , ordP V1p x ordP V0p m pm ¯2 = k2 pm + j2 , ordP V3p x ordP V2 y¯ m m ¯y¯ = k4 pm + j1 + j2 , ordP V5p y¯2 ordP V4p x

= = =

k1 pm + j1 , k3 pm + 2j1 , k5 pm + 2j2 .

At least two of these orders must be equal, and they are less than or equal to the remaining four. Hence one of the following relations must hold: (a) (c) (e) (g) (i) (k) (m) (o)

(k0 − k1 )pm (k2 − k4 )pm (k0 − k2 )pm (k0 − k4 )pm (k1 − k2 )pm (k1 − k4 )pm (k1 − k5 )pm (k3 − k5 )pm

= j1 , = j2 , = 2j2 , = j1 + j2 , = j2 − j1 , = j2 , = 2j2 − j1 , = 2(j2 − j1 ).

(b) (d) (f) (h) (j) (l) (n)

(k1 − k3 )pm = j1 , (k0 − k3 )pm = 2j1 , (k2 − k5 )pm = j2 , (k0 − k5 )pv = 2j2 , (k3 − k4 )pm = j2 − j1 , (k4 − k5 )pm = j2 − j1 , (k2 − k3 )pv = 2j1 − j2 ,

If n < pm , then j2 < pm . This leaves just three possibilities; namely, (n) k2 = k3 ; (m) k1 = k5 + 1; (g) k0 = k4 + 1. In particular, the following result is obtained. T HEOREM 7.76 Let F = v(f (X, Y )) be an irreducible curve of degree n such that the following conditions are satisfied: (a) 3 ≤ n < pm ; (b) f (X, Y ) satisfies (7.22);

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(c) γ is a branch of F whose centre is the origin and whose tangent is the Xaxis; (d) the (L1 , P)-orders of a branch point P of F are 0, j1 , j2 . Then one of the following holds: ¯ 2 ), c 6= 0; j2 = 2j1 , and C0 = v(Y¯ + cX (i) m 1 j2 = 2 (j1 + p ), and C0 = v(Y¯ 2 ); (ii) ¯ Y¯ ). j2 = pm − j1 , and C0 = v(X (iii)

(7.33) (7.34) (7.35)

¯ Y¯ ) = H(X, ¯ Y¯ )F (X, ¯ Y¯ )). Then Let G(X,

m

m

m

¯ + z p Y¯ , GX¯ = HX¯ F + HFX¯ = Z1p + 2Z3p X 4 m m m ¯ + 2Z p Y¯ . GY¯ = HY¯ F + F FY¯ = Z p + Z p X

(7.36)

(7.37) Note that both FX¯ and FY¯ are not zero polynomials, otherwise F would be nonclassical with order sequence (0, 1, pv ) where v ≤ m. The Hessian v(G∗ (X, Y )) of the plane curve G = v(G(X, Y )), which may be reducible, is given by G∗ = GX¯ X¯ (GY¯ )2 − 2GX¯ Y¯ GX¯ GY¯ + GX¯ Y¯ (GX¯ )2 m m m m m m m m ¯ + 2Z p Y¯ ) ¯ + 2Z p Y¯ )2 − 2Z p (Z p + Z p X = 2Z p (Z p + Z p X 2

4

5

5 4 2 4 5 pm ¯ 2 pm ¯ pm pm pm ¯ + z4 Y ) + 2Z5 (Z1 + 2Z3 X + Z4 Y ) + m = 2{(Z3 Z22 − Z1 Z2 Z4 + Z5 Z12 )p m m m m m m ¯ Y¯ + Z p Y¯ 2 )} ¯ 2 + Zp X ¯ + Z p Y¯ + Z p X +(4Z3 Z5 − Z42 )p (Z1p X 5 4 3 2 m m m ≡ 2{(Z3 Z22 − Z1 Z2 Z4 + Z5 Z12 )p + (4Z3 Z5 − Z42 )p Z0p } 3

m (Z1p

2

4 pm ¯ 2Z3 X

¯ Y¯ )) (mod F (X,

pm

¯ Y¯ )). (mod F (X, ≡ −∆(Z0 , . . . , Z5 ) On the other hand, GX¯ X¯ (GY¯ )2 − 2GX¯ Y¯ GX¯ GY¯ + GY¯ Y¯ (GX¯ )2 ¯ Y¯ )). ≡ H 3 [FX¯ X¯ (FY¯ )2 − 2FX¯ Y¯ FX¯ FY¯ + FY¯ Y¯ (FX¯ )2 ] (mod F (X, It follows that H 3 [FX¯ X¯ (FY¯ )2 − 2FX¯ Y¯ FX¯ FY¯ + FY¯ Y¯ (FX¯ )2 ] m ¯ Y¯ )). ≡ −∆(Z0 , . . . , Z5 )p (mod F (X, (7.38) 3 It may be noted that the left-hand side, apart from H , gives the Hessian of F. Since FY¯ is not the zero polynomial, x ¯ is a separable variable of K(F). Hence, 2 2 2 F F − 2FX¯ Y¯ FX¯ FY¯ + FY¯ Y¯ FX ¯ ¯ d y¯ ¯ X X Y¯ , (7.39) = − d¯ x2 FY¯3 where the polynomials on the right-hand side are evaluated at (¯ x, y¯). Therefore m d2 y¯ = ∆(Z0 (¯ x, y¯), . . . , Z5 (¯ x, y¯))p /(H(¯ x, y¯)FY¯ (¯ x, y¯))3 . (7.40) d¯ x2 As F is classical with respect to L1 , this shows that ∆(Z0 (¯ x, y¯), . . . , Z5 (¯ x, y¯)) 6= 0. If R is the ramification divisor with respect to the linear series cut out on F by lines, the following result is obtained.

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L EMMA 7.77 vP (R) = pm ordP ∆(Z0 (¯ x, y¯), . . . , Z5 (¯ x, y¯)) + 3 ordP d¯ x m pm pm −3[ordP (Z2 (¯ x, y¯) + Z3 (¯ x, y¯) x ¯ + 2Z5 (¯ x, y¯)p y¯)]. (7.41) Now, a geometric interpretation of (7.41) is given. For this purpose, the following conditions are assumed satisfied: (i) z0 (x, y), . . . , z5 (x, y) are linearly independent over K; (ii) the linear system, λ0 z0 (X, Y ) + · · · + λ5 z5 (X, Y ),

cuts out a simple linear series L′2 .

It may be noted that both conditions are satisfied when n < pm , but the proof is postponed to the end of the section. As usual, it is also assumed that L′2 is fixedpoint-free. Let Z be the image curve of F under the rational transformation ϕ′ :

(1, x, y) 7→ (z0 (x, y), z1 (x, y), z2 (x, y), z3 (x, y), z4 (x.y), z5 (x, y)).

By condition (ii), Z is an irreducible curve of PG(5, K), birationally equivalent to F over K. It may be noted that Z can be viewed as the dual curve of Γ. In fact, Z is birationally equivalent to the dual curve of Γ as the Gauss map associated to Γ is the product of ϕ′ by the Frobenius transformation of PG(5, K). For more details on dual curves, see Section 7.9. As in Section 7.6, points of Z are considered as branch points, so that the term ‘point P of Z’ stands for a branch γ of Z associated to the place P. If γ is centred at A and Φ is a hypersurface of PG(5, K), then I(P, Z ∩ Φ) is used to indicate I(A, γ ∩ Φ). Apart from Z, it is appropriate to consider the cubic hypersurface Φ = v(det(∆(X0 , . . . , X5 ))).

This depends on the fact that I(P, Z ∩ Φ) = ordP det ∆(z0 (x, y), . . . , z5 (x, y) − min ordP (zi (x, y)).

In particular, Z is not contained in Φ, and B´ezout’s Theorem 3.14 gives that P 3 deg Z = I(P, Z ∩ Φ). (7.42) According to (7.30) and (7.31), the computation of the orders on the right-hand side can be carried out in the new reference system. Let ordP Zj (¯ x, y¯) = min ordP (zi (x, y)); then vP (R) = pm I(P, Z ∩ Φ) + 3 ordP d¯ x − 3ordP B(¯ x, y¯) with B(¯ x, y¯) =



Z2 (¯ x, y¯) Zj (¯ x, y¯)

pm

+



Z3 (¯ x, y¯) Zj (¯ x, y¯)

pm

x ¯+2



Z5 (¯ x, y¯) Zj (¯ x, y¯)

pm

(7.43)

y¯.

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Orders on the right-hand side can be directly computed for each of the 15 possibilities listed above. As in Theorem 7.76, only the case n < pm is treated. Summing (7.43) gives the result P pm deg Z = c − n + deg B, (7.44) where c denotes the class of F. If (7.33) holds, then ordP Z2 (¯ x, y¯) = ordP Z3 (¯ x, y¯) is strictly less than the orders of the four other Zi (¯ x, y¯). Therefore m

m

B(¯ x, y¯) = 1 + [Z4 (¯ x, y¯)/Z2 (¯ x, y¯)]p + [Z5 (¯ x, y¯)/Z2 (¯ x, y¯)]p x ¯. Hence ordP B(¯ x, y¯) = 0. Assume that (7.34) holds. This time, ordP Z5 (¯ x, y¯) is strictly less than the orders of the other ordP Zi (¯ x, y¯). Therefore m

m

B(¯ x, y¯) = [Z2 (¯ x, y¯)/Z5 (¯ x, y¯)]p + [Z4 (¯ x, y¯)/Z5 (¯ x, y¯)]p x ¯ + 2¯ y.

Since ordP y¯ < pm , so ordP B(¯ x, y¯) = j2 . Finally, if (7.35) holds, then ordP Z4 (¯ x, y¯) is strictly less than the orders of the other ordP Zi (¯ x, y¯). Therefore m m ¯ + 2[Z5 (¯ B(¯ x, y¯) = [Z2 (¯ x, y¯)/Z4 (¯ x, y¯)]p + X x, y¯)/Z4 (¯ x, y¯)]p y¯. Since ordP x ¯ < pm , so ordP B(¯ x, y¯) = j1 . As ordP d¯ x ≥ j1 − 1, the following result is obtained. L EMMA 7.78 Let n < pm . Then

 when (7.33) holds;  3(j1 − 1) −3(j2 − j1 + 1) when (7.34) holds; vP (R) ≥ pm I(P, Z ∩ Φ) +  −3 when (7.35) holds.

Since vP (R) ≥ j1 + j2 − 3, this yields the following inequality:  6 when j1 = 1 and (7.34) holds; m 4vP (R) ≥ p I(P, Z ∩ Φ) − 0 otherwise.

(7.45)

If j1 < p, then ordP d¯ x = j1 −1, and (j1 −0)(j2 −0)(j2 −j1 ) 6≡ 0 (mod p) in each of the three cases in Theorem 7.76. Hence, by Corollary 7.60, vP = j1 + j2 − 3, and equality holds in Lemma 7.78. Therefore the following result holds. L EMMA 7.79 If j1 < p, then

  0 I(P, Z ∩ Φ) = 2  1

when (7.33) holds; when (7.34) holds; when (7.35) holds.

So far, places corresponding to branches centred at affine points have been considered. However, these results hold true when the centre is an infinite point, as in Remark 7.69. Now, summing the inequality (7.45) as P ranges over all branch points of F centred at an affine point, 12(2g − 2) + 12n ≥ 3pm deg Z − 6N,

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237

where N denotes the number of linear branches P satisfying (7.34). Since vP (R) ≥ j1 + j2 − 3 = 12 (pm − 3), for any P counted in N , an upper bound on N is 3(2g − 2) + 3n N ≤2 pm − 3. Therefore the following theorem is established. T HEOREM 7.80 Let 3 ≤ n < pm . Then 4 [(2g − 2) + n]. deg Z ≤ m p −3 Equality holds if and only if F is non-singular. In this case, (7.46) reads 4n(n − 2) deg Z = . pm − 3

(7.46)

(7.47)

E XAMPLE 7.81 Let q be an even power of the characteristic p of K. When the √ degree n = 21 ( q + 1), the Fermat curve F = v(X n + Y n + 1) is a non-classical √ curve with respect to L2 , as the L2 -orders are (0, 1, 2, 3, 4, q). In fact, (7.22) holds for h = (X n + Y n − 1)(X n − Y n + 1)(X n + Y n − 1), z0 = 1, z1 = −2X, z2 = 2Y, z3 = X 2 , z4 = −2XY, z5 = Y 2 . Since the six curves v(Zi (X, Y )) have no common points, deg Z = 2n. On the other hand, F has exactly 3n inflexion points, and its branch points satisfying (7.34) are precisely its inflexion points.

As mentioned before, conditions (i) and (ii) are satisfied when n < pm . Now, a proof is given by using some of the above results. To prove (i), assume on the contrary that a non-trivial linear combination of zi (x, y) is zero; that is, λ0 z0 (x, y) + · · · + λ5 z5 (x, y) = 0, (7.48) with λi ∈ K not all zero. From B´ezout’s Theorem 3.14, Φ and Z have non-trivial intersection. Choose a branch point P of Z whose centre (z0 (P ), . . . , z5 (P )) is a point of Φ. Suppose, without loss of generality, that P arises from the branch γ under ϕ. From (7.48) it follows that neither (7.34) nor (7.35) can actually occur in the Theorem 7.76. To show this, note first that since the relationship between Zi (¯ x, y¯) and zi (x, y) are linear and invertible, (7.48) implies that ρ0 Z0 (¯ x, y¯) + · · · + ρ5 Z5 (¯ x, y¯) = 0, (7.49) with ρi ∈ K not all zeros. Therefore at least two of the six numbers ordP Zi (¯ x, y¯) are equal and less than the remaining four. This means that just one case, namely (n), can occur from the previous long list (a), . . . , (o). Hence, (7.33) is the only possibility in Theorem 7.76. In particular, C0 is a non-degenerate conic, j = 2 or j = 3, and ordP det ∆(Z0 (¯ x, y¯, . . . , Z5 (¯ x, y¯)) − ordP Zj (¯ x, y¯) = 0. But then I(P, Z ∩ Φ) = 0, a contradiction. If L′2 is composed of an involution of order r, then n > rpm , which is impossible for n < pm . Thus, (ii) is satisfied when b < pm . Hence, Theorem 7.46 holds for any non-classical curve F with order sequence (0, 1, 2, 3, 4, pm) with respect to the linear series L2 cut out on F by all conics.

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7.9 DUAL CURVES OF SPACE CURVES In Section 5.11, the dual curve of a plane curve is investigated. Now, duality for space curves is considered. Let Γ be a non-degenerate irreducible curve of PG(r, K). D EFINITION 7.82 The dual curve of Γ is the irreducible curve Γ′ of PG(r, K) containing all but finitely many points P ′ = (b0 , . . . , br ) such that the hyperplane v(b0 X0 + · · · + br Xr ) in PG(r, K) is an osculating hyperplane of Γ. Corollary 7.49 shows how to obtain the dual curve of a space curve as its image under a rational transformation. For every i = 0, . . . , r, delete the last row and the i-th column in the Wronskian determinant (7.12), and denote by Fi the resulting (r − 1) × (r − 1) determinant multiplied by (−1)r+i . Then the rational transformation ωC given by the formula ωC (x′0 , x′1 , . . . , x′r ) = (F0 , F1 , . . . , Fr ) is the Gauss map associated to Γ, and the curve Γ′ given by the point P ′ = (F0 , F1 , . . . , Fr ) is the dual curve of Γ. Note that, from the definition of the Fi , F0 x0 + · · · + Fr xr = 0,

(1) F0 Dζ x0

(1)

+ · · · + Fr Dζ xr = 0.

The geometric meaning of these equations is that the osculating hyperplane contains the tangent line. As for plane curves, the bidual Γ′′ of Γ is the dual of Γ′ . Also, ω ′ stands for the Gauss map associated to Γ′ . The argument used in Section 5.11 can be adapted to the study of duality of space curves. However, the present situation is much more involved in positive characteristic and no details are given here. Nevertheless, the main results are stated together with illustrative examples. Let ǫ0 < ǫ1 < · · · < ǫr be the orders of a nondegenerate irreducible curve Γ of PG(r, K). Denote by degi ω the inseparability degree of the Gauss map. T HEOREM 7.83 The highest power of p dividing ǫr is equal to degi ω. T HEOREM 7.84 Suppose that the dual curve Γ′ of Γ is also non-degenerate and let ǫ′0 < ǫ′1 < . . . ǫ′r be its orders. For a rational transformation δ : Γ → Γ′ whose inseparability degree is equal to q degi ω with q ≥ 1, the following conditions are equivalent: (i) Γ = Γ′′ and δ = ω ′ ◦ ω; (ii) q divides ǫ′r and, for a general point P ∈ Γ, (d)

(r−1)

Lω(P ) (Γ) ⊂ LP

(Γ),

where the integer d is defined by ǫd < ǫ′r /q ≤ ǫd+1 .

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Also, if this is the case, then ǫd+1 = ǫ′r /q, and ǫd+1 6≡ 0 (mod p). E XAMPLE 7.85 Let the irreducible plane curve F = v(f (X, Y )) be non-classical with orders 0, 1, q ′ , where q ′ = pm > 2. Then (7.21) holds, and hence the tangent to F at a general point Q is ′

′

′

ℓQ = v(z0 (Q)q + z1 (Q)q X + z2 (Q)q Y ). ′

(7.50)

gdr

Suppose that s < q and let be the linear series introduced before Theorem 7.68. The corresponding rational transformation τ is given by x′ij = xi y j with 0 ≤ i + j ≤ s, and it defines the irreducible curve Γ = τ (F) in PG(r, K). The orders of Γ are all integers in the set {u + vq ′ | u + v ≤ s}. In particular, ǫr = sq ′ , and the equation of the osculating hyperplane of Γ at the general point δ(Q) is obtained formally raising the linear polynomial in (7.50) to the s-th power and replacing X i Y j with Xij . Here, the Xij with 0 ≤ i + j ≤ s, denote the non-homogeneous coordinates in PG(r, K). ′ Let δ be the qq ′ -th Frobenius transformation. If ℓQ contains δ(Q) = Qqq for a general point Q, then (s−1)

(r−1)

Lτ (P ) (Γ) ⊂ LP

(Γ),

(s)

(r−1)

Lτ (P ) (Γ) 6⊂ LP

(Γ),

for a generic point P . Additionally, Γ′ is not degenerate provided that p > s. Also, Γ′ coincides with Γ up to a linear collineation associated to a diagonal matrix. In particular, Γ and Γ′ have the same degrees. T HEOREM 7.86 Suppose the following conditions hold: (a) ǫ0 < ǫ1 < . . . < ǫr are the orders of a non-degenerate irreducible curve Γ of PG(r, K); (b) the dual curve Γ′ of Γ is non-degenerate; (c) ǫ′0 < ǫ′1 < . . . ǫ′r are the orders of Γ′ ; (d) ǫr · ǫ′r 6≡ 0 (mod p). Then Γ′′ = Γ if and only if ǫ′r ≤ ǫr . If this is the case, then ω ′ ◦ ω is the identity map and ǫr = ǫ′r . E XAMPLE 7.87 Let q > 3. The rational curve Γ given by the point Q = (1, x, x2 , x3 , xq + x2q+3 , xq+1 + x2q+2 ) is a non-degenerate curve in PG(5, K) with order sequence (0, 1, 2, 3, q, q+1). The dual curve Γ′ is also non-degenerate and its orders are (0, 1, q, q + 1, 2q, 2q + 1). This gives an example of a curve with ǫr ǫ′r 6≡ 0 (mod p). Thus, by Theorem 7.86, Γ 6= Γ′′ . Another generalisation of the concept of dual curve of a plane curve is the
Gaussian dual, which is an irreducible curve ∆ in PG(r′ , K) with r′ = r+1 − 1, 2 obtained from Γ by the following Gauss map, via the classical Pl¨ucker embedding of the lines of PG(r, K) in PG(r′ , K).

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Let Γ be given by the point Q = (1, x1 , . . . , xr ), where x = x1 is a separable variable of K(Γ). The 2 × 2 minors of the matrix   1 x x2 ··· xr (1) (1) 0 1 Dx x2 · · · Dx xr can be arranged in a sequence of r′ + 1 elements, say, y0 = 1, y1 , . . . , yr′ . Then the irreducible curve ∆ given by the point R = (1, y1 , . . . , yr′ ) is the Gaussian dual of Γ. Also, the points Q = (1, x, . . . , xr ) and R define a rational transformation ω of K(Γ). Put Σ = K(Q) = K(Γ) and Σ′ = K(R). The first of the above minors is equal to 1, while the others are αi = Dx(1) xi ,

2 ≤ i ≤ r;

βi = xi − xαi , 2 ≤ i ≤ r; γij = xi αj − xj αi , 2 ≤ i < j ≤ r. Since γij = xi αj − xj αi = (βi + xαi )αj − (β + xaj )αi = βi αj − βj αi , it follows that Σ′ = K(α2 , . . . , αr , β2 , . . . , βr ). On the other hand, since Σ = K(x, x2 , . . . , xr ) and xi = xαi + βi for every 2 ≤ i ≤ r, so Σ = Σ′ (x) = K(α2 , . . . , αr , β2 , . . . , βr , x).

The following result is a generalisation of Theorems 5.90 and 5.91 to space curves. T HEOREM 7.88 Let ǫ0 = 0, ǫ1 = 1, ǫ2 , . . . , ǫr be the orders of Γ. Then ǫ2 = q ′ is a power of p if and only Σ/Σ′ is inseparable with inseparability degree q ′ . Proof. Let q ′ be a power of p, and let d be any power of p such that 1 ≤ d < q ′ . By Lemmas 5.83 (iii), 5.84, and Theorem 5.37, It suffices to show that the following conditions are equivalent: (i) for all 2 ≤ i ≤ r and i < j ≤ r, (ii) for all 2 ≤ i ≤ r,

Dx(d) αi = Dx(d) βi = Dx(d) γij = 0;

(7.51)

Dx(1) Dx(1) xi = Dx(d) xi = 0.

(7.52)

Now, Dx(d) βi

=

Dx(d) xi

−

Pd

(d−k) αi Dx(k) x k=0 Dx

=

(

(1)

−xDx αi (d) (d) Dx xi − xDx αi

if d = 1, if d > 1. (7.53)

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On the other hand, Dx(d) γij =

Pd

k=0

From (7.53) and (7.54), if

(Dx(k) xi Dx(d−k) αj − Dx(k) xj Dx(d−k) αi ).

(d) Dx αi

Dx(d) γij

= 0 for 2 ≤ i ≤ r and 1 ≤ d ≤ q ′ , then

= αj Dx(d) xi − αi Dx(d) xj ;  0 if d = 1, Dx(d) βi = (d) Dx xi if d > 1.

241

(7.54)

(7.55) (7.56)

By (7.55)and (7.56), it follows that (7.51) is equivalent to the condition Dx(d) αi = Dx(d) Dx(1) xj = 0, for all 2 ≤ i ≤ r and d ≥ 1, while

Dx(d) xi = 0,

when d > 1. This condition is equivalent to (7.52).

2

7.10 COMPLETE LINEAR SERIES OF SMALL ORDER It is a difficult problem to determine the smallest n such that a curve has a complete linear series of order n and positive dimension, that is, a gnr with r > 0; see also Definition 9.49. In this section, those curves are investigated for which this minimum order is either 1 or 2. By Theorem 6.81 the former possibility only occurs for rational curves. Also, by Theorems 6.81 and 7.42, if a curve has more than one g21 , it is either rational or elliptic, and in the latter but not in the former case, g21 is complete. Also, if a curve has genus 2 then it is hyperelliptic since its canonical series is a g21 . T HEOREM 7.89 An irreducible plane curve F = v(f (X, Y )) has genus 1 if and only if F is birationally equivalent to a plane non-singular cubic curve. A canonical form for f (X, Y ) is one of the following: (i) Y 2 − X 3 − uX − v with 4u3 + 27v 2 6= 0, for p 6= 2, 3; (ii) Y 2 − X 3 − uX 2 − vX − w with v 3 = u3 w, for p = 3; (iii) Y 2 + uY + X 3 + vX + w with u 6= 0, for p = 2; (iv) Y 2 + XY + X 3 + uX + v with v 6= 0, for p = 2. Proof. By Theorem 5.57, any non-singular cubic curve has genus 1. Conversely, let C be an elliptic curve, and take a place P of Σ = K(C). By Theorem 7.40, the cubic curve derived from the rational transformation associated to the complete linear series |3P| of Σ is a non-singular plane curve F. Take the coordinate system in the plane such that the line at infinity is a tangent to F at the point Y∞ corresponding to P. Then F = v(f (X, Y )) with f (X, Y ) containing only one term of degree 3, namely X 3 . Replacing X by cX and Y by dX for suitable non-zero elements c, d ∈ K, the polynomial f (X, Y ) becomes the following: Y 2 + a1 XY + a3 Y − X 3 + a2 X 2 + a4 X + a5 .

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If p 6= 2, it is possible to get rid of both terms XY and Y by replacing Y by aX + bY + c for suitable a, b, c ∈ K. So, without loss of generality, take a1 = a3 = 0. For p = 3, this leads to (ii), the condition v 3 = u3 w being necessary and sufficient for F to be non-singular. If p 6= 2, 3, the X 2 -term can also be removed by replacing X by X − 31 a2 , giving the canonical form (i). The condition 4u3 + 27v 2 6= 0 says that F is non-singular. Suppose p = 2. If a1 6= 0, then linear transformations of X and Y can be made to give a3 = a4 = 0 and a1 = 1. If a1 = 0, then a3 6= 0 and a2 = 0 can be obtained by a linear transformation of X. Also, a linear transformation in Y gives a3 = 1. The resulting canonical forms are those in (iii) and (iv); the conditions are necessary and sufficient for F to be non-singular. 2 For p 6= 2, 3, the following result states when two plane non-singular cubics of PG(2, K) are birationally equivalent. T HEOREM 7.90 (Salmon) For p 6= 2, 3, let F be a plane non-singular cubic. (i) There are four tangents to F from a point P ∈ F, other than the tangent at P, and the cross-ratio of the four tangents or, more precisely, the set of the six cross-ratios is constant. (ii) Two plane non-singular cubics are birationally equivalent if and only if they have the same cross-ratio. (iii) Two plane non-singular cubics have the same cross-ratio if and only if they have the same j-invariant, with j = j(F) =

u3 . 4u3 + 27v 2

T HEOREM 7.91 (Salmon) Two non-singular cubics are birationally equivalent if and only if they are projectively equivalent. Proof. Let P be any non-inflexion point of a non-singular plane cubic F. If ℓ is the tangent to F at P , then ℓ meets F at a second point, say P ′ . Let L be the simple, base-point-free, linear series cut out on F, up to the fixed divisor 2P ′ + P , by the conics containing P and tangent to F at P ′ ; then L = |3P |. Since deg L = 3, the Riemann–Roch Theorem 6.61 gives dim L = 2; hence L = g32 . From Theorem 7.40, the curve Γ arising from L is non-singular. By Theorem 7.33, L is cut out on Γ by lines. Hence P is an inflexion point of Γ. This shows that, if P is any place of K(F), there is a non-singular model Γ of K(F) such that the unique branch centred at P is an inflexion of Γ. Now, assume that ω is birational transformation of F to another non-singular plane cubic F ′ . Let O′ be an inflexion point of F ′ . Choose P ∈ F such a way that O′ is the image of P under ω. If P is not an inflexion point, let τ be the birational transformation of K(F) onto itself associated to the linear series L. Then ρ = ωτ −1 is a birational transformation of F to F ′ which takes the inflexion point P of Γ to O′ . Let L′ be the linear series cut out on F ′ by lines. By Corollary 6.105 (ii), ρ takes L to L′ . Therefore ρ sends every one-dimensional linear subseries of

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L to another contained in L′ . Since such a one-dimensional linear subseries is cut out by lines through a point, this shows that ρ is a collineation taking Γ to Φ′ . As deg Γ = deg F ′ , such a collineation is a projectivity of PG(2, K). 2 For more on elliptic cubics, see Sections 9.9, 9.10. Now, hyperelliptic curves are investigated. T HEOREM 7.92 Every hyperelliptic curve of genus g is birationally equivalent to a curve Γ0 of degree g + 2 with only one singular point. This singular point is a g-fold point, and the reference system can be chosen so that Γ0 = v(a0 (X)Y 2 + a1 (X)Y + a2 (X)).

(7.57)

The unique g21 is cut out by the vertical lines, apart from a fixed divisor gP, where P is the place corresponding to the g-fold point Y∞ of Γ0 . Proof. Let Γ be irreducible of genus g. There exist distinct places P1 , . . . , Pg of Σ = K(Γ) which arise from branches of Γ centred Pg at non-singular points P1 , . . . , Pg such that the linear series |E|, with E = i=1 Pi , is non-special. Now, assume Γ to be hyperelliptic, and let D = Q1 + Q2 be a divisor in the unique g21 of Σ such that the branches of Γ corresponding to Q1 and Q2 have different centres Q1 and Q2 . Since the linear series |D + E| is non-special, from the Riemann–Roch Theorem 6.61, dim |D + E| = 2. Then, since i(|E|) = −1, so i(|E − Pi |) = 0 for each i = 1, . . . , g. Therefore there is a unique canonical divisor Ci containing E − Pi . If |D + E| has a fixed place P, then P is one of the places in E because g21 has no fixed place. Also, i(|D + E − Pi |) = 1; that is, there is a unique canonical divisor Ci containing D + E − Pi . Therefore, to ensure that D + E has no fixed place, it suffices to choose Q1 different from the finitely many places appearing in 2 the canonical divisors Ci . So, suppose that D + E = gg+2 has no fixed place. Let g21 = {div(c0 + c1 x) + D | c = (c0 , c1 ) ∈ PG(1, K)}. Then there exists y ∈ K(Γ) such that

2 gg+2 = {div(c0 + c1 x + c2 y) + D + E | c = (c0 , c1 , c2 ) ∈ PG(2, K)}.

2 Since gg+2 has no fixed place, from Theorem 6.82 it follows that div(z)∞ = D+E for some z ∈ Σ. Hence

z = c0 + c1 x + c2 y with [Σ : K(z)] = g + 2. In particular, z cannot be written as d0 + d1 x with d0 , d1 ∈ K. Therefore y may be replaced by z in the above representation of |D + E|. By Theorem 7.44, z 6∈ K(x), whence it follows that Σ = K(x, z). In other words, |D + E| is simple, that is, not composed of an involution. Hence, up to birationally equivalence, Γ may be supposed to be an irreducible plane curve of 2 degree g + 2, and gg+2 is cut out on Γ by lines. ′ ′ ′ If D = Q1 + Q2 and D′ = D + div(ξ) for some ξ ∈ Σ, then D′ + E = D + E ′ + div(ξ).

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In particular, if the branches of Γ corresponding to Q′1 and Q′2 have different centres Q′1 and Q′2 , then the common point of the lines Q1 Q2 and Q′1 Q′2 is the common centre U of the branches corresponding to P1 , . . . , Pg . This is possible only when U is a g-fold point of Γ. This implies that Γ has no more singular points. In fact, its virtual genus g ∗ is equal to g, since g ∗ ≤ 21 ((g + 1)g − g(g − 1)) = g,

while g ∗ ≥ g by the remark after Definition 5.55. Without loss of generality, U may be assumed to be Y∞ . Then Γ = v(g(X, Y )) with g(X, Y ) = a0 (X)Y 2 + a1 (X)Y + a2 (X), where deg g(X, Y ) ≤ 2g + 2. The unique g21 is cut out on Γ by the vertical lines, minus the fixed divisor E. 2 C OROLLARY 7.93 A hyperelliptic curve of genus g is birationally equivalent to a curve Γ1 = v(Y 2 + h(X)Y + g(X)),

(7.58)

with h(X), g(X) ∈ K[X] and deg h(X) ≤ g + 1, deg g(X) ≤ 2g + 2.

Proof. Let Γ = v(a0 (X)Y 2 + a1 (X)Y + a2 (X)). For a generic point P = (x, y) of Γ, the map τ : x′ = x, y ′ = a0 (x) is a rational transformation of the function field K(Γ) = K(x, y) of Γ. Then, τ is birational, and the image curve Γ′ = v(g ′ (X, Y )), with g ′ (X, Y ) = Y 2 + h(X)Y + g(X), where h(X) = a1 (X) and g(X) = a0 (X)a1 (X). This proves the corollary.

2

Two cases are distinguished according as K has even characteristic or not. T HEOREM 7.94 Let K to be of zero or odd characteristic. Then (i) a curve of genus g is hyperelliptic if and only if it is birationally equivalent to a curve X = v(Y 2 − f (X)), where the polynomial f (X) ∈ K[X] has degree 2g + 1 and no square factors ; (ii) the unique g21 is cut out by the vertical lines apart from a fixed divisor ; (iii) there are finitely many places P of K(X ) such that 2P ∈ g21 . Proof. In zero or odd characteristic, the form (7.57) can be simplified further by means of the birational transformation x′ = x, y ′ = y + 21 h(x) to the form v(Y 2 − f (X)),

f (X) ∈ K[X], deg f (X) ≤ 2g + 2.

(7.59)

Suppose that f (X) = g(X)2 k(X), with g(X), k(X) ∈ K[X], where k(X) has no square factor. Then the birational transformation, x′ = x, y ′ = g(x)−1 y, reduces it to the same form (7.59) but so that f (X) has no square factors.

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Suppose deg h = 2m, and let α be a root of f (X). The birational transformation, x′ = (x − α)−1 , y ′ = yxm , reduces the curve to the form (7.59) with m odd. Example 5.59 shows that deg h = 2g + 1. An alternative proof depending on the Hurwitz formula is also possible. For a generic point P = (x, y) of Γ, let Σ = K(x, y) be the associated function field and Σ′ = K(x) its rational subfield. The field extension Σ/Σ′ has degree 2, and hence it is separable, as the characteristic of K is distinct from 2. From the calculation in Example 5.59, ω : Σ 7→ Σ′ only ramifies at the places corresponding to branches centred at the points P = (a, 0) with f (a) = 0 and at Y∞ . The ramification index is equal to 2 for each of these places. Hence, the degree of the different divisor D(Σ/Σ′ ) is equal to 2g + 2. If g(Σ) denotes the genus of Σ, then (7.6) becomes 2g(Σ) − 2 = 2(0 − 2) + 2g + 2,

whence g(Σ) = g. The places P such that 2P ∈ g21 are those arising from the branches centred at Y∞ and at the points (a, 0) with f (a) = 0. So, they are finitely many. 2 T HEOREM 7.95 Suppose the following conditions hold: (a) K is of zero or odd characteristic; (b) the polynomials g(X), h(X) ∈ K[X] have both degree 2g +1 and no square factors; (c) α0 , . . . , α2g and β0 , . . . , β2g are the roots of g(X) and h(X). Then the hyperelliptic curves Γ = v(Y 2 − f (X)),

Γ′ = v(Y 2 − g(X))

are birationally equivalent if and only if there is rational function w(X) = (aX + b)/(cX + d),

ad − bc 6= 0

which maps the set {α0 , . . . , α2g , ∞} onto the set {β0 , . . . , β2g , ∞}. Proof. For a generic point P = (x, y) of Γ, let Σ = K(x, y) be the function field of Γ. The unique g21 of Σ consists of all divisors div(c0 + c1 x) + B,

c0 , c1 ∈ K, (c0 , c1 ) 6= 0,

where B = 2P∞ and P∞ is the place corresponding to the unique branch of Γ centred at Y∞ ; see Example 5.59. Now, assume that Γ′ is a birationally equivalent to Γ. Then it may also be assumed that Σ = K(ξ, η) for a generic point Q = (ξ, η) of Γ′ . Taking into account the uniqueness of g21 , Theorem 6.17 ensures the existence of ζ ∈ Σ, for which 1 = ζ(cx + d),

ξ = ζ(ax + b)

with ad − bc 6= 0. Eliminating ζ gives ξ = (ax + b)/(cx + d). For 0 ≤ i ≤ 2g, let γi be the branch of Γ centred at the point Pi = (αi , 0). The corresponding place Pi of Σ is such that the divisor 2Pi belongs to g21 . In the

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model (Γ′ ; (ξ, η)) of Σ, the place 2Pi comes from a branch γ of Γ′ , centred at a point P , possible coincident with Y∞ . If σ is a primitive representation of Pi , then σ(ξ) = (aσ(x) + b)/(cσ(x) + d), whence either P is affine and its X-coordinate is (aαi + b)/(cαi + d), or P = Y∞ and cαi + d = 0. Hence w(αi ) ∈ {β0 , . . . , β2g , ∞}. The previous argument still works if γi is replaced by the branch of Γ centred at Y∞ , showing that w(∞) = b/d ∈ {β0 , . . . , β2g , ∞}. To show the converse, put Σ = K(x, y) with y 2 − f (x) = 0, Σ′ = K(x′ , y ′ ) with y ′2 − g(x′ ) = 0. Assume first that w(∞) = ∞. Then w(X) = aX + b. Relabel the roots of g(X) such that w(αi ) = βi for i = 0, . . . , 2g, and write Q2g f (X) = u i=0 (X − αi ), Q2g g(X) = v i=0 (X − βi ).

Then the birational transformation ω : ξ = ax + b, maps Γ onto Γ′ . In fact, since g(ξ) = v =v

η = ǫy Q2g

i=0

Q2g

i=0

Q2g

with

ǫ2 = a2g+1 v/u

(ξ − βi )

(ax + b − βi )

i=0 (ax + b − (aαi 2g+1 Q2g = va i=0 (x − ai ) 2g+1

=v

=a

+ b)

(v/u)f (x),

so η 2 − g(ξ) = a2g+1 (v/u)(y 2 − f (x)),

whence η 2 − g(ξ) = 0. Therefore not only P = (x′ , y ′ ) but also Q = (ξ, η) is a generic point of the curve v(Y 2 − g(X)). By Theorem 5.7, K(ξ, η) and K(x′ , y ′ ) are isomorphic over K. On the other hand, as ω is birational, K(x, y) and K(ξ, η) are isomorphic over K. The proof in the case w(∞) = ∞ is complete. If w(∞) 6= ∞, relabel the indices of the roots of f (X) and g(X) such that w(∞) = β0 , u(α0 ) = ∞, w(αi ) = βi , for i = 1, . . . , 2g. Then, w(X) = (β0 X + b)/(X − α0 ),

β0 αi + b = αi βi − α0 βi ,

for 1 ≤ i ≤ 2g. It follows that ω:ξ=

β0 x + b , x − α0

η=ǫ

y , (x − α0 )g+1

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with ǫ2 = vu−1 (b + α0 β0 )

Q2g

i=1

is a birational transformation of Γ onto Γ′ . In fact, Q g(ξ) = v 2g i=0 (x − βi )  2g  Y β0 x + b =v − βi x − α0 i=0

(β0 − βi ),

x − αi 1 Q2g (β0 − βi ) x − α0 i=1 x − α0 Q  v 1 2g = (b + α0 β0 ) f (x). i=1 (β0 − βi ) u (x − α0 )2g+2 = v(b + α0 β0 )

This shows that y 2 − f (x) = 0 implies η 2 − g(ξ) = 0. Now, the previous argument depending on Theorem 5.7 can be repeated to finish the proof. 2 Analogous results hold in the case of even characteristic. T HEOREM 7.96 When p = 2, a hyperelliptic curve is birationally equivalent to a curve Γ = v(Y 2 + h(X)Y + g(X))

(7.60)

as in (7.58), with no affine singular points and the leading coefficient of g(X) equal to 1. Proof. From Theorem 7.92, consider a hyperelliptic curve Γ in the form 7.60, with h(X), g(X) ∈ K[X], deg h(X) ≤ g + 1, deg g(X) ≤ 2g + 2. If P = (a, b) is an affine singular point of Γ, then, by a transformation (X, Y ) 7→ (X, Y + b), the point P = (a, 0); hence g(a) = 0. Since P is a singular point of Γ, so ∂(Y 2 + h(X)Y + g(X))/∂Y = 0 at (a, 0); hence h(a) = 0. Also, ∂(Y 2 + h(X)Y + g(X))/∂X = 0 at (a, 0). Putting g ′ (X) = dg(X)/dX, it follows that g ′ (a) = 0. Thus h(X) = (X − a)h1 (X),

g(X) = (X − a)2 g1 (X),

with h1 (X), g1 (X) ∈ K[X], deg h1 (X) < deg h(X), deg g1 (X) < deg g(X). The birational transformation x′ = x, y ′ = y/(x − a) maps Γ onto the curve Γ′ = v(Y 2 + h1 (X)Y + g1 (X)). By repeated application of this argument, a hyperelliptic curve with no affine singular point is obtained. 2 L EMMA 7.97 For p = 2, let Γ = v(Y 2 +h(X)Y +g(X)) be a hyperelliptic curve with no affine singular point. Then, under the birational transformation x′ = x,

y ′ = y + r(X),

with r(X) ∈ K[X], the image curve Γ′ of Γ has no affine singular points.

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Proof. It is shown that, if Γ′ had an affine singular point, then Γ would also have an affine singular point. Note that Γ′ = v(Y 2 + h(X)Y + f (X)), with f (X) = r(X)2 + h(X)r(X) + g(X). If P = (a, b) is a singular point of Γ′ , then both partial derivatives are zero at P . Putting r′ (X) = dr(X)/dX,

h′ (X) = dh(X)/dX,

g ′ (X) = dg(X)/dX

gives b2 + h(a)b + r(a)2 + h(a)r(a) + g(a) = 0, h(a) = 0, bh′ (a) + r′ (a)h(a) + r(a)h′ (a) + g ′ (a) = 0. Note that the third equation can also be written as (r(a) + b)h′ (a) + g ′ (a) = 0. Since ∂(Y 2 + h(X)Y + g(X))/∂Y = h(X), ∂(Y 2 + h(X)Y + g(X))/∂X = Y h′ (X) + g ′ (X), it follows that the point Q = (a, c), with c = r(a) + b, is a singular point of Γ. 2 L EMMA 7.98 In Theorem 7.96, the polynomials h(X), g(X) can be chosen such that deg g(X) ≥ 2 deg h(X) + 1.

(7.61)

Proof. Let Γ be as in Theorem 7.96. Put k = deg h(X), n = deg g(X). If n is even, then the birational transformation, x′ = x, y ′ = y + xn/2 , reduces it to the same form (7.58) with deg h(X) = k, deg g(X) < n. Lemma 7.97 ensures that no affine point becomes singular. If deg g(X) is still even, the argument can be repeated. Thus deg g(X) can be supposed to have odd degree 2m + 1. If k > m, then the birational transformation, x′ = x, y ′ = y + xk−1 , maps Γ onto the curve (7.58) with deg g(X) ≥ 2k − 1. Hence m ≥ k; that is, (7.61) holds. 2 L EMMA 7.99 In Lemma 7.96, the polynomials h(X), g(X) can be chosen so that deg g(X) = 2 deg h(X) + 1.

(7.62)

Proof. Assume that h(X) and g(X) satisfy (7.61); let deg g(X) = 2m+1. Choose an element a ∈ K for which h(a) 6= 0 6= g(a). Replacing X by X + a ensures that h(0) 6= 0 6= g(a). Now, x′ = x−1 , y ′ = yxm+1 is a birational transformation of Γ onto the curve Γ1 = v(Y 2 + h1 (X)Y + g1 (X)) with h1 (X), g1 (X) ∈ K[X] defined as follows: h1 (X) = h(X −1 )X m+1 , g1 (X) = g(X −1 )X 2m+2 . Note that deg h1 (X) = m + 1, and deg g1 (X) = 2m + 2. If g2m+2 is the leading coefficient of g1 (X), let u ∈ K be a root of the polynomial X 2 + g2m+2 .

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Then the birational transformation, x′ = x, y ′ = y + uxm+1 , maps Γ1 onto a hyperelliptic curve Γ2 as in (7.58) satisfying (7.62). If Γ2 has some affine singular points, use the argument employed in the proof of Theorem 7.96 to obtain a curve with no affine singular point. Such a curve is still of the form (7.58) with h(X), g(X) satisfying (7.62). 2 T HEOREM 7.100 For p = 2, a hyperelliptic curve of genus g is birationally equivalent to a curve Γ = v(Y 2 + h(X)Y + g(X)),

deg g(X) = 2g + 1, deg h(X) ≤ g

with the following properties: (i) Γ has no affine singular point; (ii) Γ has only one branch centred at its unique point at infinity Y∞ ; (iii) if P∞ is the place corresponding to Y∞ , then the g21 is cut out on Γ by the vertical lines minus the fixed divisor (2g − 1)P∞ ; (iv) there are finitely many places P of Σ such that 2P ∈ g21 . Proof. With the notation used in the preceding two proofs, k + 1 < n = 2m + 1 and hence Γ has a unique point at infinity, namely Y∞ . More precisely, Y∞ is a (2m − 1)-fold point of Γ. Now, it is shown that Γ has exactly one branch centred at Y∞ . Write h(X) = h0 + · · · + hk X k , with hk 6= 0, and let h(X, Y ) be the associated homogeneous polynomial; that is, h(X, Y ) = h0 Y k + · · · + hk X k .

Similarly, write g(X) = f0 + · · · + X 2m+1 , and let g(X, Y ) = f0 Y 2m+1 + · · · + X 2m+1 . Then, in homogeneous coordinates, Γ = v(Y 2 Z 2m−1 + Y Z 2m−k h(X, Z) + g(X, Z)). Take Y∞ to the origin O by the projectivity T : (X, Y, Z) 7→ (X, Z, Y ). The curve Γ is mapped to the curve Γ′ = v(Y 2m−1 Z 2 + Y 2m−k h(X, Y )Z + g(X, Y )) or, in inhomogeneous coordinates, Γ′ = v(Y 2m−1 + Y 2m−k h(X, Y ) + g(X, Y )). The mapping T takes the line at infinity to the X-axis, which is then a tangent to Γ′ at the origin.

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Now, it is shown that there is only one branch of Γ′ centred at O. Let (x(t), y(t)) be a primitive representation of a branch of Γ′ centred at the origin and tangent to the X-axis. Write x(t) = ati + · · · ,

y(t) = btj + · · · ,

with i < j.

Then y(t)2m−1 + y(t)2m−k h(x(t), y(t)) + g(x(t), y(t)) = 0.

(7.63)

Substitution gives (btj + · · · )2m−1 + (btj + · · · )2m−k h(ati + · · · , btj + · · · ) +g(ati + · · · , btj + · · · ) = 0, which shows that the terms of possible minimum degree in t are (2m − 1)j, (2m − k)j + ik, (2m + 1)i. Therefore one of the following cases occurs: (i) (2m − 1)j = (2m − k)j + ki < (2m + 1)i; (ii) (2m + 1)i = (2m − k)j + ki < (2m − 1)j; (iii) (2m − 1)j = (2m − k)j + ki = (2m + 1)i; (iv) (2m − 1)j = (2m + 1)i < (2m − k)j + ki. In fact, none of the first three cases occurs. To show that (i) is impossible, note that (2m−1)j = (2m−k)j+ki implies that (k−1)j = ki; hence i = k−1, j = k, since k − 1 and k are relatively prime. Then (2m − 1)j < (2m + 1)i, (2m − 1)k < (2m + 1)(k − 1), m < k,

contradicting that k ≤ m. Similarly for (ii), (2m+1)i = (2m−k)j+ki implies i = 2m−k, j = 2m−k+1. Thus (2m − k)j + ki < (2m − 1)j,

(2m − k)(2m − k + 1) + k(2m − k) < (2m − 1)(2m − k + 1), m < k, again a contradiction. To rule out (iii), (2m−1)j = (2m+1)i implies that i = 2m−1, j = 2m+1, as in (i), while (2m+ 1)i = (2m− k)j + ki implies that i = 2m− k, j = 2m− k + 1, as in (ii). This leaves (iv). The only positive integers i, j which can occur in (IV) are i = 2m − 1 and j = 2m + 1. Hence, Γ′ has a branch γ centred at the origin such that I(O, v(Y ) ∩ γ) = 2m + 1. But then no other branch of Γ′ is centred at the origin. This follows from Theorem 4.50, since I(O, Γ′ ∩ v(Y )) = 2m + 1.

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It remains to show that g = m. As i = 2m − 1 is odd, Theorem 4.27 gives a substitution of type t 7→ t = t + · · · , for which x(t) = x(t) = t

2m−1

,

y(t) = y(t) = αt

2m+1

+ ··· ,

with α ∈ K\{0}. Replace t by t and omit the bar over x and y. If y(t) contains some terms of odd degree, then βt2m+1+n denotes the lowest degree for which this happens. Otherwise, let β = 0. Write y(t) = αt2m+1 + · · · + βt2m+1+n + · · ·

with β ∈ K. Now, the left-hand side of (7.63) can be written as t(2m−1)(2m+1) u(t), with u(t) = (α + · · · + βtn + · · · )2m−1 + t2m−2k+1 (α + · · · + βtn + · · · )2m−k [hk + · · · + hk−r t2r (α + · · · + βtn + · · · )r + · · · + h0 t2k (α + · · · + βtn + · · · )k ] +1 + g2m t2 (α + · · · + βtn + · · · )

+ge t4m−2e+2 (α + · · · + βtn + · · · )2m+1−e + · · · + g0 t2(2m+1) (α + · · · + βtn + · · · )2m+1 .

If g(X) contains some non-zero terms with positive index, then s denotes the smallest for which this happens. Otherwise, let gs = 0. From the definition of n, du/dt = (α + · · · + βtn + · · · )2m−2 (βtn−1 + tn+1 G1 (t))

+t2m−2k (α + · · · + βtn + · · · )2m−k (hk + t2 G2 (t)) +t2m−2k+1 (α + · · · + βtn + · · · )2m−k−1 (βtn−1 + · · · )(hk + t2 G2 (t))

+t2m−2k+1 (α + · · · + βtn + · · · )2m−k (t2 G4 (t)) +(gs t2(2m−s)+2 (α + · · · βtn + · · · )2m−s (βtn−1 + tn+1 G1 (t)).

It should be noted that, if β = 0, then the term, βtn−1 + tn+1 G1 (t), vanishes by definition. In this case, du/dt has only one term with lowest degree, namely α2m−k hk t2m−2k , but for β 6= 0 there is one more, namely α2m−2 βtn−1 . By (7.63), u(t) = 0, and hence du(t)/dt = 0. This rules out the former possibility. Thus, the latter possibility occurs, and hence 2m − 2k = n − 1. Going back to Γ, its unique branch centred at Y∞ has the following primitive representation: 1 t2m−1 = t−2 αt2m+1 + · · · + βt4m−2k+2 + · · · α + · · · + βt2m−2k+3 + · · · 1 η(t) = 2m+1 , αt + · · · βt4m−2k+2 + · · · with α, β 6= 0. Since K has even characteristic, ξ(t) =

βt2m−2k + · · · dξ(t) = t−2 2 . dt (α + · · · + βt2m−2k+1 + · · · )

Hence, if P is the corresponding place of Σ, then

ordP (dξ) = ordt (dξ(t)/dt) = 2m − 2k − 2.

(7.64)

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Note that dξ(t)/dt 6= 0, and this shows that ξ is a separable variable of K(Γ). To calculate the genus g of Γ, take all branches γ of Γ with ordt dx(t)/dt 6= 0, where (x(t), y(t)) is a primitive representation of γ. Apart from the branch centred at Y∞ , all branches are centred at affine points, and hence ordt dx(t)/dt ≥ 0. So, attention is focused on branches with ordt dx(t)/dt > 0. Since Γ has no affine singular point, a necessary and sufficient condition for ordt dx(t)/dt to be positive is that the tangent line to Γ at the centre P = (a, b) of γ is the vertical line v(X −a). If this happens, then   dy(t) , with h(t) = h(x(t)), ordt dx(t)/dt = ordt h(t) dt by Theorem 5.56. Since K has even characteristic,       dy(t) dy(t) dx(t) = ordt h(t) = ordt + ordt h(t). ordt dt dt dt Let j denote the multiplicity of a root a of h(X); that is, h(X) = (X − a)j h1 (X), with h1 (X) ∈ K[X] and h1 (a) 6= 0. A primitive representation of the unique branch centred at P = (a, b) is x(t) = a + ti + · · · ,

y(t) = b + t,

where i ≥ 2. Actually, i = 2 since Γ has degree 2m + 1 and Y∞ is a (2m − 1)-fold point of Γ; then ordt h(t) = 2j. Let a1 , . . . , as with multiplicities j1 , . . . , js be the roots of h(X) . Then j1 + · · · + js = k. This, together with (7.64), shows that P ordt dx(t)/dt = 2k + 2m − 2k − 2 = 2m − 2.

By Definition 5.55, this sum is equal to the genus of Γ, and hence m = g. The 1 places P such that p 2P ∈ g2 are those arising from the branches centred at Y∞ and at the points (a, g(a)) with h(a) = 0. So, they are finitely many. 2

T HEOREM 7.101 For p = 2, let Γ and Γ′ be two hyperelliptic curves of genus g given in their canonical form as in Theorem 7.100; that is, Γ = v(Y 2 + h(X)Y + g(X)), deg g(X) = 2g + 1, n = deg h(X) ≤ g;

Γ1 = v(Y 2 + h1 (X)Y + g1 (X)), deg g1 (X) = 2g + 1, m = deg h1 (X) ≤ g. Let α0 , . . . , αn and β0 , . . . , βm be the roots of h(X) and h1 (X), with multiple roots counted. Then Γ and Γ1 are birationally equivalent if and only if each of the following conditions are satisfied: (i) n = m, and there is rational function w(X) = (aX + b)/(cX + d), with ad − bc 6= 0, which maps the multi-set {α1 , . . . , αn , ∞} onto the multi-set {β1 , . . . , βn , ∞};

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(ii) there is a rational function v(X) ∈ K(X) such that v(X)2 + v(X) =

g(X) g1 (w(X)) + . 2 h(X) h1 (w(X))2

Proof. For a generic point P = (x, y) of Γ, let K(Γ) = K(x, y). Assume that both (i) and (ii) hold. Then the birational transformation ω:

ξ = w(x),

η = v(x)h1 (w(x)) +

h1 (w(x)) y h(x)

maps Γ onto Γ1 . In fact, a direct computation shows that η 2 + h1 (ξ)η + g1 (ξ) = 0 follows from y 2 + h(x)y + g(x) = 0. To prove the converse, assume that Σ = K(ξ, η) with η 2p + h1 (ξ)η + g1 (ξ) = 0. Let γi be a branch of Γ centred at the point Pi = (αi , g(αi )); let Pi be the corresponding place of K(Γ). As in the proof of Theorem 7.95, the uniqueness of g21 implies that ξ = w(x) with w(X) = (aX + b)/(cX + d) and ad + bc 6= 0. The argument used in that proof also shows that the set {β1 , . . . , βm , ∞} contains w(αi ), for i = 1, . . . , n, and w(∞). Interchanging the roles of Γ and Γ1 , it follows that the set {α1 , . . . , αn , ∞} contains w(βi ), for i = 1, . . . , m, and w(∞). Therefore n = m and (i) holds. To show (ii), note that η = a(x) + b(x)y, with a(X), b(X) ∈ K(X) since [K(Γ) : K(x)] = 2. Since y 2 = h(x)y + g(x), so η 2 + h1 (ξ)η + g1 (ξ) = 0 can be written as follows: (b(x)2 h(x)+b(x)h1 (w(x))y+a(x)2 +b(x)2 g(x)+a(x)h1 (w(x))+g1 (w(x)) = 0. Since y 6∈ K(x), this is only possible when b(x) = h1 (w(x))/h(x) and 2  a(x) g1 (w(x)) g(x) a(x) + + . = h1 (w(x)) h1 (w(x)) h(x)2 h1 (w(x))2 Putting v(X) = a(X)/h1 (w(X)), (ii) follows.

2

T HEOREM 7.102 Let p = 2. For u, u1 ∈ K(X), put

Σ = K(x, y) with y 2 + y + u(x) = 0, Σ1 = K(ξ, η) with η 2 + η + u1 (ξ) = 0.

Suppose that both Σ and Σ1 are hyperelliptic, and that Σ1 ⊂ Σ. Then Σ = Σ1 if and only if there exist w(X) = (aX+b)/(cX+d), ad−bc 6= 0, and v(X) ∈ K(X) such that ξ = w(x),

v(x)2 + v(x) = u(x) + u1 (w(x)).

(7.65)

Proof. The hypothesis that Σ is hyperelliptic means that y 6∈ K(x); that is, no element f (x) ∈ K(x) exists for which u(x) = f (x)2 + f (x), and similarly for Σ1 . A direct calculation shows that, if (7.65) holds, then y 2 + y + u(x) = 0 implies that (y + v(x))2 + y + v(x) + u1 (ξ) = 0.

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Hence, either η = y + v(x), or η = y + v(x) + 1. In both cases, y ∈ K(ξ, η). Since K(x) = K(w(x)) = K(ξ), so Σ = Σ1 . Conversely, if Σ = Σ1 , the uniqueness of g21 in any hyperelliptic function field implies that ξ = w(x), where w(x) = (ax + b)/(cx + d), ad − bc 6= 0 and a, b, c, d ∈ K. Also, η = a(x) + b(x)y with a(x), b(x) ∈ K(x) and b(x) 6= 0. Then η 2 + η + u1 (w(x)) = 0 implies that (b(x)2 + b(x))y + a(x)2 + b(x)2 u(x) + a(x) + u1 (w(x)) = 0. Since y 6∈ K(x), this is only possible when b(x) = 1 and

(a(x))2 + a(x) = u(x) + u1 (w(x)).

Letting v(x) = a(x), the second equation in (7.65) follows.

2

T HEOREM 7.103 Let F be a hyperelliptic curve of genus g. If w is the number of distinct Weierstrass points of F, then w = 2g + 2, 1 ≤ w ≤ g + 1,

when either p = 0 or p > 2; when p = 2.

Proof. For p 6= 2, the assertion follows from Theorem 7.94 and Example 6.90. For p = 2, if the canonical form in Theorem 7.100 is adopted to represent F, then the arguments in Example 6.90 can be used to prove that w − 1 is equal to the number of distinct roots of h(X). Hence, 0 ≤ w ≤ g. 2

7.11 EXAMPLES OF CURVES In this section, q denotes a power of p such that q ≡ −1 (mod 3). Two examples are worked out to illustrate some points of the theory of space curves. In the first example, F is the irreducible plane curve considered in Example 1.40. Then F = v(g(X, Y )), with g(X, Y ) = Y + X q + XY q − 3(XY )(q+1)/3 .

Also, Hq = v(Y + X q + Y q X) is the Hermitian curve, P = (x, y) is a generic point of Hq , and Σ = K(x, y) is the associated function field. Note that Y 3 + X q 3 + (XY q )3 − 3(XY )q+1

= (Y + X q + Y q X)(Y + ǫX q + ǫ2 Y q X)(Y + ǫ2 X q + ǫY q X),

where ǫ3 = 1 but ǫ 6= 1. Putting ξ = x3 , η = y 3 , it follows that η + ξ q + ξη q − 3(ξη)(q+1)/3 = 0.

Therefore Q = (ξ, η) is a generic point of F, and the function field Σ′ = K(ξ, η) of F is a subfield of Σ. From Example 1.40, F has exactly 13 (q 2 − q + 1) singular points, each of which is an ordinary double point. By Theorem 5.57, the genus of F is   1 1 2 2 3 (q − q + 1) − 1 .

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Let L denote the linear series cut out on Hq by the linear system of plane cubics 3 F ′ = v(c0 + c1 X 3 + c2 Y 3 + XY ). Then L has no fixed place and L = g3(q+1) . ′′ ′ The associated subfield Σ = K(ξ, η, xy) of Σ contains Σ . It is shown below that Σ′ = Σ′′ ; see Lemma 7.106. 3 L EMMA 7.104 The series g3(q+1) is composed of an involution of order three.

Proof. Let P1 be the place of Σ arising from the unique branch γ of Hq centred at the fundamental point P1 = (1, 0, 0). To compute the ramification index eP1 , a representation of the place P1′ of Σ′′ lying under P1 is needed. As shown in Example 7.29, P1 has a primitive representation σ such that σ(x) = t and σ(y) = −(tq + tq

2

+1

+ · · · + aj tq+αj (q

2

−q+1)

+ · · · ).

Note that h i3 2 + · · · )3 = tq (1 + tq −q+1 + · · · ) h i3 2 2 = (t3 )q (1 + tq −q+1 + · · · )3 = (t3 )q 1 + (t3 )(q −q+1)/3 + · · · ,

−σ(y)3 = (tq + tq

2

+1

h i 2 + · · · ) = t tq (1 + tq −q+1 + · · · ) h i 2 2 = tq+1 (1 + tq −q+1 + · · · ) = (t3 )(q+1)/3 1 + (t3 )(q −q+1)/3 + · · · .

−σ(xy) = t(tq + tq

2

+1

So, the substitution s = t3 provides a primitive representation of P1′ , and hence eP1 = 3. The point P = (1, x3 , y 3 , xy) determines an irreducible curve Γ′ in PG(3, K). The branch γ1′ of Γ′ associated with P1′ has a primitive representation (x0 (s) = 1, x1 (s) = s, x2 (s) = sq u(s)3 , x3 (s) = s(q+1)/3 u(s)),

(7.66)

with u(s) = 1 + s(q

2

−q+1)/3

+ · · · + aj sαj (q

2

−q+1)/3

+ ··· .

Its centre is the point P1′ = (1, 0, 0, 0). If P2 is the place of Σ arising from the branch with centre at P2 = (0, 1, 0), the argument may proceed as above or use the fact that Γ is invariant under the following cyclic change of the coordinate system: (X0 , X1 , X2 ) → (X2 , X0 , X1 ). The ramification index eP2 is 3. Also, x0 (s) = (sq u(s)3 , x1 (s) = 1, x2 (s) = s, x3 (s) = s(q+1)/3 u(s)) is a primitive representation of the branch γ2′ of Γ′ associated with P2 . Its centre is the point P2 = (0, 1, 0, 0). Similarly, the place P3 of Σ arising from the branch centred at P3 = (0, 0, 1) has ramification index 3, and (x0 (s) = s, x1 (s) = sq u(s)3 , x2 (s) = 1, x3 (s) = sq+1)/3 u(s))

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is a primitive representation of the branch γ3′ associated with the place P3′ of Σ′ lying under P3 . The centre of γ3′ is the point P3′ = (0, 0, 1, 0). Every other point P of F is affine: P = (1, a, b). Let P be the unique place of Σ centred at P . The corresponding branch γ centred at P has a primitive representation (x(t) = a + a1 t + a2 tq + · · · ,

y(t) = b + t)

2

with a1 = −b−q and a2 = b−(q +q) − ab−q . Hence the branch γ ′ corresponding to P ′ has a representation (x0 (t) = 1, x1 (t), x2 (t), x3 (t)) with x1 (t) = a3 + 3a2 a1 t + 3aa21 t2 + a31 t3 + 3a2 a2 tq + 6aa1 a2 tq+1 + · · · , x2 (t) = b3 + 3b2 t + 3bt2 + t3 ,

x3 (t) = ab + (a + a1 b)t + a1 t2 + a2 btq + a2 tq+1 + · · · .

Its centre is the point P ′ = (1, a3 , b3 , ab). For a primitive cube root of unity ǫ, the point Pǫ = (1, ǫa, ǫ2 b) ∈ Hq if and only if P = (1, a, b) ∈ Hq . Let Pǫ be the place of Σ arising from the branch γǫ of Hq centred at Pǫ . Since q ≡ −1 (mod 3) implies that ǫq+1 = 1, it follows that the branch γǫ′ is centred at the point P ′ = (1, a3 , b3 , ab). Therefore the places lying over P ′ are P, Pǫ and Pǫ′2 . In particular, the ramification index eP is 1. 2 As a corollary, the following lemma is obtained. L EMMA 7.105 The curve Γ′ has degree q + 1. From the results above, the different D(Σ/Σ′ ) is 2(P1′ + P2′ + P3′ ). By Hurwitz’s Theorem 7.6, 2g − 2 = 3(2g ′ − 2) + 6, where g = 21 (q 2 − q) is the genus of Γ. Therefore Γ′ and F have the same genus. Since Σ′ is a subfield of Σ′′ , this gives the following result. L EMMA 7.106 The fields Σ′ and Σ′′ are the same. In particular, Γ′ is birationally equivalent to F. A geometric explanation of this is that F can be viewed as the projection of Γ′ from the point P4 = (0, 0, 0, 1) to the plane v(X3 ), where P4 6∈ Γ′ . Now, the L-orders are calculated. First, the branch point P1′ is considered. For i = 0, 1, 2, 3, let Hi = v(Xi ). From (7.66),  0 for i = 0,    1 for i = 1, ′ ′ I(P1 , Hi ∩ Γ ) = 1 (q + 1) for i = 2,    3 q for i = 3.

Hence the (L, P1′ )-order sequence is (0, 1, 13 (q + 1), q). This holds true for both P2′ and P3′ . If P = (1, a3 , b3 , ab) is another point of Γ′ , the first three (L, P )-orders are 0, 1, 2, and the last one is either q or q +1. To decide this, I(P, H ∩Γ′ ) is calculated for any plane H = v(u0 X0 + u1 X1 + u2 X2 + u3 X3 )

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with u0 = −(u1 a3 + u2 b3 + u3 ab). Then I(P, H ∩ Γ′ ) takes both values 1 and 2 for some choice of u1 , u2 , u3 . However, I(P, H ∩ Γ′ ) = 3 cannot occur because the matrix   3a2 a1 3b2 a + a1 b  3aa21 3b  a1 a31 1 0 has zero determinant. Also, I(P, H ∩ Γ′ ) > 3 implies that I(P, H ∩ Γ′ )
≥ q. On the other hand, from Lemma 7.66, I(P, H ∩ Γ′ ) ≤ q + 1. Since q+1 6≡ 0 q (mod p), Lemma 7.62 shows that the fourth L-order is q. In particular, there are finitely many Weierstrass points with L-orders (0, 1, 2, q + 1). The precise number M of the Weierstrass points is calculated from the ramification divisor R. Note that deg R = (1 + 2 + q)( 13 (q 2 − q − 8)) + 4(q + 1) = 31 (q 3 + 2q 2 + q − 12).

The contribution of each of the three fundamental points P1′ , P2′ , P3′ is 31 (q + 1) − 2. Therefore M = deg R − (q − 5) = 31 (q 3 + 2q 2 − 2q + 3).

It may be noted that M = q 2 + 1 + 2g ′ q, where g ′ is the genus of Γ′ .

7.12 THE LINEAR GENERAL POSITION PRINCIPLE The Riemann–Roch Theorem 6.61 shows how the dimension and the degree of a complete linear series are related. Some more results in this direction are stated in this section using the Linear General Position Principle. D EFINITION 7.107 Let Γ be a non-degenerate irreducible curve in PG(r, K). A hyperplane H of PG(r, K) is in linear general position if there is no subspace of PG(r, K) of dimension less than r − 1 that contains r common points of Γ and H. The main result on the Linear General Position Principle is the following. T HEOREM 7.108 Let Γ be a non-degenerate irreducible curve in PG(r, K). If Γ is not strange, then some hyperplane is in linear general position. In particular, if K has zero characteristic, then the Linear General Position Principle is valid. Since the irreducible conic in characteristic 2 is the only non-singular strange curve, see Section 7.19, the Linear General Position Principle is valid for every non-singular curve.

7.13 CASTELNUOVO’S BOUND If F is an irreducible plane curve of degree n and genus g, then g ≤ 21 (n−1)(n−2) by Definition 5.58. Castelnuovo’s bound is a generalisation of this for space curves. The following two lemmas are the main ingredients in the proof of Castelnuovo’s Bound.

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L EMMA 7.109 Let A and B be two positive divisors such that both dim |A| and dim |B| are positive. Then dim |A| + dim |B| ≤ dim |A + B| ≤ deg A + dim |B|.

Proof. It may be that |A| has a fixed divisor F ; if this is the case, replace A by D = A − F . Since dim |A| = dim |D| and dim |D + B| ≤ dim |A + B|, if suffices to prove the assertion for D in place of A. Put r = dim A = dim D. Since |D| has no fixed divisor, dim(D − P) = r − 1 for every place. Geometrically speaking, the divisors of |D| correspond to the points of PG(r, K), and those in |D − P| to the points of a hyperplane HP of PG(r, K). Since finitely many hyperplanes cannot cover the whole PG(r, K), there is a divisor D0 ∈ |D| which does not contain any place P from B. Replace D0 by D, and put P |D| = {div(c0 + ri=1 ci xi ) + D | c = (c0 , . . . , cr ) ∈ PG(r, K)}, Ps |B| = {div(c0 + i=1 cr+i xr+i ) + B | c = (c0 , cr+1 , . . . , cr+s ) ∈ PG(s, K)}, Pr+s L = {div(c0 + i=1 ci xi ) + D0 + B | c = (c0 , . . . , cr+s ) ∈ PG(r + s, K)}. First, div(xi ) ≻ −(D + B) for every 1 ≤ i ≤ r + s. Therefore L consists of effective divisors. Also, dim L = r + s. To show this, it is enough to show that 1, x1 , . . . , xr , . . . , xr+s are linearly independent over K. Taking into account that both sets {1, x1 , . . . , xr } and {1, xr+1 , . . . , xr+s } are linearly independent over K, suppose by contradiction that ξ = u0 + u1 x1 + · · · + ur xr = v1 xr+1 + · · · + vs xr+s = η,

with coefficients not all zero. Let P be a pole of ξ; then P is also a pole of η. It follows, on the one hand, that P is in D and, on the other hand, that P is in B. But this contradicts the choice of D0 . Since D + B is a common divisor of L and |D + B|, Theorem 6.23 shows that L is a subseries of |D + B|. Hence r + s = dim L ≤ dim |D + B|. To show the other bound, put C = D + B. Then C ≻ B. By Theorem 6.40, |B| = |C − D| = ||C| − |D||

is the subseries of |C| consisting of all the divisors containing D, minus its fixed divisor D. From Theorem 6.26, dim |B| ≥ dim |C| − ord D,

whence it follows that dim |D + B| ≤ deg D + dim B.

2

L EMMA 7.110 Let Γ be a non-degenerate, irreducible curve in PG(r, K) of degree n that is not strange. If D is the divisor cut out on Γ by a general hyperplane, then (i)

dim |kD| − dim |(k − 1)D| ≥ k(r − 1) + 1,

(7.67)

dim |kD| − dim |(k − 1)D| = n,

(7.68)

for every positive integer k ≤ m, where m = (ii) for k > m.

⌊ n−1 r−1 ⌋;

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Proof. Let D be the divisor cut out on Γ by a general hyperplane. Then D can be written as the sum of places corresponding to branches of Γ centred at distinct points. Assume that k ≤ m, and choose k(n − 1) + 1 places from D. Let Ω be the set of the centres of the corresponding branches of Γ. For each of the k(n − 1) + 1 points Qj ∈ Ω, a hypersurface of degree k exists which passes through Ω\{Qj } but Qj . To show this, partition the remaining k(n − 1) points of Ω into k sets S1 = {P1,1 , P1,2 , . . . , P1,n−1 } , . . . , Sk = {Pk,1 , Pk,2 , . . . , Pk,n−1 } of n − 1 points each. By the Linear General Position Principle, each is linearly independent over K. Therefore, for every 1 ≤ u ≤ k, there is a hyperplane Hu through every point in Su that does not contain Qj . The reducible hypersurface Kj of degree k whose components are the hyperplanes Hu , u = 1, . . . , k is the required hypersurface of degree k. The resulting k(n − 1) + 1 hypersurfaces Kj with Qj ranging over Ω are linearly independent over K. From this (7.67) follows. In fact, let Γ be the curve arising from Q with Q = (1, x1 , . . . , xn ), and let |(k − 1)D| = div(c0 y0 + c1 y1 + · · · + cr yr ) + (k − 1)D with r = dim |(k − 1)D|. If the hyperplane H = v(F (X0 , X1 , . . . , Xn )) cuts out D, put ξ = F (1, x1 , . . . , xn ). Similarly, if Kj = v(Fj (X0 , X1 , . . . , Xn )), put ηj = Fj (1, x1 , . . . , xn ). Let L be the linear series P P {div ( ri=0 ci yi ξ + k(n−1)+1 dj ηj ) + kD | j=1 (c0 , . . . , cr , d1 , . . . , dk(n−1)+1 ) ∈ PG(r + k(n − 1) + 1, K)}.

The r + 1 elements yi ξ together with the k(n − 1) + 1 elements ηj form a linearly independent set over K. Hence, (7.67) holds. For k > m, the same argument shows how to find k hyperplanes in PG(n, K) containing all but any one of the points of Ω. From this, (7.68) follows. 2 T HEOREM 7.111 (Castelnuovo’s Bound) Let Γ be a non-degenerate irreducible curve in PG(n, K) of degree d and genus g. If Γ is not strange, then g ≤ 12 m(m − 1)(n − 1) + mǫ,

d−1 ⌋ and d − 1 = m(n − 1) + ǫ. where m = ⌊ n−1

Proof. Let L be the linear series cut out on Γ by hyperplanes. Since L may not be complete, consider the complete linear series L′ containing L. Then L′ = |D|, where D is a divisor cut out on Γ by a hyperplane. As dim |D| ≥ n, from Lemma 7.110, it follows by induction that dim |kD| ≥ 21 k(k + 1)(n − 1) + k, k = 1, . . . , m; dim |(m + j)D| ≥ 12 k(k + 1)(n − 1) + k + jd, j ≥ 1. However, for sufficiently large j, the divisor |(m + j)D| is non-special. So, by the Riemann–Roch Theorem 6.61 dim |(m + j)D| = (m + j)d − g. Hence 1 g ≤ (j + m)d − m(m + 1)(n − 1) − m − jd 2 = 12 m(m − 1)(n − 1) + m(d − m(n − 1) − 1),

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whence the assertion follows.

2

A slightly different wording of Castelnuovo’s Bound is stated in the following theorem. T HEOREM 7.112 Let Γ be as in Theorem 7.111. If ǫ is the unique integer with 0 ≤ ǫ ≤ r − 2 and d − 1 ≡ ǫ (mod (r − 1)), then g ≤ c0 (d, r) =

d−1−ǫ (d − r + ǫ). 2(r − 1)

(7.69)

L EMMA 7.113 Let D be as in Lemma 7.110, let ǫ′ = d − m(r − 1) for ǫ′ in {2, . . . , r}, and let g = c0 (d, r). If m ≥ 2, then (i) the dimension of the linear series |2D| is 3r − 1; (ii) there exists a base-point-free (ǫ′ − 2)-dimensional complete linear series D′ of order (ǫ′ − 2)(m + 1) such that (m − 1)|D| + D′ is the canonical linear series.

7.14 A GENERALISATION OF CLIFFORD’S THEOREM Clifford’s Theorem 6.79 states that n ≥ 2r for any special gnr . Here, an alternative proof depending on the Linear General Position Principle is given, which also shows that n = 2r with n > 0 only possible for canonical or hyperelliptic curves. T HEOREM 7.114 A non-canonical special gnr , with n > 0, on a non-hyperelliptic curve C satisfies n > 2r.

Proof. Suppose gnr is not complete. Choose an effective special divisor D such ′ that |D| = gnr . Then |W − D| is a gnr ′ with r′ ≥ 0. Note that n′ > 0, since |D| is not the canonical series. From the Reciprocity Theorem 6.78 of Brill and Noether, n − 2r = n′ − 2r′ . Since n + n′ = 2g − 2, this gives r − r′ = n − g + 1. Also, r + r′ ≤ g − 1 by Lemma 7.109. Therefore n ≥ 2r and equality holds if and only if n′ = 2r′ . Since n + n′ = 2g − 2, either n or n′ is less than or equal to g − 1, and suppose that this occurs for n; so, 0 < n ≤ g − 1. Note as well that n = 2r only occurs when r + r′ = g − 1; that is, every canonical divisor is the sum of a divisor from |D| and a divisor from |W − D|. Now, assume that n = 2r for a non-hyperelliptic curve and consider the canonical curve Γ. Let H be a hyperplane in PG(g − 1, K) such that Γ ∩ H consists of 2g − 2 distinct points. Then Γ ∩ H contains n points, say P1 , . . . , Pn , such that the corresponding places define a divisor P1 + · · · + Pn in |D|. By Theorem 7.45 (iii), these points span a ( 12 n − 1)-dimensional subspace of PG(g − 1, K). Since n ≤ g − 1, either 21 n − 1 = g − 2, or H is not in a linear general position. But Theorem 7.108, together with Theorem 7.45 (i), ensures the existence of a linear general hyperplane section, and this completes the proof. 2

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This generalisation of Clifford’s Theorem 6.79 gives a refinement of the lower bound in Theorem 7.64. T HEOREM 7.115 Let N be the number of Weierstrass points of a function field Σ which is not rational, elliptic, or hyperelliptic. If its canonical series is classical, then N ≥ 2g + 6. Proof. It suffices to note in the second part of the proof of Theorem 7.64 that the bound (7.16) may be replaced by ji ≤ i − 2, by Theorem 7.114. 2 7.15 THE UNIFORM POSITION PRINCIPLE Another position principle on hyperplane sections of an algebraic curve, which is valid in zero characteristic but not for all curves in positive characteristic, is treated briefly. D EFINITION 7.116 A finite set S of points in PG(n, K) is in uniform position if, for any two positive integers k, m, all subsets of S of a size k impose the same number of conditions of hypersurfaces of degree m. In other words, the the dimension of the linear system of all hypersurfaces of degree m containing a subset of S of size k depends only on k but is independent of the choice of the subset. With this definition, the Uniform Position Principle states that the points of a general hyperplane section of an algebraic curve are in uniform position. Like the General Linear Position Principle, the Uniform Position Principle has been an important tool in investigating relationships between the genus g and the degree d of a space curves of PG(n, K). Here a few results are quoted without proofs. A classical theorem of Halphen improves Castelnuovo’s genus bound for certain curves in PG(3, C). Halphen’s Theorem holds true in positive characteristic for non-singular curves of sufficiently large degree. T HEOREM 7.117 (Halphen) In PG(3, K) let X be an irreducible non-singular curve of degree d. If either  for p = 0,  7, 7, for p > 2 and X reflexive, d≥  25, for p > 0 and X non-reflexive or d = 17 for p > 0, then X lies on a quadric surface provided that g > c1 (d, 3) = ⌊ 61 (d2 − 3d + 6)⌋ . For the definition of reflexivity of Γ, see Section 7.19. T HEOREM 7.118 Let p 6= 2. Then Γ is not reflexive if and only if ǫ2 > 2.

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Halphen’s Theorem 7.117 extends to certain curves in PG(r, K) for r ≥ 4, and it appears to be very useful when seeking a bound cα (d, r) for the genus of a curve of degree d in PG(r, K) not lying on any irreducible surface of degree less than r + α − 1. The result for the smallest case α = 1 is stated below since it plays a role in Section 10.5. L EMMA 7.119 Let Γ be an irreducible curve of PG(r, K) of degree d and genus g, and let  for r ≤ 6,  36r, 288, for r = 7, d≥  r+1 2 , for r ≥ 8. Then X lies on a surface of degree at most r − 1, provided that ( d − 1 − ǫ1 0 if ǫ1 ≤ r − 2, g > c1 (d, r) = (d − r + ǫ1 + 1) + 2r 1 if ǫ1 = r − 1,

(7.70)

where ǫ1 is the unique integer such that 0 ≤ ǫ1 ≤ r − 1 and d − 1 ≡ ǫ1 (mod r). These numbers c1 (d, r) are Halphen’s numbers. Note that (7.70) for r = 3 coincides with the formula in Theorem 7.117.

7.16 VALUATION RINGS Valuations of a field Σ over a base field F occur frequently in algebraic geometry. Although, for the theory of algebraic curves as developed in this book, this concept is not really needed, it is useful to establish the relation between valuations and places. In the case that F = K and Σ is a field of transcendence degree 1 over K, any place P of Σ gives rise to a valuation. In fact, all valuations of Σ/K are obtained in this way. D EFINITION 7.120 The group G is ordered if it is an abelian group with a transitive relation < satisfying the following properties: (i) for all α, β, γ ∈ G, α < β ⇒ α + γ < β + γ;

(7.71)

(ii) for every element α ∈ G, precisely one of the three relations α = 0,

α < 0,

0 2, let F = v(f (X, Y )), with

f (X, Y ) = V0p + V1p X + V2p Y + V3p X 2 + V4p XY + V5p Y 2 ,

where V0 , V1 , . . . , V5 ∈ K[X, Y ] are polynomials satisfying the equation V0 (4V3 V5 − V42 ) = V12 V5 + V22 V3 − V1 V2 V4 .

Show that F is non-classical with respect to lines, as its Hessian curve is v(2(4V2 V5 − V42 )p f (X, Y )). 3. For p = 2, let F = v(Y + X 3 + Y 8 ). Show that the orders of F with respect to the linear system of conics are 0, 1, 2, 3, 4, 6. 4. For p > 5, let G = v(G(X, Y )) be the non-singular curve with G(X, Y ) = 1 + X p + XY p + Y p+1 + X p−1 .

Show that G is non-classical with respect to the linear system of conics, and that (7.22) holds with h(X, Y ) = X but not with h(X, Y ) = 1. 5. Let F be the curve in Example 1.41. Show that, if p ≥ 5 and k 2 6≡ 1 (mod p), then F is classical with respect to the linear system of conics.

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6. Given g + 1 distinct elements ai ∈ K, let Σ = K(x, y) with Pg+1 y 2 + y = i=1 (x + ai )−1 .

Show that, if p = 2, then Σ has g + 1 distinct Weierstrass points.

7. Show that the Klein quartic, as in Example 7.18, is not hyperelliptic. 8. Let F be a hyperelliptic curve. In characteristic p 6= 2, deduce from Example 6.90 and Theorem 7.94 that the smallest non-gap number m1 (P) at an ordinary place of K(F) is g + 1. Extend the result to characteristic p = 2. 9. Show that two distinct irreducible curves of PG(r, K) have only finitely many common points. 10. Show that ǫm + ji (P ) ≤ ji+m (P ) for i + m ≤ r. 11. Let p ≥ 5. For a non-hyperelliptic curve F of genus 4, show the existence of a place P of K(F) such that 4 is the first non-gap at P. 12. Let (0, 1, pm ) be the order sequence of an irreducible plane curve F which is non-classical with respect to lines, and let P be any place of K(F) with order sequence (0, j1 , j2 ). Adapting the techniques of Section 7.8, show that one of the following holds: (a) j1 ≡ 0 (mod pn );

(b) j2 ≡ 0 (mod pn );

(c) j2 ≡ j1 (mod pm ).

13. Show that a plane non-singular cubic v(f (X, Y )) is supersingular over Fq , q = ph , if and only if the coefficient of (XY )p−1 in F (X, Y )p−1 is zero.

7.19 NOTES The first four sections follow Seidenberg [400, Chapter 20]. For studies of the Klein quartic, see Elkies [117] and Duursma [109]. For coverings of curves, see also [7]. Section 7.6 follows St¨ohr and Voloch [432]; see also Schmidt [387] and Laksov [289]. For rational strange curves of degree q + 1, there are three types; see Ballico and Hefez [38]. Bayer and Hefez [40] described equations and singularities of plane strange curves and gave an upper bound on the genus of a strange curve of PG(r, K) in terms of its degree. For the General Order of Contact Theorem, see Hefez and Kleiman [204]. Proposition 7.70 is due to Pardini [357] for m = 1 and to Homma [232] for arbitrary m. If singular points are allowed in Proposition 7.70, then h(X, Y ) = 1 can still be ensured under some natural conditions; see [200]. However, it is not possible in general to get rid of h(X, Y ), the curve in Exercise 2 being such an

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example; here, deg F ≤ q with q = pm . Hefez [200] and Homma [229] show that equality holds for just one type of curve. Fermat curves which are not classical with respect to L2 were investigated by Garcia and Voloch [161]; if p > 5, then the Fermat curve F = v(X n + Y n + 1) is non-classical with respect to L2 if and only if p | (n − 2)(n − 1)(n + 1)(2n − 1). The classical proof of Salmon’s Theorem 7.90 works for p 6= 2, 3; see [497, Section 9.3], [353, Chapter 15] and [216, Chapter 11]. Related to Theorem 7.103 for p = 2, Schmidt [387] exhibited a family of hyperelliptic curve with only one Weierstrass point. Henn [207] exhibited nonhyperelliptic curves with few Weierstrass points. The Linear Position Principle in positive characteristic and, more generally, the general hyperplane section of a curve, were investigated by Ballico [27, 28, 29, 30, 31, 32, 34]; see also Ballico and Migliore [39], and Ballico and Cossidente [35]. The original paper of Castelnuovo is [72]. Curves hitting Castelnuovo’s Bound, that is, those with genus equal to Castelnuovo’s number c0 (d, r) are extremal and they have several remarkable properties; see [11], [191, Chapter 3], [19, Chapter 3, Section 2]. One of these properties is stated in Lemma 7.113 since it is used in Section 10.5 where certain curves with g = c0 (d, r) are considered. The proof can be deduced from that of Theorem 7.111. For more information, see [11, p. 361 and Lemma 3.5]. The main reference for the Gaussian duality of space curves is [163], in which the authors determine when Γ coincides with its Gaussian dual Γ′ , up to a linear transformation. A third generalisation of the concept of the dual curve of an irreducible plane curve C is the dual hypersurface of an irreducible curve Γ of PG(r, K). The idea is to consider all hyperplanes of PG(r, K) containing at least one tangent line to Γ and regard them as points in the dual projective space PG(r, K)∗ ∼ = PG(r, K). Such points, but finitely many of them, lie on an irreducible hypersurface of Π which is the dual hypersurface ∆ of Γ. To find a generalisation of the notion of a reflexive curve, more is needed, namely the concept of the conormal variety C(Z) of an irreducible algebraic variety Z defined over K; see [200]. The variety C(Z) is defined to be the closure in PG(r, K) × PG(r, K)∗ of the set of all pairs (P, H), where P is a non-singular point of Z and H is a hyperplane of PG(r, K) containing the tangent space of Z at P . The dimension of C(Z) is r − 1. The dual variety of Z ′ of Z is the projection of C(Z) onto PG(r, K)∗ . A variety is reflexive when C(Z) = C(Z ′ ). From the definition, if Z is reflexive then (Z ′ )′ = Z and Z ′ is also reflexive. To identify the inequivalent elliptic curves in Theorem 7.89, see [216, Chapter 11]. A characteristic-free proof of Theorem 7.108 is found in [365]. A proof of the Uniform Position Principle in characteristic zero, due to Eisenbud and Harris, is found in [191]. Rathmann [365] extended the result to positive characteristic provided that n ≥ 4. He also showed the validity of the Uniform Position Principle for a large family of curves of PG(3, K) including all reflexive curves. It should be noted that the validity of both position principles depends on the mono-

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dromy group of the curve, which is a highly transitive permutation group. In characteristic p, the classification of highly-transitive permutation groups depends on the classification of all finite simple groups. For more results, see [191]. The general theory of algebraic varieties is found in several textbooks, notably in [410] and [503]. Good references for Section 7.17 are [301] and [19]. The result that the irreducible conic in characteristic 2 is the only non-singular strange curve is due to Samuel; see [192, Theorem IV.3.9]. The characterisation of reflexive curves in Theorem 7.118 is due to Hefez and Kleiman [200]; see also [204]. For a proof of Lemma 7.119, see [191, Theorem 3.22] and [365, Corollary 2.8]. A full account of results related to Halphen’s Theorem is found in the books, [191], [182], as well as in the papers, [79], [78]. For a historical account of Halphen’s Theorem, see [192, p. 349], [182] and [190]. Halphen’s Theorem remains essentially valid in any characteristic; see [191, Theorem 3.13] for p = 0 and [33] for p > 0. For places in the function field setting, see [428, I.1.8. Definition]. For Exercise 5, see [1]. For Exercise 10, see [123], [234], [233], and [140]. For Exercise 11 and more results on Weierstrass points of curves with low genus, see [261].

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PART 2

Curves over a finite field

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Chapter Eight Rational points and places over a finite field In this chapter, K = Fq , the algebraic closure of the finite field Fq of order q. For any r, the space PG(r, K) contains the finite projective spaces PG(r, q i ) with i ≥ 1. If Γ is an algebraic curve embedded in PG(r, K), the set of points of Γ lying in PG(r, q) is a natural geometric object to investigate. To do this in a birational spirit, the concept of an Fq -rational place, that is, a place defined over Fq , is required. As in Section 5.3, the idea is to derive this concept from that of an Fq -rational branch of Γ, even though the approach works properly if any non-singular point of Γ is the centre of a unique Fq -rational branch. The latter hypothesis is satisfied by all curves which are defined over Fq ; for an irreducible plane curve F = v(f (X, Y )), this just means that f (X, Y ) ∈ Fq [X, Y ]. It should be noted, however, that a singular point of F lying in PG(2, q) may happen not to be the centre of any Fq -rational branch of F. Therefore a bijection between Fq -rational places of K(F) and points of F lying in PG(2, q) can only be ensured for non-singular curves. The foundation of the theory of the algebraic curves defined over a finite field is developed in the first four sections. The central problem on algebraic curves over finite fields is to estimate the number of their Fq -rational branches. The study starts with the St¨ohr–Voloch Theorem 8.65 which is obtained in a way that parallels Section 7.6. An interesting consequence of the St¨ohr–Voloch Theorem 8.65 is that curves with a large number of Fq -rational branches are mostly non-classical. This will be apparent in Sections 8.6 and 8.7, where Fq -rational branches of irreducible plane curves that are non-classical with respect to lines and conics are thoroughly investigated. The study of this problem is continued in Chapters 9 and 10.

8.1 PLANE CURVES DEFINED OVER A FINITE FIELD D EFINITION 8.1 A plane curve F = v(f (X, Y )) is defined over Fq if there is a non-zero constant c ∈ K such that cf (X, Y ) ∈ Fq [X, Y ]. An irreducible plane curve F defined over Fq is Fq -rational. L EMMA 8.2 An irreducible plane curve F is defined over Fq if and only if, for every point P = (a0 , a1 , a2 ) ∈ F, its image point P q = (aq0 , aq1 , aq2 ) under the Frobenius collineation Φ is also in F. Proof. Let F = v(F ) where F ∈ K[X0 , X1 , X2 ] is a homogeneous polynomial

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of degree d. If F =

P

d−(i+j)

aij X0i X1j X2

,

then the homogeneous polynomial P q i j d−(i+j) G= aij X0 X1 X2

defines the irreducible plane curve G = v(G). Thus, P = (a0 , a1 , a2 ) ∈ F if and only if P q = (aq0 , aq1 , aq2 ) ∈ G. Now, if F is defined over Fq , then F = G and hence P q ∈ F. Conversely, if P q ∈ F then P q is a common point of F and G. If this occurs for every point P ∈ F, the irreducible curves F and G have infinitely many common points. By B´ezout’s Theorem, F = G and hence F is defined over Fq . 2 L EMMA 8.3 The irreducible plane curve F = v(F (X, Y )) is defined over Fq if and only if, for every generic point P = (x, y) of F, the image point P q = (xq , y q ) of P = (x, y) under the Frobenius collineation is also a generic point of F.

Proof. If F (X, Y ) ∈ Fq [X, Y ], then F (x, y) = 0 implies F (xq , y q ) = 0. Therefore P q = (xq , y q ) is a point of F, and hence a generic point of it. To show the converse, assume without loss of generality that some coefficient of F (X, Y ) is equal to 1. Now, F (x, y) = 0 implies that G(xq , y q ) = 0, where P P q i j F = aij X i Y j , G = aij X Y .

If P q = (xq , y q ) ∈ F, then it is a common generic point of F and the irreducible plane curve G = v(G). By Remark 5.2, F (X, Y ) = c G(X, Y ). Actually, c = 1 by the assumption. Therefore F (X, Y ) ∈ Fq [X, Y ]. 2 Let F = v(f (X, Y )) be an irreducible plane curve defined over Fq . D EFINITION 8.4 For a generic point P = (x, y) of F, the set of all elements u(x, y)/v(x, y), with u(X, Y ), v(X, Y ) ∈ Fq [X, Y ] and F (X, Y ) ∤ v(X, Y ), form a subfield Fq (F) of K(F), the Fq -rational function field of F. By Theorem 5.7, this definition is independent of the choice of the generic point. D EFINITION 8.5 (i) A rational transformation ω of Σ = K(F) which maps Fq (F) into itself is an Fq -rational transformation, and Σ′ = ω(Σ) is an Fq -rational function field, as well. (ii) If F ′ is the image of F under an Fq -rational transformation, then F → F ′ is an Fq -rational covering. This term is also used if F and F ′ are replaced by Fq -rational non-singular models X and X ′ of Σ and Σ′ = K(F ′ ). 8.2 Fq -RATIONAL BRANCHES OF A CURVE Let f =

P

ai ti ∈ K((t)). For any integer n, the n-th conjugate of f is P qn i f (n) = ai t .

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For any f, g ∈ K((t)) and integers n, m, the following properties hold: (f + g)(n) = f (n) + g (n) ;

(f g)(n) = f (n) g (n) ;

(f (n) )(m) = f (n+m) . (8.1)

Further, every f ∈ K((t)) can be written as f = g (n) with g ∈ K((t)). Hence, the n-th conjugate map κ : f 7→ f (n) is an automorphism of K((t)). Note that κ is not a K-automorphism as κ(a) 6= a for any constant a 6∈ Fqn . Now, f = f (n) if and only if f ∈ Fqn ((t)). Also, if f2 = α(f1 ) for a K-automorphism α of K((t)), (n) (n) then f2 = κακ−1 (f1 ). Note that κακ−1 is a K-automorphism of K((t)). Thus, given ξ, ξ1 , η, η1 ∈ K[[t]], if the branch representations (ξ, η) and (ξ1 , η1 ) (n) (n) are K-equivalent, then (ξ (n) , η (n) ) and (ξ1 , η1 ) are K-equivalent branch representation as well. From this, if (ξ, η) is a primitive branch representation, then (ξ (n) , η (n) ) is also a primitive branch representation. Let γ be a branch with a primitive representation (ξ, η). The branch Φn (γ) given by the primitive representation (ξ (n) , η (n) ) is the n-th Frobenius image of γ. D EFINITION 8.6 A branch γ is Fq -rational if γ has a primitive representation (ξ, η) with ξ, η ∈ Fq [[t]], that is, if ξ (1) = ξ, η (1) = η. It should be observed that, if γ is Fq -rational having a primitive representation (ξ, η) with ξ, η ∈ Fq [[t]], then other primitive representations (ξ1 , η1 ) of γ does (1) (1) not have the property that ξ1 , η1 ∈ Fq [[t]] unless there exists α ∈ AutFq K[[t]] such that ξ1 = α(ξ), η1 = α(η). Here, AutFq K[[t]], the Fq -automorphism group of K[[t]], consists of all K-automorphisms α such that α(f ) ∈ Fq [[t]] for every f ∈ Fq [[t]]. The observation follows from the following two lemmas. The first is a refinement of Theorem 4.21. L EMMA 8.7 For an Fq -rational primitive branch representation (x(t), y(t)), the subfield Fq (x(t), y(t)) of K(x(t), y(t)) contains an element ξ of order 1. Proof. Let τ be an element in Fq (x(t), y(t)) of minimal positive order. Arguing as in the proof of Theorem 4.21, both x(t) and y(t) can be expanded into power series of τ . Since they are the components of a primitive branch representation, this is only possible for ordt τ = 1. 2 L EMMA 8.8 Let ξ ∈ Fq [[t]] with ordt ξ = 1, and let α ∈ Aut(K[[t]]) with α(ξ) in Fq [[t]]. Then α ∈ AutFq K[[t]]. Proof. It suffices to prove that α(t) ∈ Fq [[t]]. Write

ξ = u 1 t + · · · + u k tk + · · · ,

with u1 , . . . , uk , . . . ∈ Fq , and α(t) = a1 t + · · · + am tm + · · · , where both u1 and a1 are non-zero. Then α(ξ) = u1 α(t) + · · · + uk α(t)k + · · · = a1 u1 t + t2 G(t).

Since α(ξ) ∈ Fq [[t]], so a1 u1 ∈ Fq , whence a1 ∈ Fq . To prove that this holds true for every am , induction on m is used. Suppose that a1 , . . . , am−1 ∈ Fq . Then α(ξ) = b1 t + · · · + bm−1 tm + (am + cm )tm + · · ·

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with b1 , . . . , bm−1 , cm ∈ Fq . Since α(ξ) ∈ Fq [[t]], so am + cm ∈ Fq , implying that am ∈ Fq . 2 Now, branches of an irreducible plane curve F defined over Fq are considered. From the definition, it may appear that a branch of F is Fq -rational if and only if its centre is a point in PG(2, q). This is false in general, even if it holds for non-singular points. T HEOREM 8.9 Let P be a non-singular point of an irreducible plane curve F defined over Fq . If P ∈ PG(2, q), then the unique branch of F centred at P is Fq -rational. Proof. After a change of coordinates over Fq , suppose that P = (1, 0, 0), and that the tangent line to F at P is not the Y -axis. By Theorem 4.6, the branch of F centred at P has a primitive representation (x(t), y(t)) with x(t) = t,

y(t) = c1 t + · · · + ck tk + · · · .

From the proof of that theorem, each coefficient ci is in the subfield of K generated by the coefficients of F (X, Y ). Since F is defined over Fq , this subfield is contained in Fq . 2 T HEOREM 8.10 Let P ∈ PG(2, q) be a singular point of an irreducible plane curve F defined over Fq . If γ is a linear branch of F with centre P and ℓ is the common tangent line to F and γ at P, then γ is Fq -rational if and only if ℓ is defined over Fq . Proof. Again, assume that P = (1, 0, 0), and that ℓ is not the Y -axis. The proof of Theorem 8.9 still works for linear branches, and gives the ‘if’ part of the assertion. Conversely, any primitive representation (x(t), y(t)) of γ is of type x(t) = at + · · · , a 6= 0,

y(t) = bt + · · ·

such that b/a is the slope of ℓ. If γ is Fq -rational, then there is a representation with a, b ∈ Fq . Hence ℓ is defined over Fq , as it passes through a point of PG(2, q) and its slope is in Fq . 2 The notation for the set of all Fq -rational branches of F is F(Fq ), and Sq for |F(Fq )|. Theorems 8.9 and 8.10 show that F(Fq ) is a contained in the set F(Fq )∗ consisting of all branches of F centred at points of PG(2, q). Hence Sq ≤ Bq where Bq = |F(Fq )∗ | and, if F is non-singular, then equality holds. However, it may happen for singular curves that Sq < Bq . This possibility is investigated in Section 9.6. Here an example is given. E XAMPLE 8.11 Let q be odd, and choose a non-square element c ∈ Fq . The irreducible cubic curve F = v(cX 2 − Y 2 + X 3 ) defined over Fq contains the point P = (1, 0, 0), which is a double point. Two branches γ + and γ − of F are centred at P , namely those with primitive representation (x(t), y(t)), where x(t) = t,

y(t) = ±νt + · · · ,

with ν 2 = c. Here, ν ∈ Fq2 \Fq and hence neither branch is Fq -rational.

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On the other hand, the following result holds. L EMMA 8.12 Let γ be a branch of an irreducible plane curve F defined over Fq . Then there is an overfield Fqn of Fq depending on γ such that γ has a primitive representation (x(t), y(t)) with x(t), y(t) ∈ Fqn [[t]].

Proof. Let P ′ = (x′ , y ′ ) be a generic point of F. By Theorem 3.27, there is an irreducible plane curve G = v(g(X, Y )) with only ordinary singularities which is birationally equivalent to F. Without loss of generality, suppose that no point of G at infinity is singular and that no vertical line is tangent to G at a singular point. Let ω : x′ = u(x, y)/v(x, y), y ′ = w(x, y)/z(x, y)

be a birational transformation that takes G to F, where P = (x, y) is a generic point of G. The subfield Fqk of K generated by Fq , together with the coefficients of the polynomials u, v, w, z ∈ K[X, Y ], contains all coefficients of g(X, Y ). A possible larger subfield Fqs of K also contains the slopes of the tangents to G at singular points. Now, let γ be any branch of F. By Theorems 8.9 and 8.10, ω −1 (γ) is Fqn rational if Fqn is the subfield generated by Fqs and the coordinates of the centre of ω −1 (γ). Since ω(ω −1 (γ)) = γ, the subfield Fqn of K has the desired property. 2 A corollary to Lemma 8.12 is the following result. T HEOREM 8.13 Let γ be a branch of an irreducible plane curve F defined over Fq . Then there is a positive integer n depending on γ such that Φn (γ) = γ.

8.3 Fq -RATIONAL PLACES, DIVISORS AND LINEAR SERIES P Let f = aij X i Y j be any polynomial in K[X, Y ]. For any integer n, the n-th conjugate of f is the polynomial P n f (n) = aqij X i Y j .

The n-th conjugate map f 7→ f (n) is an automorphism of K[X, Y ]. Here, f is irreducible if and only if f (n) is irreducible; also, (8.1) holds for any f, g in K[X, Y ]. Then, f = f (n) only when f ∈ Fqn [X, Y ], and f · f (1) · . . . · f (n−1) ∈ Fq [X, Y ]

f +f

(1)

+ ··· + f

(n−1)

∈ Fq [X, Y ]

if and only if f ∈ Fqn [X, Y ]; (8.2)

if and only if f ∈ Fqn [X, Y ]. (8.3)

Now, these results together with those obtained in Section 8.2 are applied to the function field K(F) of an irreducible plane curve F = v(F (X, Y )) defined over Fqn . Let P = (x, y) be a generic point of F. Those elements in K(F) having a representation u(x, y)/v(x, y), with u, v ∈ Fqn [X, Y ] constitute a subfield, the Fqn -rational function field Fqn (F) of F. Since any element f ∈ K(F) can be written as u(x, y) f= , with u[X, Y ], v[X, Y ] ∈ K[X, Y ], F (X, Y ) ∤ v(X, Y ), (8.4) v(x, y)

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the n-th conjugate of f can be defined: f (n) =

u(n) (x, y) . v (n) (x, y)

Note that v (n) (x, y) 6= 0 because F (X, Y ) ∤ v (n) (X, Y ). L EMMA 8.14 The definition of f (n) does not depend on the representation. Proof. It must be shown that w(n) (x, y) w(x, y) u(n) (x, y) u(x, y) = (n) . = =⇒ (n) v(x, y) z(x, y) v (x, y) z (x, y) To do this, write the first equation as u(x, y)z(x, y) − v(x, y)w(x, y) = 0.

Raising this to the q n -th power, this gives n

n

n

n

n

n

n

n

u(n) (xq , y q )z (n) (xq , y q ) − v (n) (xq , y q )w(n) (xq , y q ) = 0.

By Lemma 8.3, P = (x(n) , y (n) ) is a generic point of F. Thus,

F (X, Y )G(X, Y ) = u(n) (X, Y )z (n) (X, Y ) − v (n) (X, Y )w(n) (X, Y ),

for some polynomial G ∈ K[X, Y ], whence it follows that

u(n) (x, y)z (n) (x, y) − v (n) (x, y)w(n) (x, y) = 0,

as required.

2

L EMMA 8.15 If f = f (1) then f ∈ Fq (F). Proof. Assume first that f ∈ K(x) and write f = c · u(x)/v(x) with c ∈ K and u(X), v(X) ∈ K[X] monic polynomials for which gcd(u(X), v(X)) = 1. By Lemma 8.14, f (1) = cq · u(1) (x)/v (1) (x). From f = f (1) , u(X)v (1) (X) = u(1) (X)v(X)

and c ∈ Fq . Since gcd(u(X), v(X)) = 1 and deg u(X) = deg u(1) (X), this implies that u(X) = u(1) (X), whence u(X) ∈ Fq [X]; similarly, v(X) ∈ Fq [X]. Therefore the assertion holds when f ∈ K(x). Let n = [K(x, y) : K(x)]. Since K(F) = K(x, y), every element f ∈ K(F) can be uniquely written in the form f = c0 (x) + c1 (x)y + · · · + cn−1 (x)y n−1 ,

with c0 (x), · · · , cn−1 (x) ∈ K(x). Then (1)

(1)

(1)

f (1) = c0 (x) + c1 (x)y + . . . + cn−1 (x)y n−1 . (1)

From f = f (1) it follows that ci (x) = ci (x) for i = 0, . . . , n − 1. By the preceding assertion, this implies that ci (x) ∈ Fq (x). Then ci (X) ∈ Fq [X], and the result follows. 2 Note that (8.1) holds for any f, g ∈ K(F). Also, for every f ∈ K(F) there is a g ∈ K(F) such that f = g (n) . Hence the n-th conjugate map is an automorphism

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ι of K(F) that fixes every element in Fqn (F). The constants fixed by ι are those in Fqn . Let F = v(F (X, Y )) be an irreducible plane curve defined over Fq . For a generic point P = (x, y) of F, let K(F) = K(x, y) be the associated function field of F. Given a place P of K(F), let σ be any primitive place representation σ of P. As usual, put x(t) = σ(x), y(t) = σ(y). Then F (x(t), y(t)) = 0 and, by Lemma 8.3, F (x(n) (t), y (n) (t)) = 0. This means that the pair (x(n) (t), y (n) (t)) is a primitive representation of a branch of F = v(F (X, Y )). The corresponding place Φn (P) is the n-th Frobenius image of P. This definition is independent of the choice of σ. Now, a primitive representation of Φn (P) is given. Next, σι−1 is a monomorphism from K(F) into K((t)). This does not fix every constant, but the monomorphism τ = κσι−1

(8.5)

does. To check that τ is a primitive place representation of the Φn (P) of P, let ι−1 (x) = u(x, y)/v(x, y), that is, x = u(n) (x, y)/v (n) (x, y). Then   u(n) (x(n) (t), y (n) (t)) u(x(t), y(t)) = (n) (n) = x(n) (t). τ (x) = κ v(x(t), y(t)) v (x (t), y (n) (t)) Similarly, τ (y) = y (n) (t), and the result is established. D EFINITION 8.16 For an irreducible plane curve F defined over Fq , a place P of the function field K(F) is an Fqn -rational place if Φn (P) = P. Note that Φn (P) = P if and only if, for any primitive representation σ, also (8.5) is a primitive representation of P. Further, from Theorem 8.13, for every place P of K(F), there is a positive integer n depending on P such that Φn (P) = P. L EMMA 8.17 Let F be an irreducible plane curve defined over Fq . If the branch γ of F is Fq -rational, then the corresponding place P is also Fq -rational. Proof. With the notation above applied to the case n = 1, if (x(t), y(t)) is an Fq rational branch representation of γ, then x(1) (t) = x(t), y (1) (t) = y(t). Therefore τ = σ, showing that P is Fq -rational. 2 It is shown later that the converse of Lemma 8.17 also holds; see Theorem 8.31. T HEOREM 8.18 Let P be a non-singular point of an irreducible plane curve F defined over Fq . Then P ∈ PG(2, q) if and only if the place P of K(F) corresponding to the unique branch γ of F centred at P is Fq -rational. Proof. Suppose that P = (1, a, b) is the centre of γ, and that the tangent to F at P is not the vertical line through P . Assume first that P ∈ PG(2, q). By Theorem 8.9, the branch γ has a primitive representation (x(t), y(t)) such that x(t) = x(1) (t), y(t) = y (1) (t). Therefore P is an Fq -rational place.

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To show the converse, let σ be a primitive representation of P such that σ(x) = a + t,

σ(y) = b + ϕ,

with ordt ϕ ≥ 1. Since Φ(1) (P) = P means that σ = λτ for a K-automorphism λ of K[[t]], it follows that a + t = aq + λ(t),

b + ϕ = bq + λ(ϕ−1 ).

Since both ordt λ and ordt λ(ϕ−1 ) are positive, this shows that P ∈ PG(2, q). 2 R EMARK 8.19 Let F and G be irreducible curves defined over Fq , and suppose that F and G are birationally equivalent over K. It is not true generally that they must be birationally equivalent over Fq , as well. Consider the following counter-example. For a square q, let H√q = v(H) be the non-singular Hermitian curve defined over Fq with √ q+1

H = X0

√ q+1

+ X1

√ q+1

+ X2

.

For a generic point P = (x0 , x1 , x2 ) of H√q , let K(H√q ) be the associated function field. Now, the non-singular plane curve F = v(F (X0 , X1 , X2 )) is defined to be the image of H√q under the projectivity ω of PG(3, K) associated to the matrix   t 1 tq+1 t 1 , M =  tq+1 q+1 1 t t where t is a suitable element of order q 2 + q + 1 in the cubic extension Fq3 of Fq . More precisely, the element t is chosen such that tq

√

√ q+ q

√

+ tq+ q+1 + t = 0, √ √ √ q q+q+ q+1 + t q+1 + 1 = 0, t tq and det M 6= 0. Then

√

√ q+ q+1

+ tq+1 + t

√ q

√

q

6= 0,

√ q

√ q

F (X0 , X1 , X2 ) = X0 X1 + X1 X2 + X2 X0 . Hence F can also be regarded as a curve over Fq . If H√q and F were birationally equivalent over Fq , then they would have the same number of Fq -rational branches. √ But, as in Section 8.5, the number of Fq -rational branches of H√q is q q+1, while √ √ F has at most 12 (q q + 2q + q) such points. P The map Φn can be extended to divisors. If D = nP P, then P Φn (D) = nP Φn (P). D EFINITION 8.20 Let D be a divisor of K(F), where F is an irreducible plane curve defined over Fq . (i) If Φn (D) = D, then D is an Fqn -rational divisor. (ii) The Fqn -rational divisors form the Fqn -rational divisor group Div(Fqn (F)), a subgroup of Div(Σ).

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(iii) The subgroup of Div(Fqn (F)) consisting of all divisors of degree zero is the zero-degree Fqn -rational divisor group Div0 (Fqn (F)). P If D = nP P where all P are Fqn -rational places, then D is an Fqn -rational divisor. The converse does not hold in general. To understand why this can happen, for instance when n = 1, choose any place P of K(F), and define k to be the smallest positive integer for which Φk (P) = P occurs. Then the divisor D = P + Φ(P) + · · · + Φk−1 (P)

is Fq -rational even if P is not, that is, if k > 1. The divisor D is the closed place of P, and N (P) = deg D is the degree of the closed place P. Analogous to the trace of an element in a finite field, this observation leads to a similar concept for divisors. D EFINITION 8.21 For any Fqn -rational divisor D, the divisor, T(D) = D + Φ(D) + · · · + Φn−1 (D), is an Fq -rational divisor, the trace of the Fqn -rational divisor D over Fq . The trace map defines a homomorphism, T : Div(Fqn (F)) → Div(Fq (F)), D 7→ T(D).

It may be noted that ordP ξ = ordΦ(P) ξ (1) for P ∈ P(Σ) and ξ ∈ K(F), ξ 6= 0. Therefore T(div ξ) = div ξ + div ξ (1) + · · · + div ξ (n−1) , ξ ∈ Fqn (F), ξ 6= 0, which justifies the terminology; see Section A.4. P ROPOSITION 8.22 If n is not divisible by p, then the trace map is surjective. Proof. Let D′ be any Fq -rational divisor of K(F). Then Φk (D′ ) = D′ for every k. Choose m ∈ N such that mn ≡ 1 (mod p), and let D = m D′ . Then D is an Fq -rational divisor, and hence an Fqn -rational divisor of K(F) as well. Thus Pn−1 k Pn−1 Pn−1 T(D) = k=0 Φ (D) = k=0 mΦk (D′ ) = k=0 m D′ = D′ . This proves the assertion.

2

L EMMA 8.23 Let F be an irreducible plane curve defined over Fq . If ξ ∈ Fq (F), then both div(ξ) and dξ are Fq -rational divisor. P P Proof. With the notation above, as in (8.5), if σ(ξ) = ai ti , then τ (ξ) = aqi ti . Also, P P q i−1 dσ(ξ)/dt = iai ti−1 , dτ (ξ)/dt = iai t . Thus

ordP ξ = ordΦ(P) ξ,

ordP dξ = ordΦ(P) dξ.

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L EMMA 8.24 Let F be an irreducible plane curve defined over Fq . If ξ is a nonzero element of K(F) and div ξ is an Fq -rational divisor, then ξ ∈ Fq (F) up to a constant factor. Proof. If div ξ is an Fq -rational divisor, then div ξ = div ξ (1) . By Corollary 5.35, there is a a non-zero constant a ∈ K such that aξ (1) = ξ. Choose b ∈ K such that bq−1 = a, and replace ξ by η = bξ. Then η (1) = η, as η (1) = bq ξ (1) = bq a−1 ξ = bq a−1 b−1 η = η. Hence η ∈ Fq (F), and the assertion follows from Lemma 8.15.

2

D EFINITION 8.25 For F an irreducible curve defined over Fq , a virtual linear series L of K(F) is Fq -rational if there exist x0 , . . . , xr in Fq (F) and an Fq rational divisor B such that L consists of all divisors Pr Ac = div ( i=0 ci xi ) + B, (c0 , . . . , cr ) ∈ PG(r, K).

T HEOREM 8.26 If F is an Fq -rational curve and B an Fq -rational divisor of Fq (F), then |B| is an Fq -rational linear series.

Proof. As usual, P = (x, y) is a generic point of F, and K(F) is represented by K(x, y). Choose g, f0 , . . . , fr in K[X, Y ] for which xi = fi (x, y)/g(x, y), for i = 0, . . . , r. It may be assumed that g(X, Y ) ∈ Fq [X, Y ] by the following argument. The coefficients of g(X, Y ) generate a finite subfield Fqn of K. Put h(X, Y ) = g(X, Y )(1) · · · g(X, Y )(n−1) .

From (8.2), g(X, Y )h(X, Y ) ∈ Fqn [X, Y ]. For brevity, write h = h(x, y),

hi = hfi (x, y),

for i = 0, . . . , r, and put D = B − div(gh). By Lemma 8.23, D is still an Fq rational divisor. Then |B| = |D| and |B| consists of all divisors Pr Ac = div( i=0 ci hi ) + D, (c0 , . . . , cr ) ∈ PG(r, K).

The coefficients of the polynomials hi (X, Y ), 0 ≤ i ≤ r, together with Fq generate a finite subfield Fqk of K. The simple field extension Fqk /Fq has degree k. So, Fqk = Fq (θ), with θ a root of a separable polynomial s(X) ∈ Fq [X] of degree k, irreducible over Fq . The roots of s(X) are θj = θj where j = 0, 1, . . . , k − 1. Let d ∈ Fqk [X, Y ]. For j = 0, 1, . . . , k − 1, put dj (X, Y ) = θj d(X, Y ) + θj q d(1) (X, Y ) + · · · + θj q

k−1

d(k−1) (X, Y ).

(1)

Then dj (X, Y ) = dj (X, Y ), and hence dj ∈ Fq (X, Y ). Also, d0 (x, y) = d(x, y) + d(x, y)(1) + · · · + d(x, y)(k−1) , k−1

d1 (x, y) = θd(x, y) + θq d(x, y)(1) + · · · + θq d(x, y)(k−1) , .. . q qk−1 dk−1 (x, y) = θk−1 d(x, y) + θk−1 d(x, y)(1) + · · · + θk−1 d(x, y)(k−1) .

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Note that the determinant of the matrix  1 1  θ q θ   . ..  .. .  q θk−1 θk−1

... 1 k−1 . . . θq .. ... . k−1

q . . . θk−1

     

does not vanish as {1, θ1 , . . . , θk−1 } is a basis of Fqk over Fq . Therefore there exist e0 , e1 , . . . , ek−1 ∈ Fqk such that d(x, y) = e0 d0 (x, y) + e1 d1 (x, y) + · · · + ek−1 dk−1 (x, y).

Applying this to d(X, Y ) = hi (X, Y ) for every i = 0, 1, . . . , r, it turns out that P ci hi can be written as a K-linear combination of rk elements from Fq (x, y). Since these elements are chosen independently of c = (c0 , . . . , cr ), the assertion follows. 2 R EMARK 8.27 Theorem 8.26, together with Lemma 8.23, ensures the existence of Fq -rational linear series. In particular, the canonical series of Fq (F) is an Fq rational linear series. Also, If L is an Fq -rational linear series, and P is an Fq rational place, then the sub-series of L consisting of all divisors containing P is also an Fq -rational linear series.

8.4 SPACE CURVES OVER Fq In this section, F is an irreducible plane curve defined over Fq . Also, for a generic point P = (x, y) of F, the field Fq (F) is the Fq -rational subfield of the function field K(F) of F. Consistent with the concept of an Fq -rational linear series, some but not all space curves arising from F are considered to be Fq -rational curves. The main goal is to define and characterise the Fq -rational branches of an Fq -rational space curve in terms of Fq -rational places of K(F). D EFINITION 8.28 (i) An irreducible curve Γ of PG(r, K) is defined over Fq or is Fq -rational if the curve is given by a point Q = (x0 , x1 , . . . , xr ), with xi ∈ Fq (F) for i = 0, 1, . . . , r. (ii) If Γ is defined over Fq , then a branch γ of Γ is Fq -rational if it has a branch representation, (x0 (t), x1 (t), . . . , xr (t)), with xi (t) ∈ Fq [[t]] for i = 0, 1, . . . , r. T HEOREM 8.29 Let Γ be an irreducible curve Γ of PG(r, K) defined over Fq . For a non-singular point of Γ the following conditions are equivalent: (i) P ∈ PG(r, q); (ii) the unique branch γ of Γ with centre at P is Fq -rational;

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(iii) the place P of K(F) associated to γ is Fq -rational. Proof. As usual, let x0 = 1 and let σ be a primitive representation of P. Put σ(xi ) = xi (t) for i = 1, . . . , r. Then (1, x1 (t), . . . , xr (t)) is a primitive represen(1) (1) tation of γ. Since Γ is defined over Fq , so (1, x1 (t), . . . , xr (t)) is a primitive ′ representation of the branch γ of Γ associated to Φ(P). (i) ⇒ (ii) As P is a non-singular point of F, take x1 (t) = a1 + t. Write xi (t) = ai + ϕi (t),

with ϕi ∈ K[[t]] and ordt ϕi (t) ≥ 1. Then (1)

(1)

xi (t) = aqi + ϕi (t),

for i = 1, . . . , r. As P ∈ PG(r, q), so P is the centre of γ ′ as well. Since P is the centre of exactly one branch, this implies that γ = γ ′ . So there exists τ ∈ K[[t]] (1) (1) such that xi (t) = xi (τ ). For i = 1, this gives t = τ . But then xi (t) = xi (t) for every i = 1, . . . , r; that is, γ is an Fq -rational branch of Γ. (ii) ⇒ (iii) By hypothesis, γ has a primitive representation (1, x1 (t), . . . , xr (t)) (1) with xi (t) = xi (t) for i = 1, . . . , r. If σ is the corresponding primitive repre(1) sentation of P, then σ(xi ) = xi (t) = xi (t) for i = 1, . . . , r. As in (8.5), let (1) τ = κσι−1 . Then τ (xi ) = xi (t), for 1 ≤ i ≤ r. This means that τ = σ, and hence Φ(P) = P. (1) (iii) ⇒ (i) If τ = σ, then xi (t) = xi (t), implying that ai = aqi , for each i = 1, . . . , r. 2 R EMARK 8.30 Theorem 8.26 shows how to obtain non-singular models of F defined over Fq ; in particular, the canonical curve is such a curve provided that K(F) is neither rational, nor elliptic, nor hyperelliptic. For each non-singular model, the above three conditions are equivalent and P is an Fq -rational point. T HEOREM 8.31 If Γ is an irreducible curve of PG(r, K) defined over Fq , then a branch γ of Γ is Fq -rational if and only if the corresponding place P is Fq rational. Proof. By Theorems 7.41 and 8.26, Γ is birationally equivalent over Fq to a nonsingular model X of K(F) defined over Fq . So, if Γ = K(Q) and X = K(R) with R = (1, y1 , . . . , yr ), then the rational transformation ui (x1 , . . . , xr ) ω : yi = , vi (x1 , . . . , xr )

yi ∈ Fq (F), i = 1, . . . , r

is Fq -rational; that is, ui , vi ∈ Fq [X1 , . . . , Xr ]. Note that ω −1 is also Fq -rational. Now, assume that γ is Fq -rational. Then the image γ ′ of γ under ω is also an Fq rational branch of X . By Theorem 8.29, P is Fq -rational. To prove the converse, choose a primitive representation of γ ′ , (1, y1 (t) = a1 + ϕ1 (t), . . . , yr (t) = ar + ϕr (t)), ordt ϕi (t) ≥ 1, i = 1, . . . , r,

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such that τ (yi (t)) = σ(yi (α(t)) with an automorphism α of K[[t]]. Then aqi = ai , that is, P ∈ PG(r, q), Again, Theorem 8.29 applies, whence γ ′ is Fq -rational. Since γ = ω −1 (γ ′ ), the result follows. 2 The notation for the set of all Fq -rational places of a function field Σ defined over Fq of transcendence degree 1 is Σ(Fq ). By Theorem 8.31, there is a one-toone correspondence between Σ(Fq ) and Γ(Fq ) for each birational model Γ of Σ; in particular, |Σ(Fq )| = Sq . R EMARK 8.32 If a non-singular model X of F is defined over Fq , and places of F are identified with points of X , any closed place of K(F) is viewed as a closed point. For a point A = (a0 , . . . , ar ) ∈ PG(r, K) belonging to X and having a coordinate equal to 1, let Fqk be the subfield of K generated by Fq and the coordinates of A. Theorem 8.31 shows that the closed point of A is the set consisting of the k points j

A = {A0 , A1 , . . . , Ak−1 }, j

where Aj = (aq0 , . . . , aqr ) for j = 0, . . . , k − 1. L EMMA 8.33 For every n, the field Σ has only finitely many effective Fq -rational divisors of degree n. Proof. With the notation in Remark 8.32, if D = P1 + · · · + Pn is an Fq -rational divisor, and P is a point in D, then either P ∈ X (Fq ) or P ∈ X (Fqi )\X (Fq ) for some i with 1 < i ≤ n. If the latter case occurs then the Frobenius image Φ(P ) of P is another point in D, and this holds true for Φj (P ) with j = 2, . . . i − 1. Therefore every point appearing in D with positive weight belongs to PG(r, q m ), where m is the least common multiple of all integers j with 1 < j ≤ n. As D ranges over all Fq -rational divisors of degree n, such points form a finite set {P1 , . . . , Ps } as PG(r, q n ) has only finitely many points. Put deg Pi = vi . It suffices to show that the equation Ps i=1 ni vi = n has finitely many solutions (n1 , . . . , ns ) with non-negative integers. Actually, the number of the solutions is bounded above by (n + 1)s . 2

Using Lemmas 8.24 and 8.15, it is straightforward to show that the Fq -rational principal divisors form a subgroup Prin(Fq (F)) of Prin(Σ). A divisor class [A], that is, an element [A] of the divisor class group Pic(Σ), is an Fq -rational divisor class if [A] contains at least one Fq -rational divisor. By Theorem 8.26, if [A] is an Fq -rational divisor class, then the linear series |A| is Fq -rational. The set of all Fq -rational divisors classes form a subgroup Pic(Fq (F)) of Pic(Σ), the group of the Fq -rational divisor classes of the Fq -rational function field Fq (F). Its subgroup Pic0 (Fq (F)) consisting of all Fq -rational divisor classes of degree zero is the group of the zero-degree Fq -rational divisor classes of the Fq -rational function field Fq (F). T HEOREM 8.34 The group Pic0 (Fq (F)) is finite.

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Proof. Choose a non-constant element f ∈ Fq (F) and let g = f m for a sufficiently large positive integer m. From Theorem 5.33, deg(div(f )0 ) > 0. So deg(div(g)0 ) = m deg(div(f )0 ) = n > 0 and m is chosen so that n ≥ 2g. By Lemma 8.33, it suffices to show that any Fq rational divisor of degree zero is the difference of the effective Fq -rational divisors of degree n. Let D ∈ div0 (Fq (F)). By the Riemann–Roch Theorem 6.61, dim(div(g)0 − D)) ≥ deg(div(g)0 ) − g ≥ g ≥ 0.

Since div(g)0 − D is an Fq -rational divisor, there exists a non-zero element h in Fq (F) such that div(h) + div(g)0 − D ≻ 0. Put D1 = div(g)0 ,

D2 = div(h) + div(g)0 − D.

Both divisors D1 and D2 are Fq -rational of the same degree n; therefore the equivalence D ≡ D1 − D2 follows from the fact that D = D1 − D2 + div(h) 2 This theorem gives rise to following definition. D EFINITION 8.35 Let F be an irreducible algebraic curve defined over a finite field Fq . The order of the group Pic0 (Fq (F)) is the class number. E XAMPLE 8.36 Assume that F is rational. Let A and B be any two Fq -rational divisors of the same degree n. From Theorem 6.81, the linear series |B| has dimension n, and hence A ∈ |B|. Since |B| is also Fq -rational by Theorem 8.26, there is f ∈ Fq (F) such that A = B + divf . Therefore the class number is equal to 1. Let P0 ∈ F(Fq ). For any P ∈ F(Fq ), the divisor class [P − P0 ] is contained in Pic0 (Fq (F)). If g ≥ 1, distinct places P of F(Fq ) give rise to distinct elements of Pic0 (Fq (F)). Such elements generate a subgroup G of Pic0 (Fq (F)). The following result shows that, if q is large enough compared to g, then G is the whole group. T HEOREM 8.37 Let F be an Fq -rational curve of genus g ≥ 2 containing an Fq -rational place P0 , and let G = h[P − P0 ]i with P ranging over F(Fq ). If q ≥ (8g − 2)2 , then G = Pic0 (Fq (F)). Let e denote the smallest degree of all positive degree Fq -rational divisors of Fq (F). It is easily seen that the degree of every Fq -rational divisor D is divisible by e, that is, deg D = m · e with m ∈ Z. Actually, e = 1, but this is shown later; see Proposition 9.4. Meanwhile no use is made of it. P ROPOSITION 8.38 Let [C1 ], . . . , [Ch ] be all zero-degree Fq -rational divisor classes of Fq (F), and let C0 be a fixed Fq rational divisor of Fq (F) of degree e. Then (i) any Fq -rational divisor class [C] can be written as [C] = [vC0 ]+[Ci ], where deg C = v · e and 1 ≤ i ≤ h;

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(ii) for any v ≥ 0, the number of Fq -rational divisor classes of degree v · e is equal to the class number. Proof. Let [C] be an Fq -rational divisor class of Fq (F). Then deg C = v · e. The class v[C0 ] = [vC0 ] has the same degree, and hence [C − vC0 ] is a degree zero Fq -rational divisor class of Fq (F). Therefore [C − vC0 ] = [Ci ] for some i with 1 ≤ i ≤ h. 2 T HEOREM 8.39 The number nq (|C|) of distinct effective Fq -rational divisors in an Fq -rational divisor class [C] is q dim |C|+1 − 1 . nq (|C|) = q−1

Proof. Let r = dim |C|. By Theorem 8.26 there are linearly independent elements xi ∈ Fq (F) such that the complete linear series |C| consists of the divisors Ac = div(c0 x0 + c1 x1 + · · · + cr xr ) + C, where c = (c0 , . . . , cr ) runs over all points of PG(r, K). The divisor Ac is Fq rational if and only if the corresponding point c belongs to PG(r, q). The assertion follows from the fact that |PG(r, q)| = (q r+1 − 1)/(q − 1). 2 An infinite sequence associated to Pic(Fq (F)) is A0 , A1 , . . . , An , . . . , where An = |{A ∈ Div(Fq (F)) | A ≻ 0; deg A = n}|. Note that A0 = 1, and A1 = N1 , the number of all Fq -rational places of Fq (F). For n > 2g − 2 the number An can be calculated from the class number h of Fq (F). P ROPOSITION 8.40 For any integer n > 2g − 2 divisible by e,
h An = q n+1−g − 1 . q−1

Proof. There are h divisor classes of degree n, say [C1 ], . . . , [Ch ]. By Theorem 8.39,  1  dim |Cj |+1 |{A ∈ [Cj ] ∩ div(Fq (F)); A ≻ 0}| = q −1 . q−1 This, together with the Riemann–Roch Theorem 6.61, implies that
1 |{A ∈ [Cj ] ∩ Div(Fq (F)); A ≻ 0}| = q n+1−g − 1 . q−1 Any Fq -rational divisor of degree n of Fq (F) lies in exactly one of the above divisor classes [C1 ], . . . , [Ch ]. Therefore
Ph h q n+1−g − 1 . 2 An = j=1 |{A ∈ [Cj ] ∩ div(Fq (F)); A ≻ 0}| = q−1 Write ℓ1 = Fqn , ℓ = Fq . Then the trace map Tℓ1 |ℓ for fields also defines a homomorphism, T0 = Tℓ1 |ℓ , (8.6) T0 : Pic0 (Fqn (F)) → Pic0 (Fq (F)), [D] 7→ [T(D)]. Like the trace map for fields, such a homomorphism is surjective provided that n is not divisible by p. Therefore Proposition 8.22 holds true for T0 .

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¨ 8.5 THE STOHR–VOLOCH THEOREM In this section, the fundamental theorem of St¨ohr and Voloch on the number Sq of Fq -rational points on a non-singular curve F is obtained. The essential idea is to count those points P on a non-singular model X for which the osculating hyperplane at P contains the image P q of P under the Frobenius collineation Φ. Since every Fq -rational point has this property, an upper bound for Sq is obtained. If an explicit presentation of a non-singular model X is not available, or too complicated, it is convenient to use some simpler singular model Γ defined over Fq and count the Fq -rational branches of Γ. This can be done by using the same idea but points on the curve must be regarded as branch points, in the same way as Section 7.6. An elementary proof of a simple case of the St¨ohr–Voloch Theorem is given first. T HEOREM 8.41 Let p > 2, and let F be an irreducible plane curve of degree d defined over Fq . If F has only finitely many points of inflexion, then the number Sq of Fq -rational points of F satisfies the inequality 2Sq + N ′ ≤ d(q + d − 1),

(8.7)

′

where N counts the non-Fq -rational points Q ∈ F such that the tangent line at Q contains the image Qq of Q under the Frobenius collineation. Proof. If F = v(F (X, Y )), the tangent to F at a non-singular point P = (1, a, b) is v((X − a)FX (a, b) + (Y − b)FY (a, b)). Let G(X, Y ) = (X − X q )FX + (Y − Y q )FY

(8.8)

and put G = v(G). Suppose that F (X, Y ) does not divide G(X, Y ). Then the curves G and F have no common components. Since GX = FX + (X − X q )FXX + (Y − Y q )FXY , GY = FY + (X − X q )FXY + (Y − Y q )FY Y , the tangent to F at P ∈ PG(2, q) is also tangent to G at the same point. Hence the intersection number I(P, F ∩ G) ≥ 2; this remains valid when P is a singular point of F. Also, a direct calculation shows that I(P, F ∩ G) ≥ 2 for a point P = (0, a, b) of F with a, b ∈ Fq . For any other common point Q of F and G, the tangent to F at Q contains the Frobenius image Qq . Hence, if there are N ′ points of the latter type, then B´ezout’s Theorem implies (8.7). In the case that F (X, Y ) divides G(X, Y ), write G(X, Y ) = H(X, Y )F (X, Y ). Then HX F + HFX HY F + HFY

= GX = FX + (X − X q )FXX + (Y − Y q )FXY , = GY = FY + (X − X q )FXY + (Y − Y q )FY Y .

(8.9)

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Let P = (x, y) be a generic point of F. Either x or y is a separable variable of K(F). Without loss of generality, suppose that this is x. Then FY (x, y) 6= 0. Since G(x, y) = 0, from (8.8), yq − y FX (x, y) =− q . FY (x, y) x −x

(8.10)

Also, from (8.9), FX (x, y) FX (x, y) + (x − xq )FXX (x, y) + (y − y q )FXY (x, y) = . FY (x, y) FY (x, y) + (x − xq )FXY (x, y) + (y − y q )FY Y (x, y)

(8.11)

Eliminating x − xq and y − y q from (8.10) and (8.11) shows that (x, y) is a zero of the polynomial, (FY )2 FXX − 2FX FY FXY + (FX )2 FY Y .

(8.12)

Since P = (x, y) is a generic point of F, it follows that F (X, Y ) divides (8.12). By Theorem 1.36, all non-singular points of F are inflexions. 2 Now, the general case is considered. The notation and terminology from Section 7.6 are used. In addition, Γ is defined over Fq , and the simple base-point-free linear series L is assumed to be Fq -rational. The symbol Γ(Fq ) stands for the set of all Fq -rational branches of Γ. As in Section 7.6, the term ‘point P ’ is adopted to denote the branch of Γ associated to the place P of K(Γ). In doing so, Φ(P ) denotes the Frobenius image of P , and the point (x0 (P )q , . . . , xr (P )q ) is the centre of Φ(P ). Further, P ∈ Γ(Fq ) means that the point P , viewed as a branch of Γ, is Fq -rational. Bearing this in mind, P ∈ Γ(Fq ) does not merely mean that the centre of P is in PG(r, q), even if this is true when the centre is a non-singular point of Γ. Let ζ be a local parameter at P. The osculating hyperplane to Γ at P contains the centre of Φ(P ) if and only if ∆ = 0, where x0 (P )q x1 (P )q ... xr (P )q (j0 ) (j0 ) D(j0 ) x0 (P ) D x (P ) . . . D x (P ) 1 r ζ ζ ζ (j1 ) (j1 ) (j1 ) D x (P ) D x (P ) . . . D x (P ) . (8.13) 0 1 r ∆= ζ ζ ζ .. .. .. . . . . . . (jr−1 ) (j ) (j ) D x0 (P ) D r−1 x1 (P ) ... D r−1 xr (P ) ζ

ζ

ζ

As in Section 7.6, this motivates the study of the Wronskian-type determinant ν ,...,νr−1

Wζ 0

(x0 , . . . , xr ) = det M (ν0 , . . . , νr−1 ),

where ζ denotes a separable variable, ν0 , . . . , νr−1 are non-negative integers, and   xq0 xq1 ... xqr (ν ) (ν )  D(ν0 ) x0 Dζ 0 x1 ... Dζ 0 xr  ζ     (ν1 ) (ν1 ) (ν1 ) D x D x . . . D x .  0 1 r M (ν0 , . . . , νr−1 ) =  ζ ζ ζ  .. .. ..     . . ... . (ν ) (ν ) (ν ) Dζ r−1 x0 Dζ r−1 x1 ... Dζ r−1 xr

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P ROPOSITION 8.42 There exists a sequence of increasing non-negative integers ν0 , . . . , νr−1 with ν0 ≥ 0 such that ν ,...,νr−1

Wζ 0

(x0 , . . . , xr ) 6= 0.

If the νi are chosen minimally in lexicographical order, then there exists an integer I with 0 < I ≤ r such that  ǫi for i < I, νi = ǫi+1 for i ≥ I.

Proof. Let I be the smallest integer such that the row (xq0 , xq1 , . . . , xqr ) is a lin(ǫ ) (ǫ ) (ǫ ) ear combination of the vectors (Dζ i x0 , Dζ i x1 , . . . , Dζ i xr ) with i = 0, . . . , I. Then {ν0 , . . . , νr−1 } = {ǫ0 , . . . , ǫn }\{ǫI }. Since ν0 = 0, so I > 0. 2 E XAMPLE 8.43 Let p = 5. Take F = v(Y 5 + Y − X 3 ) and Γ = {(1, x, y, y 2 )} as in Examples 7.50 (ii) and 7.53. Let q = p2 . Then det M (0, 1, 2) = det M (0, 1, 3) = det M (0, 1, 4) = 0, 1 x25 y 25 y 50 2 1 x y y = 2x4 (1 − x12 )3 . det M (0, 1, 5) = 2 2 0 1 3x x y 0 0 2x10 x − x10 y

The minimal choice of the νi has a stronger significance.

C OROLLARY 8.44 If m0 , . . . , mk are any increasing integers with m0 ≥ 0 such that the row vectors of the matrix M (m0 , . . . , mk ) are linearly independent, then νi ≤ mi for i = 0, . . . , k. Proof. The matrix M (ν0 , . . . , νk−1 ) has rank k + 1, whence νk−1 < mk ; that is, νk ≤ mk . 2 D EFINITION 8.45 The St¨ohr–Voloch divisor of L is ν ,...,ν

r−1 (x0 , . . . , xr )) + (ν1 + . . . + νr−1 ) div(dζ) + (q + r)E, S = div (Wζ 0 P where E = eP P and eP = − min{ordP x0 , . . . , ordP xr }.

By the lemma below, the integers ν0 , . . . , νr−1 and the St¨ohr–Voloch divisor depend only on the linear series L. D EFINITION 8.46 The integers ν0 , . . . , νr−1 are the Frobenius orders of the linear series L, or, equivalently, of the curve Γ, and (ν0 , . . . , νr−1 ) is the Frobenius order sequence. Since Frobenius orders of L are also L-orders, νi ≤ deg Γ for i = 0, 1, . . . , r − 1. P L EMMA 8.47 (i) If yi = aij xj with (aij ) ∈ GL(r + 1, q), then ν ,...,νr−1

Wζ 0

ν ,...,νr−1

(y0 , . . . , yr ) = det(aij ) Wζ 0

(x0 , . . . , xr ).

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(ii) If h ∈ Fq (Γ), then

ν ,...,νr−1

Wζ 0

ν ,...,νr−1

(hx0 , . . . , hxr ) = hq+r Wζ 0

(x0 , . . . , xr ).

(iii) If η is another separable variable, then ν ,...,νr−1

Wην0 ,...,νr−1 (x0 , . . . , xr ) = (dζ/dη)ν1 +...+νr−1 Wζ 0

(x0 , . . . , xr ).

Proof. This is similar to that of Lemma 7.51.

2

Corollary 5.35 together with the definition of the genus (5.55) shows that deg S = (ν1 + . . . + νr−1 )(2g − 2) + (q + r)n.

(8.14)

P ROPOSITION 8.48 The coordinate functions x0 , . . . , xr may be chosen so that x1 − xq1 ... xr − xqr (ν ) D(ν1 ) x1 ... Dζ 1 xr ζ (ν2 ) (ν ) ν ,...,νr−1 ... Dζ 2 xr . (8.15) (x0 , . . . , xr ) = Dζ x1 Wζ 0 .. . . . . ... (νr−1 ) (νr−1 ) D x1 ... Dζ xr ζ

Proof. Dividing x0 , . . . , xr by x0 is equivalent to taking x0 = 1. Now, the result is obtained by putting ν0 = 0. 2 (ν )

R EMARK 8.49 By (5.20) and Remark 5.80, Dζ i xqj = 0 if νi is not a multiple of q. Hence, if νi < q, then ν0 , . . . , νi are the first i + 1 orders of the irreducible curve of PG(r − 1, K) given by the point R = (x1 − xq1 , . . . , xr − xqr ). Such a curve is defined over Fq . P ROPOSITION 8.50 (i) If ν is a Frobenius order of L less than q, then every integer p-adically less than ν is also a Frobenius order of L. (ii) If νi < p, then (ν0 , . . . , νi ) = (0, . . . , i). Proof. This follows from Corollary 7.62.

2

Now, the weight vP (S) of the place P in the St¨ohr–Voloch divisor S is examined. Given a local parameter ζ ∈ K(Γ) at P, if the components x0 , . . . , xr are divided by ζ eP , then eP becomes zero. Therefore ν ,...,νr−1

vP (S) = vP (Wζ 0

(x0 , . . . , xr )) ≥ 0;

in particular, S is an effective divisor. Further, P is in Supp(S) if and only if xq0 (P ) xq1 (P ) ... xqr (P ) x0 (P ) x1 (P ) ... xr (P ) (ν1 ) (ν1 ) D(ν1 ) x (P ) Dζ x1 (P ) ... Dζ xr (P ) = 0. (8.16) 0 ζ .. .. .. . . ... . (νr−1 ) (ν ) (ν ) r−1 r−1 Dζ x0 (P ) Dζ x1 (P ) ... Dζ xr (P )

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Let (j0 , . . . , jr−1 ) be the (L, P)-order sequence. In the case that (ν0 , . . . , νr−1 ) = (j0 , . . . , jr−1 ), P ∈ Supp(S) if and only if Φ(P ) is centred at a point lying in the osculating hyperplane of Γ at P . If νi < ji for some i, then P ∈ Supp(S). Also, if the centre of P is in PG(r, q), the determinant in (8.16) vanishes as its first two rows coincide. Hence, Supp(S) contains every place that corresponds to a branch centred at a point in PG(r, q). In particular, all Fq -rational places of K(Γ) are in Supp(S). D EFINITION 8.51 The linear series L or, equivalently, the curve Γ is Frobenius classical if the Frobenius order sequence (ν0 , . . . , νr−1 ) is (0, . . . , r − 1). R EMARK 8.52 If Γ is Frobenius non-classical, then Γ is also non-classical in general. Exceptions only occur when the integer I in Definition 8.42 satisfies both conditions: I + 1 ≡ 0 (mod p), and I < r. This follows from Propositions 8.42 and 8.50. An exception for ν1 = ǫ2 = q = 2 is the linear series g42 cut out by lines on the irreducible plane quartic F = v((X 2 + X)(Y 2 + Y ) + 1) defined over F2 . T HEOREM 8.53 The curve Γ is Frobenius non-classical if and only if, for infinitely many places P of K(Γ), the osculating hyperplane at P contains the centre of Φ(P ). Proof. Let

xq0 x0 D(ǫ1 ) x 0 ζ ξ= .. . (ǫr−1 ) Dζ x0

xq1 x1 (ǫ1 )

Dζ

x1

.. .

(ǫr−1 )

Dζ

x1

(ǫ1 ) Dζ xr . .. . (ǫ ) Dζ r−1 xr xqr xr

... ... ... ... ...

If Γ is Frobenius classical, then ξ = 0. Hence, ∆ = 0, with ∆ as in (8.13), for every P of Γ. Conversely, ∆ = 0 implies that ξ has infinitely many zeros. Then ξ itself is zero, and hence (ν0 , . . . , νr−1 ) 6= (0, . . . , r − 1). 2 As in Remark 8.52, Frobenius non-classical curves are not classical in general. When this is the case, they have properties established in Theorem 7.65. The following result shows how that theorem can be strengthened for Frobenius nonclassical curves. T HEOREM 8.54 Let Γ be a non-classical curve with (ǫ0 , ǫ1 , . . . , ǫr ) its order sequence. Assume that ǫr ≤ pm ≤ q = ph and that (7.17) holds for some z0 , . . . , zr in K(Γ). Then Γ is Frobenius non-classical if and only if h−m

z0 x0p

h−m

+ z1 x1p

h−m

+ · · · + zr xrp

= 0.

(8.17)

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Proof. For a primitive representation σ of P, let σ(xi ) = xi (P ) + ai t + · · · , σ(zi ) = zi (P ) + bi t + · · · , for all i. Also, let m

m

m

η = z0p xq0 + z1p xq1 + · · · + zrp xqr . Then m

m

η = z0 (P )p X0 (P )q + · · · + zr (P )p Xr (P )q + tG(t). This shows that the osculating plane at P contains the centre of Φ(P ) if and only if P is a zero of η. Suppose that Γ is Frobenius non-classical. Then η = 0 as it has infinitely many zeros. Therefore (8.17) holds, because η is the pm -th power of the left-hand side in (8.17). Conversely, (8.17) implies η = 0. Hence, for every point P of Γ, the osculating plane at P contains the centre of Φ(P ). By Theorem 8.53, Γ is a Frobenius non-classical curve. 2 E XAMPLE 8.55 Let p = 5 and q = p2 . Take F and Γ as in Examples 7.50 (ii) and 8.43. Then z0 + z1 X 5 + z2 Y 5 + z3 Y 10 = Y 2 − X 6 + 2Y 6 + Y 10 = (Y 5 + Y − X 3 )2 = 0. Therefore Γ is Frobenius non-classical. Now, some numerical results follow. For P ∈ Γ, let (j0 , . . . , jr ) be the (L, P)order sequence. By convention in this chapter and Section 7.6, P is the place of K(Γ) corresponding to the branch point P ∈ Γ. P ROPOSITION 8.56

(i) If P ∈ Γ(Fq ), then P vP (S) ≥ ri=1 (ji − νi−1 ),

with equality if and only if

det

  ji 6≡ 0 (mod p), νk

0 ≤ k ≤ r − 1, 1 ≤ i < r. (ii) If P 6∈ Γ(Fq ), then vP (S) ≥ If det

Pr−1 i=1

(ji − νi ).

  ji ≡ 0 (mod p), νk

with 0 ≤ k ≤ r − 1, 0 ≤ i ≤ r − 1, then strict equality holds.

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Proof. The argument is similar to the proof of Theorem 7.55 with τ a primitive representation of P. (i) Since P is Fq -rational, the osculating plane at P is over Fq ; that is, it is HP = v(a0 X0 + · · · + ar Xr ), with a0 , . . . , ar ∈ Fq . As in the proof of Theorem 7.47, it is possible to replace xi by yi ∈ Fq (Γ) such that y0 = 1, τ (yi ) = tji + · · · , ν ,...,νr−1 (y0 , . . . , yr ), as given in with ji ≥ 1 for i = 1, . . . , r. Then ordP Wζ 0 (8.15), may be computed as follows. Since ν0 = 0 and ji > 0 for i = 1, . . . , r, ν ,...,νr−1 (1, σ(y1 ), . . . , σ(yr )) ordt (Wt 0    ji ji −νk t + ··· = ordt det νk   ji tj1 +···+jr −ν0 −...−νr−1 + . . . . = ordt det νk The result follows. (ii) There is a non-singular (r + 1) × (r + 1) matrix (aij ) with entries in K, but not necessarily in Fq , such that Pr yi = j=0 aij xj , τ (yi ) = tji + · · · , i = 0, . . . , r. Pr Let hi = j=0 aij xqj . In contrast to (i), it may not be true that hi = yiq . Nevertheless, ordP hi ≥ 0 for each i. As h0 ... hr (ν0 ) D(ν0 ) y ... Dt yr 0 t (ν ) (ν1 ) ν ,...,νr−1 ... Dt 1 y r (x0 , . . . , xr ) det(aij ) = Dt y0 Wζ 0 .. .. . ... . (νr−1 ) (ν ) n−1 D y0 ... Dt yr t Pr = 0 (−1)i hi di , where di is the minor of hi in the matrix, i = 0, . . . , n, so vP (S) ≥ min{ordt τ (d0 ), . . . , ordt τ (dr )}. Now, by the same computations as above, ordt τ (di ) ≥ j0 + · · · + jr − ji − ν0 − · · · − νr−1 . If   ji det ≡ 0 (mod p), νk i,k=0,...,r−1 then

This proves (ii).

ordt τ (dr ) > j0 + · · · + jr−1 − ν0 − · · · − νr−1 .

2

Now, numerical relations between the Frobenius orders ν0 , . . . , νn−1 and the Hermitian orders j0 , . . . , jn of an Fq -rational place P are considered.

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P ROPOSITION 8.57 Let P ∈ Γ(Fq ). If m0 , . . . , mr−1 are integers satisfying both the conditions that 0 ≤ m0 < · · · < mr−1 and   ji − j1 det 6≡ 0 (mod p), mk i=1,...,r; k=0,...,r−1 then νi ≤ mi for each i. Proof. The best choices for the integers mi are the orders of the rational map PG(1, K) → PG(r − 1, K) defined as follows: (1, x) 7→ (1, xj2 −j1 , . . . , xjr −j1 ) = (xj1 , xj2 , . . . , xjr ).

It may therefore be supposed that m0 = 0 and that   ji det 6≡ 0 (mod p). mk i=1,...,r; k=0,...,r−1 Assume again that x0 = 1 and σ(xi ) = tji + . . . for each i. Thus, as in the proof of Proposition 8.56 (i),   ji m ,...,mr−1 tj1 +···+jr −m0 −···−mr−1 + . . . 6= 0. (x0 , . . . , xr ) = det Wζ 0 mk Hence νi ≤ mi for all i, by the minimality of the νi .

2

Some useful corollaries to Propositions 8.56 and 8.57 are now stated. C OROLLARY 8.58 If P ∈ Γ(Fq ), then νi ≤ ji+1 − j1 for each i, vP (S) ≥ r j1 . In the case that mi = i, this gives the next result. Q C OROLLARY 8.59 Let P ∈ Γ(Fq ). If the integer r≥k>s≥0 (jk − js )/(k − s) is not divisible by p, then νi = i for every i, and Pr vP (S) = n + i=0 (ji − i). When mi = ǫi , Proposition 8.57 reads as follows.

C OROLLARY 8.60 Suppose P ∈ Γ(Fq ). If   ji − j1 det 6≡ 0 ǫk i=1,...,r; k=0,...,r−1

(mod p),

then νi ≤ ǫi for each i. C OROLLARY 8.61 If the Frobenius order sequence (ν0 , . . . , νr−1 ) of Γ is not classical, then every P ∈ Γ(Fq ) is an L-osculating point. Proof. If there were P ∈ Γ(Fq ) with (L, P)-order sequence (0, . . . , r), then Corollary 8.58 would imply that νi = i for all i. 2

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C OROLLARY 8.62 If the Frobenius order sequence (ν0 , . . . , νr−1 ) is other than the sequence (ǫ0 , . . . , ǫr−1 ), then every P ∈ Γ(Fq ) is an L-Weierstrass point. Proof. If there is an L-ordinary point, then νi ≤ ǫi+1 − ǫ1 by Corollary 8.58; hence νi = ǫi by Proposition 8.42. 2 C OROLLARY 8.63 If L is complete and Γ has some Fq -rational point, then νi = i for each i < n − 2g. Proof. Let P ∈ Γ(Fq ). Since L is complete, so ji = i for i ≤ d − 2g. Hence, by Corollary 8.58, νi = i for each i < n − 2g. 2 C OROLLARY 8.64 If some branch point P is Fq -rational, then νi ≤ i + n − r. Proof. Since jr ≤ n, so ji ≤ i + n − r for each i, and Corollary 8.58 gives the result. 2 T HEOREM 8.65 (St¨ohr–Voloch) Let Γ be a curve with the following properties: (a) Γ is an irreducible curve in PG(r, K) defined over Fq ; (b) n is the degree and g is the genus of Γ; (c) Sq is the number of Fq -rational points of Γ; (d) (ν0 , . . . , νr−1 ) is the Frobenius order sequence of the linear series L = gnr cut out by hyperplanes. Then the following results hold: Sq ≤ r−1 {(ν1 + · · · + νr−1 )(2g − 2) + (q + r)n};

(i)

(8.18)

(ii) in stronger form, Sq ≤ r−1 {(ν1 + · · · + νr−1 )(2g − 2) + (q + r)n − with

 Pr  k=1 (jk − νk−1 ) − r, A(P ) = P r−1  k=0 (jk − νk ),

P

A(P )},

(8.19)

for P ∈ Γ(Fq ); otherwise.

Proof. By Corollary 8.58, vP (S) ≥ r for each P ∈ Γ(Fq ). Since S is effective, so Sq ≤ deg(S)/r. Now, from (8.14), the first result (8.18) follows. Using Proposition 8.56 gives (8.19). 2 C OROLLARY 8.66 If Sq > r−1 {(r − 1)(g − 1)+ (q + r)n}, then each Fq -rational point is an L-osculating point. E XAMPLE 8.67 (i) Let C be an irreducible plane curve of degree n. By Lemma 3.24, the genus g ≤ 21 (n − 1)(n − 2).

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(a) If L1 = gn2 is the linear series cut out by on C by lines in PG(2, K), then Sq ≤ 21 {ν1 (2g − 2) + n(q + 2)}.

(8.20)

Sq ≤ 21 {n(n − 3) + (q + 2)n} = 21 n(n − 1 + q).

(8.21)

Sq ≤ 21 {2n(n − 3) + (q + 2)n} = 21 n(2n − 4 + q).

(8.22)

Here either (ν0 , ν1 ) = (0, ǫ1 ) with ǫ1 = 1 or (ν0 , ν1 ) = (0, ǫ2 ), where ǫ2 is the order of contact of C with the tangent at a generic point; so ǫ2 ≤ n. Also, ǫ2 = 2 or ǫ2 = pv for some integer v. If not every point of C is an inflexion and p 6= 2, then (ν0 , ν1 ) = (0, 1) and If, under the same condition that C is classical and p = 2, then (b) Let L2 =

5 g2n

be the linear series cut out on C by conics in PG(2, K). Then

Sq ≤ 51 {(ν1 + ν2 + ν3 + ν4 )(2g − 2) + 2n(q + 5)}. gn2 ;

(8.23) 5 g2n ,

Suppose that C is classical with respect to then, with respect to the order sequence is (0, 1, 2, 3, 4, ǫ5) for some ǫ5 ≤ 2n. If p 6= 2, 3, then ǫ5 = 5 or ǫ5 = pv . 5 If C is classical with respect to g2n , then (ν0 , ν1 , ν2 , ν3 , ν4 ) = (0, 1, 2, 3, 4),

Sq ≤

1 5 {10n(n −

3) + (q + 5)2n} = 52 n{5(n − 2) + q}.

(c) Now, let C be defined over Fq and suppose that n ≥ 3. If n ≤ s double points, then Sq ≤ 52 n{5(n − 2) + p} − 4s.

1 2p

(8.24) and C has (8.25)

This is obtained from (8.23). By Corollary 7.61, the order sequence is classical as d = 2n ≤ p. Also the genus of the curve is g ≤ 21 (n − 1)(n − 2) − s.

Then (8.23) gives (8.25). (ii) Now take the Hermitian curve √

H√q = v(X0

q+1

2 g√ q+1

and the complete linear series L1 = tangent at P = (a0 , a1 , a2 ) to H√q is √

√ q+1

+ X1

),

cut out by the lines of the plane. The

√ q

q

√ q+1

+ X2

√ q

v(a0 X0 + a1 X1 + a2 X2 ). This meets H√q at Q = (b0 , b1 , b2 ), where √ q+1

b0 with

√ q+1

a0

√

√ q+1

+ b1

√

+ a1

q

a0 b 0 +

q+1

√ q a1 b 1

√ q+1

+ b2

√ q+1

+ a2

√

= 0, = 0,

q

+ a2 b2 = 0.

Substitution for b2 gives √ √ q q

√

q

√

q

(a1 b0 − a0 b1 )(aq1 b0 − aq0 b1 ) = 0.

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√ Thus the tangent at P meets H√q with multiplicity q at P and with multiplicity one at P q = (aq0 , aq1 , aq2 ). In particular, Q = P q . So, with respect to L1 , at a √ √ generic point, j2 = q, whereas at an Fq -rational point, j2 = q + 1. Therefore the order sequence for H√q is √ (ǫ0 , ǫ1 , ǫ2 ) = (0, 1, q) √ and the Frobenius order sequence is (ν0 , ν1 ) = (0, q). For a non-singular curve √ C of degree q + 1 with a linear series giving these νi , the genus √ √ g = 12 q( q − 1), and Theorem 8.65 shows the following: Sq ≤ r−1 {(2g − 2)(ν0 + ν1 ) + (q + r)n} √ √ √ = 21 {(q − q − 2) q + (q + 2)( q + 1)} √ = q q + 1. The upper bound is achieved by H√q , To prove this, change the coordinate system over Fq such that H√q becomes √

√

v(Y q + Y + X q+1 ). Now, Y∞ is an Fq -rational point of H√q . Each vertical line v(X − c) with c ∈ Fq meets H√q in q affine points, each an Fq -rational point of H√q . In fact, if c ∈ Fq , √ √ √ then u = c q+1 ∈ F√q , and the polynomial Y q + Y − u has exactly q roots in Fq .

There are only a few curves over Fq whose number of Fq -rational places can be calculated by using a simple formula. However, certain Frobenius non-classical curves are of this kind. T HEOREM 8.68 (Hefez–Voloch) Let X be an irreducible non-singular curve of degree n defined over Fq . If ν1 ≥ 2, then Sq = n(q − n + 2). Proof. The idea is to compute vP (S) − vP (R) for any place P of Γ. Up to a change of the coordinate functions within Fq (X ), the centre P is an affine point, and the hyperplane v(X1 ) does not contain the tangent line to X at P . Then a local parameter at P is ζ = x1 − x1 (P ). Since ν1 > 1, 1 xq1 xq2 ... xqr 1 x1 x2 ... xr 0 q 0 vP (S) = = ordP ((x1 − x1 ) det V ), .. .. . . V 0 0 (i)

where V denotes the (r − 2) × (r − 2) matrix with general entry Dζ xj . On the other hand, since ν1 > 1, from Proposition 8.42, νi = ǫi+1 for i = 1, . . . , r − 1. Thus, (7.14) becomes the following: 1 x1 x2 ... xr (1) (1) 0 1 Dζ x2 ... Dζ xr 0 vP (R) = 0 = ordP (det V ). . .. .. . V 0 0

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Hence vP (S) − vP (R) = ordP (det V ). Now, it is shown that ordP (x1 − Since ν1 > 1, the matrix  1  1 0

xq1 )

xq1 x1 1

=



1 0

xq2 x2 Dζ x2

has rank 2. In particular,

for P ∈ Fq (X ), for P 6∈ Fq (X ). ... ... ...

(8.26)

(8.27)

 xqr xr  Dζ xr

(x1 − xq1 )Dζ xi = (xi − xqi )

for i = 2, . . . , r. So, if ordP (x1 − xq1 ) > 0, then ordP (xi − xqi ) > 0 for i = 2, . . . r, and hence P ∈ Fq (X ). If Dζ (x1 − xq1 ) = 1, then P 6∈ Fq (X ). Therefore (8.27) holds. From (8.26) and (8.27), Sq = ord(S) − ord(R). By (7.13) and (8.14), the result follows. 2 R EMARK 8.69 Let F be the DLS curve, that is, the irreducible plane curve considered in Examples 5.24 and 7.73; then F is defined over Fq . More precisely, F is a Frobenius non-classical plane curve with ǫ = ν = 2q0 . This follows from Theorem 8.54 for z0 (X, Y ) = X q0 +1 + Y q0 , z1 (X, Y ) = X, z2 (X, Y ) = 1. As noted before, Y∞ is the unique singular point of F, and it is the centre of only one branch of F. Theorem 8.68 does not hold true for F, since F has exactly q 2 +1 Fq -rational branch points, but n(q − n + 2) = (q + 2q0 )(−2q0 + 2) 6= q 2 + 1. E XAMPLE 8.70 Let q = ph be a power of p > 3. For a power d = pm with m < h and for a, b ∈ K with b 6= a2 , the plane curve F = v(f (X, Y )), with f (X, Y ) = X d Y d − (Y + a)X d − (X + a)Y d + XY + a(X + Y ) + b,

has the following properties: (i) F has two singular points, namely X∞ and Y∞ , both ordinary d-fold points; (ii) the tangent lines to F at X∞ are ℓ = v(Y − λ) and at Y∞ are ℓ′ = v(X − λ), with λ ranging over the solutions of λd − λ − a = 0; (iii) I(X∞ , ℓ ∩ F) = I(Y∞ , ℓ′ ∩ F) = 2d; (iv) F has no linear components at either X∞ or Y∞ ; (v) the genus of F is (d − 1)2 .

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From (ii), (iii), (iv), Exercise 4 in Chapter 1 implies the irreducibility of F. For a generic point P = (x, y) of F, the birational transformation, ω:

x1 = x,

x2 = y,

x3 = xy,

defines an irreducible curve Γ of PG(3, K). Since fY = −(X d − X − a) is not the zero polynomial, x is a separable variable in the function field K(F) of F. Also, (1)

Dx(1) y = −

(2)

yd − y − a Dx y Dx y , Dx(2) y = d , Dx(3) y = d . d x −x−a x −x−a x −x−a

(8.28)

(ǫ )

The generalised Wronskian W = det(Dx j ) arising from ω is 1 x y xy (1) (1) 0 1 D y y + xD x x y W = (2) (1) (2) 0 0 Dx y Dx y + xDx y (3) (2) (3) 0 Dx y + xDx y 0 Dx y 1 x y 0 (1) 0 y 1 Dx y = . (2) (1) 0 0 Dx y Dx y (3) (2) 0 0 Dx y Dx y



From (8.28), W = 0; hence Γ is a non-classical curve of PG(3, K). The orders are 0, 1, 2, ǫ3, with ǫ3 = pv for some v ≥ 1 by Theorem 7.65(i). Note that f (X, Y ) can be written in the form, f (X, Y ) = z0 (X, Y )d + z1 (X, Y )d X + z2 (X, Y )d Y + z3 (X, Y )d XY, (8.29) where z0 (X, Y ) = XY − c(X + Y ) + e, z1 (X, Y ) = −Y + c, z2 (X, Y ) = −X + c, z3 (X, Y ) = 1,

with cd = a and ed = b. Since z2 (x, y) = −x + c is a separable variable of K(F), so ǫ3 = d by Theorem 7.65 (i). Now, take a, b ∈ Fq , and view F as a curve defined over Fq . Then Γ is Frobenius non-classical if and only if q = d2 , aq + a = 0, b = bd . For this, the determinant (8.48) with (8.28), this is xq − x ′ 1 W = 0

Since so

νi = i, for i = 0, 1, 2, 3, is calculated. By yq − y (1) Dx y (2) Dx y

xq y q − xy q y (1) Dx y

.

xd y d − (y + a)xd − (x + a)y d + xy + a(x + y) + b = 0,

W ′ = (xy)q − (y d − a)xq − (xd − a)y q + xd y d − a(xd + y d ) + b = [(xy)q/d − (y − c)xq/d − (x − c)y q/d + xy − c(x + y) + e]d .

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Thus, a necessary and sufficient condition for Γ to be Frobenius non-classical is g(x, y) = 0, where g(X, Y ) = (XY )q/d − (Y − c)X q/d − (X − c)Y q/d + XY − c(X + Y ) + e.

Let G = v(g(X, Y )). Note that, replacing q/d by d and (c, e) by (a, b) as well, g(X, Y ) becomes f (X, Y ). Hence, G is also irreducible. From this discussion, F is Frobenius non-classical if and only if P = (x, y) is a common generic point of F and G. From Remark 5.2, F = G. Lemma 1.6 implies that f (X, Y ) = kg(X, Y ) with k ∈ K\{0}. Finally, a straightforward calculation shows that this occurs if and only if q = d2 , b = bd , ad + a = 0. The St¨ohr–Voloch Theorem applied to ω gives the inequality  2 2 for q = d2 , b = bd , ad + a = 0; 3 (d + 1)((d − 1) − 1), Sq ≤ 2 2 2(d − 1) − 2 + 3 d(q + 3), otherwise.

(8.30)

If F is defined over Fq , then the following result can be added to Theorem 7.68. T HEOREM 8.71 With the notation in Theorem 7.68, the curve F is Frobenius nonclassical if and only if there exists t(X, Y ) in K[X, Y ] such that m

m

m

t(X, Y )f (X, Y ) = z0 (X, Y )p +z1 (X, Y )p X q +. . .+zr (X, Y )p Y qs . (8.31) Also, Remark 7.69 remains valid. Despite the characterisation given in Theorems 7.68 and 8.71 it is hard to find Frobenius non-classical curves. What emerges is that they are rare but important curves that can have many Fq -rational points. In the next two sections, the two simplest cases, s = 1, 2, are considered. The case s = 1 refers to lines and the case s = 2 to conics.

8.6 FROBENIUS CLASSICALITY WITH RESPECT TO LINES In this section, classicality and Frobenius classicality are intended with respect to the linear series L1 = gn2 cut out by lines on an irreducible curve F of degree n defined over Fq . Also, if a branch of F centred at the point P and P is the associated place of Σ = K(F), then j0 = 0 < j1 < j2 denote the L1 -orders of P. For s = 1, Theorem 8.71 reads as follows. T HEOREM 8.72 Let F = v(f (X, Y )) be an irreducible plane curve defined over Fq , that is non-classical, and so (7.21) holds. Then F is Frobenius non-classical if and only if there exists t(X, Y ) in K[X, Y ] such that m

m

m

t(X, Y )f (X, Y ) = z0 (X, Y )p + z1 (X, Y )p X q + z2 (X, Y )p Y q .

(8.32)

R EMARK 8.73 For every square q, the Hermitian curve H√q = v(X

√ q+1

+Y

√

q+1

+ 1)

is Frobenius non-classical. By Theorem 8.68, the number of its Fq2 -rational points √ is equal to q q + 1.

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The following result comes from the discussion in Example 8.67. T HEOREM 8.74 If Sq is the number of Fq -rational points of F and q ′ = ν1 , then Sq ≤ 12 {q ′ (2g − 2) + (q + 2)n] ≤ 21 n[(n − 3)q ′ + q + 2}.

(8.33)

Non-classical plane curves with respect to lines are investigated in Section 7.7. Now, some more material is added on Frobenius non-classical plane curves. P ROPOSITION 8.75 Let F be a non-singular plane curve of degree n ≥ 3 defined over Fq such that F is Frobenius non-classical with ν1 = q ′ . (i) If q ′ > 2, then F is a non-classical, and ν1 = ǫ2 ≤ q. (i) If F is a strange curve, then it has genus 0. Proof. Let (0, 1, ǫ) be the order sequence of F. From Proposition 8.42, 1 ≤ ν1 ≤ ǫ. If ν1 > 2 then ν1 = ǫ2 and hence ǫ > 2. Therefore F is non-classical. In particular, ǫ = pm with m ≥ 2 when p = 2. Let P = (x, y) be a generic point of F. It may be assumed that either x, or y is a separable variable of K(F). From (7.21) and (8.32), m

m

m

z0p + z1p x + z2p y = 0, m z0p

+

m z1p xq

+

m z2p y q

= 0.

(8.34) (8.35)

Assume on the contrary that pm > q. Let pk = pm /q. Then, if k ≥ 1, (8.35) is equivalent to the equation k

k

k

z0p + z1p x + z2p y = 0.

(8.36)

This and (8.34) define a system of linear equations with solution (x, y). If (8.34) and (8.36) are dependent, then ziq−1 = zjq−1 when 0 ≤ i, j ≤ 2 and zi 6= 0, zj 6= 0. The latter condition means that zi = λzj with (q − 1)-st root of unity λi ∈ K. If this occurs, (8.34) can be written as α0 + α1 x + α2 y = 0 with α0 , α1 , α2 ∈ K not all zero. But then F must be a line, a contradiction. If (8.34) and (8.36) are independent, then x = ξ p , y = η p , with ξ, η ∈ K(F). But this is impossible since at least one of the elements x and y is a separable variable of K(F). To show the last assertion in the proposition, assume that F is a strange curve. Then z0 , z1 and z3 are linearly dependent over K, which implies that m

m

m

z0p α0 + z1p α1 + z2p α2 = 0 with α0 , α1 , α2 ∈ K not all zero. Therefore 1 x y 1 xq y q = 0, α0 α1 α2

whence α(x − xq ) = y − y q with α ∈ K\{0}. Therefore F is a component of the plane curve G = v(α(X − X q ) − (Y − Y q )). If α ∈ Fq , then G splits into q lines, and hence F itself a line. If α 6∈ Fq , let β q = α with β ∈ K. Then α 6= β and G is

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an irreducible curve of degree q. This implies that F = G. Since the point (0, 1, β) is an (q − 1)-fold point of G, Exercise 6 of Chapter 1 shows that F has genus 0. 2 For Frobenius non-classical curves of degree n < q with ν1 > 2, Corollary 8.58 extends to any branch centred at a point of PG(2, q), that is, for any branch in F(Fq )∗ . P ROPOSITION 8.76 Let F be an irreducible plane curve of degree n defined over Fq , and suppose that F is Frobenius non-classical. If ν1 > 2, then ν1 ≤ j2 − j1 ,

∗

for every branch in F(Fq ) . Proof. If the tangent to γ lies in PG(2, q) then γ is an Fq -rational branch by Theorem 8.10 and the assertion follows from Corollary 8.58. From Proposition 8.75, F is non-classical with order sequence (0, 1, ǫ2 ) with ǫ2 = ν1 and ǫ2 = pm . Here ν1 = pm < q by Proposition 8.75. Let (x(t), y(t)) be a primitive representation of a branch γ of F centred at a point P of PG(2, q); it may be assumed that P is the origin. Then, take ordt x(t) = ordt y(t) = j1 ; otherwise the tangent to γ is either v(X) or v(Y ) and hence it lies in PG(2, q). From equations (7.21) and (8.32), m

m

m

m0 (t)p + m1 (t)p x(t) + m2 (t)p y(t) = 0, q/pm

m0 (t) + m1 (t)x(t)

q/pm

+ m2 (t)y(t)

(8.37)

= 0,

(8.38)

where zi (t) = zi (x(t), y(t)) and mi (t) = t−uP zi (t), with uP = min{ordt zi (t)} for i = 0, 1, 2. Let µi = mi (0). From (8.38), µ0 = 0; more precisely, (8.38) implies that ordt m0 (t) > j1 . Then either µ1 or µ2 is different from zero, and µ1 6= 0 without loss of generality. If µ2 is zero, then (8.37) implies that j1 = pm ordt m0 (t), while (8.37) would give j1 q = pm ordt m0 (t). This contradiction shows that µ2 6= 0. Now write (8.37) in the form m

m

m

m

m0 (t)p + (m1 (t) − µ1 )p x(t) = −m2 (t)p y(t) − µp1 x(t). m

m

As ordt m0 (t)p ≥ (j1 + 1)pm and ordt [(m1 (t) − µ1 )p x(t)] ≥ pm + j1 , this gives the relation m

m

ordt (−m2 (t)p y(t) − µp1 x(t)) ≥ pm + j1 . Since µ2 6= 0, it follows that

m

m

ordt(−µp2 y(t) − µp1 x(t)) ≥ pm + j1 .

m

m

Therefore the intersection number of γ with the line ℓ = v(−µp2 Y − µp1 X) is at least pm + j1 . Hence j2 ≥ pm + j1 = ν1 + j1 . 2 T HEOREM 8.77 Let F be a non-singular plane curve of degree n defined over Fq such that F is Frobenius non-classical with ν = q ′ .

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(i) If q ′ > 2, then q−1 √ q+1≤n≤ ′ q −1

(ii) If q ′ =

and q ′ ≤

√ q.

√ q, then F is projectively equivalent to the Hermitian curve √

H√q = v(X0

q+1

√ q+1

+ X1

√ q+1

+ X2

).

Proof. Since F is non-singular, its genus g = (n − 1)(n − 2)/2 by Theorem 5.57. From (9.46), which is the Hasse–Weil Bound for non-singular plane curves, as in Section 9.6, √ Sq ≤ q + 1 + (n − 1)(n − 2) q. On the other hand, from Theorem 8.68, Sq = n(q − n + 2), whence the lower bound on n in (i) follows. Put q = ph . Since q ′ > 1, from Theorem 8.111, F is non-classical and 1 < ν1 ≤ ǫ2 = pm ≤ ph .

Then Theorem 7.65, together with Remark 7.70, shows that m

m

m

F (X0 , X1 , X2 ) = z0p X0 + z1p X1 + z2p X2 , with homogeneous polynomials z0 , z1 , z2 ∈ Fq [X0 , X1 , X2 ] of the same degree d and n = dpm + 1. On the other hand, as F is Frobenius non-classical, from the proof of Theorem 8.54 there is a homogeneous polynomial G ∈ K[X0 , X1 , X2 ] such that h−m

GF = z0 X0p

h−m

+ z1 X1p

h−m

+ z2 X2p

.

h−m

So, deg F ≤ deg(GF ). Therefore n ≤ d + p , whence the upper bound on n in (i) follows. √ Eliminating n from the first inequality in (i) gives q ′ ≤ q. If equality holds, then d = 1, and a direct computation shows that this implies (ii). 2 Now, it is shown that (ii) remains true for singular curves of low degree. T HEOREM 8.78 If F is an irreducible plane curve of degree n defined over Fq , with q square, such that √ (a) F is Frobenius non-classical with ν = q, √ (b) n < 2 q and F is not rational, then (ii) in Theorem 8.77 holds. Proof. The idea is to consider the dual curve D of F. First it is shown that D and F are birationally equivalent over Fq . The second step consists in proving that D is a non-singular plane curve which is still Frobenius non-classical with the same ν1 = q ′ . After that, the assertion follows from Theorem 8.77 (ii). From Proposition 8.75, either q = 4 and F is classical, or F is not only Frobenius √ non-classical but also non-classical with order sequence (0, 1, q). In the former

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case, F is a non-singular plane cubic curve, and the assertion follows from Theorem 8.77 (ii). √ Therefore F may be taken as non-classical with order sequence (0, 1, q). If Q = (x, y) is a generic point of F, from (7.17) and (8.17), √ q

√ q

√ q

z0 + z1 x + z2 y = 0, √

z0 + z1 x

q

+ z2 y

√ q

(8.39)

= 0,

(8.40)

where at least one of the elements zi is a separable variable. Let γ be a branch of F centred at the point A = (a, b) ∈ F, corresponding to the place P of K(F). If ordP zj = min ordP zi with i ∈ {0, 1, 2}, then, by (8.39), the tangent to γ is v(m0 (a, b)X0 + m1 (a, b)X1 + m2 (a, b)X2 ).

By Proposition 8.75 is not a strange curve. Hence z0 , z1 , z2 are linearly independent over K. The dual curve D of F is the image curve of F under the rational transformation ω:

x′0 = z0 , x′1 = z1 , x′2 = z2

defined over Fq . Let n′ denote the degree of D, and suppose that ω is not birational. ′ Then the linear series g2n consisting of all divisors P

Ac = div (c0 z0 + c1 z1 + c2 z2 ) + B,

where B = eP P and c = (c0 , c1 , c2 ) ∈ PG(2, K), is composed of an involution. Therefore, for infinitely many points A ∈ F, the tangent ℓ to F at A is tangent to F at another point A′ , as well. Since both I(A, F ∩ ℓ) and I(A′ , F ∩ ℓ) are at √ √ least q, this implies that n ≥ 2 q, a contradiction. Now, it is shown that every branch γ ′ of D is linear. To do this, take a primitive representation σ of P of K(F), and let σ(x) = x(t), σ(y) = y(t) where x(t), y(t) ∈ K[[t]]. Then the branch γ has the primitive representation x(t) y(t)

= a + a 1 t + . . . + a k tk + · · · , = b + b 1 t + . . . + b k tk + · · · .

(8.41)

Put zi (t) = σ(zi ) and σ(mi ) = mi (t). Then a homogeneous primitive representation of the branch γ ′ of D associated to P is (m0 (t), m1 (t), m2 (t)). From (8.39), √

z0 (t) whence

√

m0 (t)

q

q

√

+ z1 (t)

q

√

+ m1 (t)

√

x(t) + z2 (t)

q

q

y(t) = 0,

√

x(t) + m2 (t)

q

y(t) = 0.

(8.42)

When the formal power series mi (t) is written in the form (0)

mi (t) = µi

(1)

(k)

+ µi t + · · · + µi tk + · · · ,

for i = 0, 1, 2, then (8.42) gives the following: (0) √

(0) √

(0) √

[(µ0 ) q + (µ1 ) q x(t) + (µ2 ) q y(t)] √ (1) √ (1) √ (1) √ + t q [(µ0 ) q + (µ1 ) q x(t) + (µ2 ) q y(t)] + ··· √ (k) √ (k) √ (k) √ + tk q [(µ0 ) q + (µ1 ) q x(t) + (µ2 ) q y(t)] + · · · = 0;

(8.43)

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  (0) √ (0) √ (0) √ here (µ0 ) q , (µ1 ) q , (µ2 ) q is the centre of the branch γ ′ . This shows that the line ℓ, where (0) √q

(0) √

(0) √

X0 + (µ1 ) q X1 + (µ2 ) q X2 ), (8.44) √ √ meets F in P with multiplicity at least q; that is, I(P, ℓ ∩ F) ≥ q. √ If there were another branch of D with the same centre, then I(R, ℓ ∩ F) ≥ q √ would hold for another branch R of F, contradicting the hypothesis that n < 2 q. (1) Now suppose that γ ′ is a non-linear branch of D. Then µi = 0 for i = 0, 1, 2, and equation (8.43) implies that ℓ = v((µ0 )

(0) √

(0) √

(0) √

[(µ0 ) q + (µ1 ) q x(t) + (µ2 ) q y(t)] √ (2) √ (2) √ (2) √ + t2 q [(µ0 ) q + (µ1 ) q x(t) + (µ2 ) q y(t)] + ··· √ (k) √ (k) √ (k) √ + tk q [(µ0 ) q + (µ1 ) q x(t) + (µ2 ) q y(t)] + · · · = 0.

(8.45)

This shows that the line ℓ given by (8.44) meets F at P with multiplicity at least √ √ 2 q. But this is impossible, again by the hypothesis n < 2 q. Hence, D has only linear branches and just one is centred at each point. Therefore D is a non-singular plane curve. To prove that D is non-classical, (8.40) is used. From (8.40), √

m0 (t) + m1 (t)x(t)

q

√

+ m2 (t)y(t)

q

= 0,

(8.46)

G(t) = 0.

(8.47)

whence √

√

q

q

m0 (t) + m1 (t)x0 + m2 (t)y0 + t √ q

√ q

√ q

This shows that the line ℓ = v(X0 + a X1 + b X2 ) is the tangent to D at the √ centre of γ ′ , and that j2 ≥ q. Thus D is non-classical. Since either x or y is a separable variable of K(F), from Theorem 7.65 (iii) it follows that the orders of D √ are 0, 1, q. Also, ℓ passes through the point which is the image of the centre of γ ′ under the Frobenius collineation. This follows from (8.39). By Theorem 8.77 (ii), D is the Hermitian curve H√q as in (ii), up to a change of the coordinate system over Fq . Since the dual curve of the Hermitian curve coincides with itself, F itself is birationally equivalent to the Hermitian curve, by means of a birational transformation defined over Fq . To finish the proof, it suffices to show that F is a non-singular plane curve. From (8.47) it follows that F has only linear branches. In fact, if the branch γ were not linear, then (8.47) would imply for the corresponding place P of K(F) that √ ordP [m0 (t) + ξ0 m1 (t) + η0 m2 (t)] ≥ 2 q; that is, the line ℓ = v(X0 + ξ0 X1 + η0 X2 ) would have intersection multiplicity at √ √ least 2 q with the branch γ ′ of D, contradicting that deg D = q + 1. If the point (1, aq , bq ) were also the centre of another branch of F, then ℓ would √ be a bitangent of D, which would therefore have degree at least 2 q, again contra√ dicting that deg D = q + 1. Therefore each point of F is the centre of a single linear branch. Thus F is non-singular, and this completes the proof. 2

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Since a non-singular point P of a classical curve F is an inflexion if and only if j2 (P) 6= 2j1 (P), the classical idea to distinguish inflexions from other nonsingular points can be extended to branches. For branches centred at points of PG(2, q), this produces a partition of F(Fq )∗ into two subsets S1 S2

= {γ ∈ F(Fq )∗ | j2 = 2j1 }, = {γ ∈ F(Fq )∗ | j2 6= 2j1 },

(8.48)

where (j0 = 0, j1 , j2 ) is the order sequence at the place arising from the branch γ. This gives rise to important constants, Mq and Mq′ . N OTATION 8.79 (i) Mq is the number of branches γ ∈ S1 , each one counted j1 (P) times; (ii) Mq′ is the number of branches γ ∈ S2 , each one counted j1 (P) times. Here, Bq = Mq + Mq′ . As stated in Theorems 8.9, 8.10 and Remark 8.11, F may have branches centred at points in PG(2, q) that are not Fq -rational branches of F. In other words, Bq ≥ Sq but it is not necessary that Bq = Sq . Estimates for Bq are obtained in Section 9.6. For any Frobenius classical curve, Proposition 8.75 implies that Mq = 0, and hence Sq = Mq′ . On the other hand, if F is classical and has degree n, then Mq′ ≤ 3n(n − 2).

(8.49)

As shown in Section 8.7, for Frobenius non-classical curves with respect to conics, it is easier to estimate 2Mq + Mq′ rather than Sq . Then Sq can be estimated from that bound and some bound on Mq′ . Also, if F is classical, then the inequality √ (8.49) can be used. The latter occurs when q ≥ 25, n ≤ q and √ 2Mq + Mq′ ≥ n(q − q + 1). (8.50) This is a consequence of the following result. T HEOREM 8.80 Let F be an irreducible plane curve of degree n defined over Fq √ with q odd and q ≥ 25. If n ≤ q + 1 and (8.50) holds, then F is either classical or projectively equivalent to the Hermitian curve √

H√q = v(X0

q+1

√ q+1

+ X1

√ q+1

+ X2

).

Proof. Suppose that F is non-classical. Then Corollary 7.60 implies that j1 j2 (j2 − j1 ) ≡ 0 (mod p),

(8.51)

for any branch γ of F centred at a point P of PG(2, K). First, the case where F is Frobenius classical is investigated. From Theorem 8.56, vP (S) ≥ (j2 − 1) + j1 . If γ ∈ S2 , this implies that vP (S) ≥ 2j1 . If γ ∈ S1 , it follows from (8.51) that j1 ≡ 0 (mod p). From this and Theorem 8.56 (ii), vP (S) > j2 + j1 − 1, whence vP ≥ 3j1 .

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Therefore (2g − 2) + (q + 2)n = deg S ≥ 3Mq + 2Mq′ ≥

3 2

(2Mq + Mq′ ) .

(8.52)

Since n(n − 3) ≥ 2g − 2, it follows from (8.52) and (8.50) that √ (n − 3) + (q + 2) ≥ 23 (q − q + 1); √ √ that is, 2n ≥ (q − 3 q + 5). But this cannot actually occur when both n ≤ q + 1 and q ≥ 25. Suppose that F is Fq -Frobenius non-classical. It is shown first that n(2g − 2) + 3n ≥ deg(R) = (1 + ǫ)(2g − 2) + 3n ≥ 2Mq + Mq′ .

(8.53)

To prove the first inequality in (8.53), it must be shown that n ≥ ǫ + 1. If F has an Fq -rational point, this follows from Corollary 8.58 as ν1 = ǫ. Otherwise F(Fq )∗ consists of singular points each being the centre of at least two branches. Take two branches γ and γ ′ of F centred at P ∈ F, and let ℓ be the tangent to γ. Then n ≥ I(P, ℓ ∩ γ) + I(P, ℓ ∩ γ ′ ) ≥ ǫ + 1.

To prove the second equality in (8.53), note that from Theorem 7.55, vP (R) ≥ (j2 − ǫ2 ) + (j1 − 1) = j2 − j1 − ǫ2 + 2j1 − 1. If γ ∈ S1 then j2 ≡ 0 (mod p) by (8.51), and hence the strict inequality holds. This, together with Proposition 8.76 and ǫ2 = ν1 , implies that vP (R) ≥ 2j1 . If γ ∈ S2 , the above argument shows that vP (R) ≥ 2j1 − 1 ≥ j1 . Therefore (8.53) holds. √ Since n(n − 3) ≥ 2g − 2 by Lemma 3.24, (8.53) shows that n ≥ q + 1. This √ √ and the hypothesis n ≤ q + 1 imply that n = q + 1. Hence, n(n − 3) = 2g − 2 √ and ǫ2 = ν1 = q. By Lemma 3.24, F is non-singular and the assertion follows from Theorem 8.77 (ii). 2 T HEOREM 8.81 For a polynomial f (X) ∈ Fq [X] having some simple root, let F = v(Y n − f (X)) with p ∤ n be an irreducible plane curve. If p > 2, then the following hold. (i) F is non-classical if and only if n ≡ 1 (mod p) and there exist polynomials g(X), h(X) ∈ K[X] with deg g(X) ≥ deg h(X) such that f (X) = g(X)p X + h(X)p .

(ii) F is Frobenius non-classical if and only if

(8.54)

s

g(X)p X q + h(X)p = (g(X)p X + h(X)p )p

m′

,

(8.55)

′

where m is given by the system,  n = ps m + 1, m 6≡ 0 (mod p), q − 1 = (ps m′ − 1)(ps + m), m′ ≡ 6 0 (mod p).

(8.56)

(iii) If F is Frobenius non-classical, then deg f (X) = n. If, in addition, the multiplicities of the roots of f (X) are relatively prime to p, then all roots of f (X) belong to Fq .

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Proof. First, K(F) = K(x, y) with y n − f (x) = 0. Here, x is a separable variable of K(F) by the hypothesis p ∤ n. The non-classicality of F is expressed by the (2) condition Dx y = 0. By (5.24) and Lemma 5.80, differentiating y n − f (x) twice with respect to x gives the equation 2ny n−1 Dx(2) y +

(n − 1)f ′ (x)2 = f ′′ (x), nf (x) (2)

where f ′ (x) is f ′ (X) evaluated at x, and similarly for f ′′ (x). So, Dx y = 0 is equivalent to that (n − 1)f ′ (x)2 = nf ′′ (x)f (x). Calculating both sides at a simple root of f (X), this equation holds if and only if n ≡ 1 (mod p) and f ′′P (x) = 0. Since x is not a constant, f ′′ (x) = 0 implies f ′′ (X) = 0. Write f (X) = ai X i . Now, f ′′ (X) = 0 occurs if and only if every ai with i 6≡ 0, 1 (mod p) vanishes. Hence, f (X) = g(X)p X + h(X)p with g(X), h(X) ∈ Fq [X]. Further, deg f (X) is not divisible by p if and only if deg g(X) ≥ deg h(X). Therefore (i) is proved. To prove (ii), the following hypotheses may be assumed: Y n = g(X)p X + h(X)p ; n ≡ 1 (mod p); deg g(X) ≥ deg h(X).

(8.57)

Using the first equation, the Frobenius classicality of F expressed by the relation y q − y = (xq − x)Dx(1) y

becomes the following: y q−1+n = g(x)p xp + h(x)p .

(8.58)

q−1+n

So, if F is Frobenius non-classical, then y ∈ K(x). No power of y with exponent smaller than n is in K(x), since [K(F) : K(x)] = n. Hence n divides q −1+n, and so q −1 = λn for a positive integer λ. Since n ≡ 1 (mod p), there is a positive integer m such that n = ps m + 1 with m a positive integer not divisible by p. Thus λ ≡ −1 (mod p). So, λ = pr m′ − 1 with m′ a positive integer not divisible by p. Hence, q = psr mm′ + pr m′ − ps m. From this, it follows that r = s, otherwise either m or m′ would be divisible by p. Then m ≡ m′ (mod p). Therefore s

y q−1+n = (y n )p

m′

s

= (g(x)p x + h(x)p )p

m′

,

(8.59)

whence s

g(x)p xq + h(x)p = (g(x)p x + h(x)p)p

m′

.

Since x is not a constant, this shows (8.55). Conversely, if (8.55) holds, then F is Frobenius non-classical. This completes the proof of (ii). Comparison of degrees in (8.55) gives the equation p deg g(X) + q = (p deg g(X) + 1)ps m′ . Since q − 1 = (ps m′ − 1)(ps m + 1), it follows that deg g(X) = ps−1 m. Thus deg f (X) = p deg g(X) + 1 = ps m + 1 = n,

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which proves (iii).

2

Note that, writing f (X) = a0 X n + a1 X n−1 + . . . + an , a change of the indeterminate X of type X ′ = aX + b can be made with a, b ∈ Fq such that f ′ (X) = f (aX +b) has no term of degree n−1. Therefore, deg g(X) > deg h(X) may be assumed in Theorem 8.81 without loss of generality. Now, the cases q = p2 or q = p3 in Theorem 8.81 are considered more closely. Assume that both (8.54) and (8.55) hold. If q = p2 , then s = m = m′ = 1 can only occur in (8.56). Also, deg g(X) = ps−1 m = 1. Hence h(X) is a constant polynomial. It turns out that F is projectively equivalent over Fq to the Hermitian curve H√q . If q = p3 , there are examples which are not Fermat curves. Since p3 − 1 = (p − 1)(p(p + 1) + 1),

from (8.56), s = m′ = 1, m = p + 1, and deg g(X) = p + 1. Since it is assumed that deg g(X) > deg h(X), so (8.55) gives the equation 2

g(X)X p + h(X) = g(X)p X + h(X)p . Comparing the coefficients gives that 2

g(X) = αX p+1 + β p ,

h(X) = β p X p + βX + δ

(8.60)

with α, δ ∈ Fp , α 6= 0, and β ∈ Fp3 . For instance, α = β = 1 and δ = 0 is a good choice, giving rise to the Frobenius non-classical curve 2

F = v(Y p

+p+1

− ((X p+1 + 1)p X + (X p + X)p )).

(8.61)

It may be noted that this curve is non-singular. By Theorem 8.68, the number Sq of its Fq -rational places is (p2 + p + 1)(p3 − p2 − p + 1). R EMARK 8.82 Let q = p3 . Up to birational equivalence over Fq , the curve F from (8.61) is the only Frobenius non-classical curve of the form v(Y n − f (X)) over Fq . This follows from the fact that (8.56) is only possible when s = m′ = 1 and m = p + 1.

8.7 FROBENIUS CLASSICALITY WITH RESPECT TO CONICS This section continues the study of irreducible curves F which are non-classical with respect to the linear series L2 cut out by conics. The L2 -orders are assumed to be 0, 1, 2, 3, 4, pm with p > 2. In particular, F is classical with respect to L1 . The notation is that of Section 7.8. In addition, F is defined over Fq and it is assumed that pm < q = ph . The hypothesis on F to be Frobenius non-classical is expressed by the existence of a polynomial t(X, Y ) ∈ K[X, Y ] such that t(X, Y )f (X, Y ) = m m m z0 (X, Y )p + z1 (X, Y )p X q + z2 (X, Y )p Y q m m m +z3 (X, Y )p X 2q + z4 (X, Y )p X q Y q + z5 (X, Y )p Y 2q .

(8.62)

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Equivalently, m

m

m

m

m

m

v0p + v1p xq + v2p y q + v3p x2q + v4p xq y q + v5p y 2q = 0. As in the proof of Theorem 8.77, in (8.63), h ≥ m. Then h−m

v0 + v1 xp

h−m

+ v2 y p

h−m

+ v3 x2(p

)

h−m

+ v4 (xy)p

h−m

+ v5 y 2(p

)

(8.63)

= 0. (8.64)

In terms of osculating conics, as in Lemma 7.74, Frobenius non-classicality is expressed in the following theorem. T HEOREM 8.83 If L2 is Frobenius non-classical, then the conic C0 passes through the image Aq = (1, aq , bq ) of the centre A = (1, a, b) of γ under the Frobenius collineation. In addition, if A ∈ PG(2, q), then C0 meets γ with multiplicity greater than pm . Proof. Take ξ, η in K[[t]] such that ξ(tq ) = x(t)q and η(tq ) = y(t)q . From (8.63), (0)

(0)

m

(0)

m

(0)

m

m

[(µ0 )p + (µ1 )p ξ(tq ) + (µ2 )p η(tq ) + (µ3 )p ξ(tq )2 (0) m (0) m +(µ4 )p ξ(tq )η(tq ) + (µ5 )p η(tq )2 ]+ m (1) m (1) m (1) m (1) m tp [(µ0 )p + (µ1 )p ξ(tq ) + (µ2 )p η(tq ) + (µ3 )p ξ(tq )2 + m (1) (1) pm +(µ4 ) ξ(tq )η(tq ) + (µ5 )p η(tq )2 ] + · · · (k) m (k) m (k) m (k) m kpm + t [(µ0 )p + (µ1 )p ξ(tq ) + (µ2 )p η(tq ) + (µ3 )p ξ(tq )2 m (k) pm (k) +(µ4 ) ξ(tq )η(tq ) + (µ5 )p η(tq )2 ] + · · · = 0. Since x(t) = a + tG(t) and y(t) = b + tH(t), it follows that ξ(tq ) = aq + tq G1 (t),

η(tq ) = bq + tq H1 (t).

Hence (0)

m

(0)

m

(0)

m

m

(0)

2

(µ0 )p + (µ1 )p aq + (µ2 )p bq + (µ3 )p (aq ) (0)

m

(0)

m

2

+(µ4 )p aq bq + (µ5 )p (bq ) = 0. This shows that that C0 passes through the point Aq = (1, aq , bq ). Also, since pm < q, (1)

m

(1)

m

(1)

m

m

(1)

2

(µ0 )p + (µ1 )p aq + (µ2 )p bq ) + (µ3 )p (aq ) (1)

m

(1)

m

2

+(µ4 )p aq bq + (µ5 )p (bq ) = 0. If aq = a, bq = b, then this equation is s1 (a, b) = 0. By Theorem 7.74 (ii), the second assertion follows. 2 For the rest of this section, F is a Frobenius non-classical curve with Frobenius orders 0, 1, 2, 3, pm. The St¨ohr–Voloch Theorem applied to L2 reads as follows: Sq ≤ 15 {(6 + pm )(2g − 2) + (q + 5)2n}.

(8.65)

The aim is to find other good upper bounds on the number Sq of Fq -rational branches of such a curve. The idea is to exploit Frobenius classicality with respect to conics to compute the possible (L1 , P)-orders, rather than counting Fq -rational branches only. If there are only a few possibilities for the (L1 , P)-orders, and the

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weights vP (S) can be calculated using the previous numerical formulas, this idea can work effectively, and the St¨ohr–Voloch divisor arising from the linear series L1 cut out by lines can bring about an improvement to (8.65). By Theorem 7.76, the hypothesis n < pm ensures that only three cases can occur for (L1 , P)-orders, and in each case j2 is calculated from j1 . Now, these three cases are investigated further. P ROPOSITION 8.84 Let n < pm . If (7.33) holds and the centre A = (1, a, b) of γ is not in PG(2, q), then the Frobenius image Aq = (1, aq , bq ) does not lie on the tangent to γ; also, (a − aq )v21 − (b − bq )v11 6= 0.

Proof. By Theorem 8.83, C0 passes through Aq = (1, aq , bq ). On the other hand, when (7.33) holds, C0 is a non-degenerate conic whose tangent line at A = (1, a, b) coincides with the tangent of γ. 2 P ROPOSITION 8.85 Let n < pm . If (7.34) holds and the centre A = (1, a, b) of γ is not in PG(2, q), then the Frobenius image Aq = (1, aq , bq ) lies on the tangent to γ; also, (a − aq )v21 − (b − bq )v11 = 0. Proof. If (7.34) holds, then C0 is a degenerate conic and consists of the tangent line of γ counted twice. Thus, the assertion is an immediate consequence of Theorem 8.83. 2 P ROPOSITION 8.86 If (7.35) holds, then the centre of γ is not in PG(2, q). Proof. In this case, C0 splits into two distinct lines, one of which coincides with the tangent of γ. This shows that C0 meets γ with multiplicity j1 + j2 = pm . Hence, the result follows from the second assertion in Theorem 8.83. 2 Now, it is shown that Frobenius non-classicality sharply limits the possible values of j1 . In the following theorem, Z is the curve introduced in Section 7.8, P is the branch of Z associated to a branch γ of F, and P is the corresponding place. T HEOREM 8.87 There is a hyperplane H in PG(5, K) such that I(P, Z ∩ H) ≥ j1 ph−m . Proof. Without loss of generality, let γ be centred at an affine point A = (1, a, b). h−m h−m Let α, β ∈ K such that α = ap and β = bp . The intersection multiplicity of Z with the hyperplane H = v(X0 + αX1 + βX2 + α2 X3 + αβX4 + β 2 X5 )

(8.66)

at P is I(P, Z ∩ H) = ordP (v0 + αv1 + βv2 + α2 v3 + αβv4 + β 2 v5 ). Putting ξ = x − a, η = y − b gives j1 = min{ordP ξ, ordP η}. From (8.64),

v0 + αv1 + βv2 + α2 v3 + αβv4 + β 2 v5 h−m h−m h−m h−m h−m ) ) +ξ p v1 + η p v2 + ξ 2(p v3 + (ξη)p v4 + η 2(p v5 = 0,

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whence ordP (v0 + αv1 + βv2 + α2 v3 + αβv4 + β 2 v5 ) ≥ j1 ph−m , which proves the result.

2

It is worth mentioning the following result that is valid for singular curves. P ROPOSITION 8.88 If F is singular, then deg Z ≥ 2ph−m . Proof. Let A = (1, a, b) be a singular point of F, and let H be the hyperplane (8.66). If A is the centre of only one branch of F, it must have order greater than 1, and the assertion follows from Proposition 8.87. Suppose that A is the centre of two branches γ1 and γ2 of F. The corresponding branches P1 and P2 of Z are distinct. Therefore deg Z ≥ I(P1 , Z ∩ H) + I(P2 , Z ∩ H). From Proposition 8.87, I(Pi , Z ∩ H) ≥ ph−m for i = 1, 2, and the assertion follows. 2 If n < ph−m also holds, then Proposition 8.87 together with Theorem 7.80 gives the following sharp bound for j1 . P ROPOSITION 8.89 If n ≤ min{pm , ph−m }, then j1 ≤ 3. From this, an upper bound on Sq is obtained after giving an explicit formula for vP (S) that actually holds without any limitation on n with respect to pm and ph−m . Since vP (S) remains invariant under the change of coordinates (7.27) only when all coefficients are from Fq , it is necessary to know how vP (S) changes in general under (7.27). For this, take a local parameter ζ at P, and adopt the usual simpler notation u′ in place of Dζ (u) for u ∈ K(F). Since (x − xq )y ′ − (y − y q )x′

= [a − aq + v11 x ¯ + v12 y¯ − (v11 x¯ + v12 y¯)q ][v21 x ¯′ + v22 y¯′ ] q q −[b − b + v21 x ¯ + v22 y¯ − (v21 x¯ + v22 y¯) ][v11 x ¯′ + v12 y¯′ ]

= [(a − aq )v21 − (b − bq )v11 ]¯ x′ + [(a − aq )v22 − (b − bq )v12 ]¯ y′ +(v11 v22 − v12 v21 )(¯ xy¯′ − y¯x ¯′ ) − (v11 x ¯ + v12 y¯)q (v21 x ¯′ + v22 y¯′ ) +(v21 x ¯ + v22 y¯)q (v11 x ¯′ + v12 y¯′ ),

it follows that vP (S) =

ordP x¯′ + ordP {[(a − aq )v21 − (b − bq )v11 ] + [(a − aq )v22 − (b − bq )v12 ]y¯′ /¯ x′ + (v11 v22 − v12 v21 )(¯ xy¯′ /¯ x′ − y¯) − (v11 x ¯ + v12 y¯)q (v21 + v22 y¯′ /¯ x′ ) q + (v21 x ¯ + v22 y¯) (v11 + v12 y¯/¯ x′ )}.

T HEOREM 8.90 Let n ≤ min{pm − 4, ph−m }.

(8.67)

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(i) If j1 6= 3 for every branch of F, then (q − 1)n − 2(2g − 2) . 2

(8.68)

(q − 1)n − 2(2g − 2) + 12n . 2

(8.69)

Sq ≤ (ii) If p > 3, then Sq
3, then Sq ≤

R EMARK 8.94 Let q ≥ 49 be a square power of p with p > 2. The irreducible plane curve G = v(g(X0 , X1 , X2 )), with √ 2 q

g(X0 , X1 , X2 ) = X22 X0

√ 2 q

+ X12 X2

√ q+1

−2(X0 X1

√ q

√ 2 q

+ X02 X1 √ q+1

X2 + X0

√ q

√ q

√ q+1

X1 X2 + X0 X1 X2

),

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is classical with respect to lines, but non-classical with respect to conics, since the √ L2 -orders are 0, 1, 2, 3, 4, q. In fact, (7.22) holds for f (X, Y ) = G(1, X, Y ) when h = 1, z0 = X 2 , z1 = −2XY, z2 = −2X, z3 = Y 2 , z4 = −2Y, z5 = 1. Since (8.62) does not hold, G is Frobenius classical with respect to conics. Also, G has three singular points, namely the vertices of the fundamental triangle. Each of them is a double point and is the centre of only one branch. The (L1 , P)-orders √ of the corresponding place P of such a branch are 0, 2, 2 q. The genus of G is √ 1 2 (q − q). Now, the irreducible plane curve F = v(f (X0 , X1 , X2 )) is defined to be the image of F under the projectivity of PG(3, K) associated to the matrix   α 1 αq+1  αq+1 α 1 , (8.74) q+1 1 α α

where α is an element of order q 2 + q + 1 in the cubic extension Fq3 of Fq . Equivalently, f (αX0 + X1 + αq+1 X2 , αq+1 X0 + αX1 + X2 , X0 + αq+1 X1 + αX2 ) = g(X0 , X1 , X2 ).

Now, G and F have the same L1 - and L2 -orders. In particular, F is classical with respect to lines, but non-classical with respect to conics. Although the projectivity (8.74) is defined over Fq3 , the curve F is defined over Fq ; this is not trivial. It might be thought that F is also Frobenius classical with respect to conics. Actually, this is false. In fact, F is the image of the Hermitian curve, √

H√q = v(X0

q+1

√ q+1

+ X1

√ q+1

+ X2

),

√ under a birational transformation defined over Fq . In particular, Sq = q q + 1. √ This is consistent with (8.65) only for pm = q, yielding that the last Frobenius √ √ order is q. This example also shows that the hypothesis n ≤ q − 4 in Theorem √ 8.90 cannot be weakened to n ≤ 2 q + 2. To obtain an upper bound on Sq , begin with (7.43), evaluate ordP B(¯ x, y¯) using vP (S), and calculate deg S. P ROPOSITION 8.95 Let n < pm . (i) If a, b ∈ Fq , then ordP B ≤



vP (S) − (3j1 − 1) vP (S) − (j1 − 1)

(ii) If either a 6∈ Fq or b 6∈ Fq , then  vP (S) − (j1 − 1) ordP B ≤ vP (S) + 1

if (7.33) holds, if (7.34) holds.

(8.75)

if (7.33) holds, if (7.34) or (7.35) holds.

(8.76)

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321

Proof. Let a, b ∈ Fq . From (8.67), vP (S) ≥ j1 + j2 − 1. By Proposition 8.86, either (7.33) or (7.34) occurs. In the former case, ordP B = 0 and j2 = 2j1 ; in the latter case, ordP B = j2 and 2j2 − j1 = pm . From this, (i) is obtained. The argument is similar for (ii). 2 Summing up the inequalities in Proposition 8.95 gives the inequality P P P ordP B ≤ vP (S) − 2Mq − Mq′ − ′ (j1 − 1) + N, (8.77) P ′ where Mq and Mq′ are as in Definition 8.79, while the summation is over all branches of F of type (7.33), and N is the number branches of type (7.34) or (7.35). An upper bound on N is N ≤ 6(2g − 2 + d)/(pm − 3).

(8.78)

To show (8.78), from the inequality vP ≥ j2 + j1 − 3, it follows that P P (j2 + j1 − 3) ≤ vP (R) = 3(2g − 2) + 3n.

On the other hand, P P′′ m (j2 + j1 − 3) = N0 (pm − 3) + (p + 3j1 − 6)/2,

where N0 is the number of branches of type (7.33) while the summation over the N − N0 branches of type (7.34). Therefore P (j2 + j1 − 3) ≥ N0 (pm − 3) + (N − N0 )(pm − 3)/2.

P′′

is

Hence

P

(j2 + j1 − 3) ≥ 21 N (pm − 3),

and (8.78) follows. It should be noted that, since 2g − 2 ≤ n(n − 3) and n ≤ pm − 1, the expression on the right of (8.78) is at most 6n. Hence (8.77) can be put in a more manageable but somewhat weaker form as follows: P P ordP B ≤ vP (S) − 2Mq − Mq′ + 6n. (8.79) This together with (7.44) gives the inequality

pm deg Z < c + (2g − 2) + [(q + 1)n − 2Mq − Mq′ ] + 6n.

(8.80)

Also, from Theorem 8.87, j1 ≤

c + (2g − 2) + [(q + 1)n − 2Mq − Mq′ ] + 6n . q

(8.81)

Since c ≤ n(n − 1) and 2g − 2 ≤ n(n − 3), from (8.80) and (8.81), the following result is obtained. P ROPOSITION 8.96 Let n < pm . If (8.50) holds, then √ pm deg Z ≤ n(2n + q + 2), √ j1 ≤ n(2n + q + 2)/q.

(8.82) (8.83)

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√ = q, that is,

Now, the case that q is a square and pm  √ (i) the order sequence is (0, 1, 2, 3, 4, q), √ (ii) the Frobenius order sequence is (1, 2, 3, q), √ is investigated in detail. Since n ≤ q − 1, √ √ n(2n + q + 2)/q ≤ 3 − 3/ q < 3.

(8.84)

By (8.83), this gives the following result. √ P ROPOSITION 8.97 If n < q, and both (8.50) and (8.84) hold, then (i) j1 = 1 for branches of type (7.34) while j1 ≤ 2 for those of type (7.33) and (7.35); (ii) for branches of type (7.35), there are only two cases: √ (a) n = q − 2 and j1 = 2; √ (b) n = q − 1 and j1 = 1. It should be noted that the linearity of the branches of type (7.34) depends on the condition that p > 2. A first consequence of Proposition 8.97 is the following result. P ROPOSITION 8.98 Let τ1 and τ2 be the numbers of branches of type (7.34) and (7.35). Then 3 deg Z = 2τ1 + τ2 ;

3 deg Z = 2τ1 ,

√ when n < q − 2.

(8.85) (8.86)

Proof. From Proposition 8.97, j1 ≤ 2 < p. So, by Corollary 7.60 the classical formula vP (R) = j1 + j2 − 3 holds. Also, ordP d¯ x = j1 − 1. From the discussion after (7.43), the exact value of ordP B is known, namely  if the branch is of type (7.33),  0 j2 if the branch is of type (7.34), ordP B =  j1 if the branch is of type (7.35). This, together with (7.42), (7.43) and Lemma 7.79, gives (8.85). The second assertion is a consequence of (8.85) and Proposition 8.97 (ii). 2

P ROPOSITION 8.99 If each of the conditions (8.86), (8.50), (8.84) is satisfied, then √ deg Z ≤ 2n < 2( q − 2). √ Proof. Since n < q − 2, it follows from Proposition 8.97 that no branch of type (7.35) exists. Hence, for a branch γ, 3vP (S) − vP (R)  6j1    0 √ = q + 3j   √ 1  q

if γ if γ if γ if γ

is of type (7.33) centred at a point in PG(2, q), is of type (7.33) centred at a point not in PG(2, q), is of type (7.34) centred at a point in PG(2, q), is of type (7.34) centred at a point not in PG(2, q).

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Since P 3vP (S) − vP (R) = 3(2g − 2) + 3(q + 2)n − 3(2g − 2) − 3n = 3nq + 3n,

it follows that

√ 3nq + 3n = 6Mq + 3Mq′ + τ1 q, whence 2(nq − 2Mq − Mq′ + n) =

√

q deg Z,

by Proposition 8.98. Now, from (8.50), the assertion follows.

2

T HEOREM 8.100 If each of the conditions (8.86), (8.50), (8.84) is satisfied, then (i) F is non-singular ; √ (ii) n = 12 ( q + 1); (iii) the number of Fq -rational points of F is √ √ √ Sq = q + 1 + 41 ( q − 1)( q − 3) q. √ Proof. Suppose that F has a singular point. Since ph−m = q, Proposition 8.88 √ implies that deg Z ≥ 2 q; this contradicts Proposition 8.99. As F is non-singular and classical with respect to lines, (7.20) applies. Since the inflexions of F are exactly the points of type (7.34), it follows that √ τ1 = 6n(n − 2)/( q − 3). √ Hence, by Proposition 8.98, deg Z = n(n − 2)/( q − 3). Therefore the first assertion follows from Proposition 7.80. To prove the second it suffices to show that every inflexion point of F lies in PG(2, q) and apply Theorem 8.91. Assume on the contrary that F has an inflexion point P outside PG(2, q). If ℓ is the tangent to F at P , then Theorem 7.78 shows that ℓ contains the Frobenius image P q of P . Since P 6= P q , B´ezout’s Theorem gives the inequality √ n = |F ∩ ℓ| ≥ 21 ( q + 1) + 1, √ 2 contradicting n = 12 ( q + 1). √ For the Fermat curve F = v(X n + Y n + 1) with n = 12 ( q + 1), described in Example 7.81, √ √ √ Mq′ = 23 ( q + 1). Mq = 41 ( q + 1)(q − q − 2) This shows that the conditions in Proposition 8.100 can be fulfilled. In fact, this Fermat curve is, up to projectivities in PG(2, q), the only non-singular plane curve of degree n defined over Fq that satisfies both conditions in Proposition 8.100. Therefore the following result holds. √ T HEOREM 8.101 If (8.86), (8.50) and (8.84) hold, then n = 21 ( q + 1) and F is the Fermat curve.

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Theorem 8.101 shows that an irreducible plane curve defined over Fq is fully determined by three conditions, namely (8.86) on the degree, (8.50) on the number of Fq -rational points, and (8.84) on the Frobenius order sequence. Actually, the following proposition shows that the latter condition is independent of the other two only for small values of q. P ROPOSITION 8.102 Let q ≥ 25 be a power of an odd prime p such that if p = 3 then q is a square. Let G be an irreducible plane curve of degree n ≥ 3 and defined over Fq . If condition (8.50) is satisfied and √  q+1 for q > 232 , q 6= 36 , 55 ,      for q = 36 , 23 (8.87) n < 49 for q = 55 ,   √  min (q − 5 q + 45)/20 ,    √  for q ≤ 232 , (q − 5 q + 57)/24 then q is square and (8.84) holds.

Proof. By Theorem 8.80, G is classical with respect to lines. Consequently, the L2 -order sequence of G is (0, 1, 2, 3, 4, ǫ) and the Frobenius L2 -order sequence is (0, 1, 2, 3, ν) with either ν = 4, or ν = ǫ. To show that ν = ǫ, suppose on the contrary that ν = 4. Since 5Sq = 5(Mq + Mq′ ) ≥ 25 (2Mq + Mq′ ), from (8.65), √ 10(n − 3) + (q + 5)n ≥ 52 (q − q + 1), √ whence n ≥ (q − 5 q + 45)/20, a contradiction. Hence ǫ = ν ≥ 5. If equality holds, then either p = 3, or p = 5. From Remark 8.52, ǫ is a power of p for p ≥ 5, apart from just one possibility for p = 3, namely when ǫ = ν = 6. The latter case cannot actually occur under the hypothesis (8.87). In fact, for ν = 6, (8.65) implies that √ 12(n − 3) + (q + 5)2 ≥ 25 (q − q + 1), √ so that n ≥ (q − 5 q + 57)/24 and hence q > 232 by (8.87). From the inequality √ √ (q − 5 q + 57)/24 ≤ n ≤ q, √ it follows that q − 29 q + 57 ≤ 0, whence 529 < q < 722. This is a contradiction as no power of 3 lies in this interval. √ √ To complete the proof it must be shown that neither ν > q nor n < q. √ Suppose that ν > q, that is, ν 2 = pe q with e ≥ 1. Since p ≥ 3 and q is a √ √ square for p = 3, this implies that ν > 2 q. On the other hand, ǫ ≤ 2n < 2 q, a contradiction. √ Suppose that ν < q; that is, q = pe ν 2 , with e ≥ 1. Since n(n − 3) ≥ 2g − 2, √ n < q, the hypothesis (8.50), together with (8.65), gives the following: √ √ (6 + ν)( q − 3) + (q + 5)2 ≥ (6 + ν)(n − 3) + (q + 5)2 ≥ 25 (q − q + 1). (8.88)

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In particular, √ q − 5 q − 15 . 2(6 + ν) √ √ From (8.88) it follows that 2ν q − 6ν ≥ q − 17 q + 21, whence √ √ √ 2ν − 6/ pe ≥ q − 17 + 21/ q. n−3≥

(8.89)

Therefore √ 6 21 17 − √ − √ e ≥ ν( pe − 2). q p

(8.90)

Since ν ≥ 5, this gives pe < 36. As q is a square for p = 3 and ν ≥ 5 is a power of p, only five possibilities are left; namely, (pe , ν, q) ∈ {(9, 9, 729), (5, 5, 125), (5, 25, 3125), (7, 7, 343), 11, 11, 1331)}. But, for these, (8.89) implies that n ≥ 23, 6, 49, 13, 37, contradicting the hypothesis (8.87).

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R EMARK 8.103 If G is a non-singular plane curve defined over Fq with q an odd √ √ square such that G has degree n = ( q + 1)/2 and Sq = q + 1 + n(n − 3) q Fq -rational points, then G is the Fermat curve, up to a projectivity over Fq . This result, together with Theorems 8.101 and 8.102, has the following corollary. T HEOREM 8.104 Let q ≥ 25 be a power of an odd prime p such that, if p = 3, √ then q is a square. Let G be an irreducible plane curve of degree n < q − 2 √ defined over Fq . If both (8.50) and (8.87) hold, then n = 21 ( q + 1) and G is the Fermat curve, up to a projectivity over Fq . R EMARK 8.105 For the Fermat curve Fn = v(X n + Y n + 1) of degree n with n 6≡ 0 (mod p), viewed as an irreducible curve over Fq , the values of n for which Fn is Frobenius non-classical with respect to L2 are known. T HEOREM 8.106 Let q = ph with p > 5. Then Fn is Frobenius non-classical with respect to conics if and only if one of the following holds: (i) n − 1 ≡ 0 (mod p), and Fn is Frobenius classical with respect to lines; (ii) n − 2 ≡ 0 (mod p), and n = 2(ph − 1)/(pr − 1) with r a divisor of h and r < h; (iii) 2n − 1 ≡ 0 (mod p), and n = (ph − 1)/[2(pr − 1)] with r a divisor of h such that h/r is even.

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The St¨ohr–Voloch Theorem in its stronger form (8.19) can be applied to Fn for n < p to show that Sq ≤ 21 (M − 1)n(n − 3) + M −1 {sn(q + M ) − 3n(A − dB)}, where A = 61 {(n − s − 1)s(s − 1)(s + 4) + 14 s(s − 1)(s − 2)(s + 5)}, M = 21 (s + 2)(s + 1) − 1,

B = sn − M, d = number of Fq -rational points of Fn lying on the coordinate axes,

with 1 ≤ s ≤ n − 3, sn ≤ p. R EMARK 8.107 For the number Sp of Fp -rational points of the Fermat curve, F = v(X n + Y n + c) with c ∈ Fp and n | p − 1, the St¨ohr–Voloch Theorem in its stronger form (8.19) gives the following result. T HEOREM 8.108 If (n − 21 )4 ≥ p − 1, then Sp ≤ 4n4/3 (p − 1)2/3 . The constant 4 in the bound can be improved to 3 · 2−(2/3) . R EMARK 8.109 The number Nqk of Fqk -rational points of Fn , when N = (q k−1 + · · · + q + 1)/n,

is an integer and p is sufficiently large with respect to N is as follows: Nqk = n2 [(q − 2) + (d − 1)(d − 2)] + 3n; here d = (N/n, t + 1), q ≡ t (mod N/n) with 0 < t < N/n, and !(t−1)(N/n−d) 2 p p > t+1 +1 . sin(nπ/2N )) 8.8 THE DUAL OF A FROBENIUS NON-CLASSICAL CURVE The notation is the same as Section 7.9. Also, Γ is defined over Fq ; then the Gaussian dual ∆ is also defined over Fq . T HEOREM 8.110 If ν1 > 1, then K(Γ)/K(∆) is a purely inseparable extension with inseparability degree q ′ and with q ′ ≤ q. Proof. From the definition of Frobenius orders, ν1 > 1 if and only if the matrix,   1 xq xq2 ... xqr  1 x , x2 ... xr (1) (1) 0 1 Dx x2 . . . Dx xr

has rank 2. Hence, ν1 > 1 yields the equations, (x − xq )Dx(1) xi = xi − xqi ,

i = 2, . . . , r.

(8.91)

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Since xi = xαi + βi , it follows that (x − xq )αi = xαi + βi − (xαi + βi )q ,

whence xq (αqi − αi ) = βi − βiq . Thus

βiq − βi . αqi − αi As K(Γ) = K(∆)(x) = K(α2 , . . . , αr , β2 , . . . , βr , x), the assertion follows. xq =

2

This, together with Theorem 7.88, has the following corollary. T HEOREM 8.111 If ν1 > 1, then ǫ2 = q ′ with q ′ a power of p not exceeding q. Frobenius non-classicality has implications for the geometry of dual curves of plane curves. T HEOREM 8.112 Let F = v(f (X, Y )) be an irreducible plane curve defined over Fq which is Frobenius non-classical. If ǫ2 < q, the dual curve D of F is non-reflexive. Proof. By Theorem 5.91, it is assumed that p > 2. From Remark 8.52, F is not only Frobenius non-classical but also non-classical with order sequence (0, 1, q ′ ) with ǫ2 = ν1 = q ′ . Since q > q ′ , the argument at the beginning of the proof of Theorem 8.78 can be used. If Q = (x, y) is a generic point of F, from (7.65) and (8.17), ′

′

z0 + z1 xq/q + z2 y q/q = 0,

(8.92)

where at least one of the elements zi is a separable variable. Since F is neither a line nor a strange curve by Exercise 3 of this chapter, so K(F) = K(D). Without loss of generality, suppose that z2 6= 0, and let u = z0 /z2 , v = z1 /z2 with d = q/q ′ . Then (8.92) reads: y d + uxd + v = 0. Therefore D is non-classical, and hence is non-reflexive by Theorem 5.90.

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8.9 EXERCISES 1. For any two positive integers m, n, both prime to p > 2, the irreducible plane curve F = v(X n + Y m + 1) is viewed as a curve defined over Fq . Show that F is Frobenius classical if and only if m = n = (pk − 1)/(ps − 1), where s divides k. 2. Show by direct computation that, if ν1 > 1 and p > 2, then ǫ2 > 2. 3. Prove that, if F is a strange curve defined over Fq , then ν1 = 1. 4. Let p > 2. Assume that f (X) ∈ Fq [X] has only simple roots and that its degree n is not divisible by p. Show that, if the irreducible plane curve F = v(Y n − f (X)) defined over Fq is Frobenius non-classical, then the roots of f (X) belong to Fq .

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5. Let F be the curve considered in Example 1.41 and Exercise 3 in Chapter 7 regarded as a curve defined over Fq with q = ph and p ≥ 5. (a) If k ≤ 21 (q −1) and k ≡ 1 (mod p), show that F is Frobenius classical with respect to the linear system of conics. (b) If q ≡ 1 (mod k) and k ≡ −1 (mod p), show that F is Frobenius classical with respect to linear system of lines but it can be Frobenius non-classical with respect to the linear system of conics. The second case in (b) occurs if and only if (q − 1)/k = pn + 1 with 2n | h and u, v ∈ Fqn . For instance, this happens for h = 2, k = p + 1, n = 1, and also for h = 6, k = (p2 − p + 1)(p3 − 1), n = 1. 6. Let F = v(f (X, Y )) be a Frobenius non-classical plane curve defined over Fq , let Q = (x, y) be a generic point of K(F) such that x is a separable variable of K(F), and let ǫ2 = q ′ . (a) If q > q ′ , show that ′

′

′

′

Dx(q ) y Dx(2q +1) y − Dx(2q ) y − Dx(q +1) y = 0. (b) If q = q ′ , show that (xq −x)(Dx(q) y Dx(2q+1) y −Dx(2q) y Dx(q+1) y) = (Dx(1) y −(Dx(1) y)q )Dx(2q) y. (c) For q = q ′ and p > 2, prove that the dual curve D of F is non-reflexive (2q) if and only if Dx y = 0. 7. Let C be a plane curve, possibly reducible, defined over Fp of degree d < p, and suppose that C has no linear component defined over Fp . Prove the following properties of the number Rp of points of C lying in PG(2, p): (a) Rp ≤ 21 d(d + p − 1);

(b) If Rp ≥ 12 d(d + p − 1) − (d − 1) then C is irreducible. 8. Let C = v(f (X, Y )), where, with c ∈ Fp a non-square,

f (X, Y ) = (Y + cX)(p−1)/2 + Y (p−1)/2 − X (p−1)/2 − 1.

Show that, if p ≡ 1 (mod 4), then Rp = 3(p − 1)2 /8.

9. Let g(X) = (X + c)(p−1)/2 + X (p−1)/2 , with c ∈ Fp a non-square, and let C = v(f (X, Y )), with g(X) − g(Y ) . X −Y Show that, if p ≡ 3 (mod 4), then Rp = (p − 3)(3p − 5)/8. f (X, Y ) =

10. Show that the number of points N1 on an elliptic cubic, for which the number n of rational inflexions is n = 0, 1, 3, 9, satisfies the following: (a) if n = 0, then N1 ≡ 0 (mod 3);

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(b) if n = 1, then N1 ≡ ±1 (mod 3); (c) if n = 3, then N1 ≡ 0 (mod 3); (d) if n = 9, then N1 ≡ 0 (mod 9). 11. Show that, for the curve in Exercise 6 of Chapter 4, the number of F3 -rational points is 13: three are non-singular points and ten arise from branches centred at singular points. 12. Let F be a plane curve defined over Fq containing every point in PG(2, q), and let ϕij = Xiq Xj − Xjq Xi for 0 ≤ i, j ≤ 2. Prove the following. (a) deg F ≥ q + 1, and equality holds if and only if F splits into the q + 1 lines of PG(2, q) through a point P . (b) Each of the homogeneous polynomials over Fq below defines an irreducible plane curve F = v(F ) of degree q + 2 containing every point of PG(2, q): (i) F = c(X0 + X1 )ϕ01 + X0 ϕ20 + X2 ϕ12 , for any q, where T 3 − cT − c ∈ Fq [T ] is irreducible; (ii) F = cX1 ϕ01 + X0 ϕ20 + X2 ϕ12 , for q ≡ 1 (mod 3), where T 3 − c ∈ Fq [T ] is irreducible; (iii) F = (X1 + cX2 )ϕ01 − X0 ϕ20 − X2 ϕ12 = 0, for p = 3, where T 3 + cT + 1 ∈ Fq [T ] is irreducible. Conversely, if F = v(F (X0 , X1 , X2 )) is an irreducible plane curve of degree q + 2 defined over Fq containing every point of PG(2, q), then there exists a homogeneous coordinate system (X0 , X1 , X2 ) over Fq such that F is as in (i), (ii) or (iii). If irreducible plane curves defined over an algebraic extension of Fq are allowed, then there exist curves of degree q+1 containing all points of PG(2, q).

8.10 NOTES The counter-example in Remark 8.19 comes from [91]. Theorem 8.31 comes from [415]. Theorem 8.37 is due to Voloch, see [496], who also pointed out the existence of ′ ′ curves with G 6= Pic0 (Fq (F)). The Hermitian curve F = (Y q + Y − X q +1 ) of genus g = 12 q ′ (q ′ − 1) regarded as a curve over q = q ′4 is such an example. For any divisor n of (q − 1)2g , this allows the existence of Fq -rational curves of genus n(g − 1) + 1 containing at least n(q 3 + 1) Fq -rational points. A related result of ¨ Ozbudak [355] is the following bound valid for any Fq -rational curve F of genus g:  Sq ≤ max 2 + (q − 1)⌊ 21 ex(G)⌋, 21 Nq (g)(ex(G) − 2)(ex(G) − 1) ,

where ex(G) is the exponent of G and Nq (g) is as in Section 9.4. Explicit computations for Pic0 (Fq (F)) of particular families of curves are found in Stichtenoth and Xing [430] and Blache, Cherdieu, and Sarlabous [53].

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Lachaud and Martin-Deschamps [287] proved that, if the irreducible curve F of genus g defined over Fq has an Fq -rational point, then |Pic0 (Fq (F))| ≥ (g q − 1)(q − 1)/(q + g + gq). The St¨ohr–Voloch Theorem 8.65 is the central result in the book and comes from [432]. A geometric interpretation of Sq in Example 8.70 is found in [270]. Theorems 8.68 and 8.77 are due to Hefez and Voloch; see [205]. Theorem 8.81 comes from Garcia [137], where some particular families of Frobenius non-classical curves are investigated. Related to Proposition 8.42, further properties of the integer I are found in Garcia and Homma [144]. These authors proved that I > m if and only if Γ is extremely strange of order m; that is, the intersection of the m-osculating spaces of Γ at all but finitely many points is an (m − 1)-dimensional projective subspace. Theorem 8.78 comes from [219]. Section 8.7 is based on [218]; see also [217], [12]. For a similar but weaker result to Theorem 8.104 for curves of degree one less, √ namely, n = q − 2, see [12]. Theorem 8.91 comes from Giulietti [168]. The result that the curve F in Remark 8.94 is the image of Hq is shown in [89] and [90]. The characterisation of the Fermat curve in Theorem 8.100 is found in [88]. The result in Remark 8.103 is shown in [88]; that in Remark 8.109 is shown in [280]. Theorems 8.106 and 8.108 are due to Garcia and Voloch [162]. Remark 8.107 is related to Waring’s problem for finite fields on the least integer m for which every element of Fp is a sum of this number of n-th powers; see [162]. The improvement on Theorem 8.108 is due to Matterei [325]; here, recent results on Waring’s problem obtained by using character sums are also surveyed. The number of Fq -rational points of Fermat curves F = v(aX n + bY n + c) was investigated by Anuradha and Katre [18]; see also [16] and [333]. Section 8.8 is based on Hefez and Voloch [205]. Motivated also by applications of curves to public-key cryptography, as described in Koblitz’s book [274], and to certain linear codes, much work has been done on low-genus curves defined over Fq and, more generally, over any field of positive characteristic. Projective classifications, the number of Fq -rational points, automorphism groups and zero-divisor class groups (Jacobians) of hyperelliptic and other curves of low genera, including computational aspects, have been investigated in great detail in a series of papers by Anuradha, Auer, Barcelo, Barreiero, Blache, Cardona, Cherdieu, Choie, Deng, Enge, Estrada, Gebhardt, Gutierrez, Hernandez, Ibukiyama, Katsura, Keller, Lomont, Lopez, Luengo, Manin, Menezes, Moreno, Mulan, Mu˜noz, Nart, Oort, Pujol´as, Ritzenhaler, Sadorni, Sevilla, Shaska, Takizawa, van der Geer, van der Vlugt, V¨olklein, Yasin, D. Zinoviev, V. Zinoviev, and others. For this, see [23], [53], [76], [80], [81], [99], [100], [184], [185], [120], [124], [125], [164], [210], [211], [312], [245], [246], [266], [315], [322], [337], [68], [344], [345], [411], [412], [413], [445], [475], [487], [498], [510]; see also

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Hurt’s book [242]. For Exercise 6, see [205]. For Exercise 12, see [447, 448]. For Exercises 8 and 9 see [370]. For Exercise 7, see [70]. For Exercise 5, see [1]. The result in Remark 8.103 is shown in [88], that in Remark 8.105 is shown in [162], and that in Remark 8.109 is shown in [280]. For a similar but weaker result to Theorem 8.104 for curves of degree one less, √ namely, n = q − 2, see [12]. Section 8.8 is based on [205]. For Exercise 12, see Tallini [447, 448].

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Chapter Nine Zeta functions and curves with many rational points In this chapter as in the last, K = Fq , the algebraic closure of Fq . Let Γ be an irreducible algebraic curve of PG(r, K) defined over Fq and equipped with the places of its function field Σ. By definition, as in Section 7.1, Γ arises from an irreducible plane curve F defined over Fq . If F = v(f (X, Y )) with f (X, Y ) in Fq [X, Y ], then Σ = K(x, y) with f (x, y) = 0. The aim here is to investigate birational properties of Γ using its Fq -rational points and , in particular, the number of its Fq -rational places. For this purpose, a non-singular model X of Σ is chosen which is birationally equivalent over Fq to F. So, X is an irreducible non-singular curve in an appropriate projective space PG(r, K), and places of Σ are identified with points of X . Consequently, divisors are finite sums of points of X . Since X can also be viewed as an irreducible algebraic curve over Fqi for every finite extension Fqi of Fq , the number Nqi of the Fqi -rational points of X is defined, and every point of X is an Fqi -rational point for some i. It is conceivable, at least intuitively, that from the infinite sequence N1 , N2 , . . . , Ni , . . ., with Ni = Nqi ,

(9.1)

it is possible to obtain information about X , including properties apparently independent of the behaviour of the sets F(Fqi ) of the Fqi -rational points of X , such as the class number and the fundamental equation. To move in this direction, the technical tool is the zeta function.

9.1 THE ZETA FUNCTION OF A CURVE OVER A FINITE FIELD The Riemann zeta function ζ(s) =

P∞

n=1

n−s

is important in algebraic number theory. It is convergent for Re(s) > 1 and extends to a meromorphic function of s with a simple pole s = 1. Equivalently, the Riemann zeta function can also be written as an infinite product, the Euler product Q ζ(s) = p (1 − p−s )−1 , where p ranges over all primes of Q. A similar function for the curve X is the formal product
Q −s −1 , P 1 − N (P )

(9.2)

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where P ranges over all closed points of X and N (P ) = q deg P . The aim is to give an expression for this product in terms of the Fq -rational divisors of X . D EFINITION 9.1 For a complex variable s, the zeta function of X , viewed as a curve over Fq , is P ζ(X , s) = D N (D)−s , (9.3)

where D runs over all effective Fq -rational divisors of X , and N (D) = q deg D is the norm of D. P Note that, if D = nP P , then Q N (D) = P N (P )nP . Also, N (P ) = q i if P ∈ PG(r, q i )\PG(r, q i−1 ).

T HEOREM 9.2 For Re(s) > 1, the series (9.3) is convergent to the function F0 (q) = F (q −s ) + where

hq 1−g q (1−s)m h , − e(1−s) (q − 1)(1 − q ) (q − 1)(1 − q −es )

(i) F (q −s ) is a polynomial in q −s of degree at most 2g − 2; (ii) h is the class number of X ; (iii) g is the genus of X ; (iv) e is the smallest positive degree of divisors in Div(Fq (X ));  (v) 0, for g = 0, m= 2g − 2 + e, for g ≥ 1. Proof. The minimality of e implies that the degree of any Fq -rational divisor of X is divisible by e, but it does not imply that every multiple of e is the degree of some Fq -rational divisor of X . In fact, e = 1, but this is shown later, in Proposition 9.4; meanwhile no use is made of it. The Fq -rational effective divisors of X are partitioned into divisor classes, two such divisors D1 and D2 being equivalent if and only if there exists u ∈ Fq (F) such that D1 = D2 + div u. Each class consists of all Fq -rational divisors of a complete linear series |C| for an Fq -divisor C of X ; that is, it coincides with the set |C| ∩ Div(Fq (X )). By Theorem 8.39, the number nq (|C|) of such Fq -rational divisors is equal to (q ℓ(C) − 1)/(q − 1) with the usual notation ℓ(C) = dim |C| + 1. From the Riemann–Roch Theorem 6.61,  0, if deg C < 0;     1, if C is the zero divisor;    0, if C is a non-zero divisor and deg C = 0 ; ℓ(C) = g − 1, if C is a non-canonical divisor, deg C = 2g − 2;     g, if C is a canonical divisor;    deg C − g + 1, if deg C > 2g − 2.

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By Remark 8.27, X has some Fq -rational canonical divisor. Hence e divides 2g−2. In the summations below, |C| runs over all Fq -rational linear series of X containing some effective Fq -rational divisor. Now, P P P ζ(X , s) = D N (D)−s = |C| D∈|C| N (D)−s P P P = |C| D∈|C| q −s deg D = |C| n(|C|)q −s deg C . Hence P ζ(X , s) = |C| q −s deg C (q ℓ(C) − 1)/(q − 1) P P = (q − 1)−1 deg C≥0 q ℓ(C)−s deg C − (q − 1)−1 deg C≥0 q −s deg C . If Re(s) > 1, then P P P P h −s deg C −s deg C −esv = deg C=ve ∞ =h ∞ = , deg C≥0 q v=0 q v=0 q 1 − q −es whence P h ζ(X , s) = (q − 1)−1 deg C≥0 q ℓ(C)−s deg C − . (q − 1)(1 − q −es ) If g = 0, then ℓ(C) = deg C + 1, and h = 1 by Example 8.36. So, for g = 0, P∞ P 1 ζ(X , s) = (q − 1)−1 v=0 deg |C|=ve q deg C+1−s deg C − (q − 1)(1 − q −es ) P∞ 1 = q(q − 1)−1 v=0 q e(1−s)v − (q − 1)(1 − q −es ) q 1 . = − (q − 1)(1 − q e(1−s) ) (q − 1)(1 − q −es ) For g ≥ 1, P ζ(X , s) = (q − 1)−1 0≤deg C≤2g−2 q ℓ(C)−s deg C P h +(q − 1)−1 deg C>2g−2 q ℓ(C)−s deg C − (q − 1)(1 − q −es ) P(2g−2/e −esv Ph ℓ(Cive ) q = (q − 1)−1 v=0 i=1 q P∞ h , + h(q − 1)−1 v=(2g−2)/e+1 q e(s−1)v−g+1 − (q − 1)(1 − q −es ) where |Cive | denotes an Fq -rational divisor class of degree v · e. Hence ζ(X , s) converges to hq 1−g q (1−s)m h F0 (q) = F (q −s ) + , − e(1−s) (q − 1)(1 − q ) (q − 1)(1 − q −es ) with P(2g−2/e −esv Ph ve q − i=1 q ℓ(C ) , F (q −s ) = (q − 1)−1 v=0 which is a polynomial in q −s of degree at most 2g − 2. 2 Theorem 9.2 shows that ζ(X , s) has an analytic continuation to the whole complex plane. Its poles of first order are of two types, namely 2πi 2πi su = u, sv = 1 − v, e log q e log q where u, v range over Z.

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T HEOREM 9.3 For Re(s) > 1, the product (9.2) is absolutely convergent to the function ζ(X , s). In particular, the product is independent of the order of its factors. Proof. For any integer N ≥ 1,
Q P∞ Q −s −1 −ns . = N (P )≤N N (P )≤N (1 − N (P ) ) n=0 N (P )

Since the product on the right-hand side has finitely many factors and each factor is an absolutely convergent series, it follows by multiplication that Q P P −s −1 = N (D)≤N N (D)−s + N (D)>N N (D)−s , N (P )≤N (1 − N (P ) )

where the first summation on the right-hand side is over all Fq -rational divisors with N (D) ≤ N while the second is over all effective Fq -rational divisors D with N (D) > N which contain no closed point with N (P ) > N . Then Q P N (P )≤N (1 − N (P )−s )−1 − N (D)≤N N (D)−s P ≤ N (D)>N N (D)−Re(s) .

The sum on the right-hand side can be viewed as the error term of the convergent series ζ(X , s), and hence it tends to zero. This proves the convergence of (9.2). The absolute convergence follows from the inequality P∞ P∞ −ns ≤ n=1 N (D)−n(Re(s)) . n=1 N (D) 2 Theorem 9.3 implies that ζ(X , s) has no zeros for Re(s) > 1. Also, it is the key in the proof of the following proposition which has already been enunciated in Section 8.3 as well as in the proof of Theorem 9.2. P ROPOSITION 9.4 The curve X has an Fq -rational divisor of degree 1. Proof. With the notation of Theorem 9.2, it suffices to prove that e = 1. By the minimality of e, the divisor D is Fq -rational of degree e if and only if D is a closed point of degree e; that is, D = P + Φ(P ) . . . + Φe−1 (P ), for some Fqe -rational point P of X . Note that the Frobenius images P¯i = Φi (P ) of P , for i = 1, . . . , e − 1, are also Fqe -rational points of X . Let ¯ , s) = Q ¯ (1 − N (P¯ )−s )−1 ζ(X P

be the zeta function of X regarded as a curve over Fqe . Then Q  −se deg P¯i −1 ¯ , s) = Q ¯ (1 − q −se deg P¯ )−1 = Qe . ζ(X ) i=1 P P¯i (1 − q Since deg P = e deg P¯i , it follows that
Q ¯ , s) = Qe (1 − q −s deg P )−1 = ζ(X , s)e . ζ(X i=1

P

¯ , s) have a pole of the same order 1 at s = 1, this implies As both ζ(X ; s) and ζ(X that e = 1. 2 Proposition 9.4 makes it possible to simplify Theorem 9.2.

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T HEOREM 9.5 The zeta function of X can be written as ζ(X , s) =

L(q −s ) , (1 − q −s )(1 − q 1−s )

where L(q −s ) =

P2g

j=0

aj q −js ,

with a0 = 1, a2g = q g .

Proof. The proof is carried out in three parts according as g is 0, 1, ≥ 2. For g = 0, ζ(X , s) = For g = 1, ζ(X , s) =

q 1 1 − = . 1−s −s −s (q − 1)(1 − q ) (q − 1)(1 − q ) (1 − q )(1 − q 1−s )

1 q−1

X

q ℓ(C)−s deg C +

deg C=0

h hq 1−s − 1−s (q − 1)(1 − q ) (q − 1)(1 − q −s )

q hq 1−s h h−1 + + − q − 1 q − 1 (q − 1)(1 − q 1−s ) (q − 1)(1 − q −s ) 1 + (h − q − 1)q −s + q · q −2s . = (1 − q 1−s )(1 − q −s ) =

For g ≥ 2, ζ(X , s) = + −

1 q−1 1 q−1

X

q ℓ(C)−s deg C +

deg C=0

X

1 q−1

q ℓ(C)−s deg C +

deg C≥2g−2

q ℓ(C)−s deg C

1≤deg C 4g 4 (g − 1)2 , then take r = result. To obtain the lower bound (9.18), Galois theory is also needed.

√ q to give the 2

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P ROPOSITION 9.23 If q is a square, then √ Nn = q n + O( q n ), where, for a constant c, √ O( q n ) √ n = c. n→∞ q lim

Proof. Choose a separable variable u ∈ K(X ). From (VII) in Section A.2, K(u) has a normal extension Σ′ that is also a normal extension of K(X ). Let G and H be the corresponding Galois groups with H a subgroup of G. In geometric terms, if X ′ is an irreducible non-singular curve of PG(r′ , K ′ ) which is a model of Σ′ , then both X and the projective line ℓ = v(X) with generic point P = (u) are quotient curves of X ′ ; more precisely, X = X ′ /H and ℓ = X ′ /G. Here, G and H are linear collineation groups of PG(r′ , K ′ ) which preserve X ′ . Although the curve X ′ and the group G need not be defined over Fqn , but only over some finite extension of Fqn , here Fqn may be replaced by any of its finite extensions. So, X ′ is defined over Fqn , and G is a subgroup of PGL(r′ + 1, q n ). Let Φn be the Frobenius collineation of PG(r′ , K). For any γ ∈ G, the product γ ′ = Φn γ −1

is also a collineation of PG(r′ , K ′ ) which preserves X ′ . Let Nn (X ′ , γ) be the number of all fixed points of γ ′ on X ′ . Such points of X ′ lie in an s-dimensional subspace Πγ of PG(r′ , K) consisting of all fixed points of γ ′ in PG(r′ , K). Note that Πγ need not consist of points with coordinates in Fqn since Πγ is the image of PG(s, q n ) by a linear collineation belonging to PGL(r, q in ) for some i ≥ 1. Again, for the present purpose, the finite extension Fqin may be replaced by Fqn . Then the fixed points of γ ′ are Fqn -rational points of the Fqn -rational curve X ′ . To apply the St¨ohr–Voloch Theorem 8.65 in the same way as in the proof of Proposition 9.22, a further replacement of Fqn by a larger finite field may be needed √ 4 in order to ensure that q n > 4g ′ (g ′ − 1)2 . With this, Nn ≤ q n + 1 + 2g q n , and so √ (9.19) Nn (X ′ , γ) ≤ q n + 1 + 2g ′ q n . If Q is a point of ℓ, and P1 , . . . , PN are the distinct points of X ′ lying over Q in the covering X ′ → ℓ, then Q is an Fq -rational point of ℓ if and only if Φn (P1 ) coincides with γ(P ) for some γ ∈ G. Since the points P1 , . . . , PN form an orbit of G, either N = |G| and the orbit is long, or |N | is a divisor of |G|, with |N | < |G|, and the orbit is short. The latter case only occurs for at most g − 1 + |G| points Q, by Theorem 11.57 and Remark 11.61. Since the number of Fq -rational points on ℓ is q n + 1, it follows that P ′ ′ n (9.20) γ ′ ∈G Nn (X , γ ) = |G|(q + 1) − c, where c > 0 and c only depends on g and G but is independent of q n . From (9.19) and (9.20), √ Nn (X ′ , γ) = q n + O( q n ).

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The above argument still works if ℓ is replaced by X , and it shows that √ n P ′ ′ γ ′ ∈H Nn (X , γ ) = |H|Nn + O( q ).

Therefore

√ Nn = q n + O( q n ).

2

The Hasse–Weil Bound also provides an estimate for the number Bn of closed Fq -rational points of a given degree n. The idea is to use the equation P Nn = d|n d · Bd , (9.21)

where the sum is extended over all positive divisors d of n. The M¨obius Inversion Formula inverts (9.21) to the equation P (9.22) n · Bn = d|n µ(n/d) · Nd , where µ : N → {0, 1, −1} is the M¨obius function. Since n > 1, P d|n µ(n/d) = 0. Substituting (9.10) into (9.22) gives the following result.

P ROPOSITION 9.24 For any n ≥ 2,   P2g P Bn = n−1 d|n µ(n/d) q d − i=1 ωin ,

where ωi , . . . , ω2g are the reciprocals of the roots of the L-polynomial Lq (t) of X . From Proposition 9.24, an upper bound on Bn is obtained which only depends on q n and g. T HEOREM 9.25 (i) For every n ≥ 2,  √ n √  √ n n q q −1 q g Bn − q ≤ · + 2g √ < (2 + 7g) · . n q−1 q−1 n n (ii) For any n satisfying the condition 2g + 1 ≤

p √ q n−1 ( q − 1),

(9.23)

the curve X has at least one closed point of degree n. (iii) If n ≥ 4g + 3 then Bn ≥ 1. Proof. (i) The assertion for n = 1 follows readily from the Hasse–Weil Bound (9.14). For n ≥ 2, Proposition 9.25 gives the following: P P2g P Bn − q n /n = n−1 d|n,d 0 holds, provided that  √  √ n q q −1 qn q · > + 2g √ . (9.24) n q−1 q−1 n If g = 0, then (9.24) holds for all n ≥ 1. So, it may be assumed that g ≥ 1. A direct calculation shows that (9.24) can be written as follows: √ √ q n ( q − 1) q . (9.25) 0. (iii) If n ≥ 4g + 3, then p √ n−1 √ √ √ 2g + 1 < 22g+1 ( q − 1) ≤ 2 ( 2 − 1) ≤ q n−1 ( q − 1), whence it follows that Bn > 0.

2

It is of interest to compare the bounds (8.18) and (9.14), at least for the series gn2 cut out by lines in the case of a plane non-singular curve of degree n. √ L EMMA 9.26 When n ≥ 21 q + 3, the bound given by Theorem 8.65 is better than that given by Theorem 9.18. Proof. For g = follows:

1 2 (n

− 1)(n − 2), the bounds given by (8.18) and (9.14) are as 1 2 n(n

+ q − 1), √ q + 1 + (n − 1)(n − 2) q.

Then 1 2 n(n

(SV) (HW)

√ + q − 1) ≤ q + 1 + (n − 1)(n − 2) q

√ ⇐⇒ n2 + n(q − 1) ≤ 2q + 2 + 2(n2 − 3n + 2) q √ √ √ ⇐⇒ n2 (2 q − 1) − n(q + 6 q − 1) + 2(q + 2 q + 1) ≥ 0.

(9.26)

The discriminant of (9.26) is √ √ √ δ = (q + 6 q − 1)2 − 8(2 q − 1)(q + 2 q + 1) √ = (q − 2 q + 3)2 .

Hence (9.26) holds when √ √ (q + 6 q − 1) − (q − 2 q + 3) n≤ =2 √ 2(2 q − 1) or √ √ √ (q + 6 q − 1) + (q − 2 q + 3) ( q + 1)2 n≥ = √ √ 2(2 q − 1) 2 q−1 9 1√ 5 = q+ + , √ 2 4 4(2 q − 1) √ which holds for n ≥ 12 q + 3.

2

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9.3 REFINEMENTS OF THE HASSE–WEIL THEOREM The Hasse–Weil Bound 9.18 is a powerful result even though it only depends on two parameters, the genus g of the curve X and the order q of the finite field over which X is defined. Even though the bound is sharp for an infinite number of values of q and g, see Chapter 10, some refinements of the bound are possible, and this is the aim of this section. √ An immediate refinement derives from the fact that the term 2g q in the Hasse– Weil Bound (9.18) may be replaced by its integer part. Therefore √ |N1 − (q + 1)| ≤ ⌊2g q⌋. (9.27) A non-trivial improvement of (9.27) occurs when q is a non-square. T HEOREM 9.27 (Serre Bound) √ |N1 − (q + 1)| ≤ g⌊2 q⌋.

(9.28)

Proof. For g = 0, equality holds in (9.28). So g ≥ 1 is assumed and the Lpolynomial (9.7) is considered. Its roots ω1 , . . . , γ2g are algebraic integers, that is, they belong to the subring A of C consisting of all complex numbers which are roots of polynomials with integer coefficients. All integers are in A, and A ∩ Q = Z. (9.29) √ By the Hasse–Weil Theorem 9.19, |ωi | = q for i = 1, . . . , 2g. Also, by the last statement in Proposition 9.9, these roots may be ordered so that ωi ωg+i = q for i = 1, . . . , g. So, with z the complex conjugate of z, ωi = ωg+i = q/ωi ,

for 1 ≤ i ≤ g.

Let √ αi = ωi + ωi + ⌊2 q⌋ + 1; √ βi = −(ωi + ωi ) + ⌊2 q⌋ + 1. Then αi , βi are real algebraic integers. Also, since |ωi | = αi > 0,

βi > 0,

√ q,

for i = 1, . . . , g.

(9.30)

The monic minimal polynomial of the field extension Q(ω1 , . . . , ωg ) of Q is Q2g L⊥ (t) = i=1 (t − ωi ),

whose roots are the reciprocals of the roots of L(t). Since L(t) ∈ Z[t], the same holds true for L⊥ (t). Let σ be any embedding of Q(ω1 , . . . , ωg ) in C. If σ(ωi ) = ωj , then σ(ωi ) = σ(q/ωi ) = q/σ(ωi ) = σ(ωi ) = ωj . Therefore σ induces permutations on both the sets {α1 , . . . , αg } and {β1 , . . . , βg }. Hence both algebraic integers Qg Qg α = i=1 αi , β = i=1 βi

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are left invariant by all embeddings of Q(ω1 , . . . , ωg ) into C. This implies that α, β ∈ Q ∩ A. Hence, by (9.29), α, β ∈ Z. From (9.30), α ≥ 1 and β ≥ 1. Since the arithmetic mean of positive real numbers is greater than or equal to the geometric mean of the same numbers, Qg Pg 1/g ≥ 1, g −1 i=1 αi ≥ ( i=1 αi ) whence

g≤ From (9.10),

Pg

i=1

P2g

i=1

P2g √ √ (ωi + ωi ) + g⌊2 q⌋ + g = i=1 ωi + g⌊2 q⌋ + g.

ωi = (q + 1) − N1 . Hence

√ N1 ≤ q + 1 + g⌊2 q⌋.

Similarly, the inequality g −1 implies that

Pg

i=1

Qg 1/g ≥1 βi ≥ ( i=1 βi )

√ N1 ≥ q + 1 − g⌊2 q⌋.

2

T HEOREM 9.28 In the Serre Bound, equality is attained if and only if the Lpolynomial of X is √ Lq (t) = (1 ± ⌊2 q⌋t + qt2 )g , (9.31) with + or − according as the bound is upper or lower. Proof. The arithmetic mean of positive real numbers xi equals to their geometric mean if and only if all xi are equal. From the proof of Theorem 9.27, if equality holds in the upper bound (9.28) then α1 = · · · = αg and hence ω1 = · · · = ωg . Let ω = ω1 . Then √ g(ω + q/ω) = q + 1 − N1 = −⌊2 q⌋, whence (9.31) follows with a plus sign. For the converse, (9.31) with + implies that √ −⌊2 q⌋ = ω1 + q/ω1 = · · · = ωg + q/ωg . √ From (9.10), N1 − (q + 1) = g⌊2 q⌋. A similar argument using β shows the assertion for the lower bound in (9.28) with a minus sign. 2 R EMARK 9.29 The Klein quartic over F8 , as in Example 9.14, shows that the upper Serre Bound is sharp. For other curves attaining the Serre Bound, see Exercise 4 and Section 12.5. Now, another general formula for N1 is given. It may appear complicated but can provide good estimates for appropriate choices of the polynomials.

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T HEOREM 9.30 (Serre’s Explicit Formula) Take any sequence c1 , c2 , . . . of real numbers and, for any m ≤ 1, let Pm λm (t) = i=1 cm tm , fm (t) = 1 + λm (t) + λm (t−1 ). (9.32)

Then     g m X X 1 1 √ 1 fm (αi )− (Nn −N1 )cn √ n . λm √ N1 = λm ( q)+λm √ +g− q q q n=1 i=1 (9.33)

Proof. Let √ αi = ωi / q,

i = 1, . . . , 2g,

(9.34)

and observe that |αi | = 1 by the Hasse–Weil Theorem 9.19. As in the proof of Theorem 9.27, these elements may be ordered in such a way that αg+i = αi = α−1 i ,

for

i = 1, . . . , g.

From Theorem 9.10,  g  X √ n 1 1 Nn n √ n = q +√ n− αi + n . q q αi i=1

(9.35)

Note that fm (t) ∈ R for all t ∈ C with |t| = 1. Multiply (9.35) by cn and write the result as follows:   g X √ n 1 1 N1 1 n cn αi + n − (Nn − N1 )cn √ n . (9.36) cn √ n = cn q + cn √ n − q q α q i i=1

Summing the equations (9.36) for n = 1, . . . , m gives the result.

2

A useful corollary is the following result. P ROPOSITION 9.31 (i) If c1 , . . . , cm is a sequence of non-negative real numbers not all zero, and the polynomial fm (t) defined in (9.32) is non-negative for all t ∈ C with |t| = 1, then √ λm ( q) g p p + + 1, (9.37) N1 ≤ λm ( q −1 ) λm ( q −1 ) with λm (t) as defined in (9.32).

(ii) Equality holds in (9.37) if and only if Nn = N1 , for n = 1, . . . , m, and Pg f (αi ) = 0. m i=1 p Proof. Note that λm ( q −1 ) 6= 0, as the sequence c1 , . . . , cm contains a non-zero element. Since N1 ≤ Nn , Serre’s Explicit Formula implies that p p √ λm ( q −1 )N1 ≤ λm ( q) + λm ( q −1 ) + g. p 2 Division by λm ( q −1 ) gives the result.

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Before showing two important consequences of Proposition 9.31, an equation similar to (9.33) is stated. With the notation of (9.9) and (9.34), let αj = cos ϕj + i sin ϕj for j = 1, . . . , 2g. Then (9.10) can also be written as follows: √ Pg Nn = q n + 1 − q n j=1 2 cos nαj .

(9.38)

If Bd denotes the number of Fq -rational closed points of X of degree d, then P Nn = d|n d Bd . For a sequence c1 , c2 , . . . of real numbers, define the trigonometric series P∞ f (α) = 1 + 2 j=1 cj cos jα, and, for every positive integer n, the power series P∞ Ψn (t) = j=1 cjn tjn . From this, Pg

j=1

f (αj ) +

P∞

d=1

√ √ d Bd Ψd (t) = g + Ψ1 (1/ q) Ψ1 ( q),

(9.39)

which is Weil’s Explicit Formula. Now, two cases are examined, related to the maximal curves having the Suzuki and Ree simple groups as their automorphism groups. P ROPOSITION 9.32 Let q = 2q02 with q0 = 2s and s ≥ 1. (i) If g = q0 (q − 1), then N1 ≤ q 2 + 1. (ii) If N1 = q 2 + 1, then (1 + 2q0 t + qt2 )g ; (1 − t)(1 − qt) √ Nn = 1 + q n − g q n κ,

Z(X , t) =

where

 −2   √    − 2 κ= √0    2   2

Proof. Let m = 2, c1 =

1 2

if n ≡ 4 if n ≡ 1, 7 if n ≡ 2 if n ≡ 3, 5 if n ≡ 0

(mod (mod (mod (mod (mod

8), 8), 4), 8), 8).

√ 2, c2 = 14 . Then λ2 (t) =

√1 t 2

+ 14 qt2 .

If |t| = 1, write t = cos ϕ + i sin ϕ, and calculate f2 (t). The result is that f2 (t) = ( √12 + cos ϕ)2 ,

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and hence f2 (t) ≥ 0 for |t| = 1. Applying (9.37) gives N1 ≤ q 2 + 1. Suppose that equality holds and write αj = cos φj + i sin φj . Then 2 Pg Pg  1 √ + cos φj = 0, j=1 f2 (αj ) = j=1 2

which implies that cos φ1 = . . . = cos φg = − √12 . Therefore α1 = · · · = αg = − √12 + i √12 ,

αg+1 = · · · = α2g = − √12 − i √12 .

Direct computation shows the remaining parts.

2

R EMARK 9.33 The DLS curve satisfies the hypothesis of Proposition 9.32. P ROPOSITION 9.34 Let q = 3q02 with q0 = 3s and s ≥ 1. (i) If g= then N1 ≤ q 3 + 1.

3 2

q0 (q − 1)(q + q0 + 1),

(ii) If N1 = q 3 + 1, then 2

(1 + 3q0 t + qt2 )q0 (q −1) (1 + qt2 )q0 (q−1)(q+3q0 +1)/2 , (1 − t)(1 − qt) Nn = 1 + q n √ −q0 q(q − 1){(q + 3q0 + 1) cos 12 πn + 2(1 + q) cos 65 πn}.

Z(X , t) =

Proof. The same argument in the proof of Proposition 9.32 is used, but with a different choice of the sequence c1 , c2 , . . ., namely, √ √ 3 3 7 1 , c2 = , c3 = , c4 = . c1 = 2 12 6 12 The calculations give the following result: for t = cos φ + i sin φ, √ f2 (t) = 31 ( 3 + 2 cos ϕ)2 cos2 ϕ. √ Pg This and j=1 f2 (αj ) = 0 imply that either αj = ± 21 i or αj = − 23 ± 12 i. √ Therefore ωj + ωj can only assume two values, namely, 0 and − 3q. √ Now, let k denote the number of j with 1 ≤√j ≤ g such that ωj + ωj = − 3q. Then (9.10) applied to N1 gives q 3 − q = m2 3q, whence k = q0 (q 2 − 1). This also shows that ωj + ωj = 0 occurs g − k = 12 q0 (q − 1)(q + 3q0 + 1) times. Therefore, up to re-labelling the indices, √ √ ωg+1 = · · · = ωg+k = − 12 q i, ω1 = · · · = ωk = 12 q i, √ √ √ √ ωg+k+1 = · · · = ω2g = ( −2 3 − 21 i) q. ωk+1 = · · · ωg = ( −2 3 + 12 i) q, From this and equation (9.10), Z(X , t) and Nn can be obtained. R EMARK 9.35 The DLR curve satisfies the hypotheses of Proposition 9.32.

2

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9.4 ASYMPTOTIC BOUNDS Proposition 9.31 is also useful for investigating the asymptotic behaviour of Nn as the genus goes to infinity. D EFINITION 9.36 (i) Let Ng (q) = max N1 , taken over all irreducible nonsingular algebraic curves of genus g defined over Fq . (ii) Let A+ (q) = lim supg→∞ Ng (q)/g. (iii) Let A− (q) = lim inf g→∞ Ng (q)/g. √ The Hasse–Weil Bound implies that A+ (q) ≤ 2 q. This bound was improved using the Hasse–Weil Bound for N1 and N2 and the inequality N1 ≤ N2 . T HEOREM 9.37 (Ihara Bound)

p A+ (q) ≤ 12 ( 8q + 1 − 1).

Refining the ideas in this, a better bound can be obtained. T HEOREM 9.38 (Drinfeld–Vl˘adut¸ Bound) √ A+ (q) ≤ q − 1. Proof. For a fixed positive integer m, consider the sequence c1 , . . . , cm , where n cn = 1 − , for n = 1, . . . , m. m Then the functions in (9.32) are as follows:  m  m  X n n t t −1 1− λm (t) = t = + 1 − t ; (9.40) m (1 − t)2 m n=1 fm (t) = 1 + λm (t) + λm (t−1 ) =

2 − (tm + t−m ) . m(t − 1)(t−1 − 1)

(9.41)

Since t−1 = t for |t| = 1, equation (9.41) implies that fm (t) ≥ 0 for all t ∈ C. From Proposition 9.31 applied to the sequence c1 , . . . , cm , ! √ λm ( q) 1 1 N1 p p . (9.42) + 1+ ≤ g λm ( q −1 ) g λm ( q −1 ) From (9.40),

p lim λm ( q −1 ) =

m→∞

p p q −1 1 p ; (1 − q −1 ) = √ q−1 (1 − q −1 )2

so, for any ǫ > 0, there exists m0 such that √ λm0 (q −1/2 ) < ( q − 1)−1 + 12 ǫ.

Now, choosing g0 such that

1 g0

! √ λm ( q) p 1+ < λm ( q −1 )

1 2

ǫ,

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(9.42) implies that √ N1 /g < q − 1 + ǫ √ for every g ≥ g0 . Therefore A+ (q) ≤ q − 1.

2

On the other hand, A+ (q) ≥

√ q − 1 for q square.

(9.43)

Therefore the Drinfeld–Vl˘adut¸ Bound is sharp for q square. D EFINITION 9.39 (i) A tower T over Fq , or an Fq -tower, is an infinite sequence of irreducible non-singular curves together with surjective and separable maps defined over Fq , · · · → Xk+1 → Xk → X2 → X1 , such that lim g(Xk ) = ∞,

k→∞

where g(Xk ) is the genus of Xk . (ii) The limit of the tower T is λ(T ) = lim

k→∞

|Xk (Fq )| . g(Xk )

(iii) A tower is asymptotically good if λ(T ) > 0, and is recursive if it uses the same equation to construct all curves in the tower. Note that A+ (q) ≥ λ(T ) for any Fq -tower T . Therefore, to show that equality holds over fields of square order in the Drinfeld–Vl˘adut¸ Bound, it suffices to construct an Fq2 -tower T satisfying λ(T ) = q − 1. As illustrative examples, two such Fq2 -towers are presented by explicit algebraic equations. E XAMPLE 9.40 λ(T ) = q − 1 for the Fq2 -tower T defined as follows: (i) X1 is the line v(X1 ), that is, an irreducible non-singular model of the rational function field K(x1 ); (ii) X2 is the non-singular plane curve v(Y 2 +Y −X q+1 ), that is, a non-singular model over Fq of the function field K(x1 , z2 ) with z2q + z2 = xq1 ; (iii) X3 is a non-singular model over Fq of the function field K(x1 , x2 , z2 , z3 ) with z2 ; z2q + z2 = xq1 , z3q + z3 = xq+1 , x2 = 2 x1 (iv) X4 is a non-singular model over Fq of the field K(x1 , x2 , x3 , z2 , z3 , z4 ) with z3 z2q + z2 = xq+1 , z3q + z3 = xq+1 , z4q + z4 = xq+1 , x3 = ; 1 2 3 x2

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(v) X5 , X6 , . . . continue in this way. E XAMPLE 9.41 The limit λ(T ) = q − 1 for the Fq2 -tower T defined as follows: (i) X1 is the line v(X1 ), that is, an irreducible non-singular model of the rational function field K(x1 ); (ii) X2 is a non-singular model over Fq2 of the field K(x1 , x2 ) with xq2 + x2 = xq1 /(x1q−1 + 1);

(iii) X3 is a non-singular model over Fq2 of the field K(x1 , x2 , x3 , x4 ) with xq2 + x2 =

xq1 , x1q−1 + 1

xq3 + x3 =

xq2 , x2q−1 + 1

xq4 + x4 =

xq3 ; x3q−1 + 1

(iv) the curves X4 , X5 , . . . continue in this way. The map Xk+1 → Xk , that is, the rational transformation K(Xk+1 ) = K(x1 , . . . , xk , xk+1 ) → K(Xk ) = K(x1 , . . . , xk ) is given by (x1 , . . . , xk , xk+1 ) 7→ (x1 , . . . , xk ). The tower is recursively defined by the following equation: y q + y = xq /(xq−1 + 1). T HEOREM 9.42 For arbitrary q, there exists a positive constant c such that A+ (q) > c log q.

(9.44)

As a consequence, A+ (q) > 0 for all q. E XAMPLE 9.43 Let q > p, and let m = (q − 1)/(p − 1). Then A+ (q) ≥ λ(T ) ≥ for the Fq -tower T defined as follows:

2 q−2

(i) X1 is the line v(X1 ), that is, an irreducible non-singular model of the rational function field K(x1 ); (ii) X2 is the non-singular plane curve v(Y m + (X + 1)m − 1), that is, a nonm singular model over Fq of the field K(x1 , x2 ) with xm 2 + (x1 + 1) = 1; (iii) X3 is a non-singular model over Fq of the field K(x1 , x2 , x3 ) with m xm 3 + (x2 + 1) = 1,

m xm 2 + (x1 + 1) = 1;

(iv) the curves X4 , X5 , . . . continue in this way.

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R EMARK 9.44

(i) For small q, the following lower bounds are known: A+ (2) ≥ 92 ;

(ii)

A+ (3) ≥ 31 ;

A+ (q 3 ) ≥

A+ (5) ≥ 12 .

2(q 2 − 1) . q+2

(iii) If l is a prime number such that q > 4l + 1 and q ≡ 1 (mod l), then p A+ (q l ) ≥ ( l(q − 1) − 2l)/(l − 1).

The significance of A− (q) is that for it to be greater than some constant c it is necessary that, for every sufficiently large g, there is a curve of genus g defined over Fq with at least cg points. T HEOREM 9.45 For every q, the constant A− (q) is positive. T HEOREM 9.46 There exists a constant d such that, for all q, A− (q) ≥ d log q. For q square, this can be improved. T HEOREM 9.47

√ q−1 , if q is an even square; 2 + (log 2)/(log q) √ A− (q) ≥ q−1    , if q is an odd square. 2 + (log 4)/(log q)    

9.5 OTHER ESTIMATES Ad hoc methods can produce estimates on N1 that are better than both the Hasse– Weil and the St¨ohr–Voloch Bounds in some special cases. Two such methods for producing upper bounds on N1 derive from the gonality of the curve X and the first non-gap at an Fq -rational point. For any ξ ∈ Fq (X ), the Fq -rational transformation x′ = x, y ′ = 0 defines an Fq -rational covering of degree d = [K(X ) : K(x)]. P ROPOSITION 9.48 For an Fq -rational covering of degree d, N1 ≤ d(q + 1).

(9.45)

Proof. The projective line ℓ, which is a non-singular model of K(x), has q + 1 rational points. If P is an Fq -rational point of X then the point of ℓ lying under P is an Fq -rational point of ℓ. Although the points of X lying over Q′ ∈ ℓ need not be Fq -rational, the bound is obtained by counting the points of X lying over the Fq -rational points of ℓ. 2 D EFINITION 9.49 (i) The L¨uroth semigroup S of an irreducible curve F is the additive integer semigroup containing all integers n for which K(F) has a base-point-free linear series of degree n.

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(i) The gonality of F is the least positive integer t such that F has a linear series of degree t and dimension 1, necessarily without base point. Equivalently, the gonality is the smallest degree of a non-trivial covering of a rational subfield of K(F) by K(F). (iii) If F is defined over Fq , the Fq -gonality of F is the least positive integer tq (F) such that F has an Fq -rational linear series of order tq (F) and dimension 1. R EMARK 9.50 Knowledge of the L¨uroth semigroup gives worthwhile information about the curve; see Section 7.10. (i) The gonality of X is 1 if and only if X is rational. (ii) The gonality of X is 2 if and only if X is either elliptic or hyperelliptic. (iii) If N1 ≥ 1 and F is birationally equivalent over to an irreducible plane curve of degree m, then the gonality is m − 1. Proposition 9.48 can be stated in terms of Fq -gonality. C OROLLARY 9.51 tq (X ) ≥ N1 /(q + 1). On the other hand, the following results hold. P ROPOSITION 9.52

(i) tq (X ) ≤ g + 1.

(ii) If N1 > 0 and deg E ≥ tq (X ) − 1, then ℓ(E) ≤ deg E + 2 − t. The result below shows that, as long as N1 > 0, the parameter x may be chosen so that d ≤ g for g ≥ 2, and this choice can be made in such a way that the number ′ ′ d(P∞ ) of Fq -rational points lying over the infinite point P∞ of ℓ is very small. L EMMA 9.53 Let X have genus g ≥ 2. If X has an Fq -rational point P, then ′ there exists x ∈ Fq (X ) for which [K(X ) : K(x)] ≤ g and d(P∞ ) ≤ 2. Proof. From the Riemann–Roch Theorem 6.61, there is a canonical divisor E of X such that E ≻ (g − 1)P . By Remark 8.27, take E to be Fq -rational. Write E = (g − 1)P + A and choose a point Q ∈ Supp A. Since A is an Fq rational divisor and D is the closed point D = Q + Φ(Q) + . . . + Φk−1 (Q) of degree k, then A ≻ D. Let B = (g − d)P + D. Since E ≻ B, the index of speciality i(B) is positive. From the Riemann–Roch Theorem 6.61, dim B ≥ 1. Since B is an Fq -rational divisor, there exists x ∈ Fq (X ) such that div(x)∞ = deg B = g; this proves the first assertion. ′ ′ Now, P∞ is the point of ℓ lying under P . Therefore d(P∞ ) ≤ 2, and equality holds if and only if k = 1, that is, Q ∈ X (Fq ). 2 As a corollary of Lemma 9.53, the following upper bound on N1 is obtained. P ROPOSITION 9.54 If g ≥ 2, the following hold:

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(i) N1 ≤ qd + 2; (ii) if n is a non-gap at an Fq -rational point of X , then N1 ≤ qn + 1.

9.6 COUNTING POINTS ON A PLANE CURVE If F is a non-singular plane curve of degree n defined over Fq , then X can be identified with F, and the Hasse–Weil Bound (9.14) reads as follows: √ √ q + 1 − (n − 1)(n − 2) q ≤ Sq ≤ q + 1 + (n − 1)(n − 2) q, (9.46) where n is the degree of F and Sq = N1 is the number of all Fq -rational points of F. Now, (9.46) is extended to irreducible singular plane curves defined over Fq . If a curve is not identified with its non-singular model, then there is some ambiguity in the definition of an Fq -rational point. For a non-singular plane curve F defined over Fq , there is a one-to-one correspondence between the Fq -rational points of F and the Fq -rational places of the associated function field K(F); see Theorem 8.18. If F is singular and X is a non-singular model of F, defined over Fq as well, then F and X have the same function field and hence they have the same number N1 of Fq -rational points, but N1 is, in general, different from the number Rq of points of F which lie in PG(2, q). To compare Rq with Sq = N1 , certain other parameters for F must be defined, some of which have appeared previously: Rq = number of points P ∈ F that lie in PG(2, q);

Sq = N1 = number of Fq -rational points of F; Rq∗ = number of points P ∈ PG(2, q) which are centres of Fq -rational branches; eq = number of points P ∈ F in PG(2, q) R

with each r-fold point counted with multiplicity r; Bq = number of branches of F centred at points of PG(2, q);

Eq = number of singular points of F; bP = number of Fq -rational branches of F with centre at P ∈ PG(2, q);

cP = number of all branches of F with centre at P ∈ PG(2, q); mP = multiplicity of a point P ∈ F.

eq , consider the To see, in a simple case, the differences between Rq , Sq , Bq , R three singular plane cubics, F2 , F1 , F0 , with respectively two, one, and zero tangents over Fq at the singular point P ; these are cubics with a node, a cusp, and an isolated double point at P . Then Table 9.1 is straightforward to verify. From now on, F is any irreducible plane curve of degree n and genus g defined over Fq . To prove the desired result q + 1 − (n − 1)(n − 2) ≤ Rq ≤ q + 1 + (n − 1)(n − 2), it must be shown that (9.46) holds true when N1 is replaced by Rq .

(9.47)

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Table 9.1 Numbers of points on singular cubics

eq R

Rq

Sq

Bq

F2

q

q+1

q+1

q+1

F1

q+1

q+1

q+1

q+2

F0

q+2

q+1

q+3

q+3

Let F(Fq )′ be the set of all points P ∈ PG(2, q) lying on F. With this notation, P ′ Sq = P ∈F (Fq )′ bP , Rq = |F(Fq ) |.

Also, Rq∗ ≤ Sq and equality holds if and only if no two distinct Fq -rational branches of F have the same centre. The starting point are upper and lower bounds for Sq − Rq . L EMMA 9.55 An upper bound for Sq − Rq is the following:

Sq − Rq ≤ 21 (n − 1)(n − 2) − g. (9.48) P P Proof. Sq − Rq = P ∈F (F )′ (bP − 1) ≤ P ∈F (F )′ (cP − 1) q q P P ≤ P ∈F (Fq )′ (mP − 1) ≤ P ∈F (Fq )′ 21 mP (mP − 1) P ≤ P ∈F 12 mP (mP − 1) ≤ 12 (n − 1)(n − 2) − g.

The first equality follows from the definitions. The first two inequalities follow from the fact that bP ≤ cP ≤ mP ; see (iii) of Theorem 4.36. The last inequality follows from Lemma 3.24 and Definition 5.55. 2 L EMMA 9.56 A lower bound for Sq − Rq is the following: Rq∗

Rq − Sq ≤ 12 (n − 1)(n − 2) − g.

(9.49) Rq∗ .

Proof. Since ≤ Sq it follows that Rq − Sq ≤ Rq − On the other hand, Rq − Rq∗ ≤ Eq , as any point P ∈ F(Fq )′ for which bP = 0 and cP > 0 is a singular point of F. Therefore Rq − Sq ≤ Rq − Rq∗ ≤ Eq ≤ 21 (n − 1)(n − 2) − g.

(9.50) 2

T HEOREM 9.57 Let F be an irreducible plane curve defined over Fq of degree n and genus g. If Rq is the number of points of PG(2, q) lying on F, then √ (i) |Rq − (q + 1)| ≤ g⌊2 q⌋ + 21 (n − 1)(n − 2) − g; √ (ii) |Rq − (q + 1)| ≤ 21 (n − 1)(n − 2)⌊2 q⌋; (Serre Bound) √ (iii) |Rq − (q + 1)| ≤ (n − 1)(n − 2) q. (Hasse–Weil Bound)

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Proof. (i) The Serre Bound (9.28), together with Lemmas 9.55 and 9.56, implies the following: |Rq − (q + 1)| ≤ |Rq − Sq )| + |Sq − (q + 1)| √ ≤ 21 (n − 1)(n − 2) − g + g⌊2 q⌋.

(ii) From (i) and the inequality g ≤ 21 (n − 1)(n − 2) of Lemma 3.24, √ |Rq − (q + 1)| ≤ g(⌊2 q⌋ − 1) + 21 (n − 1)(n − 2) √ ≤ 21 (n − 1)(n − 2)(⌊2 q⌋ − 1) + 21 (n − 1)(n − 2) √ = 21 (n − 1)(n − 2)⌊2 q⌋. (iii) This is a consequence of (ii).

2

Under some additional hypotheses, equality can be attained in the above bounds. T HEOREM 9.58 (i) Equality occurs in Lemma 9.56 if and only if every singular point of F is an isolated double point lying in PG(2, q). (ii) Equality occurs in Lemma 9.55 if and only if every singular point of F is a node. Proof. (i) Assume that equality holds in (9.49). From (9.50), Sq = Rq∗ . Hence every point P ∈ F(Fq )′ is the centre of at most one Fq -rational branch of F. Therefore Rq − Sq gives the number of singular points P ∈ F(Fq )′ which are not the centre of an Fq -rational branch of F. Equality in (9.50) implies that Rq − Sq = Eq . Hence, every singular point P of F lies in PG(2, q), and P is not the centre of any Fq -rational branch of F. In particular, at least two branches of F are centred at P . Now, P 1 P ∈F 2 mP (mP − 1) ≥ g.

Since equality in (9.50) also implies that Eq = 21 (n − 1)(n − 2) − g, every singular point P of F is an ordinary double point. Therefore every singular point of F is an isolated double point lying in PG(2, q). To prove the converse, suppose that every singular point of F is an isolated double point. In particular, F has only ordinary singularities and hence, by Theorem 5.57, Eq = 12 (n − 1)(n − 2) − g. As Rq − Rq∗ counts the points in F(Fq )′ that are not the centre of any Fq -rational branch, it follows that Eq = Rq − Rq∗ . Thus only the non-singular points in F(Fq )′ are the centres of Fq -rational branches of F. Hence Sq = Rq∗ . Therefore Rq − Sq = Rq − Rq∗ = Eq = 12 (n − 1)(n − 2) − g.

(ii) Assume that equality holds in (9.48). Then, from the proof of Lemma 9.55, P P P ∈F (Fq )′ (cP − 1). P ∈F (Fq )′ (bP − 1) =

Since bP ≤ cP for all P ∈ F(Fq )′ , this implies that bP = cP ; that is, every branch of F centred at a point in F(Fq )′ is Fq -rational. Also, from the proof of Lemma 9.55, P P 1 1 P ∈F (Fq ) 2 mP (mP − 1) = P ∈F 2 mP (mP − 1),

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which shows that every singular point of F is in F(Fq )′ . From the proof of Lemma 9.55, every singular point of F is a double point. Again from the proof of Lemma 9.55, bP = cP = mP and hence vP = 2 for every singular point P of F. This shows that each such point P is a node. Conversely, if the singularities of F are nodes, then Eq = 12 (n − 1)(n − 2) − g, every singular point of F belongs to F(Fq )′ , and bP = cP = mP for every point P ∈ F(Fq )′ . Therefore Sq − Rq = Eq and hence Sq − Rq = 21 (n − 1)(n − 2) − g.

2

The following corollaries show that when equality occurs either in Lemma 9.55 or in Lemma 9.56, q cannot be too small compared to n. C OROLLARY 9.59 If equality holds in Lemma 9.55, then (i) (n − 1)(n − 2) − 2g ≤ Sq ;

√ (ii) (n − 1)(n − 2) − 2g ≤ q + 1 + g⌊2 q⌋. Proof. From Theorem 9.58 (ii), every singular point of F is a node, and hence Eq = 12 (n − 1)(n − 2) − g. Therefore (n − 1)(n − 2) − 2g = 2Eq ≤ Sq . This, together with Theorem 9.57 (ii), gives part (ii). 2 C OROLLARY 9.60 Let F be an irreducible singular plane curve of degree n ≥ 3 defined over Fq for which equality holds in Lemma 9.56. (i) q ≥ n − 2 + (Sq − 2g − 2)/(n − 2). (ii) If g = 0, then q ≥ n − 1. (iii) If n is odd and n ≥ 5, then q ≥ n − 1 − 2g/(n − 3). Proof. (i) If P is any point in PG(2, q), the pencil ΠP of lines of PG(2, q) through P covers every other point in PG(2, q) just once. Now, suppose P is a singular point of F. From Theorem 9.58(ii), P is an isolated double point of F. Also, from the proof of Lemma 9.55, Sq = Rq∗ . For any line ℓ through P , B´ezout’s Theorem 3.14 becomes the following: P Q∈ℓ∩F (Fq )′ I(P, ℓ ∩ F) ≤ n. Summing over all lines in the pencil ΠP , this gives the inequality P P ℓ∈ΠP Q∈ℓ∩F (Fq )′ I(P, ℓ ∩ F) ≤ n(q + 1).

Counting this double sum in a different way, P P P ℓ∈ΠP I(P, ℓ ∩ F) Q∈ℓ∩F (Fq )′ I(P, ℓ ∩ F) = ℓ∈ΠP P P + Q∈Sing(F )\P I((Q, ℓ ∩ F) + Q∈F (Fq )′ \Sing(F ) I(Q, ℓ ∩ F),

where Sing(F) is the set of all singular points of F. Also, since no Fq -rational branch of F is centred at a singular point of F and the singularities of F are double points, P ℓ∈ΠP I(P, ℓ ∩ F) = 2(q + 1), P Q∈Sing(F ),Q6=P I((Q, ℓ ∩ F) = 2(Eq − 1).

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Also, since Sq = Rq∗ ,

P

Q∈F (Fq )′ \Sing(F )

Therefore

I(Q, ℓ ∩ F) = Sq .

2(q + 1) + 2(Eq − 1) + Sq ≤ n(q + 1). As 2Eq = (n − 1)(b − 2) − 2g when equality holds in Lemma 9.56, so (i) follows. (ii) If g = 0, then Sq = q + 1 and (i) implies that q ≥ n − 2 + (q − 1)/(n − 2). Since q ≥ 2 and n > 2, part (ii) follows. (iii) Since F is an irreducible plane curve, B´ezout’s Theorem 3.14 implies that no line ℓ contains more than ⌊ 21 n⌋ singular points of F. As before, fix a singular point P and consider the pencil ΠP of lines of PG(2, q) through P . Each line contains at most ⌊ 21 n⌋ − 1 singular points of F other than P . As already noted, Eq = 21 (n − 1)(n − 2) − g. Therefore the inequality
⌊ 21 n⌋ − 1 (q + 1) + 1 ≥ Eq = 21 (n − 1(n − 2) − g

follows. If n is odd and n ≥ 5, then ⌊ 12 n⌋ = 21 (n − 1), which implies that q ≥ n − 1 − 2g/(n − 3).

Note that, if n is even and n ≥ 4, then ⌊ 21 n⌋ = 21 n, whence q ≥ n − 2 − (2g − 2)/(n − 2), which is weaker than (i).

2

R EMARK 9.61 The above results are sharp for curves of low degree and genus. For g = 2 this is illustrated by two examples. Let f (X0 , X1 , X2 ) = X02 (X12 + 2X22 ) + X0 (X13 + 2X1 X22 + X23 ) + 3X13 X2 + 3X1 X23 , g(X0 , X1 , X2 ) = X02 (X12 − X22 ) + X0 (X12 X2 − X23 ) + (6X34 + 6X12 X22 − 4X24 ). Then the parameters are given in Table 9.2. Table 9.2 Two curves

Curve

q

n

g

Rq

Sq

Bound on Sq

Bound on Rq

F = v(f )

5

4

2

13

12

0 ≤ Sq ≤ 12

0 ≤ Rq ≤ 13

F = v(g)

13

4

2

1

2

2 ≤ Sq ≤ 26

1 ≤ Rq ≤ 27

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The following Theorems 9.62 and 9.63 provide bounds on Bq . T HEOREM 9.62 Let F be an irreducible plane curve of degree n and genus g defined over Fq . (i) If F is either classical, or non-classical but Frobenius classical, then Bq ≤ 21 {(2g − 2) + (q + 2)n},

(9.51)

except for classical but Frobenius non-classical curves with ν1 = 2, in which case Bq ≤ 12 {2(2g − 2) + (q + 2)n}.

(9.52)

(ii) If F is Frobenius non-classical,

Bq ≤ 21 {ν1 (2g − 2) + (q + 2)n}.

(9.53)

Proof. From (8.56)(i), vP (S) ≥ 2 for every P ∈ F(Fq ), that is, for every Fq rational branch γ of F. By (8.14), it suffices to extend this result to every branch γ of F centred at a point in PG(2, q). This is done after stating a useful formula for vP (S). Let Q = (x, y) be a generic point of F. For any place P of K(F), let (0, j1 , j2 ) the order sequence at P with respect to lines. Then j1 = min{ordP x, ordP y}. Let γ be the branch of F corresponding to P. Without loss of generality, suppose that its centre A is affine. Assume that A = (1, a, b) lies in PG(2, q), and denote by ℓ the tangent to γ. As vP (S) is invariant under Fq -linear transformations, A may be taken to be the origin, and it may be assumed that ordP x ≤ ordP y, where P is the place of K(F) arising from γ. Then ℓ = v(m21 X − m11 Y ),

¯ Y¯ ) so with m11 (6= 0), m21 ∈ K. Now introduce a new coordinate system (X, that the tangent ℓ becomes the X-axis. This change of coordinates from (X, Y ) to ¯ Y¯ ) is given by the linear substitutions (X, ¯ + m12 Y¯ , X = m11 X

¯ + m22 Y¯ . Y = m21 X

Define x¯, y¯ by the equations, x = m11 x ¯ + m12 y¯,

y = m21 x ¯ + m22 y¯.

Then F(K) = K(¯ x, y¯). Choose a local parameter ζ at P, and write ν for ν1 . Now, (ν)

(ν)

(ν)

(ν)

(ν)

(ν)

Dζ x = m11 Dζ x¯ + m12 Dζ y¯, Dζ y = m21 Dζ x¯ + m22 Dζ y¯, whence (ν)

(ν)

(ν)

(ν)

(ν)

(ν)

(x − xq )Dζ y = (x − xq )(m21 Dζ x ¯ + m22 Dζ y¯),

(y − y q ) Dζ x = (y − y q )(m11 Dζ x ¯ + m12 Dζ y¯).

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Therefore (ν)

(ν)

(x − xq )Dζ y − (y − y q )Dζ x (ν)

(ν)

(ν)

(ν)

= (x − xq ) (m21 Dζ x ¯ + m22 Dζ y¯) − (y − y q )(m11 Dζ x ¯ + m12 Dζ y¯) (ν)

(ν)

= [m11 x ¯ + m12 y¯ − (m11 x ¯ + m12 y¯)q ] (m21 Dζ x¯ + m22 Dζ y¯) (ν)

(ν)

− [m21 x ¯ + m22 y¯ − (m21 x¯ + m22 y¯)q ] (m11 Dζ x ¯ + m12 Dζ y¯) (ν)

(ν)

=x ¯[(m11 m21 − m21 m11 )Dζ x ¯ + (m11 m22 − m21 m12 )Dζ y¯] (ν)

(ν)

+¯ y[(m12 m21 − m22 m11 )Dζ x¯ + (m12 m22 − m22 m12 )Dζ y¯] + · · · (ν)

(ν)

= (m11 m22 − m21 m12 )[¯ xDζ y¯ − y¯Dζ x ¯] + · · · , where each omitted expression has order at least j1 q at P. Hence, if vP (S) < j1 q, then (ν)

(ν)

vP (S) ≥ ordt (¯ xDζ y¯ − y¯Dζ x ¯).

(9.54)

Now it is shown that, if the branch γ of F is centred at a point in PG(2, q), then vP (S) ≥ 2j1 .

(9.55)

(ν)

(ν)

If ν1 = ν = 1, then Dζ y¯ = Dζ y¯, and hence ordP Dζ y¯ ≥ j2 − 1. From (9.54), vP (S) ≥ min{2j1 , j2 + j1 − 1}. Since j2 > j1 , the assertion follows. If ν1 = ν = 2, the above argument shows that vP (S) ≥ min{2j1 , j2 + j1 − 2}, whence the assertion follows for j2 > j1 + 1. The case j2 = j1 + 1 cannot actually occur since x ¯Dζ y¯ = y¯Dζ x ¯. Suppose now that F is Frobenius non-classical with ν1 = ǫ = pm > 2. By (9.54), to prove the result, it suffices to prove that ordP y¯ ≥ pm + j1 . However, this has already been done in the proof of Proposition 8.75. 2 T HEOREM 9.63 Let F be a non-classical irreducible plane curve of degree n and genus g defined over Fq . If F is Frobenius non-classical, and has only tame branches, then Bq ≥ (q − 1)n − (2g − 2).

(9.56)

Also, equality holds if and only if every non-linear branch of F is centred at a point in PG(2, q). Proof. As in the proof of Theorem 8.68, the idea is to compute vP (S)− vP (R) and apply (7.13) together with (8.14). Put ν = ǫ = pm > 2. From (7.17) and (8.17), m

m

z1p (xq − x) + z2p (y q − y) = 0, m

(1)

m

(1)

z1p Dζ x + z2p Dζ y = 0.

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These relations allow the simplification of both the Wronskian determinants 1 x y (1) D(1) x Dζ y (1) (1) (ǫi ) ζ 0 D x D y det(Dζ xj ) = = (pm ) ζ ζ (pm ) D x Dζ y (pm ) (pm ) ζ 0 Dζ x Dζ y q 1 x y x −x yq − y (ν ) xq yq det(Dζ i xj ) = 1 = (pm ) (pm ) Dζ x Dζ y (pm ) (pm ) 0 Dζ x Dζ y m

m

,

.

Multiplying the second column by z2p and adding to it z1p times the first column, the second column becomes (pm )

m

[0, z1p Dζ m

(pm )

in both cases. Putting u = z1p Dζ

(pm )

m

x + z2p Dζ m

(pm )

x + z2p Dζ

y]T

y, it follows that

vP (S) = ordP (xq − x) + ordP u − pm ordP z2 ,

(1) vP (R) = ordP (Dζ x)

(9.57)

m

+ ordP u − p ordP z2 .

(9.58)

Hence (1)

vP (S) − vP (R) = ordP (xq − x) − ordP (Dζ x).

(9.59)

Up to a non-degenerate linear transformation (X, Y ) 7→ (uX + vY, wX + zY ), with u, v, w, z ∈ Fq , it may be supposed that the tangent to the branch γ is not the vertical line through the centre. Then x(t) = a + tj1 + · · · ,

y(t) = b + bts + · · · ,

with s ≥ j1 . Since γ is tame, p does not divide j1 . Hence (1)

(1)

j1 − 1 = ordP (Dζ x) ≤ ordP (Dζ y).

Now ordP (xq − x) > 0 if and only if a ∈ Fq . If this is the case, then ordP (xq − x) = j1 ,

and (1, a, b) ∈ PG(2, q). To prove this last statement, it must be checked that a ∈ Fq implies b ∈ Fq . Since F is Frobenius non-classical, (1)

(1)

(xq − x)Dζ y − (y q − y)Dζ x = 0. (1)

(1)

If b 6∈ Fq , then ordP (y q − y) = 0 and hence ordP (Dζ x) > ordP (Dζ y); but (1)

(1)

this contradicts that ordP (Dζ x) ≤ ordP (Dζ y). Since ordP (xq − x) = 0 only if a 6∈ Fq , the following formula is obtained and can be viewed as an extension of (8.27) to singular branches:  j1 , if the centre of γ is in PG(2, q); ordP (xq − x) = (9.60) 0, otherwise.

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From (9.59) and ((9.60),



1, if the centre of γ is in PG(2, q); −(j1 − 1), otherwise. Since deg S − deg R = (q − 1)n − (2g − 2), this implies that P Bq = (q − 1)n − (2g − 2) + (r − 1), where the sum is over all branches γ of F whose centre (1, a, b) is not in PG(2, q). This completes the proof of Theorem 9.63. 2 vP (S) − vP (R) =

√

√

E XAMPLE 9.64 (1) The Hermitian curve H√q = v(X q+1 + Y q+1 + 1) attains the upper bound (9.53) and the lower bound (9.56). (2) Another example which illustrates Theorems 9.62 and 9.63 for q = p3 and p odd, is the dual curve C of the plane curve F in (8.61). The main properties of C are as follows: (i) C is a singular plane curve defined over Fq birationally equivalent to C; (ii) C has degree (p2 + p + 1)(p + 1), and genus g = (p2 + p)(p2 + p − 1)/2; (iii) C is a Frobenius non-classical plane curve with ǫ2 = ν1 = p2 ; (iv) C has only one non-linear branch; it is centred at a point in PG(2, q) and has order p + 1. Apply Theorem 9.63: Bp3 = (p2 + p + 1)(p + 1)(p3 − 1) − (p2 + p + 1)(p2 + p − 2) = (p2 + p + 1)(p − 1)(p3 + 2p2 + p − 1). When p = 3, this gives B27 = 1222, N27 = 208. (3) Now, let q = p = 2, and F = v((X 2 +X)(Y 2 +Y )+1). As noted in Remark 8.52, F is classical but Frobenius non-classical with ǫ2 = ν1 = 2. Here, F has two points in PG(2, q), namely X∞ and Y∞ , both singular. Also, X∞ is a double point and both branches of F centred at X∞ are Fq -rational; the similar property holds for Y∞ . Therefore F has only linear and hence tame branches. Since F has genus g = 1, so B2 = 4, as in (9.51). (4) To illustrate Theorem 9.63, put q = 22e+1 , q0 = 2e , with e ≥ 1, and consider the Deligne–Lusztig–Suzuki irreducible plane curve, or DLS curve for short, F = v(Y q + Y + X q0 (X q + X)), of genus q0 (q − 1). It has several interesting properties; see also Section 12.2. First, F is Frobenius non-classical with ǫ = ν = q0 ; in fact, both (7.65) and (8.54) hold for z0 (x, y) = xq/q0 +1 + y q/q0 , z1 (x, y) = x, z2 (x, y) = 1. Also, F has only one singular point, namely Y∞ . More precisely, Y∞ is the centre of exactly one branch of F, and hence Bq = q 2 + 1. This branch P has order r = q0 and class s = q; in particular, P is a non-tame branch. Theorem 9.63 fails in the sense that here equality does not hold in (9.56); the unique singular branch of F is centred at a point in PG(2, q), but q 2 + 1 > q 2 − qq0 − q + q0 + 2 = (q − 1)(q + q0 ) − (2g − 2).

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An improvement on the bound in Proposition 9.45 is the following result. P ROPOSITION 9.65 If C is an irreducible Fq -rational curve of degree n > 1, then Rq ≤ (n − 1)q + ⌊n/2⌋. Proof. For each point P ∈ C lying in PG(2, q) choose an Fq -rational line tP such that I(P, C ∩ tP ) > 1. If P is non-singular, then tP is the tangent line to C at P . If P is singular, then any Fq -rational line through P has the required property. The number uP of common points of C and tP is at most n − 1. Define k to be the smallest value of uP as P ranges over all points of C lying in PG(2, q). Then Rq ≤ (n − 1)q + k, as through P there are exactly q + 1 lines and one of these lines is tP . A point P is said to see a tangency at Q ∈ C if tQ contains P ; here P = Q is allowed. The points of C lying in PG(2, q) can see at least k Rq tangencies; so there must be some point P ∈ C lying in PG(2, q) which sees at least k tangencies. Counting the points of C lying in PG(2, q) by viewing from P shows that Finally, since

Rq ≤ q(n − 1) + n − k.

min{(n − 1)q + k, (n − 1)q + n − k} ≤ (n − 1)q + ⌊n/2⌋, the assertion follows.

2

Some applications require information on the number of points of PG(2, q) lying on a plane curve not defined over Fq . An upper bound on this number is as follows. L EMMA 9.66 If an irreducible plane curve of degree n is defined over Fqk but not over Fq , then the number N of its points lying in PG(2, q) does not exceed n2 . Proof. By B´ezout’s Theorem 3.14, two distinct irreducible curves of degree n may intersect in up to n2 points. So, there can be no two such curves through n2 + 1 points. Thus an irreducible plane curve defined over Fqk and containing n2 + 1 points in PG(2, q) is unique and is defined over Fq . Hence N ≤ n2 . 2 In conclusion, two more bounds are established; these are applied in Chapter 13. T HEOREM 9.67 Let Cn be a plane curve of degree n defined over Fq with no Fq -linear components, and let Tq be the number of rational simple points of Cn . If √ q > n − 1, (9.61) then

Tq < n(q + 2 − n).

(9.62)

Proof. (i) Cn irreducible. Suppose (9.62) does not hold. Then, from (9.46), which is the Hasse–Weil Bound for plane curves, √ n(q + 2 − n) ≤ Tq ≤ Sq ≤ q + 1 + (n − 1)(n − 2) q.

Hence

√ √ (n − 1)( q + 1)( q + 1 − n) ≤ 0,

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√

and so q ≤ n − 1, contradicting (9.61). (ii) Cn reducible. Suppose that the components of Cn are F1 , . . . , Fm , and let Fi have degree ni P (i) (i) and Tq simple points in PG(2, q) for i ∈ Nm . Since Tq ≤ Tq , it suffices to show that, for each i, P

(i) Tq

P

Tq(i) < ni (q + 2 − n).

(9.63)

Then Tq ≤ < ni (q + 2 − n) = n(q + 2 − n). Two cases must be distinguished, according as a component is Fq -rational or not. (a) Fi is an Fq -rational component. By hypothesis, ni ≥ 2 and, since Cn could not have just one non-Fq -rational linear component, n ≥ ni + 2 ≥ 4. Suppose that (9.63) is false. Then, again by (9.46), √ ni (q + 2 − n) ≤ Tq(i) ≤ q + 1 + (ni − 1)(ni − 2) q √ ⇐⇒ q(ni − 1) − q(ni − 1)(ni − 2) ≤ ni (n − 2) + 1 √ √ ⇐⇒ q(ni − 1)( q − ni + 2) ≤ ni (n − 2) + 1 =⇒ (n − 1)(ni − 1)(n − ni + 1) ≤ ni (n − 2) + 1 =⇒ (n − 1)(ni − 1)(n − ni + 1) < ni (n − 1) =⇒ (ni − 1)(n − ni + 1) < ni

=⇒ 3(ni − 1) < ni ⇐⇒ 2ni < 3, (i)

a contradiction. So Tq < ni (q + 2 − n). (b) Fi is a non-Fq -rational component. (i) By Lemma 9.66, Tq ≤ n2i . Now, √ ( q − 1)2 ≥ 0 √ ⇐⇒ 2 q + 1 ≤ q + 2 =⇒ 2n − 1 < q + 2 =⇒ n + ni < q + 2 ⇐⇒ ni < q + 2 − n ⇐⇒ n2i < ni (q + 2 − n) (i) 2 =⇒ Tq < ni (q + 2 − n). √ 2 R EMARK 9.68 It is not true that n ≤ q + 1 + (n − 1)(n − 2) q for all values of n and q. T HEOREM 9.69 Let D2t be a plane curve of degree 2t defined over Fq and let Cn be a component of D2t , with degree n ≥ 3 if Cn is Fq -rational. Let Tq be the number of rational simple points of Cn and let s be the number of its rational singular points, of which d are double points. If √ q > 2t + n − 1, (9.64)

then

Tq + d < 12 n(q + 2 − t).

(9.65)

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Proof. (i) Cn an Fq -rational component with n ≥ 3. Suppose that (9.65) is false. Then, from (9.46), √ 1 2 n(q − t + 2) ≤ Tq + d ≤ Tq + s ≤ q + 1 + (n − 1)(n − 2) q √ =⇒ n(q − t + 2) ≤ 2(q + 1) + 2(n − 1)(n − 2) q √ ⇐⇒ (n − 2)q − 2(n − 1)(n − 2) q ≤ n(t − 2) + 2 √ √ ⇐⇒ (n − 2) q[ q − 2(n − 1)] ≤ n(t − 2) + 2 =⇒ (n − 2)(2t + n − 1)(2t − n + 1) < n(t − 2) + 2

⇐⇒ (n − 2)[4t2 − (n − 1)2 ] < n(t − 2) + 2 =⇒ (n − 2)[4t2 − (2t − 1)2 ] < n(t − 2) + 2 (since 2t ≥ n)

⇐⇒ (n − 2)(4t − 1) < n(t − 2) + 2 ⇐⇒ (3n − 8)t + n < 0, which is impossible for n ≥ 3. (ii) Cn not Fq -rational. Suppose the result is false. Then, by Lemma 9.66, 2 1 2 n(q − t + 2) ≤ Tq + d ≤ Tq + s ≤ n =⇒ q − t + 2 ≤ 2n =⇒ (2t + n − 1)2 − (t − 2) < 2n

=⇒ (2t + n − 1)2 − (2t − 1)2 < 2n ⇐⇒ n(4t + n − 2) < 2n

⇐⇒ 4t + n − 2 < 2 ⇐⇒ 4(t − 1) + n < 0, which is impossible as t ≥ 1, n ≥ 1.

2

9.7 FURTHER APPLICATIONS OF THE ZETA FUNCTION The zeta function is a useful tool also in the study of the group Pic0 (Fq (X )) of all zero-degree Fq -rational divisors classes of X . This becomes apparent below in the fundamental role of the L-polynomial in the proof of two major results on Pic0 (Fq (X )), previously stated without proof in Section 6.7. Here, the notation of Sections 8.3 and 8.4 is changed as follows. If l = Fqn with n ≥ 1, then Div(l) (X ) = the l-divisor group Div(l(F)); (l)

Div0 (X ) = the zero-degree l-divisor group Div0 (l(F)); Pic(l) (X ) = the l-divisor class group Pic(l(F)); (l)

Pic0 (X ) = the zero-degree l-divisor class group Pic0 (l(F)). (l)

By Theorem 8.34, Pic0 (X ) is a finite abelian group. Its order, that is, the class number h of X , is closely related to the zeta function of X via the L-polynomial Lq (t).

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T HEOREM 9.70 h = Lq (1). Proof. If g = 0, then Lq (1) = 1 and h = 1 by Example 8.36. If g ≥ 1, from (9.8) L(1) = h. 2 Theorem 6.94 was stated without proof. Now, using the zeta function, a proof is given in the case that K is the algebraic closure of Fq . An essential tool in the proof is the algebraic number field Q(ω, . . . , ω2g ) generated by the inverse roots of the L-polynomial of X . As suggested in Section 7.16, see especially Example 7.123, function fields of transcendence degree 1 are closely analogous to algebraic number fields, although number-theoretic methods have no direct counterpart in function field theory. A detailed discussion of these methods is beyond the scope of this book. The concepts and results used in the proofs below mostly concern the theory of divisibility. Two more pieces of notation are needed: (l)

(l)

Pic0 (X , d) = the subgroup of Pic0 (X ) of all elements of order d; (l)

(l)

Pic0 (X , d∞ ) = the subgroup of Pic0 (X ) of all elements of order a power of d. The following lemma is the main ingredient in the proof. L EMMA 9.71 Let d be a prime different from p. Then, for the curve X of genus g, there exists a finite extension k of Fq such that the following condition is satisfied by all finite extensions l of k, where l1 is an extension of l of degree dr : (l )

(l)

[Pic0 1 (X , d∞ ) : Pic0 (X , d∞ )] = d2gr .

(9.66)

(l)

Proof. The group Pic0 (X , d) is finite and abelian, in which every element has (l) order a power of d. Thus |Pic0 (X , d)| = ds(l) . By Theorem 9.70, the order of the (l) (l) group Pic0 (X ) containing Pic0 (X , d) is Q2g Lqm (1) = i=1 (1 − ωim ),

where l = Fqm ; that is, m is the degree of the extension l/Fq . Therefore q s(l) is the largest power of q which divides Q2g m i=1 (1 − ωi ).

Replacing l by l1 in this argument shows that s(l1 ) is the largest power of q dividing Q2g Q2g r mdr ) = i=1 (1 − ωim )(1 + ωim + · · · + (ωim )d −1 ). i=1 (1 − ωi Choose n such that ωin ≡ 1 (mod d) for every i = 1, . . . , 2g, and let k = q n . Then n divides m, and the condition is satisfied. 2

Lemma 9.71 can be interpreted in terms of the trace map (8.6). Let φ be the (l ) restriction of T0 to Pic0 1 (X , d∞ ). Since T0 is surjective, φ is a surjective homomorphism. Then (9.66) reads as follows: | ker φ | = d2gr . (l )

(9.67)

Now, (9.67) implies that ker φ ≤ Pic0 1 (X , d2gr ). On the other hand, ker φ con(l) tains Pic0 (X , dr ).

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To show this, let l = Fqn , l1 = F(qn )dr , and choose any [D] ∈ Pic0 (X , dr ). Then Φn ([D]) = ([D]) for the n-th Frobenius map on divisor classes, and hence [0] = dr ([D]) = [D] + Φn ([D]) + · · · + (Φn )d

which shows that [D] ∈ ker φ. Therefore (l)

r

−1

([D]) = T0 ([D]),

(l )

Pic0 (X , dr ) ≤ ker φ ≤ Pic0 1 (X , d2gr )

(9.68)

To prove Theorem 6.94, it suffices to show the following result. T HEOREM 9.72 For all positive integers r and primes d distinct from p, Pic0 (K(X ), dr ) ∼ = (Zdr )2g . Proof. Choose k, l, l1 as in Lemma 9.71. Then (l)

|Pic0 (X , dr )| ≤ | ker φ | = d2rg . This shows that, for sufficiently large k, the following condition is satisfied by all finite extensions Fl of Fk . If l1′ is a finite extension of l′ , then (l′ )

(l′ )

Pic0 (X , dr ) = Pic0 1 (X , dr ).

(9.69)

Replacing dr by d2gr , this becomes the following: (l′ )

(l′ )

Pic0 (X , d2gr ) = Pic0 1 (X , d2gr ).

(9.70)

Now, choose a sufficiently large k such that both (9.69) and (9.70) hold whenever Fl′ is a finite extension of Fk and Fl′1 is a finite extension of Fl′ . Now, from (9.68), (l′ )

(l′ )

Pic0 (X , dr ) ≤ ker φ ≤ Pic0 (X , d2gr ).

Let [D] ∈ ker φ, and l′ = Fqn′ . Then ′

′

rd [D] = [D] + Φn ([D]) + · · · + (Φn )d

r

−1

([D]) = T0 l′1 |l′ ([D]) = 0.

(l′ )

(l′ )

This implies that [D] ∈ Pic0 (X , dr ) showing that ker φ ≤ Pic0 (X , dr ). There(l′ ) fore ker φ = Pic0 (X , dr ). From (9.67), (l′ )

Pic0 (X , dr ) ∼ = (Zdr )2g . This, together with (9.69), implies the result.

2

A similar result enunciated in Chapter 6 is Theorem 6.96. Here, a proof is given in the case that K = Fq . Write Lq (t) = 1 + a1 t + · · · + ai ti + · · · + t2g , and denote by γ(X ) the highest index i with 1 ≤ i ≤ g for which ai 6≡ 0 (mod p). To prove Theorem 6.96, it suffices to show that γ(X ) is equal to the p-rank γ of X . For this, two lemmas are required. As before, ω1 , . . . , ω2g are the inverse roots of the L-polynomial Lq (t) of X . The idea is to use properties of the algebraic number field Ω = Q(ω1 , . . . , ω2g ). Let τ be a prime divisor of Ω lying over p. L EMMA 9.73 The index γ(X ) is the number of i for which ωi 6≡ 0 (mod τ ).

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Proof. Order ω1 , . . . , ω2g so that ωj 6≡ 0 (mod τ ), ωj ≡ 0 (mod τ ),

j = 1, . . . , s; j = s + 1, . . . , 2g.

From (9.9), 1 + a1 t + · · · + ai ti + · · · + qt2g = it follows that aj = (−1)j

Q

i1 0; a1 ≥ 1, au ≥ 1, ai ≥ ai−1 for i = 1, . . . , u − 1. (9.75) Let r = dim D. Then mr (P ) = m. From Section 6.6, the Weierstrass semigroup H(P ) at P ∈ X contains the elements of the increasing sequence m0 (P ) = 0, m1 (P ), . . . , mi (P ), . . . , where mi (P ) is the i-th non-gap at P . P ROPOSITION 9.83 The Frobenius linear series D of X has the following properties: (i) the (D, P )-orders at an Fq -rational point P are ji (P ) = m − mr−i (P ) for i = 0, . . . , r; (ii) if Φi (P ) 6= P for i = 1, . . . , u, then j1 (P ) = 1 and, in particular, ǫ1 = 1; (iii) the integers 1, a1 , . . . , au are D-orders, so that r ≥ u + 1;

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(iv) if Φi (P ) 6= P for i = 1, . . . , u + 1, then au is a non-gap at P ; (v) if Φi (P ) 6= P for i = 1, . . . , u, but Φu+1 (P ) = P, then au−1 is a non-gap at P . Proof. (i) Let m(P ) be a non-gap at an Fq -rational point P of X . There exists an effective divisor E for which E ≡ m(P )P and P 6∈ supp(E). From this, E + (m − m(P ))P ≡ mP . If m(P ) ≤ m, then m − m(P ) is a (P, D)-order. (ii) Since Φ, and hence Φu , is surjective on X , there is a unique point P ′ ∈ X whose image under Φu is P. From (9.74) applied to P ′ , Pu u−i (P ′ ) + P ≡ mP0 . i=1 ai Φ

For 1 ≤ j ≤ u, the hypothesis that Φj (P ) 6= P implies that Φu−j (P ′ ) 6= P . Therefore 1 is a (P, D)-order. (iii) Take a point P ∈ X such that Φj (P ) 6= P for j = 1, . . . , u. Then the assertion follow from (9.74). (iv),(v) Again, the Fundamental Equation (9.74) applies, this time to P and Φ(P ), giving the equivalence Pu mP0 ≡ i=1 ai Φu−i (P ) + Φu (P ) Pu−1 ≡ a1 Φu (P ) + i=1 ai+1 Φu−i (P ) + Φu+1 (P ), whence

au P ≡ Φu+1 (P ) + (a1 − 1)Φu (P ) + Taking (9.75) into account, the results follow.

Pu−1 i=1

(ai+1 − ai )Φu−i (P ). 2

P ROPOSITION 9.84 If N1 > q(m − au ) + 1, then jr−1 (P ) < au for every Fq rational point P of X . Proof. By (i) of Lemma 9.83, jr−1 (P ) = m − m1 (P ). On the other hand, Proposition 9.54 shows that N1 ≤ 1 + m1 (P )q. Taking the hypothesis into account, this gives the inequality jr−1 (P ) ≤ m − (N1 − 1)/q < au .

2

P ROPOSITION 9.85 (i) The order ǫr = νr−1 = au , and every Fq -rational point is in the support of the ramification divisor R of D. (ii) If N1 ≥ qau + 1, then m1 (P ) = q for every Fq -rational point P of X . Proof. (i) First, ǫr−1 ≤ jr−1 (P ) for any P ∈ X . Since au is a D-order by Proposition 9.83, this implies that ǫr = au . From (9.74), Φ(P ) lies in the (r − 1)-th osculating hyperplane. Thus νr−1 = ǫr . By Lemma 9.83 (i), jr (P ) = m for each Fq -rational point of X . Since m > au , the result follows from Theorem 7.55. (ii) Let P ∈ X (Fq ). Since m1 (P ) ≤ m1 (Q) for all but finitely many points Q of X , Lemma 9.83 (iv) shows that m1 (P ) ≤ au . On the other hand, m1 (P ) ≥ au by Proposition 9.54 and the hypothesis on the size of X (Fq ). 2 Now, return to Pic0 (K(X )). The machinery applied to any endomorphism α of JX or, equivalently, of Pic0 (K(X )) can give an equation of type (9.74) depending

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on the characteristic polynomial Pα (X) of α. To gain information on Pα (X), it is appropriate to use a representation of the ring End(JX ) on a module of dimension 2g over the l-adic integers Zl with l a prime not equal to p. Such a representation depends on the Tate group. Let A be an abelian group, written additively. Assume that A is infinitely divisible by powers of a fixed prime l, that is, lA = A. The Tate group Tl (A) of A consists of vectors (a1 , a2 , . . . , an , . . .) with denumerably many components ai ∈ A satisfying lai+1 = ai and la1 = 0. Addition in Tl (A) is carried out component-wise. It is possible to define on Tl (A) the structure of a module over the l-adic integers Zl as follows. For z ∈ Zl , let m be an integer such that m ≡ z (mod ln ). If ln a = 0, define za = ma, this being independent of the choice of m. Then an operation of z on Tl (A) is defined: z · (a1 , a2 , . . .) = (za1 , za2 , . . .). From now on A is assumed to be Pic0 (K(X ), l∞ ). This is possible since the group Pic0 (K(X ), l∞ ) is a divisible group. L EMMA 9.86 The Tate group Tl (A) is a module over Zl of dimension 2g. Proof. Let x1 , . . . , x2g be vectors of Tl (A) whose first components a1,1 , . . . , a2g,1 are linearly independent over Fl . Then these vectors are linearly independent over Zl . In fact, if a relation of linear dependence existed, it would be possible to assume that not all the coefficients are divisible by l, and hence this relation considered on the first component would contradict the hypothesis made on the ai,j . Now, it is shown by an inductive argument that these vectors xi form a basis of Tl (A) over Zl . Suppose that every element w ∈ Tl (A) can be written as a linear combination w = z1 x1 + · · · + z2g x2g (mod ln Tl (A)), (9.76) with integers zj ∈ Z. Let w = (b1 , . . . , bn , bn+1 , . . .). By definition, for the first n + 1 components, z1 (a1,1, , . . . , a1,n+1 ) + · · · + z2g (a2g,1 , . . . , a2g,n+1 ) = (b1 , . . . , bn , bn+1 ) + (0, . . . , 0, cn+1 ), for some cn+1 with ln+1 cn+1 = 0. By the choice of the vectors, there exist integers d1 , . . . , d2g such that cn+1 = d1 ln a1,n+1 + · · · + d2g ln a2g,n+1 . Replacing z1 , . . . , z2g by z1 + d1 ln , . . . , z2g + d2g ln shows that the congruence (9.76) extends from n to n + 1. This gives the result. 2 If α ∈ End(JX ) then α induces a linear map αT : Tl (A) → Tl (A),

(a1 , a2 , . . .) 7→ (αa1 , αa2 , . . .). The Z-module End(JX ) becomes a module over the l-adic integers Zl by defining (zα)x = α(zx) for x ∈ Tl (A) and z ∈ Z. This gives the desired representation of End(JX ). The main results in this area are summarised in the following theorems.

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T HEOREM 9.87 (i) If α1 , . . . , αm are elements of End(JX ) linearly independent over Z, then they are independent over Zl . (ii) The map α 7→ αT of End(JX ) into the ring of Zl -linear transformations of Tl (Pic0 (K(X ), l) is a ring isomorphism. (iii) End(JX ) is a module of finite type of dimension at most 16g 2 . T HEOREM 9.88 The characteristic polynomial of an endomorphism α of JX is equal to the characteristic polynomial of the linear transformation αT induced by α on Tl (Pic0 (K(X ), l)). T HEOREM 9.89 The coefficients of the characteristic polynomial of an endomorphism α of JX are integers.

9.9 ELLIPTIC CURVES OVER Fq Let F be a non-singular plane cubic curve defined over Fq . The points of F with respect to the operation described in the proof of Theorem 6.104 is an abelian group GF . T HEOREM 9.90 The Fq -rational points of F form a subgroup GF (Fq ) of GF . Proof. The proof is a consequence of the following facts: (i) any line containing two distinct Fq -rational points of the cubic F contains a third Fq -rational point of F; (ii) the tangent to F at an Fq -rational point which is not an inflexion contains another Fq -rational point of F. 2 n

(q ) From Theorem 6.104, GF (Fq ) ∼ = Pic0 (X ). Some further properties of the group structure are now summarised.

T HEOREM 9.91 (i) The characteristic polynomial is ψ(t) = t2 − mt + 1; √ (ii) | m | ≤ 2 q; (iii) N1 = q + 1 − m; (iv) p | m if and only if F has p-rank 0, that is, F is supersingular. From Theorem 9.18, √ √ ( q − 1)2 ≤ N1 ≤ ( q + 1)2 . In fact, the precise values that N1 can take are given by the next result. T HEOREM 9.92 (Waterhouse) There exists an elliptic cubic over Fq , q = ph , with √ precisely N1 = q + 1 − m rational points, where | m | ≤ 2 q, for only the values of m in Table 9.3.

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Table 9.3 Values of m

m (i)

m 6≡ 0 (mod p)

(ii)

m=0

(iii)

m=0

(iv) (v) (vi) (vii)

√ m=± q √ m = ±2 q √ m = ± 2q √ m = ± 3q

p

h

odd p 6≡ 1 (mod 4)

even

p 6≡ 1 (mod 3)

even even

p=2

odd

p=3

odd

Let Nq (1) denote the maximum number of rational points on any non-singular cubic over Fq and Mq (1) the minimum number. The prime power q = ph is √ exceptional if h is odd, h ≥ 3, and p divides ⌊2 q⌋. R EMARK 9.93 The only exceptional q < 1000 is q = 128. C OROLLARY 9.94 The bounds Nq (1) and Mq (1) are as follows:  √ if q is exceptional, q + ⌊2 q⌋, √ (i) Nq (1) = q + 1 + ⌊2 q⌋, if q is non-exceptional;  √ q + 2 − ⌊2 q⌋, if q is exceptional, √ (ii) Mq (1) = q + 1 − ⌊2 q⌋, if q is non-exceptional. Proof. This is an immediate consequence of Theorem 9.92.

2

√ C OROLLARY 9.95 The number N1 takes every value between q + 1 − ⌊2 q⌋ and √ 2 q + 1 + ⌊2 q⌋ if and only if (a) q = p or (b) q = p with p = 2 or p = 3 or p ≡ 11 (mod 12). C OROLLARY 9.96 For q ≤ 128, the values that N1 cannot take between Mq (1) and Nq (1) are given in Table 9.4. The values of Mq (1) and Nq (1) for q ≤ 128 are given in Table 9.6. For any integer nQand any prime divisor ℓ, let vℓ (n) be the highest power of ℓ dividing n; that is, ℓ ℓvℓ (n) is the prime decomposition of n. Also, write Z/(r) for the additive group of the ring Zr . T HEOREM 9.97 (Voloch, R¨uck) The groups GF (Fq ) of the cubics F occurring in Theorem 9.92 in cases (i) − (vii) are as follows: Q (i) GF (Fq ) = Z/(pvp (N1 ) ) × ℓ6=p (Z/(ℓrℓ ) × Z/(ℓsℓ )) , with rℓ + sℓ = vℓ (N1 ) and min(rℓ , sℓ ) ≤ vℓ (q − 1);

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Table 9.4 The values that N1 cannot take between Mq (1) and Nq (1) for q ≤ 128

q

Forbidden values

8 16 25 27 32 49 64 81

7, 11 11, 15, 19, 23 26 22, 25, 31, 34 23, 27, 29, 31, 35, 37, 39, 43 43, 57 51, 53, 55, 59, 61, 63, 67, 69, 71, 75, 77, 79 67, 70, 76, 79, 85, 88, 94, 97 106, 111, 116, 121, 131, 136, 141, 146 109, 111, 115, 117, 119, 121, 123, 125, 127, 131, 133, 135, 137, 139, 141, 143, 147, 149

128

(ii) ,(iii)

GF (Fq )

 when q 6≡ −1 (mod 4),  Z/(q + 1) Z/(q + 1) or =  Z/(2) × Z/((q + 1)/2) when q ≡ −1 (mod 4);

(iv) GF (Fq ) = Z/(N1 ); √ √ √ (v) GF (Fq ) = Z/( N1 ) × Z/( N1 ), N1 = ( q ± 1)2 ; (vi) GF (Fq ) = Z/(N1 ); (vii) GF (Fq ) = Z/(N1 ).

E XAMPLE 9.98 For F = X03 + X13 + X23 , Table 9.6 gives the number N1 of rational points of v(F ) for 2 ≤ q ≤ 179 with also the values of Nq (1) and Mq (1). To calculate N1 , note the following. 1. When q ≡ 0, −1 (mod 3), then N1 = q + 1. 2. When q = p and p ≡ 1 (mod 3), then N1 = q + 1 + A, 2

2

where 4q = A + 27B with A, B ∈ N and A ≡ 1 (mod 3). 3. N2 = N1 {2(q + 1) − N1 }. In particular, if N1 = q + 1, then N2 = (q + 1)2 .

4. When q ≡ 1 (mod 3), then 9 divides N1 , as the nine inflexions form a subgroup.

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Table 9.5 The number A when q = p and q ≡ 1 (mod 3)

q

4q

=

A2 + 27B 2

A

7 13 19 31 37 43 61 67 73 79 97 103 109 127 139 151 157 163

28 52 76 124 148 172 244 268 292 316 388 412 436 508 556 604 628 652

= = = = = = = = = = = = = = = = = =

1 + 27 25 + 27 49 + 27 16 + 108 121 + 27 64 + 108 1 + 243 25 + 243 49 + 243 289 + 27 361 + 27 169 + 243 4 + 432 400 + 108 529 + 27 361 + 243 196 + 432 625 + 27

1 −5 7 4 −11 −8 1 −5 7 −17 19 13 −2 −20 −23 19 −14 25

9.10 CLASSIFICATION OF NON-SINGULAR CUBICS OVER Fq For plane non-singular cubics defined over Fq , Salmon’s Theorem 7.91 states that birational and projective equivalence over Fq are the same. This is not true when equivalences over Fq are considered. Nevertheless, the proof of Salmon’s Theorem 7.91 gives some ideas for the case where F and F ′ are two non-singular plane cubic curves each having at least one point in PG(2, q). The first part of that proof shows indeed how to find a nonsingular plane cubic Γ, birationally equivalent over Fq to F, that has an inflexion point O in PG(2, q). From the second part, if there is a birational transformation T over Fq taking F to F ′ and O to O′ , then the non-singular models Γ and Γ′ are projectively equivalent over Fq . This motivates the following definition: if F, F ′ and a birational transformation T : K(F) → K(F ′ ) are defined over Fq , then F and F ′ are Fq -isomorphic. Let Aq be the total number of Fq - isomorphism classes and Pq the total number of projective equivalence classes over Fq Here, ni , for i = 0, 1, 3, 9, is the number of projective equivalence classes over Fq with exactly i rational inflexions. Hence Aq = n9 + n3 + n1 ,

Pq = n9 + n3 + n1 + n0 .

The number of inequivalent types of plane non-singular cubic with a fixed number of Fq - rational points can also be given. Let Aq (m) and Pq (m) be the number

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Table 9.6 Number N1 of points on the cubic v(X03 + X13 + X23 )

q N1 Nq (1) Mq (1)

2 3 5 1

3 4 7 1

4 9 9 1

5 6 10 2

7 9 13 3

8 9 14 4

9 10 16 4

11 12 18 6

13 9 21 7

16 9 25 9

17 18 26 10

q N1 Nq (1) Mq (1)

19 27 28 12

23 24 33 15

25 36 36 16

27 28 38 18

29 30 40 20

31 36 43 21

32 33 44 22

37 27 50 26

41 42 54 30

43 36 57 31

47 48 61 35

q N1 Nq (1) Mq (1)

49 63 64 36

53 54 68 40

59 60 75 45

61 63 77 47

64 81 81 49

67 63 84 52

71 72 88 56

73 81 91 57

79 81 83 63 82 84 97 100 102 63 64 66

q 89 97 101 103 107 109 113 121 125 N1 90 117 102 1117 108 108 114 144 126 Nq (1) 108 117 122 124 128 130 135 144 148 Mq (1) 72 79 82 84 88 90 93 100 104

127 108 150 106

128 129 150 108

q N1 Nq (1) Mq (1)

173 174 200 148

179 180 206 154

131 132 154 110

137 138 161 115

139 117 163 117

149 150 176 126

151 171 176 128

157 144 183 133

163 189 189 139

167 168 193 143

169 171 196 144

of inequivalent non-singular cubics, under Fq -isomorphism and projective equivalence over Fq , whose number of Fq -rational points is exactly q + 1 − m. So P P Pq = m Pq (m). Aq = m Aq (m), In the formulas below, both certain Legendre and Legendre–Jacobi symbols are used:   x   1 if x ≡ 1 (mod 3), 0 if x ≡ 0 (mod 3), =  3 −1 if x ≡ −1 (mod 3);     1 if c ≡ 1 (mod 4), −4 0 if c ≡ 0 (mod 2), =  c −1 if c ≡ −1 (mod 4);    1 if c ≡ 1 (mod 3),  −3 0 if c ≡ 0 (mod 3), =  c −1 if c ≡ −1 (mod 3). T HEOREM 9.99

(i) Aq = 2q + 3 +



−4 q



+2



 −3 ; q

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(ii) Pq = 3q + 2 +



−4 q



+



−3 q

2

+3



 −3 . q

It is necessary to define the class number of an integral quadratic form. Let S = {f (X, Y ) = aX 2 + bXY + cY 2 | a, b, c ∈ Z; a > 0}.   A B Consider G = SL(2, Z) acting on S. For T in G with T = and C D AD − BC = 1, let T(f ) = a(AX + CY )2 + b(AX + CY )(BX + DY ) + c(BX + DY )2 . With ∆(f ) = b2 − 4ac, also ∆(T(f )) = ∆(f ). So, all quadratic forms in the same orbit have the the same discriminant. The class number H(∆) is the number of orbits of G on quadratic forms in S with discriminant ∆. T HEOREM 9.100 Let (C1 ), (C2 ) be the following two conditions: (C1 ) : c > a and −a < b ≤ a; (C2 ) : c = a and 0 ≤ b ≤ a. The class number, H(∆) = |{(a, b, c) ∈ Z3 | b2 − 4ac = ∆; a > 0; (C1 ) or (C2 ) is satisfied}|. For 0 < −∆ ≤ 103, the class number H(−∆) is given in Table 9.7. L EMMA 9.101 Let q = ph . (i) For p ≡ 1 (mod 3), there exists a unique solution for s such that (s, p) = 1, s ≡ q + 1 (mod 9), s2 − 4q = −3x2 for some x ∈ Z. (ii) For p ≡ 1 (mod 4), there exists a unique solution for s such that (s, p) = 1, s ≡ q + 1 (mod 9), s2 − 4q = −4x2 for some x ∈ Z. Define the integers a0 , a1 as follows. When q ≡ 1 (mod 3), ( the unique if p ≡ 1 (mod 3),  solution for s in Lemma 9.101 (i) √ a0 = q √ q if p ≡ 6 1 (mod 3); 2 3

when q ≡ 1 or 4 (mod 12), ( the unique  solution for s in Lemma 9.101 (ii) √ a1 = q √ q 2 3

if p ≡ 1 (mod 4),

if p 6≡ 1 (mod 4).

T HEOREM 9.102 The number Aq (m) of Fq - isomorphism classes of non-singular √ plane cubics over Fq , q = ph , with q + 1 − m points, where | m | ≤ 2 q, is given in Table 9.8. In all other cases, Aq (m) = 0.

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Table 9.7 Values of the class number

−∆ 3 4 7 8 11 12 15 16 19 20 23 24 27 28 31 32 35

−∆

H(∆) 1 1 1 1 1 2 2 2 1 2 3 2 2 2 3 3 2

36 39 40 43 44 47 48 51 52 55 56 59 60 63 64 67 68

H(∆)

−∆

3 4 2 3 4 5 4 2 2 4 4 3 4 5 4 1 4

71 72 75 76 79 80 83 84 87 88 91 92 95 96 99 100 103

H(∆) 7 3 3 4 5 6 3 4 6 2 2 6 8 6 3 3 5

Table 9.8 Number of Fq - isomorphism classes of cubics

(i) (ii)

Aq (m)

(m, p) = 1

H(m2 − 4q)

0 √ ±√2q ± 3q

H(−4p) 1 1

h odd (a) (b) p = 2 (c) p = 3

(iii)

m

h even

  0 1 − −4  p  √ (b) ± q 1 − −3 p √ (c) ±2 q Dp h    i −4 1 p + 6 − 4 −3 − 3 where Dp = 12 p p (a)

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For the projective classification over Fq , define the functions B(m), C(m) as follows. Let B(m) be the number of isomorphism classes of elliptic curves F with 3 dividing N1 (F) = q + 1 − m; also, let C(m) be the number of isomorphism classes of elliptic curves F with N1 (F) = q +1−m. The group GF [3] ∼ = Z3 ×Z3 , where GF [3] is the subgroup of elements of order a power of 3 in the group GF (Fq ) of the curve. T HEOREM 9.103 The number Pq (m) of projectively distinct non-singular plane √ cubics over Fq , q = ph , with q + 1 − m points, where | m | ≤ 2 q, is as follows: Pq (m) = Aq (m) + B(m) + 3C(m) − ǫ(m), where

 2,    3, ǫ(m) = 4,    0,

if m = a0 or a1 with a0 6= a1 ; if m = a0 = a1 and p = 2; if m = a0 = a1 and p 6= 2; otherwise.

9.11 EXERCISES 1. Show the following corollary of Weil’s Explicit Formula (9.39). If f (α) ≥ 0 for all α ∈ R and cn ≥ 0 for all n ≥ 1, then √ Ψ1 ( q) g p p N1 ≤ + + 1; Ψ1 ( q −1 ) Ψ1 ( q −1 ) equality holds if and only if Pg j=1 f (αj ) = 0,

P∞

d=2

dBd Ψd (t) = 0.

2. Let F be a plane curve of degree n over Fq which splits into n rational lines. Prove the following upper and lower bounds on the number Rq of points of PG(2, q) lying on F: nq − 12 n(n − 3) ≤ Rq ≤ nq + 1.

Show also that the lower limit is achieved if and only if no three of the n lines are concurrent while the upper limit is achieved if and only if the n lines are concurrent. 3. For n ≥ 1, let hFqn (X ) be the class number of X . Also, for a prime d, let ( lcm (d − 1, . . . , di − 1, . . . , d2g − 1) when d 6= p, δ(d) = lcm (d − 1, . . . , di − 1, . . . , dg − 1) when d = p. Prove the following. (a) If d is a prime dividing hFq (X ) and if d | n(q n − 1)/(q − 1), then also d | hFqn (X )/hFq (X ).

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(b) If d ∤ hFq (X ) and n is relatively prime to δ(d), then d ∤ hFqn (X ). (iii) If n is the smallest integer for which d | hFqn (X ), then n | δ(d). 4. Let p ≡ 3 (mod 4) and q = p2n for n odd. For λ 6= −3, 1, 0, let Fλ be the irreducible Fq -rational quartic of genus 3 with Fλ = v(X04 + X14 + X24 − (λ + 1)(X02 X12 + X12 X22 + X22 X02 )).

Show that Fλ attains the Serre Bound; that is, Sq = q + 1 + 6pn . Compare Chapter 10, Exercise 1. 5. Let F be an irreducible Fq -rational curve of genus 3 with more than 2q + 6 Fq -rational points. Show that one of the following holds: (a) q = 8, N1 = 24, and F is birationally equivalent over F8 to the curve v(X40 + X14 + X24 + X02 X12 + X12 X22

+X02 X22 + X02 X1 X2 + X12 X0 X2 + X22 X0 X1 ); (b) q = 9, N1 = 28, and F is birationally equivalent over Fq to the Hermitian curve v(X04 + X14 + X24 ). 6. Show that an F5 -rational irreducible curve has N1 ≤ 13. 7. Show that the irreducible F5 -rational curve v(X 4 + X 3 Y 3 − X 2 − XY 5 + Y 5 + 2Y ) of genus 3 has N1 = 13. 8. Show that an irreducible F8 -rational curve has N1 ≤ 25. 9. Let X = v(Y 3 + Y 2 + 1 − X 3 Y ) be defined over F2 . Show that (a) X is non-singular and has genus 3;

(b) N1 = 3, N2 = 11, N3 = 9;

(c) L2 (t) = 1 + 3t2 + 6t4 + 8t6 = (1 + 2t2 )(1 + t2 + 4t4 ); (d) the 2-rank of X is 2. 10. Over F2 , let F = v(Y 6 + Y 5 + Y 4 + Y 3 + Y 2 + Y + 1 − X 3 (Y 2 + Y )). Show that (a) F has only one singular point X∞ ;

(b) X∞ is an ordinary triple point, and each of the branches of F centred at X∞ is F2 -rational; (c) F has genus g = 7;

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(d) N1 = 3, N2 = 9, N3 = 9, N4 = 45, N5 = 33, N6 = 69, N7 = 129; (e) L2 (t) = 1 + 2t2 + 9t4 + 16t6 + 32t8 + 72t10 + 64t12 + 128t14 = (1 + 2t2 )(1 − 2t2 + 4t4 )(1 + t2 + 4t4 )2 ; (f) the 2-rank of F is 4;

(g) X , as given in Example 9, is covered over F2 by F. 11. Let F = v(Y q − Y − f (X)) be an Artin–Schreier curve defined over Fqm , where f (X) = λ1 X i1 + λ2 X i2 + . . . + λs X is ,

λj ∈ Fqm .

Show that F has at most q m+1 points lying in AG(2, q m ).

12. Define the symmetric polynomial σ2 ∈ Fq [X0 , . . . , Xr−1 ] as follows: P σ2 (X0 , . . . , Xr−1 ) = Xi Xj , summed for 0 ≤ i < j ≤ r − 1, and let

S(X) = σ2 (X, X q , . . . , X q

r−1

).

For r ≥ 2, let F = v(f (X, Y )) be the irreducible plane curve defined over Fq , where f (X, Y ) = Y q

r−1

+ . . . + Y q + Y − S(X).

Show that F has genus g = 21 q r−1 (q r−1 − 1) and that the number Nr of its Fqr -rational points is 1 + q 2r−1 . 13. Let F = v(f (X, Y )) be the irreducible plane curve defined over Fq , where f (X, Y ) = Y q

3

1 2

+q2 +q+1 5

− (X q

3

3

+q

+ Xq

2

+1

).

2

Show that F has genus g = q(q + q − q − 2q − 1) and that the number N4 of its Fq4 -rational points is (q 2 + 1)(q 5 + 1). 14. For q = pm with p > 3, a Picard curve over Fq is a non-singular planar curve, F = v(Y 3 − p4 (X)), where p4 (X) is a polynomial of degree 4 with distinct roots. For j ∈ N, let  the coefficient of X i in p4 (X)q−1−j/3 for 3 | j, cij = 0 for 3 ∤ j. Show that  1 (mod p) for 3 | (p − 2), Sq ≡ 1 − cq−1,q−1 − c2q−1,q−1 − cq−1,2q−1 (mod p) for 3 ∤ p − 2. 15. For p > 2 and c ∈ Fq \{0}, let F = v(Y 2 − X 2g+1 + c) be the irreducible hyperelliptic curve of genus g with gcd(2g + 1, p) = 1. Show that the prank of F is zero if and only if there exists n ∈ N such that pn ≡ −1 (mod 4g + 2).

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16. Let F be a non-singular plane cubic defined over Fq that has an inflexion point in PG(2, q). Prove the following. (a) A canonical form for F over Fq is

v(Y 2 Z + a1 XY Z + a3 Y Z 2 − X 3 − a2 X 2 Z − a4 XZ 2 − a6 Z 3 ).

(b) Let b2 = a21 + 4a2 , b4 = a1 a3 + 2a4 , b6 = a23 + 4a6 , b8 = a21 a6 − a1 a3 a4 + 4a2 a6 + a2 a23 − a24 , c4 = b22 − 24b4 , c6 = −b32 + 36b2 b4 − 216b6, ∆ = −b22 b8 − 8b34 − 27b26 + 9b2 b4 b6 . Then the discriminant ∆ does not vanish and, for p 6= 2, 3, 1728∆ = c34 − c26 .

(c) The j-invariant of F is equal to c34 /∆.

(d) For p 6= 2, the cubic F becomes the curve

v(Y 2 Z − 4X 3 − b2 X 2 Z − 2b4 XZ 2 − b6 Z 3 )

with 4b8 = b2 b6 − b24 .

(v) If µ is given by 4µ = 27j/(j − 1728), then F is equianharmonic for µ = 0, j = 0 and harmonic for µ = ∞, j = 1728. 9.12 NOTES The general theory of zeta functions of an arbitrary fields of functions was developed by Schmidt in [384]; see also [374]. The presentation in Section 9.1, as in several other books, follows Hasse [195]. An introduction to algebraic varieties over a finite field is found in Joly [253]. Theorem 9.17 is proved in [21, Proposition 5]. The proof of Proposition 9.22 comes from [432]. Proposition 9.23 was proved by Bombieri [54], [55] via the approach initiated by Stepanov [418] and Schmidt [388]; see also [389] and [467]. The original proof of Theorem 9.18 is due to Weil [504]. For the M¨obius Inversion Formula, see, for example, [310, Theorem 3.24], Curves with many Fq -rational points have important applications to coding theory, and especially to algebraic-geometry codes. The methods for constructing and investigating such curves vary widely. Linear equations and quadratic forms were used by van der Geer and van der Vlugt [478, 480] and by Kawakita and Miura [265]. An elementary approach to the construction of curves of Kummer type with many Fq -rational points is due to van der Geer and van der Vlugt [482]; see also Garcia and Quoos [146], Garcia and Garzon [143] and Kawakita [262]. On the other hand, sophisticated methods from algebraic geometry are often effective. Inspired by Serre’s work, Lauter [296, 297, 298, 299] and Howe and

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Lauter [237] used Galois descent and Honda–Tate theory, endomorphisms of Jacobians of curves, Hermitian modules, and ray class fields. For constructions involving fibre products of Artin–Schreier curves, see van der Geer and van der Vlugt [476, 477, 479], Stepanov [420], Stepanov and Shalalfekh [422]. For cyclotomic function fields and Drinfeld modules of rank 1, see Neiderreiter and Xing [347] and [350, Chapter 3]. Hurt’s monograph [242] provides an overview of these methods and related results, including applications to coding theory. For survey papers, see van der Geer and van der Vlugt [479, 480], van der Geer [470, 471, 472], Garcia [141], Hajir and Maire [186], Garcia and Stichtenoth [153], and Voight [490]. Related to Remark 9.29, Fuhrmann and Torres [133] showed that, if q is square, then no curve of genus g, with √ 1 1 √ 2 q), 4 ( q − 1) < g < 2 (q − attains the Serre Bound. Also, Serre showed the non-existence of curves of genus g > 2 that miss his bound by just one. In other terms, if a curve of defect k is an Fq -rational curve for which N1 = q + 1 + ⌊2g⌋q − k with k ≥ 1, then no curve of genus g > 2 is a curve of defect 1. Serre also provided a list of possible zeta functions for curves of defect k ≤ 6 and genus g. Using this list, together with results on abelian varieties, Lauter [298] found that the Serre Bound can only be obtained if q2 − q g≤ √ . √ ⌊2 q⌋ + ⌊2 q⌋2 − 2q

The non-existence of various curves of defect 2 and genus g > 2 was also shown; for instance, no such curve exists when q = 22s , s > 1. On the positive side, Lauter [299] showed that, for all finite fields Fq , either there exists a curve of genus 3 and defect k with k ≤ 3, or a curve is within three of Serre’s lower bound. In [294] a method is given for improving the Serre Bound for a wide range of small genera, namely, √ √ g ≤ (q 2 − q)/(⌊2 q⌋ + ⌊2 q⌋2 − 2q),

for q = 8, 32, 213, 27, 243, 125 and 22s , s > 1. For more results, see [237] and, especially for a survey on curves of low defect and genus g ≤ 5, see [242]. Proposition 9.37 implies that p √ (9.77) g ≥ sup{(N1 − 1)λm ( q −1 ) − λm ( q)}, taken over all λm (t) with m ∈ N. This bound was optimised by Oesterl´e; see [391]: the right-hand side of (9.77) equals √ (N1 − 2) q cos α0 + q − (N1 − 1) , √ q + 1 − 2 q cos α0 where α0 is defined as follows. If k be the unique integer for which √ √ k q + 1 ≤ N1 ≤ q k+1 + 1 and

u=

√

q k+1 − (N1 − 1) √ √ . (N1 − 1) q − q k

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Then α0 ∈ [0, 1) and is the unique solution of the equation

cos 12 (k + 1)X + u cos 21 (k − 1)X = 0

in the interval [⌊π/(k + 1)⌋, π/k). For irreducible singular space curves, Z´un˜ iga-Galindo [520] refined the HasseWeil Bound 9.14 in terms of their singular points. ¨ Using certain hyperplane configurations in AG(r, q), Ozbudak and Stichtenoth [356] constructed curves whose number of Fq -rational points is close to Oesterl´e’s bound. The problem of determining the exact value of Nq (g) for small g was initiated by Serre [405, 407], giving Nq (1) and Nq (2). A computer-assisted search by Luengo and Lopez [315] provided irreducible plane curves defined over F3 of genus 4 ≤ g ≤ 8 that have many F3 -rational points, giving the following lower bounds:  12 for g = 4,      13 for g = 5, 14 for g = 6, N3 (g) ≥   16 for g = 7,    17 for g = 8.

On the other hand, N3 (5) ≤ 14 and N3 (7) ≤ 17. For lower bounds on N3 (g) for g ≥ 51, see also [349]. The lower bound (9.43) was shown by Ihara [247], and by Tsfasman and Vl˘adut¸ [460, 462]. The constructive but not explicit proofs of these results are based on modular curves and class field theory; see [247], [462], [336], [405], [519], [392], [508], [350], [118], [283], [119]. In 1995, Garcia and Stichtenoth [148] gave the first explicit example of a sequence of curves over Fq with square q attaining the Drinfeld-Vl˘adut¸ Bound. The tower in Example 9.40 comes from that paper; see also Elkies [116]. The Fq2 -tower in Example 9.41 is recursive; see Garcia and Stichtenoth [150]. Theorem 9.42 is due to Serre [405]. In the case that q is a proper power of p, that is, q > p, this also follows from Example 9.43, due to Garcia, Stichtenoth and Thomas [157]. In Remark 9.44, the lower bounds in (i) for q = 2, 3, 5 are due to Schoof [392] and Xing [508]. For improvements, see Niederreiter and Xing [348], Angles and Maire [15] and Temkine [452]. The result in (ii) is due to Zink [519]; that in (iii) is due to Perret [363]. For improvements to (iii), see Temkine [452] and [308]. Following the paper by Garcia and Stichtenoth [148], much work has been done on towers, especially on those given explicitly, by Beelen, Bezera, Elkies, Garcia, Li, Ling, Maharaj, R¨uck, Stichtenoth, van der Geer and van der Vlugt, Wulftange, Zaytsev and Zieve; see [142], [483], [156], [41], [42], [43], [44], [50], [51], [110], [309], [320], [507], [514] [153], [311]. For a survey, see [155]. Surveys on A+ (q) are found in [142], [350, Chapter 5], [421, Chapters 6, 9], and in [460, Chapter 4]. Theorems 9.45, 9.46 and 9.47 on A− (q) come from [119]. The L¨uroth semigroup S was introduced by Heinzer and Moh [331]. Greco and Raciti [181] considered irreducible plane curves F of degree d ≥ 4 with only

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391

ordinary singularities. They showed that if (a − 1)d + 1 ≤ n ≤ ad − (a2 + 1) for a ≥ 1 then n 6∈ S. For the gonality of a curve of degree m, see Homma [229]. The connection between Clifford’s Theorem 6.79 and gonality was investigated by Hartshorne [193]. For Proposition 9.52, see [360]. Proposition 9.54 is due to Lewittes [307]. Theorems 9.57, 9.58 and Corollaries 9.59, 9.60 are from [300]; see also [20] and [24]. Theorems 9.62 and 9.63 come from [220]. Proposition 9.65 is due to Sziklai [442]. For an upper bound on the difference between the number of points on a curve over a Fq and on a certain covering of the curve, see [22]. For more results similar to Theorems 9.67 and 9.69, see [216]. For curves with no rational points, see [511]. Bounds on the number of Fq -rational points have widely been used in the study of polynomials in one variable over finite fields. The main reference is [310]. Some applications on the Hasse–Weil Bound 9.18 to graph theory and coding theory are treated in Cohen [83]. The proof of Theorem 9.72 comes from Frey [130]. Theorems 9.75 and 9.76 and related proofs are due to Stichtenoth; see [425]. For proofs of the results on the theory of divisibility in Section 9.7, see [56]; see also [352]. A proof of Theorem 9.79 via Castelnuovo’s inequality can be found in [114, Chapter V. 5.4]. The propositions on the Frobenius linear series in Section 9.8 come from [134]. The presentation of the general theory of the abelian varieties is beyond the scope of this book. For background on this topic, see the books by Mumford [338, 339], Lang [292], and Milne [330]; see also the papers by Tate [449], Mazur [326], and Waterhouse and Milne [501]. Theorems 6.94 and 6.96 proved in Section 9.7, describing the structure of Pic0 (K(X )), extend to any abelian variety. For the fact that Pic0 (K(X ), l∞ ) is a divisible group, see [313, Theorem XI.3.2]. Proofs, for abelian varieties, of the last results in Section 9.8 can be found in [292, Chapter VII]. (F ) For computations in Pic0 p (F) of curves F of type v(Y 3 − γX 5 + δ), see [53]. Theorem 9.92 giving the possible numbers of rational points on an elliptic curve is due to Waterhouse [500]; see Ughi [466] for another proof. For the bounds on N1 , see Zimmer [517, 518]. Theorem 9.97 giving the group structure of the set of rational points is due independently to R¨uck [377] and to Voloch [491]. The formulas for the numbers of non-isomorphic and projectively distinct cubics with a certain number of rational points are due to Schoof [390]; for the prime case, see also Deuring [101]. For Theorem 9.91, see Schoof [390] and Tate [450]. For a summary of non-isomorphic cubics, see also Menezes [328, Chapter 7]. For the theory of singular plane cubics, see [216, §11.4]. For more general accounts of elliptic curves, see Cassels [71], Husem¨oller [243], Koblitz [273], Silverman [416], Tate [450]. For classical accounts of plane cubics, see Enriques [122], Hilton [212], Salmon [380], Seidenberg [400], Walker [497].

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For the projective classification of plane cubics, including the cases of characteristic 2 and 3, see [216, Chapter 11]; the values of n0 , n1 , n3 , n9 are given in Tables 11.28 to 11.30. For Exercises 4 and 5, see [23]; for Exercise 6, see [295]; for Exercise 7, see [242, Chapter 2, Section 6]; for Exercise 8, see [381]; for Exercises 12 and 13, see [152]; for Exercise 11, see [183]; for Exercise 14, see [124]; for Exercise 15, see [276].

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PART 3

Further developments

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Chapter Ten Maximal and optimal curves In the study of curves with many Fq -rational points the main problem is to determine Nq (g), the largest number of Fq -rational points that an irreducible Fq -rational curve of genus g can have. The asymptotic behaviour of Nq (g) with respect to g together with the Drinfeld–Vl˘adut¸ Bound (9.38) are discussed in Section 9.3, where √ some examples with Nq (g) ≈ ( q −1)g for a fixed square q and for g large enough are also constructed. In general, Nq (g) is known for only a few pairs (q, g). The aim of this chapter is threefold: (1) to describe the theory of Fq -maximal curves, that is, curves of genus g de√ fined over Fq whose number Sq of Fq -rational points is q + 1 + 2g q, the Hasse–Weil Bound of Theorem 9.18; (2) to give the known examples of maximal curves and some of their characterisations; (3) to describe some examples of Fq -optimal curves, that is, curves of genus g defined over Fq whose number of Fq -rational points equals Nq (g). √ The Hasse–Weil Bound implies that Ng (q) ≤ q + 1 + 2g q. If equality holds, then the Fq -optimal curve is Fq -maximal, and Fq -maximal curves of genus g > 0 can only exist for q square. An Fq -maximal curve is, by definition, Fq -optimal. For g = 0, every curve is Fq -optimal, as the number of Fq -rational points of the projective line over Fq is q + 1, while Nq (0) ≤ q + 1 by the Hasse–Weil Bound 9.18. Non-trivial examples of maximal and optimal curves appear in other chapters: (1) the Hermitian curve, Example 1.39, is both Fq -optimal and Fq -maximal for q = p2e ; (2) the DLS curve, Example 5.24, is both Fq -optimal and also Fq4 -maximal for q = 22e+1 with e ≥ 1; (3) the DLR curve, Section 12.4, is Fq -optimal and Fq6 -maximal for q = 32e+1 with e ≥ 1. Notation and terminology are the same as in Chapter 9. To avoid trivial cases, it is assumed that g > 0, unless otherwise stated. First, maximal curves are investigated. So, the underlying field is Fq2 .

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10.1 BACKGROUND ON MAXIMAL CURVES A characterisation of maximal curves can be deduced from the Hasse–Weil Theorem 9.19 and (9.10) for n = 1. T HEOREM 10.1 When X be an irreducible non-singular curve of genus g defined over Fq2 , then the following conditions are equivalent: (i) X is Fq2 -maximal, that is, N1 = q 2 + 1 + 2qg; (ii) ωi = −q for i = 1, 2, . . . , 2g; (iii) LFq2 (t) = (t + q)2g . Also, if X is Fq2 -maximal and Nm is the number of its Fq2m -rational points, then Nm = q 2m + 1 + (−1)m−1 2gq m ,

for m = 1, . . ..

(10.1)

Theorem 10.1 shows that, if the curve is maximal then its zeta function has particularly simple form. Consequently, the zeta function via its enumerator Lq2 (t), the corresponding fundamental equation, the Frobenius linear series, and the automorphisms combine to produce significant results. Beginning with L-polynomials, Theorem 9.17 together with Theorem 10.1 has the following corollary. T HEOREM 10.2 If X ′ is an Fq2 -rational curve covered by an Fq2 -maximal curve X , with an Fq2 -rational covering, then X ′ is also Fq2 -maximal. There are Fq2 -maximal curves with simple equations such as those of Kummer type Kn = v(Y q+1 − f (X)),

(10.2)

and those of Artin–Schreier type A = v(Y q − Y − f (X)),

(10.3)

as well as the Hermitian curve Hq = v(Y q + Y − X q+1 ).

(10.4)

E XAMPLE 10.3 (i) For q odd, the irreducible plane curve 1 E(q+1)/2 = v(Y q + Y − X (q+1)/2 ) 1 ) K(E(q+1)/2

(10.5)

is the subfield K(x2 , y) is covered by the Hermitian curve Hq , since of K(Hq ) = K(x, y). Note that K(x2 , y) is a proper subfield of K(Hq ) with 1 y q + y − xq+1 = 0, as the genus g ′ of E(q+1)/2 is smaller than 21 (q 2 − q), the genus 1 of Hq . Therefore [K(Hq ) : K(E(q+1)/2 )] = 2, and the rational transformation 1 ω : (x, y) 7→ (x2 , y) provides a two-fold covering of E(q+1)/2 by Hq . From 1 1 Lemma 12.1, g ′ = 4 (q − 1)2 . Since x2 , y ∈ Fq2 (Hq ), then E(q+1)/2 , ω and

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Table 10.1 Families of curves F = v(F ) containing Fq2 -maximal curves

F Dr

F Xr + Y r + 1

1 Em

Y q + Y − Xm

Tp

X q+1 − (Y + Y p + Y p + . . . + Y q/p )

T3′

′′

T3 K

2

(Y + Y 3 + · · · + Y q/3 )2 − X q − X

Y + Y 3 + · · · + Y q/3 + cX q+1 , cq−1 = −1

Y q+1 − f (X)

Y q − Y − f (X)

A F0

X (q+1)/3 + X 2(q+1)/3 + Y q+1

G

Y q − Y X 2(q−1)/3 + ωX (q−1)/3 , ω q+1 = −1

F0′ Cn

Y X (q−2)/3 + Y q + X (2q−1)/3 X nY + Y n + X

Cn,k

X nY k + Y n + X k

Cim

X mi+m + X mi + Y q+1

Xr

Y 2 + a1 Y 2

r

r−1

+ · · · + ar−1 Y 2 + Y + X q+1

the associated two-fold covering are Fq2 -rational, as well. From Theorem 10.2, 1 E(q+1)/2 is an Fq2 -maximal curve. See also Example 10.35. (ii) The corresponding example for q even is the irreducible plane curve T2 = v(X q+1 + Y + Y 2 + Y 4 + · · · + Y q/2 ),

1 4 q(q−2) by

(10.6)

Lemma 12.1. Since K(T2 ) is the subfield K(x, y 2 +y) whose genus is of K(Hq ), the same argument shows that T2 is two-fold covered by the Hermitian curve Hq , and hence it is an Fq2 -maximal curve. See also Example 10.36. (iii) See Example 10.33 for the curve F0 = v(F0 ) with F0 = X (q+1)/3 + X 2(q+1)/3 + Y q+1 ,

when q ≡ 2 (mod 3); its genus is 16 (q 2 − q + 4). (iv) See Example 10.34 for the curve F0′ = v(F0′ ) with

F0′ = Y X (q−2)/3 + Y q + X (2q−1)/3 ,

when q ≡ 2 (mod 3); its genus is 61 (q 2 − q − 2). (v) For curves of genus 61 (q 2 − q), examples are T3′ = v(T3′ ) with T3′ = T(Y )2 − X q − X,

T(Y ) = Y + Y 3 + · · · + Y q/3 ,

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when q ≡ 0 (mod 3) as in Example 10.37, and G = v(G) with G = Y q − Y X 2(q−1)/3 + ωX (q−1)/3 ,

ω q+1 = −1,

when q ≡ 1 (mod 3) as in Exercise 11.

As in Section 11.5, every non-trivial automorphism group G of X gives rise to a covering of X . In Example 10.3 (i), F = Hq /hαi with α : (X, Y ) 7→ (−X, Y ). Such a covering and the corresponding quotient curve X ′ = X /G are Fq2 -rational if G is an Fq2 -automorphism group; that is, G is the restriction to X of a subgroup of PGL(r, q 2 ). Each of the initial examples, (a), (b), (c), of Fq2 -maximal curves has a large Fq2 -automorphism group with many non-conjugate subgroups; see Chapter 12. From this, the existence of numerous Fq2 -rational maximal curves is deduced. As in Sections 12.2, 12.3 and 12.4, the genera of such quotient curves can often be computed using the techniques developed in Chapter 11; see especially Theorems 11.57 and 11.72, but the problem of finding an explicit equation has been solved so far only in a few cases; see, for instance, Theorem 12.28. E XAMPLE 10.4 For a power ℓ of p, let q = ℓ3 . Then the irreducible plane curve F = v(f (X, Y )), with 2

f (X, Y ) = Y ℓ − Y − X ℓ 1 2 2 (ℓ

2

−ℓ+1

,

(10.7)

2

is an Fq2 -maximal curve of genus g = − 1)(ℓ − ℓ). Observe first that F belongs to the family of curves studied in Section 12.1. From Lemma 12.1, F has only one singular point, namely X∞ = (0, 1, 0). More precisely, X∞ is an (ℓ − 1)-fold point and is the centre of a unique branch. In particular, this branch is Fq2 -rational. Therefore the number of all Fq2 -rational points of F is equal to the number of points of F lying in PG(2, q). To count such points, note first that X∞ is the unique point of F at infinity and that the Y -axis meets F in exactly ℓ2 points; the latter are the points P = (0, b) with b ∈ Fℓ2 and hence they lie in AG(2, q 2 ). Also, every other point of F is of 2 2 type Q = (u, v) with u 6= 0. Let ξ = uℓ −ℓ+1 . Since uq −1 = 1 if and only if u ∈ Fq2 \{0}, from the identity q 2 − 1 = (ℓ3 − 1)(ℓ + 1)(ℓ2 − ℓ + 1),

it follows that u ∈ Fq2 \{0} and ξ ℓ+1 ∈ Fq \{0} are equivalent conditions. For 2 each such ξ, the equation Y ℓ − Y = ξ has ℓ2 distinct solutions in Fq2 . Counting, therefore, gives ℓ2 (ℓ2 + 1)(ℓ + 1)(ℓ2 − ℓ + 1) points. Since q 2 + 1 + 2gq = 1 + ℓ2 + ℓ2 (ℓ2 + 1)(ℓ + 1)(ℓ2 − ℓ + 1),

so F is Fq2 -maximal.

T HEOREM 10.5 (i) For ℓ = 2, the curve F in Example 10.4 is a quotient curve of the Hermitian curve H8 . (ii) For ℓ = 3, the curve F in Example 10.4 is not a quotient curve of the Hermitian curve H27 ; by Theorem 11.36, H27 is not even a Galois covering of F.

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Table 10.2 Known Fq2 -maximal curves of large genera

Genus g 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Condition on q

1 2 q(q − 1) 1 2 4 (q − 1) 1 4 q(q − 2) 1 2 6 (q − q + 4) 1 2 6 (q − q) 1 2 6 (q − q) 1 2 6 (q − q − 2) 1 6 (q − 1)(q − 2)

1 6 q(q − 3) 1 2 8 (q − 2q + 5) 1 2 8 (q − 1) 1 8 q(q − 2) 1 8 (q − 1)(q − 3) 1 8 (q − 1)(q − 3) 1 8 q(q − 4)

Curves Hq = Dq+1

q ≡ 1 (mod 2)

1 E(q+1)/2

q ≡ 0 (mod 2)

T2

q ≡ 2 (mod 3)

F0

q ≡ 1 (mod 3)

G

q ≡ 0 (mod 3) q ≡ 2 (mod 3)

T3′

F0′

q ≡ 2 (mod 3)

1 E(q+1)/3

q ≡ 0 (mod 3)

T3

q ≡ 3 (mod 4)

(∗)

q ≡ 1 (mod 4)

(∗)

q ≡ 0 (mod 4)

(∗)

q ≡ 1 (mod 4)

′′

1 E(q+1)/4

q ≡ 3 (mod 4)

1 E(q+1)/4 , D(q+1)/2

q ≡ 0 (mod 2)

X2

For (∗) see Remark 10.64.

The classification of maximal curves is currently out of reach. However, for larger values of g for which there exists an Fq2 -maximal curve, it seems that there are few curves: see Table 10.2. This has been shown so far for g ≥ ⌊ 61 (q 2 − q + 4)⌋. The proof is postponed to Section 10.5.

10.2 THE FROBENIUS LINEAR SERIES OF A MAXIMAL CURVE The starting point is Theorem 10.1, showing that the Fundamental Equation (9.74) for an irreducible, non-singular Fq2 -maximal curve X is as follows: qP + Φ(P ) ≡ (q + 1)P0 ,

(10.8)

where P0 ∈ X (Fq2 ) and Φ is the Frobenius map. Let D = |(q + 1)P0 | be the Frobenius linear series of X , and put r = dim D. From (10.8), dim |qP | = r − 1 for every P ∈ X . By the Weierstrass Gap Theorem 6.89 and Corollary 6.75, an

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immediate consequence of (10.8) for the non-gap sequence of X at a point P ∈ X is the following: 0 < m1 (P ) < · · · < mr−1 (P ) ≤ q < mr (P ).

(10.9)

Further, Propositions 9.81 to 9.85 applied to an Fq2 -maximal curve, that is, for u = 1, m = q + 1, a0 = 1, a1 = q, provide a number of basic facts that are collected in the next proposition. P ROPOSITION 10.6 With the notation above, the following hold: (I) if P and Q are Fq2 -rational points, then (q + 1)P ≡ (q + 1)Q, and q + 1 is a non-gap at each P ∈ X (Fq2 ); (II) there exists P1 ∈ X (Fq2 ) such that both q + 1 and q are non-gaps at P1 ; (III) the linear series D is complete, base-point-free, simple and defined over Fq2 ; it gives rise to an Fq2 -rational curve Γ of PG(r, q) that is Fq2 -birationally equivalent to X ; (IV) the (D, P )-orders ji at an Fq2 -rational point P are the terms of the sequence 0 < q + 1 − mr−1 (P ) < · · · < q + 1 − m1 (P ) < q + 1; that is, jr−i (P ) + mi (P ) = q + 1 for i = 0, . . . , r − 1; (V) if P 6∈ X (Fq2 ), then j1 (P ) = 1 and so ǫ1 = 1; (VI) the integer q is a D-order, and so r ≥ 2; (VII) if P ∈ X (Fq4 )\X (Fq2 ) then q − 1 is a non-gap at P ; if P 6∈ X (Fq4 ) then q is a non-gap at P ; (VIII) if P is an Fq2 -rational point of X , then jr−1 (P ) < q; (IX) ǫr = νr−1 = q, so Γ is Frobenius non-classical, and every Fq2 -rational point of X is in the support of the ramification divisor R of Γ; (X) if N1 ≥ q 3 + 1, then m1 (P ) = q for every Fq2 -rational point P of X . Some refinements are worthwhile. (XI) If P ∈ X is not an Fq2 -rational point, then 0 ≤ q − mr−1 (P ) < · · · < q − m1 (P ) < q are (D, P )-orders at P ; in particular, jr (P ) = q. Arguing as in the proof of Proposition 9.83 (i) shows that, if m(P ) is a non-gap at a point P 6∈ X (Fq2 ), then there exists an effective divisor E for which E ≡ m(P )P and P 6∈ Supp(E). From this, E + (q − m(P ))P + Φ(P ) ≡ qP + Φ(P ) ≡ (q + 1)P0 .

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If m(P ) ≤ q, then q − m(P ) is a (D, P )-order. Assume on the contrary that jr (P ) 6= q. Then jr (P ) = q + 1, and hence (q + 1)P ≡ qP + Φ(P ) by (10.8). But then P ≡ Φ(P ) which is not possible since the genus of X is positive. A similar argument shows that (II) is true for any P1 ∈ X (Fq2 ). (XII) If P is an Fq2 -rational point then both q and q + 1 are non-gaps at P ; in particular, j1 (P ) = 1 for every Fq2 -rational point P . As (IX) implies that r ≤ q, this may be refined as follows. (XIII) Either r = q − (g − 1) or r ≤ 21 (q + 1). In fact, from the Riemann–Roch Theorem 6.61 it follows that either D is nonspecial and r = q + 1 − g or it is special and Clifford’s Theorem 6.79 implies that r ≤ 12 (q + 1). By (IV) and (XII), j1 (P ) = 1 for every P ∈ Γ; that is, the curve Γ has only linear branches. This does not imply a priori that Γ is non-singular. So the following result is an improvement. T HEOREM 10.7 (i) The irreducible Fq2 -rational curve Γ associated to the Frobenius linear series D of an Fq2 -maximal curve X is non-singular. (ii) The curves X and Γ are isomorphic over Fq2 . Proof. Choose f0 , . . . , fr ∈ Fq2 (X ) such that D consists of all divisors Ac = div (c0 f0 + · · · + cr fr ) + (q + 1)P0 with c = (c0 , . . . , cr ) ∈ PG(r, K). By (III), let Γ be the irreducible curve arising from the point (f0 , . . . , fr ); in other words, the Fq2 -rational transformation ω : x0 = f0 , . . . , xr = fr defines an Fq2 -rational morphism π : X → Γ. Since every branch of Γ is linear, it must be shown that no point of Γ is the centre of two or more branches. Let P ∈ X and π(P ) be a branch point of Γ. After multiplying (f0 , . . . , fr ) by a suitable element of Fq2 (X ), it may be assumed that P is neither a pole of any fk nor a zero of all the fk . Since ǫr = q by (XI), Theorem 7.65 applied to pm = q ensures the existence of z0 , . . . , zr in Fq2 (X ) satisfying the following three properties: (i) P is neither a pole of any zk nor a zero of all the zk ; (ii)

z0q f0 + . . . + zrq fr = 0;

(10.10)

(iii) the osculating hyperplane LP at the branch point π(P ) of Γ is v(z0 (P )q X0 + · · · + zr (P )q Xr ).

(10.11)

Since Γ is Frobenius non-classical, Theorem 8.54 gives for q 2 = ph that z0 f0q + · · · + zr frq = 0.

(10.12)

Suppose, on the contrary, that π(P ) and π(Q) have the same centre A ∈ Γ. Then the osculating hyperplane to Γ at π(P ) contains the centre of π(Q). From (10.8),

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Φ(P ) = Q, and A ∈ PG(r, q 2 ). After a suitable change of the coordinate system of PG(r, q 2 ), let A = (1, 0, . . . , 0). The coordinate functions fk also change, but it is still true that P is not a pole of any fk , and that P is not a zero of some coordinate function, say f0 ; the same holds for zk . So take f0 = 1. If τ is a primitive place representation of the branch π(P ), then π(P ) has a primitive branch representation x0 (t) = 1, x1 (t) = τ (f1 ) = a1 t + · · · , . . . , xr (t) = τ (fr ) = ar t + · · · ,

with some ak 6= 0 as π(P ) is linear. For k = 0, 1 . . . , r, write zk (t) = τ (zk ) = zk (P ) + uk (t)

with ord uk (t) ≥ 1. Apply τ to (10.10):

ord(z0 (P )q + · · · + zr (P )q xr (t)) = ord(u0 (t)q + · · · + ur (t)q xr (t)).

Hence z0 (P ) = 0. From (10.12), u0 (t) ≥ q > 1. This together with the preceding equation, implies that ord(z0 (P )q + · · · + zr (P )q xr (t)) ≥ q + 1.

Therefore jr (P ) = I(P, Γ ∩ LP ) = q + 1. But then I(Q, Γ ∩ LP ) = 0, which contradicts the hypothesis that the centre of π(Q) lies in LP . 2 C OROLLARY 10.8 Let X be an Fq2 -maximal curve of genus g and, for some point P0 ∈ X (Fq2 ), suppose that there exist positive integers a, b ∈ H(P0 ) such that all non-gaps at P which are at most q + 1 can be written as λa + µb for non-negative integers λ, µ. Then H(P0 ) = ha, bi and g = 21 (a − 1)(b − 1). Proof. Choose x, y ∈ Fq2 (X ) such that div(x)∞ = aP0 and div(y)∞ = bP0 . Since q, q + 1 ∈ H(P0 ) by (XII), the integers a, b are coprime, and so let a < b. Therefore K(X ) = K(x, y) and Fq2 (X ) = Fq2 (x, y). Let f (X, Y ) ∈ Fq2 [X, Y ] be an irreducible polynomial such that f (x, y) = 0. Its Weierstrass normal form, see Theorem 6.92, is of type P f (X, Y ) = X a + β Y b + ai+bj 2, it has classical gap sequence at a general point if and only if the canonical curve of X is classical; see Remark 7.56. By Corollary 7.61, this is the case whenever p > 2g − 2. The following example shows that (XVI) can occur. E XAMPLE 10.10 The Klein quartic F = v(X0 X13 + X1 X23 + X2 X03 ), as in Example 7.18, is a non-hyperelliptic curve of genus 3 and is a canonical curve. Also, for p > 2g − 2 = 4, F has classical gap sequence at a general point. Now, let F be viewed as an Fp2 -rational curve. By Theorem 10.78, F is Fq2 -maximal if and only if q ≡ 6 (mod 7); see also Corollary 10.82 (ii). Note that the classical Dirichlet Theorem states that there exist infinitely many primes satisfying p ≡ 6 (mod 7). In particular, F is a classical Fp2 -maximal curve for an infinite number of primes p. It was noted in the proof of Theorem 10.9 that all D-orders but one can be deduced from (XI) whenever the non-gaps at an ordinary point are known. Let J denote the missing D-order; that is, at an ordinary point P ∈ X , ǫJ−1 = q − mr−J (P ) < ǫJ < ǫJ+1 = q − mr−(J+1) (P ) .

(10.13)

From Proposition 8.42, the Frobenius D-orders are obtained from the D-orders by removing one of them, say ǫI . In fact, the integers I and J coincide. P ROPOSITION 10.11 The Frobenius orders of D are the D-orders apart from ǫJ with J as in (10.13). Proof. Let P be an ordinary point of X and suppose that P 6∈ X (Fq2 ). Choose ξt ∈ K(X ), for t = 1, . . . , r − 1, such that div ξt = Dt − mt (P )P with Dt ≻ 0 and P 6∈ Supp Dt . Also, choose ξ, v ∈ K(X ) for which div ξ = qP + Φ(P ) − (q + 1)P0 ,

div v = ǫJ P + D − (q + 1)P0 ,

with D ≻ 0 and P ∈ 6 Supp D. From (XI), ( −(q + 1)P0 + ǫi−1 P + Φ(P ) + Dr−i , i ≤ J, div(ξ ξr−i ) = −(q + 1)P0 + ǫi P + Φ(P ) + Dr−i , i > J.

(10.14)

Let xi−1 = ξ ξr−i for i = 1, . . . , r − 1. From (10.14), div xi−1 + (q + 1)P0 ∈ D. More precisely, D consists of all divisors Pr−1 Pj−1 Ac = div( i=0 ci xi + cj vj + i=j ci+1 xi ) + (q + 1)P0

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with c = (c0 , . . . , cr ) ∈ PG(r, K). Since these divisors are cut out on Γ by hyperplanes, it also follows from (10.14) that every hyperplane ∆ of PG(r, K) with I(P, ∆ ∩ Γ) > J passes through Φ(P ). On other hand, the hyperplane ∆′ arising from v does not have this property. Indeed, if Φ(P ) ∈ ∆′ , then Φ(P ) ∈ Supp D and hence div v = ǫJ P + Φ(P ) − (q + 1)P0 + D′ ,

with D′ ≻ 0 and P 6∈ Supp D′ . Thus there exists w ∈ K(X ) such that

div w = ǫJ P + Φ(P ) − (qP + Φ(P )) + D′ = (q − ǫj )P + D′ .

But this implies that q−ǫJ is a non-gap at P 6∈ X (Fq2 ), contradicting the definition of J. So, the assertion is proved. Note that, in terms of the Frenet frame of X at P , see Section 7.6, this assertion can be reworded as saying that Φ(P ) ∈ (ΠJ \ΠJ−1 ) for any ordinary non-Fq -rational point P of X . Using Proposition 8.42, the result is established. 2 P ROPOSITION 10.12 Let W be the set of all Weierstrass points of an Fq2 -maximal curve X . (i) If g > q − (r − 1), then X (Fq2 ) ⊂ W. (ii) If S is the St¨ohr–Voloch divisor of X , then Supp S ⊂ W ∪ X (Fq2 ). Proof. (i) By (XVI) and Exercise 8 in Chapter 7, the curve X is neither elliptic nor hyperelliptic. Let K be its canonical curve, and take an ordinary point P of X . Then the orders ǫ0 = 0 < · · · < ǫj < . . . < ǫg−1 of K are also the orders of K at P . From Section 7.6, the gaps at P are exactly the integers ǫj + 1, and hence H(P ) consists of all non-negative integers distinct from 1, . . . , ǫj + 1, . . . , ǫg−1 + 1. Also, for any integer i with 1 ≤ i ≤ q, Lemma A.6 implies that   q+i−1 6≡ 0 (mod p). q Suppose that q + i 6∈ H(P ) for an integer i with 1 ≤ i ≤ q. Then q + i − 1 is an order of K at P . From Lemma 7.62, q is also an order of K at P . Hence q + 1 is a gap at P . By (I), P 6∈ X (Fq2 ). Therefore, if P ∈ X (Fq2 ), then q + 1, . . . , 2q ∈ H(P ). Assume that P ∈ X (Fq2 ). Then (XII) shows that {q, q + 1, . . .} ⊂ H(P ). Since P is an ordinary point, from (10.9) it follows that the number of gaps at P is equal to q − (r − 1). By the Weierstrass Gap Theorem 6.89, g = q − (r − 1). But by the hypothesis, this contradicts the assumption that P ∈ X (Fq2 ). (ii) Let Di denote the i-th row vector in the generalised Wronskian determinant (7.12) of the coordinate functions f0 , . . . , fr of Γ; that is, Di = (D(ǫi ) f0 , D(ǫi ) f1 , . . . , D(ǫi ) fr ) ,

i = 0, 1, 2, . . . , r .

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Also, let E be the first row vector of the Wronskian, det M (ν0 , . . . , νr−1 ), from which the St¨ohr–Voloch divisor S is derived, as in Section 8.5; that is, 2

2

2

2

E = (f0q , f1q , f2q , . . . , frq ). If J is defined as in (10.13), Proposition 10.11 shows that ν0 < ν1 < · · · < νJ−1 < ǫJ < νJ < · · · < νr−1

is the order sequence of D. Let P 6∈ W ∪ X (Fq2 ). By the last remark in the proof of Proposition 10.11, Φ(P ) ∈ ΠJ \ΠJ−1 . Therefore E is a linear combination of D0 , . . . , DJ but not of D0 , . . . , DJ−1 . Hence det M (ν0 , . . . , νr−1 ) does not alter if its first row E is replaced by DJ . In other words, det M (ν0 , . . . , νr−1 ) = det(D0 , . . . , Dr ). Now, on the contrary, let P ∈ Supp S. This assumption means that P is a zero of det M (ν0 , . . . , νr−1 ). Then P is also a zero of det(D0 , . . . , Dr ); that is, (j0 (P ), . . . , jr (P )) 6= (ǫ0 , . . . , ǫr ). But then P ∈ W ∪ X (Fq2 ), by (XI). 2 R EMARK 10.13 Let X have classical gap sequence at a general point. Then, from (XVI), the first r − 1 positive non-gaps at any ordinary point are g + 1, . . . , g + r − 1 = q.

Thus the orders of D are 0, . . . , r−2, ǫr−1, q with r−1 ≤ ǫr−1 < q. By Proposition 10.11, the Fq2 -Frobenius orders are 0, . . . , r−2, q. A direct computation shows that P deg S = N1 + deg R. It may be that the stronger equation S = P ∈X (F 2 ) P + R q also holds, but no proof has yet been found. The following result regards hyperelliptic curves which are Fq2 -maximal. P ROPOSITION 10.14 Let X be an Fq2 -maximal curve of genus g ≥ 2. (i) If X is hyperelliptic, then q ≤ 2r − 2. (ii) If there exists P ∈ X such that jr−1 (P ) = q − 1, then X is hyperelliptic. Proof. If X is hyperelliptic, then m1 (P ) = g + 1 for any ordinary point P ; see Exercise 8 in Chapter 7. Then mr−1 (P ) = g + r − 1, by the Weierstrass Gap Theorem 6.89. From (XI), it follows that mr−1 (P ) = q; hence g = q − r + 1. On the other hand, N1 ≤ 2(q 2 + 1) and the maximality of X shows that 2g ≤ q. Therefore (i) holds. Let P ∈ X (Fq2 ) with jr−1 (P ) = q − 1. Since jr (P ) = q − 1, this and (IV) imply that m1 (P ) = 2. As g ≥ 2, this proves that X is hyperelliptic. Finally, let P 6∈ X (Fq2 ) with jr−1 (P ) = q−1. Then (10.8) implies the existence of a divisor E for which (q − 1)P + E ≡ qP + Φ(P ) with P 6∈ Supp(E). Hence E ≡ P + Φ(P ). This shows that the linear series |P + Φ(P )| has positive dimension. Since g ≥ 2, Theorem 6.81 (i) implies that dim |P + Φ(P )| = 1. Therefore, by Theorem 7.42, X is hyperelliptic. 2 P ROPOSITION 10.15 Suppose that jr−1 (P ) = r − 1 for every point P ∈ X . Then (r − 1)r(g − 1) = (q + 1)(q − r) .

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Proof. Under the hypothesis, a point P ∈ X is a D-Weierstrass point of X if and only if P is an Fq2 -rational point of X . Also, if RD is the ramification divisor of X , then vP (RD ) = 1 for P ∈ X (Fq2 ); see Theorem 7.55. Since X is Fq2 -maximal, the assertion follows from Definition 7.52. 2

10.3 EMBEDDING IN A HERMITIAN VARIETY Although it should not matter which model of a curve is considered, some models are better than others in revealing birationally invariant properties. For Fq2 -maximal curves, the model arising from the Frobenius linear series D appears to be a natural candidate, even though a slight change is necessary when the smallest linear subseries L of D containing all divisors qP + Φ(P ) is properly contained in D, that is, when m = dim L < dim D = r. As explained later, this replacement prevents the model from having higher-dimensional strangeness; that is, it ensures that there is no point lying in an infinite number of osculating hyperplanes. Therefore, throughout this section, let X be the curve given by Theorem 10.7. The idea is to use the geometric properties of the results of Section 10.2 to find geometric characterisations of X as an irreducible non-singular curve lying on certain hypersurfaces. Such a characterisation is the main aim in this section. Up to an isomorphism over Fq2 , Theorems 10.21 and 10.31 establish that Fq2 -maximal curves are the irreducible Fq2 -rational curves of degree q+1 lying on a non-degenerate Hermitian variety of PG(m, K) with 2 ≤ m ≤ r. The Hermitian variety of PG(n, K), with K = Fq , is a hypersurface which is the natural generalisation of the Hermitian curve to higher dimensions; canonically, it is Hn,q = v(X0q+1 + . . . + Xnq+1 ). By Theorem 10.7, the curve X satisfies the Linear General Position Principle. If m ≥ 3 and m is small, then Castelnuovo’s Bound provides an upper bound on the genus g, which has applications for larger g in the case of maximal curves; see Sections 10.5, 10.6. If m = 3 and g is large, such a curve X also lies on a low degree surface, by Halphen’s Theorem and its generalisations; see Section 7.15. In the known examples, as at the end of this section, X is a component of degree q + 1 of the intersection of the Hermitian surface with an Fq2 -rational quadric or cubic surface in PG(3, K). The characterisation, together with Castelnuovo’s Bound and Halphen’s Theorem, is the main ingredient in the classification of Fq2 -maximal curves with dim D = 3. For more details, see Section 10.6. In the characterisation, the idea is to show that an Fq2 -maximal curve X coming from its Frobenius divisor lies on a non-degenerate Hermitian variety of PG(r, K). This is demonstrated under the hypothesis that X coincides with its dual curve up to the Frobenius collineation (X0 , . . . , Xr ) 7→ (X0q , . . . , Xrq ) of PG(r, K). When this hypothesis fails, a weaker result is obtained, namely that the dual curve lies

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on a degenerate Hermitian variety Hm,q of PG(r, K). However, it is still possible to recover from the dual curve a non-singular model X ′ of X lying on a nondegenerate Hermitian variety in some PG(m, K) with 2 ≤ m < r. The starting point is the rational transformation ω : x′0 = z0 , . . . , x′r = zr , with z0 , z1 , . . . , zr satisfying (10.10), (10.11), (10.12), which defines a morphism π ⋆ since the osculating hyperplanes of X at distinct points are distinct. (r−1) By (10.11), ρ ◦ π ⋆ is the Gauss map P 7→ LP , where ρ is the Frobenius collineation (X0 , . . . , Xr ) 7→ (X0q , . . . , Xrq ). So, consider the irreducible Fq2 rational curve X ′ = π ⋆ (X ) of PG(r, K). Note that X ′ is the dual curve of X up to ρ; see Section 7.9. Also, X ′ may be degenerate in the sense that it may be contained in a proper subspace Πm = PG(m, K) of PG(r, K). If this is the case and m is chosen to be minimal, then there is an (r − m)-dimensional subspace Π′r−m of PG(r, K) which is the intersection of the osculating hyperplanes to X at general points P ∈ X , except for finitely many points P . Here, no point of X lies in Π′r−m . To show this, let R ∈ Π′r−m and assume on the contrary that R ∈ X . Choose a point Q ∈ X such that Q 6= R but the osculating hyperplane LQ to X at Q contains Π′r−m . Since the common points of LQ and X are only two, namely Q and Φ(Q), it follows that Φ(Q) = R. Hence Q is uniquely determined by R. But this is a contradiction, as Q can be chosen in many different ways. 2 2 The Frobenius collineation (X0 , . . . , Xr ) 7→ (X0q , . . . , Xrq ) fixes Πm . Take a 2 projective frame in PG(r, q ) in such a way that Πm = v(Xm+1 , . . . , Xr ). Then zm+1 = 0, . . . , zr = 0 and the coordinate functions of π ⋆ : X → Πm are (z0 , . . . , zm ). Hence, by (10.11), the osculating hyperplane to X at Q is q LQ = v(γ0q X0 + · · · + γm Xm ),

where C = (γ0 , . . . , γm ) is the centre of the branch π ⋆ (Q). The linear series D′ cut out on X ′ by hyperplanes of Πm consists of all divisors Ac = div(c0 z0 + · · · + cm zm ) + E,

where c = (c0 , . . . , cm ) ∈ Πm and, as usual, P E= eP P, eP = − min{ordP z0 , . . . , ordP zm }.

L EMMA 10.16 (i) The irreducible Fq2 -rational curve X ′ = π ⋆ (X ) of Πm has degree q + 1. (ii) The Fq2 -rational linear series D′ cut out on X ′ by hyperplanes of Πm contains all divisors qP + Φ(P ) with P ∈ X .

Proof. Choose a point P0 = (α0 , . . . , αr ) ∈ X (Fq2 ). If αi = 0 for i = 0, . . . , m, then P0 lies in the osculating hyperplane at a general point of X ; with the above notation, P0 ∈ Π′r−m . But this has already been shown to be impossible. Therefore αi 6= 0 for some i with 0 ≤ i ≤ m.

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Now, consider the hyperplane, H = v(αq0 X0 + · · · + αqm Xm ),

of PG(r, K) which can also be regarded as a hyperplane of Πm . For a point P ∈ X , let C = (γ0 , . . . , γm ) be the centre of the branch π ⋆ (P ) of X ′ . If C ∈ H then q αq0 γ0 + · · · + αqm γm = 0, and so γ0q α0 + · · · + γm αm = 0. By (10.11), the osculating hyperplane to X at P passes through P0 . Since P0 ∈ X (Fq2 ), this is only possible when P = P0 . Thus H ∩ X ′ contains no point other than the centre of π ⋆ (P0 ). The next step is to show that I(π ⋆ (P0 ), X ∩ H) = q + 1; that is, ordP0 (αq0 w0 + · · · + αqr wr ) = q + 1 ,

where wi = zi /zk ,

ordP0 (zk ) = min{ordP0 (z0 ), . . . , ordP0 (zr )}.

(10.15)

After an Fq2 -linear transformation of PG(r, K), let P0 = (1, 0, . . . , 0). Then it must be shown that ordP0 (w0 ) = q + 1. An Fq2 -linear transformation may be chosen so that f0 = 1,

ordP0 fi < ordP0 fi+1 ,

for i = 0, . . . , r − 1.

(10.16)

Then the unique branch γ of X centred at P0 has a primitive representation

x0 (t) = 1, x1 (t) = a1 t + · · · , xk (t) = ak tjk + · · · , xr (t) = ar tq+1 + · · · ,

where (0, 1, . . . , jk , . . . , q + 1) is the (D, P0 )-order sequence of X . If ρ is a primitive representation of the place arising from γ, then (10.12) gives the following: w0 (t) = −{w1 (t)(a1 t + · · · )q + · · · + wr (t)(ar tq+1 + · · · )q } .

(10.17)

Therefore it must be shown that both a1 6= 0 and ordt w1 (t) = 1. From (10.10), w0 (t)q + w1 (t)q (a1 t + · · · ) + · · · + wk (t)q (ak tjk + · · · ) + · · · + wr (t)q (ar tq+1 + · · · ) = 0.

Also, from the definition of wi , it follows that ordt wi (t) = 0 for at most one index i. Since 1 = j1 < j2 < · · · < jr = q + 1 and jr−1 < q, the only possibility is i = r, whence the assertion follows. Thus, (q + 1)π ⋆ (P0 ) is obtained as the intersection divisor of X ′ with a hyperplane of Πm . Therefore deg X ′ = q + 1. Also, a1 6= 0 implies that the branch π ⋆ (P ) of X ′ is linear. This argument applied to a point P ∈ X \X (Fq2 ) in place of P0 shows that the hyperplane H cuts out the divisor qπ ⋆ (P ) + Φ(π ⋆ (P )) on X ′ . 2

By Lemma 10.16, the invariants jm , ǫm of X ′ are equal to the corresponding invariants jr , ǫr of X . In the proof of Lemma 10.16, every branch of X ′ is linear. From that proof, zm+1 = 0, . . . , zr = 0, since Πm = v(Xm+1 , . . . , Xr ). In particular, (10.10) and (10.12) become the following: q z0q f0 + · · · + zm fm = 0,

q z0 f0q + · · · + zm fm = 0.

(10.18)

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The osculating hyperplane at the branch point π ⋆ (P ) of X ′ is v(f0 (P )q X0 + · · · + fr (P )q Xr ).

(10.19)

Therefore the proof of Theorem 10.7 may be used without alteration to prove that X ′ has no singular points. This, together with Theorem 10.7, gives the following result. T HEOREM 10.17 Every Fq2 -maximal curve is isomorphic over Fq2 to both the following curves: (i) the curve X arising from its Frobenius divisor; (ii) the dual curve X ′ of X . R EMARK 10.18 All proofs in this section remain valid for g = 0, that is, for the projective line ℓ = v(Y ) over Fq2 . The Frobenius divisor D of ℓ has dimension r = q + 1 and consists of all effective divisors of degree q + 1. The corresponding curve X is the normal rational curve of PG(q + 1, K) defined by the coordinate functions fi = xi with i = 0, . . . , q + 1 and K(F) = K(x). Given any point P (a) = (1, a, . . . , aq+1 ) ∈ X , the hyperplane v(aq

2

+q

2

X0 − aq X1 − aq Xq + Xq+1 )

of PG(q + 1, K) cuts out the divisor qP + Φ(P ) on X . To show this, note that 2 Φ(P (a)) = P (aq ), and that x0 (t) = 1, . . . , xi (t) = (a + t)i , . . . , xq+1 (t) = (a + t)q+1 is a primitive representation of the unique branch centred at P (a). Then the result follows from the identities aq

2

+q

2

2

− aq (a + t) − aq (a + t)q + (a + t)q+1 = tq (a − aq + t) , aq

2

+q

2

2

2

− aq aq − aq (aq )q + (aq+1 )

q2

=0.

Therefore the smallest linear series containing all divisors qP + Φ(P ) with P ∈ X is cut out by the three-dimensional linear system of all hyperplanes v(u0 X0 + u1 X1 + uq Xq + uq+1 Xq+1 ); that is, m = 3 in this case. For q > 2, it follows that m < r. No other such example is known to exist. The next step is to show that X ′ lies on a non-degenerate Hermitian variety of Πm . Since D′ is a linear subseries of the complete linear series D, Theorem 6.17 implies the following result. L EMMA 10.19 There exists a non-zero element η ∈ K(X ) and a non-singular matrix C = (cij ) with entries in K(X ) such that Pr (10.20) ηzi = j=0 cij fj , for i = 0, 1, 2, . . . , r.

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Since both D and D′ are Fq2 -rational linear series, from the proof of Theorem 6.17 it also follows that η ∈ Fq2 (X ) and that the entries in C are elements of Fq2 . Note that cij = 0 for m + 1 ≤ i ≤ r and the matrix C has rank m + 1. Without loss of generality, let f0 = 1, z0 = 1. (10.21) Then, from (10.20) with i = 0, η = 1, c00 = 1, c0j = 0 for j = 1, . . . , r. L EMMA 10.20 If (10.21) holds, then the matrix C in Lemma 10.19 is Hermitian; that is, cqij = cji for all i, j. Proof. The method is to substitute zi in (10.12) by (10.20) and re-write the result in the form Pr Pr Pr Pr 1 + i=1 ( j=0 cij fj )fiq = 1 + j=0 ( i=1 cij fiq )fj Pr Pr = 1 + i=1 ( j=0 cji fjq )fi = 0. This shows that the osculating hyperplane HP of X at a general point P is Pr Pr v(X0 + j=0 (cj1 fjq (P ))X1 + · · · + j=0 (cjr fjq (P ))Xr ). Pr Pr But then ziq (P ) = j=0 (cji fjq (P )) for i = 1, . . . , r. Hence ziq − j=0 cji fjq has an infinite number of zeros. By Theorem 5.33, for i = 1, . . . , r, P ziq = rj=0 cji fjq . Pr On the other hand, (10.20) implies that ziq = j=0 cij fjq . Since f1 , . . . , fr are linearly independent over K, the same holds for f1q , . . . , frq . Therefore cij = cqji for all i, j with 0 ≤ i, j ≤ r. 2 L EMMA 10.21 Up to a projectivity over Fq2 , the curve X ′ lies on the non-degenerate Hermitian variety Hm,q of Πm : q+1 Hm,q = v(X0q+1 + · · · + Xm ),

(10.22)

Proof. From Lemma 10.20, cij = 0 for i = m + 1, . . . , r implies cij = 0 for j = m + 1, . . . , r. Therefore (10.20) becomes the following: Pr i = 0, 1, 2, . . . , m, cij = cqji . (10.23) zi = j=0 cij fj , Here, the (m + 1) × (m + 1) matrix C = (cij ) is non-singular and Hermitian. Its inverse matrix D is also Hermitian; hence P fi = rj=0 dij zj , i = 0, 1, 2, . . . , m, dij = dqji . (10.24) Substitution of these fi in (10.12) gives Pm q i,j=0 dij zi zj = 0, which shows that X ′ lies on the hypersurface Pm H′ = v( i,j=0 dij Xiq Xj ) of Πm , with dij = dqji and dij ∈ Fq2 . The result follows since H′ is a nondegenerate Hermitian variety projectively equivalent over Fq2 to Hm,q . 2 Therefore the following result is obtained.

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T HEOREM 10.22 (Natural Embedding Theorem) Every Fq2 -maximal curve X of genus g ≥ 0 is isomorphic over Fq2 to a curve of PG(m, K) of degree q + 1 lying on a non-degenerate Hermitian variety Hm,q defined over Fq2 . R EMARK 10.23 The dimension m is less then or equal to the dimension r of the Frobenius linear series D of X . In particular, if r = 2, then also m = 2 and X is isomorphic over Fq2 to the Hermitian curve Hq . R EMARK 10.24 From the proof of Theorem 10.22, the osculating hyperplane of X ′ at any point P ∈ X ′ coincides with the tangent hyperplane at P to the nondegenerate Hermitian variety Hm,q in which X ′ lies. As any point of PG(m, K) is the common point of only finitely many tangent hyperplanes of Hm,q , the same holds for the osculating hyperplanes at X ′ . Therefore the Fq2 -maximal curve X in the Natural Embedding Theorem 10.22 has no higher-dimensional strangeness. The Natural Embedding Theorem 10.22 and Castelnuovo’s Bound (7.69) give the following result. C OROLLARY 10.25 Let X be an Fq2 -maximal curve of genus g. If the dimension of the Frobenius linear series of X is equal to r, then  (2q − (r − 1))2 − 1   , if r is even;  8(r − 1) g ≤ c0 (q + 1, r) ≤ F (r) = (10.25) (2q − (r − 1))2    , if r is odd. 8(r − 1)

It may be noted that F (r) ≤ F (s) for s ≤ r. Thus, F (r) ≤ F (3) ≤ 41 (q − 1)2 . Next it is shown that the property given in Theorem 10.22 characterises Fq2 maximal curves. For this purpose, assume from now on that Y is an Fq2 -rational irreducible, possibly singular, non-degenerate curve of PG(m, K) having the following property: deg Y = q + 1 and Y ⊂ Hm,q .

(10.26)

q+1 f0q+1 + · · · + fm = 0.

(10.27)

Then the aim is to prove that Y is an Fq2 -maximal curve. It is also shown that Y is non-singular and that the linear series cut out on Y by hyperplanes is contained in the Frobenius linear series of Y. Since Y is not necessarily non-singular, points of Y are considered as branch points; that is, a point P of Y indicates the branch of Y associated to the place P of K(Y). This convention comes from Section 7.6. Alternatively, choose a nonsingular model of Y given by an Fq2 -rational irreducible non-singular curve Z that is birationally equivalent to Y over Fq2 . Then there exists a morphism defined over Fq2 , and the branches of Y arise from points of Z. If f0 , . . . , fM are the coordinate functions of Y with f0 , . . . , fM ∈ Fq2 (Y), the hypothesis (10.26) means that For P ∈ Y, a primitive branch representation of P is

x0 = f0 (t), . . . , xm = fm (t).

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Let Q = (a0 . . . , am ) be the centre of P , Then P∞ fi (t) = j=0 aij tj ∈ K[[t]], a00 = a0 , . . . , am0 = am . Hence (10.27) implies that Pm P∞ q+1 = 0. j=0 aij ) i=0 ( The tangent hyperplane HQ to Hm,q at Q = (a0 , . . . , am ) is v(aq0 X0 + · · · + aqm Xm ). The first step is to show that Lemma 10.7 (ii) holds for Y. L EMMA 10.26 The linear series R cut out on Y by hyperplanes contains the divisor qP + Φ(P ) for every P ∈ Y. Proof. It is shown that HQ cuts out the divisor qP + Φ(P ) on Y. From (10.27), P∞ P∞ (10.28) ( j=0 a0j tj )q f0 + · · · + ( j=0 amj tj )q fm = 0 . Writing out the terms of lower order in t, Pm q q+1 q q+2 q+1 Pm q Pm g(t) = 0 . i=0 ai1 + t i=0 ai0 ai1 + t i=0 ai0 fi + t Hence the intersection number I(P, H ∩ Y) is at least q and equality holds if and Q Pm only if i=0 aqi1 ai0 6= 0. Pm Now it is shown that, if P ∈ Y(Fq2 ), then i=0 aqi1 ai0 = 0. From (10.28), Pm q+1 Pm q q i=0 ai0 + t j=0 ai0 ai1 + t h(t) = 0 . Pm q Thus i=0 ai0 ai1 = 0. Since P Pm q q q ( m i=0 ai0 ai1 ) = i=0 ai0 ai1 when P ∈ Y(Fq2 ), the assertion follows. Since deg(Y) = q + 1, this implies that HQ cuts out the divisor (q + 1)P on Y. Therefore the lemma is established for every P ∈ Y(Fq2 ). For the case P 6∈ Y(Fq2 ), it must also be shown that the image point of Q q ) lies in HQ . This under the Frobenius collineation (X0 , . . . , Xm ) 7→ (X0q , . . . , Xm occurs when Pm q2 +q = 0. i=0 ai Since this relation is a consequence of (10.28), the result follows. Therefore HQ cuts out the divisor qP + Φ(P ) on Y for every P 6∈ Y(Fq2 ). 2 Now, Lemma 10.26 makes it possible to use the proof of Theorem 10.7 to show that Y is an Fq2 -rational irreducible non-singular curve. If m = 2, then Y is the Hermitian curve and so Y is Fq2 -maximal. Therefore let m ≥ 3. The approach used for the proof is based on the relationship between the Wronskian determinant of Y and the projection Y¯ of Y to an (m − 1)-dimensional subspace of PG(m, K). More precisely, let π : PG(m, K) → Πm−1 , (10.29) (f0 , . . . , fm ) 7→ (f0 , . . . , fm−1 ), be the projection Y¯ of Y from the point Um = (0, . . . , 0, 1) to the hyperplane v(Xm ). It may happen that Y and Y¯ are not birationally equivalent over Fq2 . However, it is always possible to avoid this situation by changing the coordinate system in PG(m, K).

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L EMMA 10.27 (i) In PG(m, K) there is a point P with coordinates in Fq2 that satisfies each of the following conditions: (a) P ∈ / Hm,q ; (b) no tangent line to Y at an Fq2 -rational point passes through P ; (c) no chord through two Fq2 -rational points of Y passes through P. (ii) If ℓ is an Fq2 -rational line through an Fq2 -rational point R of Y, then each point of ℓ ∩ Y is an Fq2 -rational point of ℓ. Proof. (i) Take an Fq2 -rational point Q ∈ Y. Since the number of Fq2 -rational points of Y is at most q 2 + 1 + 2gq ≤ q 3 + 1, there are at most q 3 chords joining Q to another Fq2 -rational point of Y. But, since m ≥ 3, the number of Fq2 -rational lines through Q is at least q 4 + q 2 + 1 and hence one of these lines is neither a line contained in Hm,q , nor a tangent line to Y at Q, nor a chord through Q and another Fq2 -rational point of Y. Now, any Fq2 -rational point P outside Hm,q is a suitable choice for P . (ii) Assume on the contrary that ℓ meets Y in a non-Fq2 -rational point S of ℓ. Then ℓ is the line joining S and the image point S ′ of S under the Frobenius 2 q2 ). Therefore ℓ is contained in the collineation (X0 , . . . , Xm ) 7→ (X0q , . . . , Xm osculating hyperplane of Y at S. Hence the common points of ℓ and Y are only two, namely S and S ′ . But this contradicts the hypothesis that R ∈ ℓ ∩ Y. 2 Take a point P as in (i) of this lemma. Since the projective group PGU(m+1, q) preserving Hm,q acts transitively on the set of all points of PG(m, q 2 )\Hm,q , it follows that a projectivity of PG(m, q 2 ) exists which preserves Hm,q and takes P to the point Um = (0, . . . , 0, 1). Lemma 10.27 now ensures that Y and Y¯ are birationally equivalent over Fq2 . Choose a separable variable ζ of K(Y), and consider the Wronskian matrices of type investigated in Section 7.6:   Dζǫ0 f0 Dζǫ0 f1 ... Dζǫ0 fm−1   .. .. .. W(f0 , . . . , fm−1 ) = det  , . . ... . ǫm−1 ǫm−1 ǫm−1 fm−1 f 1 . . . Dζ f0 Dζ Dζ   ǫ0 ǫ0 ǫ0 Dζ f 0 Dζ f 1 . . . Dζ f m   .. .. .. (10.30) W(f0 , . . . , fm ) = det  ; . . ... . ǫm ǫm ǫm Dζ f 0 Dζ f 1 . . . Dζ f m (i)

here Dζ is the i-th Hasse derivative with respect to a separable variable ζ in K(Y). From Lemma 10.26, ǫ0 = 0, ǫ1 = 1, ǫm = q. So (10.30) becomes the following:   f0 f1 ... fm  Dζ f0 Dζ f1 ... Dζ f m     Dζǫ2 f0 Dζǫ2 f1 ... Dζǫ2 fm    W(f0 , . . . , fm ) = det   . (10.31) .. .. ..   . . . . . .  ǫm−1  ǫ ǫ  Dζ f0 Dζm−1 f1 . . . Dζm−1 fm  Dζq f0 Dζq f1 ... Dζq fm

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L EMMA 10.28 div(W(f0 , . . . , fm )) q = div(W(f0 , . . . , fm−1 )) − q div(fm ) + div(f0 Dζq f0q + . . . + fm Dζq fm ).

Proof. If the columns of the matrix in (10.31) are c0 , c1 , . . . , cm , for cm put q q f0q c0 + f1q c1 + · · · + fm−1 cm−1 + fm cm .

Then q fm W(f0 , . . . fm )  f0  Dζ f0  = det  .  ..

f1 Dζ f1 .. .

Dζq f0

Dζq f1

... ... ··· ...

fm−1 Dζ fm−1 .. .

Dζq fm−1

q+1 f0q+1 + . . . + fm q q f0 Dζ f0 + . . . + f m Dζ fm .. .

q f0q Dζq f0 + . . . + fm Dζq fm



  . 

Each element in the last column is 0 apart from the last: this follows from (10.27) by derivation. Also, the q-th Hasse derivative of the same relation gives the relation q q Dζ fm + f0 Dζq f0q + · · · + fm Dζq fm = 0, f0q Dζ f0 + · · · + fm

and this completes the proof. 2 P Let Rm = vP (Rm )P be the ramification divisor of the linear series cut out on Y by hyperplanes of PG(m, K). Let P ∈ Y and choose a local parameter ζ at P as a separable variable. Then vP (Rm ) = ordP (W(f0 , . . . , fm )) .

(10.32)

Similarly, let Rm−1 be the ramification divisor of the linear series cut out on Y by hyperplanes of PG(m, K). L EMMA 10.29 If P ∈ Y and ζ is a local parameter of Y at P, then vP (Rm−1 ) = ordP (W(f0 , . . . , fm−1 )) .

Proof. By Definition 7.52, vP (Rm−1 ) = ordP (W(f0 , . . . , fm−1 )) + (ǫ0 + ǫ1 + · · · + ǫm−1 )ordP (dζ) + m¯ eP , where e¯P = − min{ordP (f0 ), . . . , ordP (fm−1 )}.

Here, e¯P = 0. In fact, if e¯P > 0 and eP = 0, then Um = (0, . . . , 0, 1) lies on Y, contradicting (10.27). Since ζ is a local parameter at P , so ordP (dζ) = 0, and the result follows. 2 L EMMA 10.30 ordP (f0 Dζq f0q

+ ··· +

q fM Dζq fM )

=

(

1 0

when P ∈ Y(Fq2 ), when P 6∈ Y(Fq2 ).

Proof. The proof of Lemma 10.26 also provides a proof of the following result. (i) For any point P ∈ Y,

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(a) P 6∈ Y(Fq2 ) if and only if (b) P ∈ Y(Fq2 ) if and only if On the other hand,

Pm

i=0

Pm

i=0

aqi1 am1 6= 0; aqi1 am1 = 0 and

Pm

i=0

aq+1 6= 0. i1

q f0 Dζq f0q + · · · + fm Dζq fm P∞ P∞ = ( j=0 a0j tj )(aq01 + tq g1 (t)) + · · · + ( j=0 amj tj )(aqm1 + tq g2 (t)) .

This implies the following. (ii)

q ) = 0 if and only if (a) ordP (f0 Dζq f0q + · · · + fm Dζq fm

P

ai0 aqi1 6= 0; P q (b) ordP (f0 Dζq f0q + · · · + fm Dζq fm ) = 1 if and only if ai0 aqi1 = 0 and P q+1 ai,1 6= 0.

Now, comparison of (i) with (ii) proves Lemma 10.30.

2

T HEOREM 10.31 Every irreducible curve Y defined over Fq2 satisfying (10.26) is a non-singular Fq2 -maximal curve. Proof. By (7.13), P vP (Rm ) = (ǫ0 + ǫ1 + · · · + ǫm )(2g − 2) + (m + 1)(q + 1) , P vP (Rm−1 ) = (ǫ0 + ǫ1 + · · · + ǫm−1 )(2g − 2) + m(q + 1) .

Hence

P

(vP (Rm ) − vP (Rm−1 )) = q(2g − 2) + q + 1. P ordP (fM ) = q + 1, finish 2

Lemmas 10.28, 10.29, 10.30, (10.32) and the relation, the proof.

R EMARK 10.32 In (10.26) the condition ‘degree q + 1’ may be replaced by ‘minimum degree with respect to the property of lying on a non-degenerate Hermitian variety of PG(m, K) defined over Fq2 ’. To show this, let f0 , f1 , . . . , fM be the coordinate functions of an irreducible non-degenerate curve Γ of PG(m, K) lying on the Hermitian variety Hm,q in PG(m, K); then (10.27) holds. If (x0 = f0 (t) = a0 + h0 (t), . . . , xm = fm (t) = am + hm (t)),

ordP hi (t) ≥ 1,

is a primitive representation of a branch of Γ centred at the point Q = (a0 , . . . , am ), then (10.27) implies that ordt (aq0 f0 (t) + · · · + aqm fm (t)) = ordt (h0 (t)q f0 (t) + · · · + hm (t)q fm (t)),

whence

ordt (aq0 f0 (t) + · · · + aqm fm (t)) ≥ q.

This shows that the tangent hyperplane HQ = v(aq0 X0 + . . . + aqm Xm ) to Hm,q at Q meets Y at the branch γ with multiplicity at least q. If P 6∈ PG(r, q 2 ), then its image 2

2

P ′ = (aq0 , . . . , aqm )

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2

q ) is another under the Frobenius collineation (X0 , . . . , Xm ) 7→ (X0q , . . . , Xm q+1 q+1 common point of Γ and HP . In fact, a0 + . . . + am = 0 implies that 2

2

aq0 aq0 + . . . + aqm aqm = 0. Since Y is a non-degenerate curve, it is not contained in HQ . Thus Y must have degree at least q + 1. As an illustration of this material, it is shown how the known examples of Fq2 maximal curves with r = m = 3 are embedded in a non-degenerate Hermitian surface of PG(3, K). In this way, an independent proof of the maximality of these curves is obtained. E XAMPLE 10.33 Let q ≡ 2 (mod 3), and fix a primitive cube root of unity ǫ in Fq2 . For i = 0, 1, 2, let Fi = v(Fi ) be the Fq2 -rational irreducible plane curve with Fi (X, Y ) = ǫi X (q+1)/3 + ǫ2i X 2(q+1)/3 + Y q+1 , and let K(x, y), with Fi (x, y) = 0, be its function field. Let Γi be the Fq2 -rational irreducible curve of PG(3, K), birationally equivalent over Fq2 to Fi , defined by the coordinate functions, f0 = x, f1 = x2 , f2 = y 3 , f3 = xy . Note that the three curves Γi are projectively equivalent in P G(3, K). In fact, the projectivity over Fq2 induced by the matrix   i ǫ 0 0 0  0 ǫ2i 0 0  (i)  T4 =  0 0 1 0 0 0 0 ǫi maps Γ0 to Γj with j ≡ 2i (mod 3). Now, it is shown that Γi is an Fq2 -rational irreducible non-singular curve of degree q + 1 lying, up to a projectivity, on the Hermitian surface H3,q = v(X0q+1 + X1q+1 + X2q+1 + X3q+1 ).

(10.33)

To do this, note that the classical identity a3 + b3 + c3 − 3abc = (a + b + c)(a + ǫb + ǫ2 c)(a + ǫ2 b + ǫc) , with a = Y q+1 ,

b = X (q+1)/3 ,

c = X 2(q+1)/3 ,

becomes the identity (X (q+1)/3 + X 2(q+1)/3 + Y q+1 ) ×(ǫX (q+1)/3 + ǫ2 X 2(q+1)/3 + Y q+1 )

×(ǫ2 X (q+1)/3 + ǫX 2(q+1)/3 + Y q+1 ) = X q+1 + X 2(q+1) + Y 3(q+1) − 3X q+1 Y q+1 .

(10.34)

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This yields the equation xq+1 + x2(q+1) + y 3(q+1) − 3xq+1 y q+1 = 0,

whence f0q+1 + f1q+1 + f2q+1 − 3f3q+1 = 0. Therefore Γi lies on H3,q , up to the projectivity (x0 , x1 , x2 , x3 ) 7→ (x0 , x1 , x2 , wx3 ),

wq+1 = −3.

Also, Γi is contained in the cubic surface S3 = v(X33 + w3 X0 X1 X2 ) ⊂ PG(3, K). More precisely, the intersection of H3,q and S3 splits into the three curves, Γ0 , Γ1 , Γ2 , each of degree q + 1. By Theorem 10.31, each Γi is a non-singular maximal curve defined over Fq2 . From Theorem 12.28 (IV)(b), its genus is 61 (q 2 − q + 4). E XAMPLE 10.34 For a similar but non-isomorphic example, again let q ≡ 2 (mod 3), and fix a primitive cube root of unity ǫ ∈ Fq2 . For i = 0, 1, 2, let Fi′ = v(Fi′ ) be the Fq2 -rational irreducible plane curve with Fi′ (X, Y ) = ǫi Y X (q−2)/3 + Y q + ǫ2i X (2q−1)/3 , and K(x, y) with Fi′ (x, y) = 0 be its function field. Let Γ′i be the Fq2 -rational irreducible curve of PG(3, K), birationally equivalent to Fi′ over Fq2 , with coordinate functions f0 = x, f1 = x2 , f2 = y 3 , f3 = −3xy.

The three curves Γ′i are projectively equivalent in PG(3, K). In fact, the projectiv(i) ity induced by the matrix T4 in Example 10.33 maps Γ′0 to Γ′i . Arguing as in Example 10.33, it is shown now that Γ′i is an Fq2 -rational irreducible non-singular curve of degree q + 1 lying on H3,q as in (10.33). From the identity (10.34), with a = Y q,

b = Y X (q−2)/3 ,

c = X (2q−1)/3 ,

it follows that y 3 xq−2 + y 3q + x2q−1 − 3xq−1 y q+1 = 0, whence y 3 xq + y 2q x2 + x2q+1 − 3xq+1 y q+1 = 0.

Hence f2 f0q + f2q f1 + f1q f0 − 3f3q+1 = 0 and this shows that Γ′i lies on the surface ′ Sq+1 = v(X0q X1 + X1q X2 + X2q X0 − 31 X3q+1 ).

Also, Γ′i lies on the cubic surface

S3 = v(X33 + 27X0 X1 X2 ).

′ Hence the intersection of Sq+1 and S3 splits into the three curves, Γ′0 , Γ′1 , Γ′2 , each of degree q + 1.

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′ To prove that Sq+1 is projectively equivalent to H3,q , choose a root α of the 2 q+1 polynomial X + X + 1. Then αq +q+1 = 1, and hence α ∈ Fq3 . From this,

αq+1 + αq but αq+2 + αq

2

+1

2

+q+1

+ α = 0,

αq

2

+q+2

+ αq+1 + 1 = 0;

2

+ αq 6= 0 as (αq+2 + αq +1 + αq )q−1 = α−1 . Also, the matrix   2 α 1 αq +1   2 M3 = αq +1 α 1  2 1 αq +1 α

is non-singular. Choose an element µ ∈ Fq2 satisfying the equation −3µq+1 = aq

3

+q+1

+ αq

2

+1

+ αq ,

and define T to be the projectivity of PG(3, K) associated to the non-singular matrix   2 α 1 αq +1 0 2  1 αq +1 α 0  . M4 =  αq2 +1 α 1 0  0 0 0 −µ ′ Then T−1 maps Sq+1 to H3,q , and S3 to the cubic surface S3∗ = v(F3 ), with

F3 = (X03 + X13 + X23 ) + T(aq+1 )(X02 X1 + X12 X2 + X22 X0 ) +T(α)(X02 X2 + X12 X0 + X22 X1 ) +(3 + T(αq−1 ))X0 X1 X2 − αq−1 µ3 X33 , 2

where T(u) = u + uq + uq is the trace of u ∈ Fq3 over Fq . Also, αq−1 µ3 ∈ Fq2 , and this shows that S3∗ is defined over Fq2 . Now, Γ′i is mapped by T−1 to an Fq2 rational irreducible curve Ci′ of degree q + 1 lying on H3,q . By Theorem 10.31, Ci′ is a non-singular Fq2 -maximal curve. From Theorem 12.28 (V) with d = 3, its genus is 16 (q 2 − q − 2); see also Section 7.11. i E XAMPLE 10.35 Let q be odd. For i = 0, 1 and m = 21 (q + 1), let Em = v(Ei ) be the Fq2 -rational irreducible plane curve, with

Ei (X, Y ) = Y q + Y + (−1)i X (q+1)/2 and K(x, y) with Ei (x, y) = 0 its function field. ′′ Let Γi be the Fq2 -rational irreducible curve of PG(3, K), birationally equivalent to Ei over Fq2 , with coordinate functions f0 = 1, f1 = x, f2 = y, f3 = y 2 .

′′

′′

The curves Γ1 and Γ0 are projectively equivalent, since the projectivity induced by the matrix   1 0 0 0 0 ǫ 0 0  T4 =  0 0 1 0 , 0 0 0 1

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′′

with ǫ(q+1)/2 = −1, maps Γ0 to Γ1 . The polynomial identity (Y q + Y − X (q+1)/2 )(Y q + Y + X (q+1)/2 ) = Y 2q + 2Y q+1 + Y 2 − X q+1 implies that y 2q + 2y q+1 + y 2 − xq+1 = 0 and so f3q + f3 + 2f2q+1 − f1q+1 = 0. ′′ This shows that Γi lies on the surface H′ = v(X3q X0 + X3 X0q + 2X2q+1 − X1q+1 ), ′′

which is a non-degenerate Hermitian surface of PG(3, K). Also, Γi lies on the quadric cone Q = v(X22 − X0 X3 ), ′′ ′′ and hence the intersection of H′ and Q splits into the curves Γ0 and Γ1 . By The′′ orem 10.31, Γi is a non-singular Fq2 -maximal curve. From Theorem 10.41, its genus is 14 (q − 1)2 ; see also Theorem 12.28 (I).

E XAMPLE 10.36 Let q = 2t , and put T(Y ) = Y + Y 2 + · · · + Y q/4 + Y q/2 . For each i = 0, 1 ∈ F2 ⊂ Fq2 , let Ci = v(Ti ) be the Fq2 -rational irreducible plane curve with Ti (X, Y ) = T(Y ) + X q+1 + i, and K(x, y) with Ti (x, y) = 0 its function field. It may be noted that C0 = T2 , as in Example 10.3 (ii). Let Φi be the Fq2 -rational irreducible curve of PG(3, K), birationally equivalent to Ci over Fq2 , with coordinate functions f0 = 1, f1 = x, f2 = y, f3 = x2 . Since (T(Y ) + X q+1 )(T(Y ) + X q+1 + 1) = Y q + Y + X q+1 + X 2q+2 , q so y + y + xq+1 + x2q+2 = 0. Therefore Φi lies on the non-degenerate Hermitian variety H′′ = v(X2q X0 + X2 X0q + X1q+1 + X3q+1 ). Also, the quadric cone Q = v(X0 X3 − X12 ) contains Φi . Hence H′′ ∩ Q splits into Φ0 and Φ1 . Note that the curves Φ0 and Φ1 are projectively equivalent over Fq2 in PG(3, K), and hence both have degree q + 1. Again, by Theorem 10.31, they are non-singular Fq2 -maximal curves and, from Theorem 10.45, of genus 41 q(q − 2); see also Theorem 12.28 (II) (a) for p = 2. E XAMPLE 10.37 Let q = 3t , and put T(Y ) = Y + Y 3 + . . . + Y q/3 . For each i = 0, 1, 2 ∈ F3 ⊂ Fq2 , let Ci = v(Ti′ ) be the Fq2 -rational irreducible plane curve, with Ti′ (X, Y ) = T(Y )2 − X q − X + i(T(Y ) + i) , and let K(x, y) with Ti′ (x, y) = 0 be its function field. Also, C0 = T3′ , as in Example 10.3. From the identity (T(Y )2 − X q − X) ×(T(Y )2 + T(Y ) + 1 − X q − X)

×(T(Y )2 − T(Y ) + 1 − X q − X) = (Y q − Y )2 − (X q + X)(X q + X − 1)2 ,

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it follows that (x3 + x2 − y 2 + x)q + (x3 + x2 − y 2 + x) − xq+1 − y q+1 = 0 .

Let Φ′i be the Fq2 -rational irreducible curve of PG(3, K), birationally equivalent to Ci over Fq2 , with coordinate functions f0 = 1, f1 = x, f2 = y, f3 = x3 + x2 − y 2 + x .

As in the preceding examples, the three curves, Φ′0 , Φ′1 , Φ′2 , are Fq2 -projectively equivalent in PG(3, K). Also, Φ′i lies on the non-degenerate Hermitian variety H∗ = v(X0 X3q + X0q X3 − X1q+1 − X2q+1 ). Finally, the cubic surface S3′ = v(X3 X02 − X13 + X12 X0 + X22 X0 − X1 X02 )

also contains Φ′i . Therefore Φ′i has degree q + 1, and Theorem 10.31 ensures that each Φ′i is a non-singular Fq2 -maximal curve. From Theorem 12.28 (II)(b), their genus is 61 q(q − 1). 10.4 MAXIMAL CURVES LYING ON A QUADRIC SURFACE The Natural Embedding Theorem 10.22 offers a geometric approach to the study of Fq2 -maximal curves. The idea is to use the specific properties of the model of Fq2 -maximal curves described in that theorem and the successive three remarks. To do this, consider the following. C ONDITION 10.38 An Fq2 -maximal curve is an Fq2 -rational, irreducible, nonsingular curve of degree q + 1 lying on a non-degenerate Hermitian variety of PG(m, K), where 2 ≤ m ≤ r = dim D with D the Frobenius linear series of X . As in many geometric problems, low dimensions can be tackled successfully. Nevertheless, it is not apparent which of the low-dimensional results are of general type; this is the main difficulty in deciding what happens in higher dimensions. Since m = 2 only occurs when X is a Hermitian curve, the smallest dimension to consider is m = 3. Examples 10.35 and 10.36 show that there are Fq2 -maximal curves of PG(3, K) lying on a quadric. The aim of this section is to show that no other Fq2 -maximal curve of PG(3, K) has this property. Therefore let dim D = m = 3. Then the Frobenius linear series D of X is the linear series cut out on X by hyperplanes. For a point P ∈ X , the positive D-orders are j1 (P ) = 1 < j2 (P ) < j3 (P ), where j3 (P ) =

(

q + 1, when P is Fq2 -rational, q, when P is not Fq2 -rational.

(10.35)

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L EMMA 10.39 Let dim D = 3 and suppose that X lies on a quadric surface Q of PG(3, K). If q ≥ 4, then the following properties hold for any point P ∈ X : (i) j2 (P ) ∈ {2, 21 j3 (P ), 12 (j3 (P ) + 1)};

(ii) j2 (P ) > 2 if and only if the tangent line ℓP of X at P lies on Q; (iii) if j2 > 2 and P is a non-singular point of Q, then ( 1 when q is even and P is not Fq2 -rational, 2 q, j2 (P ) = 1 (q + 1), when q is odd and P is Fq2 -rational; 2 (iv) if there exists P ∈ X with j2 (P ) > 2, then Q is a cone. Proof. Let the quadric Q = v(F ), with F (X0 , X1 , X2 , X3 )

= a00 X02 + a01 X0 X1 + a02 X0 X2 + a03 X0 X3 + a11 X12 +a12 X1 X2 + a13 X1 X3 + a22 X22 + a23 X2 X3 + a33 X32 . First it is shown that Q is defined over Fq2 . Otherwise Q is distinct from its image Q′ under the Frobenius collineation 2

2

2

2

(X0 , X1 , X2 , X3 ) 7→ (X0q , X1q , X2q , X3q ).

(10.36)

F (1, f1 , f2 , f3 ) = 0,

(10.37)

Hence X would be in the intersection of two distinct quadrics, and deg X = q + 1 would be less than 5 by Corollary 7.9, contradicting the hypothesis that q ≥ 4. The coordinate functions f0 , f1 , f2 , f3 ∈ Fq2 (X ) of X may be chosen such that ordP (fi ) = ji (P ) for i = 0, 1, 2, 3. The hypothesis on X to lie on Q implies that F (f0 , f1 , f2 , f3 ) = 0, by Theorem 7.5. After a change of coordinates in PG(3, K), let the point P be a vertex of the projective frame, say P = (1, 0, 0, 0), so that the osculating plane at P is v(X3 ) and the tangent line to X at P is v(X2 , X3 ); see the proof of Theorem 7.47. Also, let f0 = 1. To use the equation the orders at P of some other elements from Fq2 (X ), namely, f12 , f1 f2 , f1 f3 , f22 , f2 f3 , f32 , are required. By direct computation, these orders are 2j1 , j1 + j2 , j1 + j3 , 2j2 , j2 + j3 , 2j3 .

(10.38)

Hence (10.37) together with P ∈ Q implies that a00 = a01 = 0. To show (i), note that j2 + 1 < j3 by (VIII) and (XI). So, from (10.38) and the inequalities, 2 ≤ j2 < j2 + 1 < j3 < j3 + 1 < j3 + j2 < 2j3 ,

(i) follows. Also, (10.38) shows that j2 > 2 if and only if a11 = 0. Now, as F (X0 , X1 , 0, 0) = a11 X12 , the last condition is satisfied if and only if the tangent line ℓP to X at P is contained in the quadric Q. From this, (ii) follows.

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Also, if j2 > 2, from (i), a11 = a02 = a12 = 0. Now, P is a non-singular point of Q if and only if a03 6= 0. Therefore 2j2 = j3 , and (iii) follows from the fact that j1 (P ) = 1 and (10.35). To show (iv), note that if P is a non-singular point of Q then Q is a cone whose vertex is the point V = (−a13 , a03 , 0, 0). Otherwise, P is singular and hence Q is a cone with vertex P . 2 L EMMA 10.40 Let q ≥ 4 and dim D = 3. Then (i) dim(2D) ≥ 8; (ii) if dim(2D) = 8, the curve X lies on a quadric surface in PG(3, K); (iii) the quadric surface in (ii) is uniquely determined and is defined over Fq2 . Proof. (i) Let P ∈ X (Fq2 ). Then m2 (P ) = q and m3 (P ) = q + 1 by (IV) and (XIV). Now, as 2m1 (P ) ≥ m2 (P ) = q and q ≥ 4, it follows that there are at least eight non-gaps in the interval [m1 (P ), 2m3 (P )]. Hence dim 2D ≥ 8. (ii) Geometrically, the linear system of all quadrics has dimension 9 and the linear series L cut out on X by quadrics is contained in 2D. So if dim(2D) ≥ 8 then L = 2D and some quadrics must contain X . (iii) See the first part in the proof of Lemma 10.39. 2 T HEOREM 10.41 Let q be odd with q ≥ 5, and let X be an Fq2 -maximal curve of genus g = 41 (q − 1)2 . Then X is birationally equivalent over Fq2 to the Fq2 1 rational irreducible plane curve E(q+1)/2 = v(E1 ) in Example 10.35, with E1 (X, Y ) = Y q + Y − X (q+1)/2 .

(10.39)

Proof. From Lemma 10.39, j2 (P ) = 2 at each point P ∈ X which is not Fq2 rational while the Fq2 -rational points P of X are of two types: (I) j2 (P ) = 2; (II) j2 (P ) = 21 (q + 1). In particular, the set of all D-Weierstrass points of X coincides with X (Fq2 ). Since the D-orders are 0, 1, 2, q, the weight of P in the ramification divisor R can be calculated from Theorem 7.55:  1  2 (q − 1), when P ∈ X (Fq2 ) and j2 (P ) = 12 (q + 1), 1, when P ∈ X (Fq2 ) and j2 (P ) = 2, vP (R) =  0, when P 6∈ X (Fq2 ).

Let n1 and n2 denote the numbers of Fq2 -rational points of type (I) and (II). From (7.13), 1 2 (q

+ 1)n2 + n1 = (q + 3)( 12 (q − 1)2 − 2) + 4(q + 1).

Since X is Fq2 -maximal,

n1 + n2 = q 2 + 1 + 12 (q − 1)2 q.

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From these two equations, n1 = q + 1 and, in particular, n1 ≥ 4. Now, the configuration of the points of type (I) is investigated. Such points are not collinear, since no non-degenerate irreducible curve of PG(3, K) of degree k contains k collinear points. By Lemma 10.39 (iv), these points lie on a cone Q with vertex T . From the proof of that lemma, the tangent line ℓP to X at a point P of type (I) is contained in Q. Therefore every line joining T to a point of type (I) is a generator of Q. If P and P ′ are distinct points of type (I), then ℓP does not contain ℓP ′ . Hence T is not a point of type (I). Also, T lies in every plane which osculates X at a point of type (I). Another property of a point P of type (I), established implicitly in the proof of Lemma 10.39, is as follows. L EMMA 10.42 The following three planes coincide: (a) the osculating plane πP of X at P ; (b) the tangent plane αP to the Hermitian surface H3,q at P ; (c) the tangent plane βP to the cone Q at P . A first consequence is that T 6∈ H3,q , as an Fq2 -rational line ℓ of α(P ) which passes through P either lies in H3,q or meets H3,q only in P . The group PGU(4, q) that fixes H3,q acts transitively on the points of PG(3, q 2 )\H3,q . Therefore a projective frame over Fq2 may be chosen so that H3,q = v(X0q X2 + X0 X2q + 2X1q+1 + X3q+1 ),

T = (0, 0, 0, 1).

(10.40)

The polar plane Π of T under the unitary polarity of H3,q is v(X3 ). So Π cuts out on H3,q the non-degenerate Hermitian curve H2,q = v(X0q X2 + X0 X2q + 2X1q+1 ),

and on Q an Fq2 -rational irreducible conic C. Another consequence of Lemma 10.42 is that C and H2,q have the same tangent at every point P of type (I). From B´ezout’s Theorem 3.14, the common points of C with H2,q are exactly the points of type (I). This shows that C and H2,q are in permutable position with respect to the two polarities arising from C and H2,q . Given a non-degenerate Hermitian curve, the irreducible conics in permutable position with it form a unique orbit under the action of the group PGU(3, q 2 ) fixing the Hermitian curve. The Fq2 -rational irreducible conic C ′ = v(X0 X2 + X12 ) provides an example of such a conic. There exists a projectivity (S) of Π preserving H2,q that takes C to C ′ . Also, δ can be lifted to an element (S)′ of PGU(4, q 2 ); then (S)′ fixes T . Hence, the cone Q may be taken as Q = v(X0 X2 + X12 ).

Now, let A = (a0 , a1 , a2 , a3 ) be any point of X . Then aq0 a2 + a0 aq2 + 2aq+1 + aq+1 = 0, 1 3

(10.41) a0 a2 = a21 .

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If a0 = 0 then A = (0, 0, 1, 0). Otherwise, let a0 = 1; then these equations imply that = 0, (aq2 + a2 )2 + aq+1 3 (q+1)/2

(q+1)/2

whence either aq2 + a2 + a3 = 0 or aq2 + a2 − a3 = 0. Comparison with Example 10.35 shows that H3,q ∩ Q is Γ1 ∪ Γ2 . From Exercise 9 in Chapter 7, either X = Γ1 or X = Γ2 . Since Γ1 and Γ2 are isomorphic over Fq2 , the result follows. 2 Theorem 10.41 together with Corollary 10.8 gives the case that m = 12 (q + 1) of the following general result. T HEOREM 10.43 Let X be an Fq2 -maximal curve of genus g, and suppose that there is a point P ∈ X (Fq2 ) with a non-gap m dividing q + 1. Then X is birat1 ionally equivalent over Fq2 to the curve Em = v(E1 ), with E1 (X, Y ) = Y q + Y − X m .

(10.42)

P ROPOSITION 10.44 Let q be even with q > 4, and let dim D = 3. If X lies on a quadric Q in PG(3, K), then (i) Q is a cone; (ii) the vertex V of Q belongs to X ; (iii) V ∈ X (Fq2 ) and j2 (V ) = 21 (q + 2). Proof. The following properties of quadrics of PG(3, K) are used, where, for a non-singular point P of Q, let βP be the tangent plane to Q at P : (a) if P ∈ X , then βP ⊃ ℓP ; (b) if ℓ and ℓ1 are distinct lines such that P ∈ ℓ ⊂ Q and ℓ1 ⊂ βP , then βP is spanned by ℓ and ℓ1 ; (c) there exist lines ℓ and ℓ1 such that P ∈ ℓ ∩ ℓ1 and Q ∩ βP = ℓ ∪ ℓ1 ; (d) if Q is non-singular, then no two tangent hyperplanes of Q at different points coincide. (i) Since X is non-degenerate, Q is irreducible. Also, Q is a cone if and only if Q is singular. Therefore let Q be non-singular, that is, a hyperbolic quadric of PG(3, K). From Lemma 10.39 (iii), j2 (Q) = 2 for each Q ∈ X (Fq2 ). Note that there exists P 6∈ X (Fq2 ) such that j2 (P ) > 2. Otherwise, by Lemma 10.15, 6(g − 1) = (q + 1)(q − 3);

but then q would be odd, a contradiction. Hence j2 (P ) = 10.39 (iii). Let Q1 ∈ X (Fq2 ). Then Q1 6∈ ℓP since X ∩ ℓP ⊂ X ∩ πP = {P, Φ(P )},

1 2q

again by Lemma

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and hence the plane H = HQ1 spanned by ℓP and Q1 is well defined. Thus H 6= πP , and the intersection divisor X · H is given by X · H = 21 P + D ,

(10.43)

Q ∩ H = ℓP ∪ ℓ .

(10.44)

where D = DQ1 is a divisor on X of degree 21 (q + 2) with Q1 ∈ Supp(D) and P 6∈ Supp(D). In addition, Lemma 10.39 (ii) ensures the existence of a line ℓ = ℓQ1 such that The line ℓ is defined over Fq2 as is Q by Lemma 10.40 (iii); the fact that Q1 is on X (Fq2 ) but not on ℓP implies that Q1 ∈ ℓ. Now it is shown that X ∩ ℓ ⊂ X (Fq2 ).

(10.45)

If there exists Q ∈ X ∩ ℓ\X (Fq2 ), then Φ(Q) ∈ ℓ as ℓ is defined over Fq2 . Thus ℓ ⊂ πQ , and hence ℓ ∩ X ⊂ {Q, Φ(Q)}. Therefore Q1 6∈ ℓ; but this is a contradiction. (A) If Q ∈ Supp(D)\{Φ(P )}, then Q ∈ X (Fq2 ) and nQ (D) = 1. To prove (A) note that from (10.45) and the relation Supp(D)\{Φ(P )} ⊂ ℓ ∩ X , it follows that Q ∈ X (Fq2 ). Now, if nQ (D) ≥ 2, then H ⊃ ℓQ since j2 (Q) = 2 and by (IV). Also, ℓ 6= ℓQ because ℓQ 6⊂ Q by Lemma 10.39 (ii). Therefore the plane H is spanned by the lines ℓ and ℓQ , and hence H is the tangent plane πQ1 to Q at Q1 . If ℓ1 is the line defined by the property that Q1 ∈ ℓ1 , then Q ∩ πQ1 = ℓ ∩ ℓ1 . From (10.44), ℓP = ℓ1 and so Q1 ∈ ℓP ; but this is a contradiction. (B) Φ(P ) 6∈ Supp(D). To prove (B) suppose, on the contrary, that Φ(P ) ∈ Supp(D). By (10.45), this assumption is equivalent to Φ(P ) ∈ ℓP . The argument in the proof of (A) can be used to deduce that H = πΦ(P ) since nΦ(P ) (D) 6= 1 and ℓP 6= ℓΦ(P ) . But this leads to a contradiction with (10.8). Therefore nΦ(P ) (D) = 1. Hence, for every Q ∈ X (Fq2 ), the divisor D in (10.43) may also be written as ′ D = DQ = Φ(P ) + DQ ,

and this may be done so that not only (10.44) remains valid but also that ′ Supp(DQ ) ⊂ X (Fq2 ),

′ deg(DQ ) = 21 q.

This gives the following: ′ (C) HQ is spanned by ℓP and Q′ , where Q′ is any point of Supp(DQ ). ′ Now, let Q1 , Q2 ∈ X (Fq2 ) such that Q2 6∈ Supp(DQ ). Then 1 ′ ′ ) = ∅; ) ∩ Supp(DQ Supp(DQ 2 1

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otherwise HQ1 = HQ2 by (C). So, 21 q must divide the number of Fq2 -rational points of X , which is a contradiction since |X (Fq2 )| = q 2 + 1 + 2gq is odd. So far it has been shown that each Q1 ∈ X (Fq2 ) gives rise to a plane HQ1 together with a line ℓ = ℓQ1 and a divisor D = DQ1 such that (10.43) and (10.44) hold, with D = Q1 + Q2 + · · · + Q(q+2)/2

being the sum of 21 (q + 2) points all Fq2 -rational. Note that Supp(D) = X ∩ ℓ. Let ℓ1 be chosen in such a way that Q1 ∈ ℓ1 and that Q ∩ πQ1 = ℓ ∪ ℓ1 .

(10.46)

Here ℓ1 is Fq2 -rational, and thus X ∩ ℓ1 ⊂ X (Fq2 ) as in the proof of (10.45). Therefore ′

X · πQ1 = 2Q1 + Q2 + · · · + Q(q+2)/2 + D′ , 1 2 (q−2)

where D is a divisor of X of degree Consider the following statement:

(10.47) ′

such that Q1 6∈ Supp(D ) ⊂ X (Fq2 ).

(D) Supp(D) ∩ Supp(D′ ) = ∅ and nS (D′ ) = 1 for each S ∈ Supp(D′ ). To show (D), let S ∈ Supp(D′ ). Suppose on the contrary that S = Qi for some i. Then πQ1 contains ℓQi which is different from ℓ as j2 (Qi ) = 2. Hence πQ1 is generated by ℓQi and ℓ. These lines also span πQi and so i = 1, contradicting that Q1 ∈ / Supp(D′ ). Finally suppose, on the contrary, that nS (D2 ) ≥ 2. Replacing ℓ by ℓ1 , the above argument shows that πS = πQ1 , whence S = Q1 , again a contradiction. Thus, to each Q1 there is associated two lines ℓ and ℓ1 such that both (10.46) and (10.47) hold, where D′ is a divisor of degree 21 (q − 2), where Supp(D′ ) ⊂ X (Fq2 ), and where Supp(D) ∩ Supp(D′ ) = {Q1 }. A hyperbolic quadric Q contains exactly two families of lines and any two lines of the same family are disjoint. This implies again that |X (Fq2 )| must be a multiple of 12 q, contradicting the Fq2 -maximality of X. (ii) As q is even, Lemma 10.15 ensures the existence of a point P ∈ X with j2 (P ) > 2. Suppose that P 6∈ X (Fq2 ); then j2 (Φ(P )) = j2 (P ) > 2. Since j2 (P )P + D ≡ (q + 1)P0 , j2 (P )Φ(P ) + Φ(D) ≡ (q + 1)P0 and so j2 (Φ(P )) = j2 (P ) > 2. Therefore both the lines ℓP and ℓΦ(P ) are contained in Q by Lemma 10.39 (ii), and hence their common point is V . Now, since V is Fq2 -rational by Lemma 10.40 (iii), so Φ(P ) 6= V , and hence ℓΦ(P ) is spanned by Φ(P ) and V . In particular, ℓΦ(P ) ⊂ πP and thus 1 = nΦ(P ) (X · πP ) ≥ j2 (Φ(P )), a contradiction. Therefore P must be Fq2 -rational and hence Q must have a singularity at P by Lemma 10.39 (iii). Then P = V and j2 (P ) = 21 (q + 2); these follow from Lemma 10.39 (i) and the assumption that q is even. 2

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T HEOREM 10.45 Let q be even with q ≥ 16, and let X be an Fq2 -maximal curve of genus g = 41 q(q − 2). Then X is birationally equivalent over Fq2 to the curve T2 = v(T2 ) in Example 10.36, with T2 (X, Y ) = Y q/2 + Y q/4 + · · · + Y 2 + Y + X q+1 .

(10.48)

Proof. From Castelnuovo’s Bound, Corollary 10.25, dim D ≤ 3. As dim D = 2 only occurs when X is a Hermitian curve, whose genus is 12 q(q−1), let dim D = 3. By Halphen’s Theorem 7.117, X lies on a quadric of PG(3, K). From (IV) and Proposition 10.44 (ii), there exists P0 ∈ X (Fq2 ) such that 12 q is a non-gap at P0 . By (XII), the same holds for q + 1. Therefore the Weierstrass semigroup H(P0 ) is generated by 12 q and q + 1. Choose x, y ∈ Fq2 (X ) such that div(x)∞ = 12 qP0 and div(y)∞ = (q + 1)P0 , and consider the Fq2 -rational linear series | 21 q(q + 1)P0 | of order 12 q(q + 1). Since 1 2 q(q

+ 1) > 2g − 2 = 12 q(q − 2) − 2,

the Riemann–Roch Theorem 6.61 implies that dim | 21 q(q + 1)P0 | = 14 q 2 + q. On the other hand, div(xq+1 )+ 12 q(q + 1)P0 , as well as each of the following 41 (q + 2)2 divisors, belongs to | 12 q(q + 1)P0 |: div(xj y i ) + 21 q(q + 1)P0 ,

0 ≤ i ≤ 21 q, 0 ≤ j ≤ q − 2i.

Therefore x and y must satisfy a non-trivial polynomial relation; that is, Pq/2−1 cy q/2 + i=0 Ai (x)y i = xq+1 ,

(10.49)

for c ∈ Fq2 \{0} and for Ai [X] ∈ Fq2 [X] with deg Ai (x) ≤ q − 2i. A lengthy calculation reduces (10.49) to the equation Pt q/2i + b = xq+1 , (10.50) i=1 ai y Pt i q/2 where ai ∈ Fq2 \{0} for all i, and the polynomial i=1 ai T + b has a root α ∈ Fq2 . Replacing y by y + α, (10.50) becomes the following: Pt q/2i = xq+1 . (10.51) i=1 ai y Replacing x by a−1 1 x and y by at−1 y give the desired equation: Pt q/2i = xq+1 . i=1 y

Therefore X is birationally equivalent to the Fq2 -rational irreducible plane curve T2 of (10.48). 2 10.5 MAXIMAL CURVES WITH HIGH GENUS

In this section, the aim is to show that the upper part of the spectrum of genera of Fq2 -maximal curves consists of the values of g in Table 10.2. Although the whole spectrum depends on the nature of q, the highest three genera in the spectrum have the same positions independently of q. Since the Hermitian curve Hq is Fq2 -maximal, its genus g = 21 q(q − 1) belongs to the spectrum, and is the maximum.

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P ROPOSITION 10.46 If X is an Fq2 -maximal curve of genus g, then g ≤ 21 q(q − 1).

Proof. Since N2 ≥ N1 , from (10.1),

q 4 + 1 − 2gq 2 ≥ q 2 + 1 + 2gq, q 4 − q 2 ≥ 2g(q 2 + q),

whence the result follows.

2

To present other results on the spectrum, let gi = gi (q) denote the i-th largest genus in the spectrum; then gi − gi−1

is the i-th hole or gap in the spectrum. Proposition 10.46 and the existence of Hq show that g1 = 12 q(q − 1). The following theorem implies the uniqueness of an Fq2 -maximal curve of genus g1 . Also, it shows that g2 = ⌊ 41 (q − 1)2 ⌋, as the curves (i) and (ii) in Example 10.3 have this value. T HEOREM 10.47 If X is an Fq2 -maximal curve of genus g, the following statements are equivalent: (i) X is isomorphic over Fq2 to the Hermitian curve Hq ; (ii) g > ⌊ 41 (q − 1)2 ⌋; (iii) r = 2; (iv) ν1 > 1. Proof. (i) ⇒ (ii), (iii), (iv) This follows from properties of Hq . (iii) ⇒ (iv) For r = 2, ν1 = νr−1 and the assertion follows from (IX). (iv) ⇒ (ii) Theorem 8.68 together with the Natural Embedding Theorem 10.22 imply that if ν1 > 1 then N1 = q 3 + 1, whence g = 21 q(q − 1). (ii) ⇒ (iii) From Corollary 10.25, F (3) = (q − 1)2 /4, and hence r = 2. (iii) ⇒ (i) This follows from the Natural Embedding Theorem 10.22. 2 The second hole in the spectrum is determined in the next result. T HEOREM 10.48 For q ≥ 7, let X be an Fq2 -maximal curve of genus g satisfying (10.38). Then the following conditions are equivalent: (i) ⌊ 16 (q 2 − q + 4)⌋ < g ≤ ⌊ 41 (q − 1)2 ⌋; (ii) dim D = 3, the curve X lies on a quadric surface in PG(3, K), and g 6= 16 (q 2 − 2q + 3),

for q ≡ 3, 5 (mod 6);

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(iii) dim D = 3, dim(2D) = 8, and

g 6= 61 (q 2 − 2q + 3),

for q ≡ 3, 5 (mod 6);

(iv) dim D = 3 and there exists P ∈ X (Fq2 ) such that ( 1 for q odd, 2 (q + 1) j2 (P ) = 1 for q even; 2 (q + 2) (v) X is birationally equivalent over Fq2 to the curve (i) or (ii) in Example 10.3 according as q is odd or even; (vi) g = 14 (q − 1)2 if q is odd and g = 14 q(q − 2) if q is even; in particular, the genus g = c0 (q + 1, 3), Castelnuovo’s number. Proof. (i) ⇒ (ii) From the hypothesis on g, Halphen’s Theorem 7.117 and (10.8) imply that dim D = 3 . Since, in Corollary 10.25, c0 (q+1, 3) = ⌊ 61 (q 2 −q+4)⌋, the Natural Embedding Theorem 10.22 together with Halphen’s Theorem shows that X lies on a quadric provided that either q ∈ / {7, 8, 9, 11, 13, 17, 19, 23} or p > 2 and X is reflexive. So, for q even, let q = 8. Then g > 61 (q 2 − q + 4) = 10. By Lemma 10.40 (i),(ii), it is enough to show that dim(2D) ≤ 8. Suppose, however, that dim(2D) ≥ 9. Then, by Lemma 7.113, g ≤ 14 (q − 1)(q − 2) = 21/2, a contradiction. Now, let q be odd with q ≥ 7, and suppose that X is not reflexive. By Theorem 7.118, ǫ2 > 2. If P ∈ X (Fq2 ), then, from Propositions 10.47 and 8.56, P3 vP (S) ≥ i=1 (ji (P ) − νi−1 ) ≥ ǫ2 + 1 ≥ 4. So the St¨ohr–Voloch Theorem applied to X implies that

(3q − 1)(2g − 2) ≤ (q + 1)(q 2 − 4q − 1).

On the other hand, 2g − 2 > 31 (q + 1)(q − 2) by hypothesis, and thus 5q + 5 < 0, a contradiction. (iii) ⇒ (ii) This follows from Lemma 10.40 (ii). (ii) ⇒ (iv) Let q be odd. Then there exists P ∈ X such that j2 (P ) > 2; otherwise g = 61 (q 2 − 2q + 3) by Proposition 10.15. If such a point P ∈ X were not in X (Fq2 ) then, by Lemma 10.39 (iii), both P and Φ(P ) would be singular points of the quadric, a contradiction. Therefore P ∈ X (Fq2 ) and hence j2 (P ) = 12 (q + 1) by Lemma 10.39 (i). If q is even, the result follows from Proposition 10.44 (ii). (iv) ⇒ (v) From (IV) and the hypothesis, m1 (P ) = 21 (q + 1) for q odd and m1 (P ) = 12 q for q even. In the odd case, q is another non-gap at P , and hence g = 41 (q − 1)2 by Corollary 10.8. A similar argument works in the even case; since g = 41 q(q − 2) and q + 1 is another non-gap at P , Corollary 10.8 applies. Finally, the implications (v) ⇒ (vi), (vi) ⇒ (i) and (v) ⇒ (iii) follow readily. 2 R EMARK 10.49 For q = 2, 3, 4, 5, the spectra of the genera of Fq2 -maximal curves are {0, 1}, {0, 1, 3}, {0, 1, 2, 6}, {0, 1, 2, 3, 4, 10}.

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10.6 CASTELNUOVO’S NUMBER In this section, certain Fq2 -maximal curves whose genus is Castelnuovo’s number c0 (q + 1, r) for 4 ≤ r ≤ 5 are considered. First the case q ≡ 1, 2 (mod 3) is investigated. In this case, c0 (q + 1, 4) = 16 (q − 1)(q − 2). The main result, see Theorem 10.54 below, provides a complete classification of Fq2 -maximal curves of genus g = 16 (q − 1)(q − 2), q ≡ 1, 2 (mod 3), q ≥ 11. Such Fq2 -maximal curves can only exist for q ≡ 2 (mod 3), and they are birationally equivalent over Fq2 to the Fq2 -rational curve 1 E(q+1)/3 = v(Y q + Y − X (q+1)/3 ) . The proof of Theorem 10.54 requires three preliminary lemmas.

(10.52)

L EMMA 10.50 For q ≡ 1, 2 (mod 3), let X be an Fq2 -maximal curve of genus g = 61 (q − 1)(q − 2). Then dim D = 4 and g = c0 (q + 1, 4). Proof. From (10.8) and Theorem 10.47, 3 ≤ r ≤ 4. Suppose on the contrary that r = 3. If ǫ2 = 2, then deg R = (3 + q)(2g − 2) + 4(q + 1), while the Fq2 -maximality of X implies that deg R ≥ q 2 + 1 + 2gq, since vP (R) ≥ 1 for every P ∈ Fq2 (X ). But then g ≥ 61 (q 2 −2q+3), contradicting the hypothesis on g. If ǫ2 > 2, then ǫ2 ≥ 5 by Lemma 7.63 and since q 6≡ 0 (mod 3). Replacing the ramification divisor R by the St¨ohr–Voloch divisor S, the previous argument again yields a contradiction. In fact, deg S = (1 + q)(2g − 2) + (q 2 + 3)(q + 1), while deg S ≥ (q 2 + 1 + 2gq)(ǫ2 + 1), by the Fq2 -maximality of X and the lower bound vP (S) ≥ ǫ2 + 1 for P ∈ Fq2 (X ), shown in the proof of Theorem 10.48. Since ǫ2 ≥ 5, so (5q − 1)(2g − 2) ≤ (q + 1)(q 2 − 6q − 3), 2 whence 2q − 3q + 13 ≤ 0 for g = 61 (q − 1)(q − 2), a contradiction. 2 The assumption that the genus of X is Castelnuovo’s number c0 (q + 1, 4) is used via Lemma 7.113 (i). Indeed, from this lemma, dim(2D) = 11 and hence the possibilities for (D, P )-orders can be determined. To show how to do this, let ji = ji (P ) and denote by ∆P the set of (2D, P )-orders. Then ∆P contains both sets ∆1 = {0, 1, 2, j3, j4 , j4 + 1, j4 + j2 , j4 + j3 , 2j4 }, (10.53) ∆2 = {j2 , j2 + 1, j3 + 1, 2j2 , j3 + j2 , 2j3 }, where j4 = q + 1 for P ∈ X (Fq2 ), and j4 = q otherwise.

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L EMMA 10.51 Let P ∈ X with j2 (P ) = 2. If r = dim D = 4, dim(2D) = 11, and q ≥ 9, then j3 (P ) = 3. Proof. The hypothesis on q together with Proposition 10.15 implies that  j4 − 2 for P ∈ X (Fq2 ), j3 < j4 − 1 for P ∈ X \X (Fq2 ).

(10.54)

Suppose j3 > 3. If P ∈ X (Fq2 ), from (10.53) and (10.54), ∆P = Σ1 ∪ {3, j3 + 1, j3 + 2} , and 2j2 , 2j3 ∈ ∆P . Hence j3 = 2j2 = 4, so that 2j3 = 8 = j4 = q + 1; thus q = 7. If P ∈ / X (Fq2 ) and j3 > 4, from (10.53) and (10.54) it follows that ∆P = ∆1 ∪ {3, 4, j3 + 1},

(j3 + 2, 2j3 ) ∈ {(q, q + 1), (q, q + 2), (q + 1, q + 2)}. Then j3 ≤ 4, a contradiction. Finally, if P 6∈ X (Fq2 ) and j3 = 4, then (10.53) together with (10.54) show that ∆P = ∆1 ∪ {3, 5, 6, 8} . Hence j4 = q = 8, and this completes the proof.

2

The previous lemma together with Lemma 10.15 gives the following result. C OROLLARY 10.52 If q ≥ 9, dim(D) = 4, dim(2D) = 11, j2 (P ) = 2 for any 1 P ∈ X , then q ≡ 1, 2 (mod 3) and g = 12 (q 2 − 3q + 8). Next, the case that j2 (P ) > 2 for some point P ∈ X is considered. L EMMA 10.53 Let P ∈ X with j2 (P ) > 2, and suppose that dim D = 4, dim(2D) = 11, q ≥ 7. (i) If P ∈ X (Fq2 ) and g > holds: (a) q ≡ 2 (mod 3), (b) q ≡ 0 (mod 3),

1 8 q(q

− 2) for q even, then one of the following

j2 (P ) = 31 (q + 1), j3 (P ) = 31 (2q + 2); j2 (P ) = 31 (q + 3), j3 (P ) = 31 (2q + 3).

(ii) If P ∈ / X (Fq2 ), then one of the following holds: (a) q ≡ 1 (mod 3), (b) q ≡ 0 (mod 3), (c) q ≡ 1 (mod 2), (d) q ≡ 0 (mod 2),

j2 (P ) = 13 (q + 2), j3 (P ) = 31 (2q + 1); j2 (P ) = 31 q, j3 (P ) = 23 q; j2 (P ) = 12 (q − 1), j3 (P ) = 21 (q + 1); j2 (P ) = 21 q, j3 (P ) = 12 (q + 2).

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Proof. Suppose first that j3 > j2 + 1. From (10.53) and (10.54), there are only three possibilities, namely, ∆P = ∆1 ∪ {j2 , j2 + 1, j3 + 1},

(j3 + j2 , 2j3 ) ∈ {(j4 , j4 + 1), (j4 , j4 + j2 ), (j4 + 1, j4 + j2 )}. The first cannot occur since j3 6= j2 + 1; in the second case,

j4 ≡ 0 (mod 3), j2 = 31 j4 , j3 = 23 j4 ;

in the third case, j4 ≡ 1 (mod 3), j2 = 31 (j4 + 2), j3 = 13 (2j4 + 1). Suppose next that j3 = j2 + 1. Then 2j2 6∈ {j3 , j3 + 1} since j2 > 2. Further, either 2j2 6= j4 + 1 or j2 = 21 (j4 + 1), j3 = 21 (j4 + 3). In the latter case, from (10.53) and (10.54), ∆P = ∆1 ∪ {j2 , j3 + 1, j4 + 2, j4 + 3}. But this implies that j4 + j2 = j4 + 3, whence j4 = 5 and so q ≤ 5. If 2j2 = j4 , then either P 6∈ X (Fq2 ) or j3 = 12 (q + 3). Again, the latter case cannot actually occur because then m1 (P ) = 12 (q − 1) by Proposition 10.6 (IV), and this would imply that dim(D) ≥ 5. Finally, assume that 2j2 6∈ {j3 , j3 + 1, j4 , j4 + 1}. Then, from (10.53) and (10.54), ∆P = {j2 , j3 + 1, 2j2 }, and j3 + j2 ∈ {j4 , j4 + 1}. If j3 + j2 = j4 + 1, then 2j2 = j4 , whence j3 + j2 = j4 . Then j2 = 12 (j4 − 1) and j3 = 21 (j4 + 1). Here P 6∈ X (Fq2 ); otherwise, j2 = 21 q, j3 = 12 (q + 2) and hence m1 (P ) = 12 q, m2 (P ) = 21 (q + 2) again by (IV). But then 2 g ≤ 81 q(q − 2), a contradiction. T HEOREM 10.54 Let q ≥ 11. (i) If q ≡ 1 (mod 3), there is no Fq2 -maximal curve of genus 61 (q − 1)(q − 2). (ii) If q ≡ 2 (mod 3), the statements below are equivalent for an Fq2 -maximal curve X of genus g : (a) g = 16 (q − 1)(q − 2);

(b) there exist P ∈ X (Fq2 ) and m ∈ H(P ) such that 3m = q + 1;

(c) X is birationally equivalent over Fq2 to the curve in (10.52).

Proof. (i) Suppose on the contrary that X is an Fq2 -maximal curve of genus g = 13 (q − 1)(q − 2),

1 3 (q − 1) · 3 + 2,

q ≡ 1 (mod 3).

Since q + 1 = it follows from Lemma 10.50 that g = c0 (q + 1, 3). Hence, Lemma 7.113 implies that dim(2D) = 11 and that | 13 (q − 4)D| is the canonical linear series on X . Then 1 + a1 i1 + . . . + a(q−4)/3 i(q−4)/3 6∈ H(P ) ,

(10.55)

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where the integersPij are (D, P )-orders, and the coefficients aj are non-negative integers such that j aj ≤ (q − 4)/3. Let P ∈ X with j2 (P ) > 2 as in Corollary 10.52. By Lemma 10.53, P is not in X (Fq2 ). Recall that m3 (P ) = q by (XIV). So, three cases are considered. Case (A): j2 (P ) = 13 (q + 2), j3 (P ) = 31 (2q + 1). From (XI), {q − m2 (P ), q − m1 (P )} ⊂ {1, 31 (q + 2), 31 (2q + 1)}.

This implies that q − m1 (P ) = 31 (q + 2), since otherwise m1 (P ) = 31 (q − 1) and hence q ≥ m4 (P ), a contradiction. Thus m1 (P ) = 13 (2q − 2). But this again leads to a contradiction as (10.55) implies that 31 (q − 7) + 31 (q + 2) + 1 = 31 (2q − 2) does not belong to H(P ). Case (B): q ≡ 1 (mod 2), j2 (P ) = 12 (q − 1), j3 (P ) = 12 (q + 1). From (10.55), 2j2 (P ) + 1 = q does not belong to H(P ), a contradiction. Case (C): q ≡ 0 (mod 2), j2 (P ) = 21 q, j3 (P ) = 12 (q + 2).

The argument used in Case (A) shows this time that either m1 (P ) = 12 q − 1 or m1 (P ) = 21 q. In the former case, q − 2 ∈ H(P ) and thus (XI) implies that j2 (P ) = 2. Since this contradicts the hypothesis, only the latter case can occur. Then m1 (P ) = 21 q and m2 (P ) = q − 1. Now, as dim(2D) = 11, so m9 (P ) = 2q. From this, 2q − m4 (P ) = 21 q + 2. In fact, 2q − mi (P ) is a (2D, P )-order for i = 0, . . . , 9, and the set of (2D, P )-orders is {0, 1, 2, 21 q, 21 (q + 2), 12 (q + 4), q, q + 1, q + 2, 23 q, 2q} .

Hence m4 (P ) = 23 q − 2. Finally, let d = 31 (q − 4)( 12 q + 1) + 1. From (10.55), d ∈ / H(P ). On the other hand, d = m4 (P ) + 16 (q − 10)m2 (P ) ∈ H(P ), a contradiction. (ii) (a) ⇒ (b) From Lemma 10.50, g = c0 (q+1, 4). Since q+1 = 31 (q−2)·3+3, Lemma 7.113 shows that dim(2D) = 11 and that 31 (q − 5)D + D′ is the canonical linear series, where D′ is a base-point-free 1-dimensional linear series of order 1 1 3 (q + 1). In particular, there exists x ∈ K(X ) with [K(X ) : K(x)] = 3 (q + 1). Let P ∈ X and assume by Corollary 10.52 that j2 (P ) > 2. If P ∈ X (Fq2 ), the result follows from Lemma 10.53 (i). Otherwise, P ∈ / X (Fq2 ), and there are two possibilities according as q is odd or even; see Lemma 10.53 (ii). Case (A): q ≡ 1 (mod 2), j2 (P ) = 12 (q − 1), j3 (P ) = 12 (q + 1). A property similar to (10.55) holds; namely, δ + 1 6∈ H(P ) for any ( 31 (q − 5)D, P )-order δ.

Hence 2j2 (P ) + 1 = q ∈ / H(P ), a contradiction.

Case (B): q ≡ 0 (mod 2), j2 (P ) = 12 q, j3 (P ) = 12 (q + 2).

From Case (C) in the proof of (i), it also follows that m1 (P ) = 12 q. Let y be an element of K(X ) such that div(x)∞ = 12 q. Since gcd( 31 (q + 1), 21 q), so K(X ) = K(x, y). From Theorem 5.61, g ≤ 16 (q − 2)2 , a contradiction.

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Finally, (b) ⇒ (c) is Theorem 10.43 for m = 13 (q + 1), and (c) ⇒ (a) follows immediately. 2 Now, Fq2 -maximal curves of genus 61 q(q − 3) are considered in the case, r = 4,

q = 3h .

Note that c0 (q + 1, 4) = 61 q(q − 3). The main result, see Theorem 10.57 below, states that such Fq2 -maximal curves are birationally equivalent over Fq2 to the Fq2 -rational curve T3 = v(Y q/3 + Y q/9 + · · · + Y 3 + Y − X q+1 ).

(10.56)

The proof of Theorem 10.57 depends on two preliminary lemmas. L EMMA 10.55 For q = 3h , let X be an Fq2 -maximal curve of genus 61 q(q − 3). Then r = dim D is either 3 or 4. If r = 4, the following hold: (i) dim(2D) = 11; (ii) there exists a complete linear series D′ of order 32 q and dimension 2 such that 13 (q − 6)D + D′ is the canonical linear series on X ; (iii) if j2 = 2, then j3 = 3; (iv) if P ∈ X (Fq2 ) and j2 > 2, then j2 = 31 (q + 3), and j3 = 13 (2q + 3); in particular, the Weierstrass semigroup at P is generated by 31 q and q + 1; (v) if q ≥ 27 and P 6∈ X (Fq2 ), then j2 = 2. Proof. This can be done using arguments and calculations similar to those in the preceding proofs for the case q ≡ 1, 2 (mod 3). 2 L EMMA 10.56 For q ≥ 27, there exists P ∈ X (Fq2 ) such that H(P ) is generated by 31 q and q + 1. Proof. By Lemma 10.55 (ii), it is enough to show the existence of P ∈ X (Fq2 ) with j2 (P ) > 2. Suppose on the contrary that j2 (P ) = 2 for every P ∈ X (Fq2 ). If P ∈ X (Fq2 ) then j4 (P ) = q + 1, and this, together with Lemma 10.55 (iii), implies that the (D, P )-orders are 0, 1, 2, 3, q + 1. Likewise, if P 6∈ X (Fq2 ), then j4 (P ) = q and, from this and Lemma 10.55 (v), the (D, P )-orders are 0, 1, 2, 3, q. Therefore, by Corollary 8.58, deg(R) = (6 + q)(2g − 2) + 5(q + 1) = |X (Fq2 )| = (q + 1)2 + q(2g − 2) , whence 2g − 2 = 16 (q − 4)(q + 1); that is, q 2 − 3q − 8 = 0, a contradiction.

2

T HEOREM 10.57 For q = 3h , let X be an Fq2 -maximal curve of genus 16 (q − 3)q. If dim D 6= 3, then dim D = 4 and X is birationally equivalent over Fq2 to the Fq2 -rational curve T3 in (10.56).

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Proof. From Lemma 10.55, r = dim D = 4. Let P ∈ X (Fq2 ) be as in Corollary 10.56. Then the Weierstrass semigroup H(P ) is generated by 13 q and q + 1. Choose x, y ∈ Fq2 (X ) for which div(x)∞ = 13 qP and div(y)∞ = (q + 1)P . Then the Frobenius linear series D consists of all divisors Ac = div(c0 + c1 x + c2 x2 + x3 x3 + c4 y) + (q + 1)P,

with c = (c0 , c1 , c2 , c3 , c4 ) ∈ PG(4, K). Since ordP (xj y i ) ≥ − 31 q(q + 1) + 1 unless either (j, i) = (q + 1, 0) or (j, i) = (0, 31 q), the complete linear series | 13 q(q + 1)P )| comprises the divisors Pq/3 Pq−3i Bb = div(b0 xq+1 + i=0 j=0 bij xj y i ) + 31 q(q + 1)P,

where b = (b0 , . . . , bij , . . .) ∈ PG(m, K) with m = 61 (q 2 + 5q) + 1. By the Riemann–Roch Theorem 6.61, dim | 13 q(q + 1)P )| = 61 (q 2 + 5q). Therefore Pq/3 (10.57) xq+1 + i=0 Ai (x)y = 0,

where Ai (X) ∈ Fq2 [X] with deg Ai (X) ≤ q − 3i and, in particular, Aq/3 (X) is a non-zero constant belonging to Fq2 . Finally, some ad hoc arguments depending on higher Hasse derivatives and changes of coordinates reduce (10.57) to the simpler (10.56). 2 Now, consider the case g = 81 (q − 1)(q − 3),

q odd.

Up to a birational transformation over Fq2 , Theorem 10.63 below states that, for p ≥ 5 and q large enough, the two curves 1 = v(Y q + Y − X (q+1)/4 ) , F = E(q+1)/4

F = D(q+1)/2 = v(X

(q+1)/2

+Y

(q+1)/2

q ≡ 3 (mod 4) ,

+ 1)

(10.58) (10.59)

are the only Fq2 -maximal curves of genus g = 81 (q − 1)(q − 3) when dim D = 5. It should be noted that the condition on dim D enters into play since, from the proof of Lemma 10.50, it can only be deduced that 4 ≤ dim D ≤ 5, whereas 1 8 (q − 1)(q − 3) is Castelnuovo’s number for dim D = 5 but not for dim D = 4. So the condition, dim D = 5, is necessary to take advantage of Lemma 7.113. It may also be noted that the two curves are not birationally equivalent even over K since their Weierstrass semigroups are different; see Exercises 6 and 7 in Chapter 6. T HEOREM 10.58 When q ≥ 11, the Fermat curve in (10.59) is the unique Fq2 maximal non-singular plane curve of degree 21 (q + 1). As dim(2D) = 14 by Lemma 7.113 (i), the possibilities for the (D, P )-order sequences at P ∈ X can be determined. The two relevant results are stated below but their proofs are omitted as being similar to those of Lemmas 10.51 and 10.53, and Corollary 10.52. L EMMA 10.59 For P ∈ X , (i) j1 (P ) = 1;

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(ii) j5 (P ) = q + 1 if P ∈ X (Fq2 ) and j5 (P ) = q if P ∈ X \X (Fq2 ). L EMMA 10.60 For q ≥ 11 and P ∈ X , let dim(D) = 5, dim(2D) = 14. (i) If j3 (P ) = 3, then j4 (P ) = 4. (ii) Let j2 (P ) = 2, j3 (P ) > 3; (a) if P ∈ X (Fq2 ), then q is odd, j3 (P ) = 21 (q + 1), j4 (P ) = 21 (q + 3);

(b) if P 6∈ X (Fq2 ), then q is even, j3 (P ) = 21 q, j4 (P ) = 21 (q + 2). (iii) If P ∈ X (Fq2 ), j2 (P ) > 2, and ( 1 2 9 (q − 2) , for q ≡ 2 g> 1 9 q(q − 3), for q ≡ 0

(mod 3), (mod 3),

then one of the following holds: (a) q ≡ 3 (mod 4), j2 (P ) = 41 (q + 1), j3 (P ) = 21 (q + 1), j4 (P ) = 43 (q + 1); (b) q ≡ 0 (mod 4), j2 (P ) = 41 (q + 4), j3 (P ) = 21 (q + 2), j4 (P ) = 41 (3q + 4). (iv) If P 6∈ X (Fq2 ), j2 (P ) > 2, then one of the following holds: (a) q ≡ 1 (mod 4), j2 (P ) = 41 (q + 3), j3 (P ) = 21 (q + 1), j4 (P ) = 41 (3q + 1); (b) q ≡ 0 (mod 4), j2 (P ) = 41 q, j3 (P ) = 12 q, j4 (P ) = 43 q; (c) q ≡ 1 (mod 3), j2 (P ) = 31 (q − 1), j3 (P ) = 13 (q + 2), j4 (P ) = 31 (2q + 1);

(d) q ≡ 0 (mod 3), j2 (P ) = 31 q, j3 (P ) = 13 (q + 3), j4 (P ) = 32 q. C OROLLARY 10.61 If q ≥ 11, dim(D) = 5, dim(2D) = 14, and j3 (P ) = 3 for 1 (q 2 − 4q + 15). every P ∈ X , then q ≡ 0, 4 (mod 5) and g = 20 C OROLLARY 10.62 If, for q ≥ 11 with q odd, X is an Fq2 -maximal curve of genus g = 81 (q − 1)(q − 3) with dim(D) = 5, then the following hold: 1 (i) X is birationally equivalent over Fq2 to the curve E(q+1)4 in (10.58) if and only if there exists P ∈ X (Fq2 ) with j2 (P ) > 2;

(ii) X is birationally equivalent over Fq2 to the curve D(q+1)/2 in (10.59) if and only if there exists P ∈ X (Fq2 ) with j2 (P ) = 2, j3 (P ) > 3. Proof. (i) From Lemma 12.1, the curve F in (10.58) has a unique branch centred at Y∞ = (0, 0, 1). This branch is Fq2 -rational and hence it corresponds to a point P ∈ X (Fq2 ). By a straightforward computation, m3 (P ) = 43 (q + 1). Hence j2 (P ) = 41 (q + 1) by (IV). Conversely, from Lemma 10.60 (iii), j4 (P ) = 43 (q + 1)

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and so m1 (P ) = 14 (q + 1) by (IV). Therefore the result follows from Theorem 10.43. (ii) The Fermat curve F in (10.58) is non-singular. By Theorem 6.51, the linear 5 system of conics cuts out on F a complete linear series gq+1 . (q+1)/2 The point P0 = (1, α, 0) of F, with α = −1, is an inflexion and the tangent F at P is ℓ = v(αX0 − X1 ). Therefore I(P0 , F ∩ ℓ) = 21 (q + 1). Hence I(P0 , F ∩ C) = q + 1, where C = v((αX0 − X1 )2 ) is the conic consisting of the 5 repeated line ℓ. Thus gq+1 contains the Fq2 -rational divisor (q + 1)P , and hence 5 D = gq+1 by Theorem 6.34. Choose a line ℓ1 through P and a line ℓ2 disjoint from P . Then ( 2 when the conic C ′ = ℓ21 , ′ I(P0 , F ∩ C ) = 1 ′ 2 (q + 1) when the conic C = ℓℓ2 . Therefore j2 (P ) = 2, j3 (P ) > 3. Conversely, (IV) and Lemma 10.60 (ii) imply that m1 (P ) = 21 (q − 1), and m2 (P ) = 21 (q + 1). Choose x, y ∈ K(X ) such that div(x)∞ = 12 (q − 1) and div(y)∞ = 21 (q + 1). Since these numbers are coprime, K(X ) = K(x, y) and there exists f ∈ Fq2 [X, Y ] such that deg f ≤ 21 (q + 1) and f (x, y) = 0. In other words, X is birationally equivalent over Fq2 to an irreducible plane curve F of degree at 12 (q + 1). If F were not singular, from Lemma 3.24 its virtual genus, g ∗ < 12 [( 12 (q + 1) − 1)( 12 (q + 1) − 2] = 81 (q − 1)(q − 3).

But then g ∗ < g, contradicting Lemma 3.28. Hence F is non-singular and the result follows from Theorem 10.58. 2 T HEOREM 10.63 If q is odd, p ≥ 5, and X is an Fq2 -maximal curve with genus g = 18 (q − 1)(q − 3) and dim(D) = 5, the following hold. (i) If q ≥ 17 and q ≡ 1 (mod 4), then X is birationally equivalent over Fq2 to the curve D(q+1)/2 in (10.59). (ii) If q ≥ 19 and q ≡ 3 (mod 4), then X is birationally equivalent over Fq2 1 either to the curve E(q+1)/4 in (10.58) or to the curve D(q+1)/2 in (10.59). Proof. As already observed, g = c0 (q + 1, 5) and thus dim(2D) = 14. In particular, by Corollary 10.61, there exists P ∈ X with j3 (P ) > 3. (i) Let q ≡ 1 (mod 4). If P ∈ X (Fq2 ), then Lemma 10.60 (ii), (iii) imply that j2 (P ) = 2. Hence the result follows from Corollary 10.62 (ii). To show that this is the only possible case, let P 6∈ X (Fq2 ). As D′ = 41 (q − 5)D is the canonical linear series by Lemma 7.113 (ii), and hence δ + 1 ∈ / H(P ) for any (K, P )-order δ. Since m4 (P ) = q by (XIV), Lemma 10.60 together with the hypothesis p ≥ 5 leads to the following two cases (A) and (B). Case (A): j2 (P ) = 41 (q + 3), j3 (P ) = 21 (q + 1), j4 (P ) = 41 (3q + 1). In this case, {q − m3 , q − m2 , q − m1 } ⊂ {1, 41 (q + 3), 12 (q + 1), 41 (3q + 1)}

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by (XIV). Thus, m1 (P ) = 21 (q − 1), m2 (P ) = 3(q − 1), m3 (P ) = q − 1.

/ H(P ), Now, δ = 41 (q −9)+ 41 (3q +1) = q −2 is a (D′ , P )-order and hence q −1 ∈ a contradiction. Case (B): q ≡ 1 (mod 3), j2 (P ) = 31 (q − 1), j3 (P ) = 13 (q + 2), j4 (P ) = 31 (2q + 1). Since 14 (q − 5) ≥ 3, in this case δ = 3j2 (P ) must be a (D′ , P )-order. Hence q∈ / H(P ), a contradiction. (ii) q ≡ 3 (mod 4). Again, it suffices to show that P ∈ X (Fq2 ), since the result then follows from Corollary 10.62. If P ∈ / X (Fq2 ), Lemma 10.60 (ii) and (iv), together with the hypothesis p ≥ 5, imply that j2 (P ) = 13 (q − 1). From Lemma 7.113 (ii) , δ + 1 ∈ / H(P ) for every ( 14 (q − 7)D, P )-order δ. On the other hand, 1 2 since 4 (q − 7) ≥ 3 it follows that 3j2 (P ) + 1 = q ∈ H(P ), a contradiction. R EMARK 10.64 There exist Fq2 -maximal curves whose genus equals Halphen’s number c1 (4, q + 1). The known examples are listed below, but in (ii) and (iii) no explicit equation is known; see Section 10.12: (i) for q ≡ 0 (mod 4), curves of genus 18 (q 2 − 2q); (ii) for q ≡ 1 (mod 4), curves of genus 81 (q − 1)2 constructed as a quotients of the Hermitian curve Hq by a subgroup of the automorphism group; (iii) for q ≡ 3 (mod 4), curves of genus 18 (q 2 − 2q + 5) constructed similarly to those in (ii).

10.7 PLANE MAXIMAL CURVES In this section Fq2 -maximal non-singular plane curves are considered. T HEOREM 10.65 For every proper divisor d of q + 1, the Fermat curve F = D(q+1)/d = v(X (q+1)/d + Y (q+1)/d + 1) is Fq2 -maximal. Proof. The rational transformation x′ = xd , y ′ = y d of the function field K(Hq ) of the Hermitian curve Hq = v(X q+1 +Y q+1 +1) is defined over Fq2 and provides a d-fold covering of K(F). The result follows from Theorem 10.2. 2 Let F be an Fq2 -maximal non-singular plane curve of degree n with n ≥ 3. From Theorem 5.57, the genus of F is g = 12 (n − 1)(n − 2). From Proposition 10.46, n ≤ q + 1, and equality holds if and only if F is projectively equivalent over Fq2 to the Hermitian curve Hq , as in Theorems 10.47 and 10.22. From now on, let n < q + 1. Upper bounds on n, and equivalently on g, are obtained from the complete linear series L1 = gn2 cut out on F by lines. Notation

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and terminology from Sections 7.7 and 8.6 are used. In particular, ǫ0 < ǫ1 < ǫ2 with ǫ0 = 0, ǫ1 = 1 are the L1 -orders and ν0 < ν1 with ν0 = 0 are the Frobenius orders of L1 . L EMMA 10.66 Let F be an Fq2 -maximal non-singular plane curve of degree n with n < q + 1. Then the following hold: (i) ǫ2 ≤ q,

ν1 = 1;

(ii) if

  ⌊(q + 2)/2⌋ when q ≥ 4 and q 6= 3, 5, 3 when q = 3, n2 (q) =  4 when q = 5, then n ≤ n2 (q);

(iii) if n = 12 (q + 1), then F is classical.

Proof. (i) For any line ℓ and P ∈ F, I(P, ℓ ∩ F) ≤ n < q + 1. Hence ǫ2 ≤ j2 ≤ n ≤ q. If ν1 > 1, then, from Theorem 8.68, q 2 + 1 + (n − 1)(n − 2)q = n(q 2 − n + 2) n2 − (q + 2)n + q + 1 = 0, whence n = 1 or q + 1, a contradiction. Hence ν1 = 1; that is, F is Frobenius classical. (ii) From the St¨ohr–Voloch Theorem 8.18, 2(q 2 + 1 + (n − 1)(n − 2)q) ≤ n(n − 3) + (q 2 + 2)n, whence n ≤ n2 (q). (iii) Suppose that j2 (P ) ≥ 3 for all P . From Corollary 8.58, vP (S) ≥ 3. Then, from the St¨ohr–Voloch Theorem 8.18, 3(q 2 + 1 + (n − 1)(n − 2)q) ≤ n(n − 3) + (q 2 + 2)n. 2 But this is impossible for n = 21 (q + 1). C OROLLARY 10.67 (i) If n ≥ 3 is the degree of an Fq2 -maximal non-singular plane curve, then either n = q + 1 or n ≤ n2 . (ii) For q 6= 3, 5, there is no Fq2 -maximal non-singular plane curve with genus g in the interval I given as follows: 1 1 for q even, 8 q(q − 2) < g ≤ 4 q(q − 2), 1 1 2 for q odd. 8 (q − 1)(q − 3) < g ≤ 4 (q − 1) , Proof. To show (ii), assume that n ≤ n2 (d). By Lemma 10.66 (i), F is Frobenius classical. From the St¨ohr–Voloch Theorem 8.18, q 2 + 2q + 1 = F (q). n≤ 2q − 1 It follows that F (q) < 21 (q + 3) for q > 5 and that F (3) = 16/5, F (4) = 25/7, F (5) = 4. Finally, a straightforward computation shows that n ∈ / I for q 6= 3, 5. 2

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R EMARK 10.68 Let 3 ≤ d ≤ n2 (q). (i) If q is odd, then the Fq2 -maximal curve in (10.59) shows that the upper bound n2 (q) = 21 (q + 1) in Corollary 10.67 is the best possible for q 6= 3, 5. (ii) The value n2 (q) = 3 is sharp for q = 3. (iii) There exists an F25 -maximal non-singular plane quintic; so n2 (q) = 4 is sharp for q = 5. (iv) From part (ii) of the lemma, n2 (q) = 3 is sharp for q = 4. For q even and q ≥ 8, it is not known if the bound, n2 (q) = 21 (q + 2), is good or bad. Refinements to the above upper bound can be obtained by similar computations depending on the St¨ohr–Voloch Theorem 8.18 applied to L1 . P ROPOSITION 10.69 If F is an Fq2 -maximal non-singular plane curve of degree n such that 3 ≤ n ≤ n2 (q), then X is classical with respect to L1 when one of the following conditions holds: (i) p > n or n 6≡ 1 (mod p); (ii) q = 4, 8, 16, 32; (iii) p ≥ 3 and either q = p or q = p2 ; (iv) p = 2, q ≥ 64, and either d ≤ 4 or ( 1 4 q − 1, for q = 64, 128, 256, n ≤ n3 (q) = 1 for q ≥ 512 and q odd; 4 q, (v) p ≥ 3, q = pv with v ≥ 3, and n ≥ n3 (q) = q/p − p + 2. R EMARK 10.70 For q = p3 , p ≥ 3, the bound n3 (q) in Proposition 10.69 is sharp, as there exists a plane Fp6 -maximal curve of degree p2 − p + 1 which is non-classical with respect to L1 ; see Corollary 10.76 and Remark 10.86. C OROLLARY 10.71 If F is as in Proposition 10.69 and non-classical with respect to L1 , then (i) q ≥ 64 for p = 2, and q ≥ p3 for p ≥ 3; (ii) ǫ22 ≤ q/p. The example in Remark 10.70 shows that Corollary 10.71 (i) is sharp for p ≥ 3. (i) For q = 8 and q ≥ 11, let p 2q 2 + 15q − 20 + 4q 4 − 40q 3 + 145q 2 − 300q + 600 . n4 (q) = 10(q − 2)

D EFINITION 10.72

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(ii) For q = pv , v ≥ 2, let

2q 2 + 3(5 − 1/p)q − 8 n4 (p, q) = 2(5 − 1/p)q − 12 p 4 4q − 8(5 − 1/p)q 3 + (113 − 50/p + 9/p2 )q 2 − 4(25 − 17/p)q + 184 + . 2(5 − 1/p)q − 12

T HEOREM 10.73 If q = 8 or q ≥ 11 and n is the degree of a plane Fq2 -maximal curve X , with 3 ≤ n < q + 1, then one of the following holds: (i) n = ⌊(q + 2)/2⌋; (ii) n ≤ n5 (q) =

(

n4 (q) when q = p, n4 (p, q) when q = pv , v ≥ 2.

10.8 MAXIMAL CURVES OF HURWITZ TYPE D EFINITION 10.74 A Hurwitz curve of degree n + 1 is a non-singular plane curve

2

Cn = v(X0n X1 + X1n X2 + X2n X0 ) ,

(10.60)

where n − n + 1 6≡ 0 (mod p). L EMMA 10.75 The Hurwitz curve Cn is covered by the Fermat curve 2

Dn2 −n+1 = v(X0n

−n+1 2

Proof. Let K(F) = K(x, y), with xn field of F. Put

2

+ X1n

−n+1

ξ = xn−1 y −1 ,

−n+1 2

+ yn

2

+ X2n

−n+1

−n+1

).

+ 1 = 0, be the function

η = xy n−1 ;

then ξ n η + η n + ξ = 0. So the function field K(ξ, η) of Cn is a subfield of K(F), and the rational transformation ω, given by x′ = ξ, y ′ = η, provides an Fq2 -rational covering of Cn by Dn2 −n+1 . 2 Consider the following condition: q + 1 ≡ 0 (mod n2 − n + 1) .

(10.61)

L EMMA 10.76 If (10.61) holds, then the curves Cn and Dn2 −n+1 are both covered over Fq2 by the Hermitian curve Hq , and hence are Fq2 -maximal. Proof. The argument of Lemma 10.75 can be adapted to show that Dn2 −n+1 is Fq2 -covered by the Hermitian curve Hq = v(X0q+1 + X1q+1 + X2q+1 ). Then Theorem 10.2 gives the result. 2 L EMMA 10.77 The Weierstrass semigroup of Cn at the point P1 = (0, 1, 0) is generated by the set S = {s(n − 1) + 1 | s = 1, . . . , n}.

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Proof. Let P0 = (1, 0, 0) and P2 = (0, 0, 1). Then div x = nP2 − (n − 1)P1 − P0 and div y = (n − 1)P0 + P2 − nP1 , so that div(xs−1 y) = ((n(s − 1) + 1)P2 + (n − s)P0 − (s(n − 1) + 1)P1 .

This shows that S is contained in the Weierstrass semigroup H(P1 ) at P1 , and hence H(P1 ) ⊃ hSi. Therefore the result follows from a result on arithmetic progressions: |N\hSi| = 21 n(n − 1). 2 T HEOREM 10.78 The Hurwitz curve Cn is Fq2 -maximal if and only if (10.61) holds. Proof. If (10.61) holds, then Cn is Fq2 -maximal by Lemma 10.76. Conversely, assume that Cn is Fq2 -maximal. Then (q + 1)P1 ≡ (q + 1)P2 by (10.8), and the case s = n in the proof of Lemma 10.77 gives that (n2 − n + 1)P1 ≡ (n2 − n + 1)P2 .

Therefore u = gcd(n2 − n + 1, q + 1) ∈ H(P1 ). From Lemma 10.77, u = A(n − 1) + B with A ≥ B ≥ 1. Now, there exists C ≥ 1 such that

(A(n − 1) + B)C = n2 − n + 1,

and so BC = D(n − 1) + 1 for some D ≥ 0. Therefore AD(n − 1) + A + BD = Bn. Here D = 0, as otherwise the left-hand side would exceed Bn. So B = C = 1,

A = n;

that is, u = n2 − n + 1.

2

C OROLLARY 10.79 The Fermat curve Dn2 −n+1 is Fq2 -maximal if and only if (10.61) holds. Proof. If (10.61) is satisfied, the result follows from Corollary 10.76. Now if Dn2 −n+1 is Fq2 -maximal, then Cn is also Fq2 -maximal by Theorem 10.2. Then the corollary follows from Theorem 10.78. 2 R EMARK 10.80 For a given positive integer n, is there a power q of a prime p such that q + 1 ≡ 0 (mod m) with m = n2 − n + 1? Since m 6≡ 0 (mod p), and p 6≡ 0 (mod m), a necessary and sufficient condition for q to have property (10.61) is that p ≡ x (mod m), where x is a solution of the congruence T w + 1 ≡ 0 (mod m), and w is defined by q = pφ(m)v+w , where φ is the Euler function. R EMARK 10.81 The Hurwitz curve Cn is Fq2 -maximal in the following cases: (i) n = 3, q = p6v+3 and p ≡ 3, 5 (mod 7); (ii) n = 4, q = p12v+6 and p ≡ 2, 6, 7, 11 (mod 13).

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By using Theorem 10.78 and Remark 10.80, this is refined as follows. C OROLLARY 10.82 The Hurwitz curves C2 , C3 , C4 are Fq2 -maximal if and only if the one of the following conditions on p and q holds. (i) C2 : q = p2v+1 and p ≡ 2 (mod 3). (ii) C3 : (a) q = p6v+1 and p ≡ 6 (mod 7);

(b) q = p6v+3 and p ≡ 3, 5, 6 (mod 7); (c) q = p6v+5 and p ≡ 6 (mod 7).

(iii) C4 : (a) q = p12v+1 and p ≡ 12 (mod 13);

(b) q = p12v+2 and p ≡ 5, 8 (mod 13);

(c) q = p12v+3 and p ≡ 4, 10, 12 (mod 13);

(d) q = p12v+5 and p ≡ 12 (mod 13);

(e) q = p12v+6 and p ≡ 2, 5, 6, 7, 8, 11 (mod 13); (f) q = p12v+7 and p ≡ 12 (mod 13);

(g) q = p12v+9 and p ≡ 4, 10, 12 (mod 13); (h) q = p12v+11 and p ≡ 12 (mod 13).

C OROLLARY 10.83 For a positive integer n, let m = n2 − n + 1. (i) If n = pe with e ≥ 1, then the curve Cn is Fq2 -maximal for q = pφ(m)v+3e . (ii) If p ≡ 3 (mod 4), n ≡ 0, 1 (mod p), m is prime and m ≡ 3 (mod 4), then the curve Cn is Fq2 -maximal for q = p(m−1)v+(m−1)/2 . Proof. (i) This is a consequence of Theorem 10.78 and the identity p3e + 1 = (pe + 1)(p2e − pe + 1).

(ii) It is enough to show that p(m−1)/2 + 1 ≡ 0 (mod m). Recall the Legendre symbol (a/p):  1 if x2 ≡ a (mod p) has two solutions in Zp , (a/p) = −1 if x2 ≡ a (mod p) has no solution in Zp . Since m ≡ 1 (mod p), here (m/p) = 1. By the Quadratic Reciprocity Law (m/p)(p/m) = (−1)((m−1)/2)((p−1)/2)

and the condition that m ≡ 3 (mod 4), it follows that (p/m) = (−1)(p−1)/2 . Now, as p ≡ 3 (mod 4), this gives (p/m) = −1. In other words, p viewed as an element in Fm is a non-square. Since −1 is also a non-square in Fm , this implies that p ≡ (−1)u2 (mod m) for an integer u with u 6≡ 0 (mod m). Hence p(m−1)/2 ≡ −1 (mod m) as m is odd and um−1 ≡ 1 (mod m). 2

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R EMARK 10.84 The condition that m ≡ 3 (mod 4) in this corollary cannot be relaxed. If n = 4, then m = 13 but, from Corollary 10.82, C4 is not F36 -maximal. R EMARK 10.85 Under the hypotheses in Corollary 10.83 (ii) with m not necessarily prime, the congruence (10.61) may be investigated by means of the group Um of units in Zm . This group has order φ(m), and p ∈ Um since m ≡ 1 (mod p). Now suppose that p, as an element of Um , has even order 2i. Then p2i ≡ 1 (mod m) and hence (pi + 1)(pi − 1) ≡ 0 (mod m). Since p has order greater than i, so pi − 1 6≡ 0 (mod m) unless both pi + 1 and pi − 1 are zero divisors in Zm . When this does not happen, then (10.61) follows for q = pφ(m)v+i . R EMARK 10.86 Let p be a prime, n = pe u with e ≥ 1 and gcd(p, u) = 1. Assume that e ≥ 2 when p = 2. Then the curves Cn and Dn2 −n+1 are both non-classical with respect to L1 , and 0, 1, pe are their L1 -orders. D EFINITION 10.87 A generalised Hurwitz curve of degree n + k is the irreducible plane curve Cn,k = v(X0n X1k + X1n X2k + X2n X0k ) , where n ≥ k ≥ 2 and p does not divide Q(n, k) = n2 − nk + k 2 . The singular points of Cn,k are P0 = (1, 0, 0), P1 = (0, 1, 0), and P2 = (0, 0, 1); each of them is the centre of a unique branch of Cn,k . The genus of Cn,k is n2 − nk + k 2 + 2 − 3 gcd(n, k) g= , 2 The Fq2 -maximality of these curves can be investigated by the same methods as for Hurwitz curves, with analogous results. L EMMA 10.88 The curve Cn,k is Fq2 -covered by the Fermat curve Dn2 −nk+k2 . Consider the condition: n2 − nk + k 2 ≡ 0

(mod q + 1) .

(10.62)

C OROLLARY 10.89 The curve Cn,k is Fq2 -maximal provided that (10.62) holds. C OROLLARY 10.90 The curve Dn2 −nk+k2 is Fq2 -maximal provided that (10.62) holds. Let P1 be the place arising from the branch of Cn,k centred at P1 . L EMMA 10.91 If gcd(n, k) = 1, then the Weierstrass semigroup at P1 is H(P1 ) = {(n−k)s+nt | s, t ∈ Z; t ≥ 0; −kt/n ≤ s ≤ (n−k)/(kt)} . (10.63)

T HEOREM 10.92 If gcd(n, k) = 1 and n2 − nk + k 2 is prime, then the curve Cn,k is Fq2 -maximal if and only if (10.62) holds. C OROLLARY 10.93 If n, k are as in Theorem 10.92, then the curve Dn2 −nk+k2 Fq2 -maximal if and only if (10.62) holds.

R EMARK 10.94 There are infinitely many pairs n, k with n > k ≥ 1 such that n2 − nk + k 2 is prime. In fact, for a prime p′ such that p′ ≡ 1 (mod 6), there exist such n, k with p′ = n2 − nk + k 2 .

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10.9 NON-ISOMORPHIC MAXIMAL CURVES In this section, a 2-parameter family of curves Xim is presented; for each fixed m, there is a large number of non-isomorphic curves all with some identical properties. With K = Fq , let Xim be a non-singular model over K of the plane curve Cim = v(X mi+m + X mi + Y q+1 ) ,

(10.64)

where m is a positive divisor of q + 1 for which d = (q + 1)/m > 3 is prime. The curve Xim is the quotient curve of the Hermitian curve Hq arising from an automorphism group of Hq of the same order d. Let D = |(q + 1)P | denote the associated complete linear series at a point P of Xim . T HEOREM 10.95 Assume 1 ≤ i ≤ d − 2. (i)

(a) The curves Xim and Xjm are K-equivalent if and only if one of the following equations holds modulo d : i ≡ j, ij ≡ 1, ij + i + j ≡ 0 , i + j + 1 ≡ 0, ij + i + 1 ≡ 0, ij + j + 1 ≡ 0 . (b) The number of K-isomorphism classes of curves Xim is given by ( 1 if d ≡ 2 (mod 3), 6 (d + 1) n(d) = (10.65) 1 (d − 1) + 1 if d ≡ 1 (mod 3). 6 (c) Each of these classes consists of six curves, apart from two exceptions of sizes 2 and 3. The corresponding indices i are as follows: (1) i1 , i2 , where i1 and i2 are the solutions of t2 +t+1 = 0 (mod d), with d ≡ 1 (mod 3); (2) 1, 12 (d − 1), d − 2.

(ii) The genus of Xim is g = 12 m(q − 2) + 1. (iii) The K-automorphism group of Xim is   Z3 ⋊ (Zq+1 × Zm ) in case (c)(1), m Aut(Xi ) = Z2 ⋊ (Zq+1 × Zm ) in case (c)(2),   Zq+1 × Zm otherwise.

(iv) When m = 2, the series D has projective dimension 21 (d + 3). There are at least six K-rational points P such that, if (j0 = 0, j1 = 1, . . . , j(d+1)/2 , j(d+3)/2 ) is the D-order sequence at P, then j(d+1)/2 = d, j(d+3)/2 = q + 1. (v) When m = 2 and q is prime, then the D-order sequence at a generic point is (0, 1, . . . , 21 (d + 1), q).

T HEOREM 10.96

m (i) The curves X0m and Xd−1 are K-isomorphic.

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(ii) X0m has genus g = 21 (m − 1)(q − 1), and is hyperelliptic when m = 2. (iii) The centre Z of the K-automorphism group AutK (X0m ) is a cyclic group of order m, and the group AutK (X0m )/Z is isomorphic to PGL(2, q). (iv) The complete linear series on X02 has projective dimension d + 1, and the D-order sequence at a Weierstrass point is one of (0, 1, 2, . . . , d, q),

(0, 1, 2, 4, 6, . . . , q − 1, q + 1).

(v) The D-order sequence of X02 is (0, 1, . . . , d, q). 10.10 OPTIMAL CURVES The previous sections contain examples of maximal curves such as the Hermitian curve Hq that is both Fq2 -maximal and Fq2 -optimal, as well as families of maximal curves. Optimal curves that are not maximal have only been classified for g = 1; see Sections 9.9 and9.10. The value of Nq (g) is known for g = 1, 2 and for various other pairs (g, q). The aim of this section is a characterisation of the DLS curve as the unique optimal curve over Fq which has genus g = q0 (q + 1) when q = 2q02 , q0 = 2s and s ≥ 1. The proof of Theorem 10.102 requires some preliminary results. From Proposition 9.32, the L-polynomial of X is Lq (t) = (t2 + 2q0 t + q)g . Hence, as already observed in Example 9.80, the Fundamental Equation (9.74) of X becomes, for P, P0 ∈ X with P0 ∈ X (Fq ): qP + 2q0 Φ(P ) + Φ2 (P ) ≡ (q + 2q0 + 1)P0 .

(10.66)

Some results from Section 9.8 to be applied to the corresponding linear series D = |(q + 2q0 + 1)P0 | are summarised below. From Section 9.8, (1) r = dim D; (2) ǫ0 = 0 < ǫ1 = 1 < · · · < ǫr are the D-orders of X ; (3) ν0 = 0 < · · · < νr−1 are the Frobenius orders of X ; (4) m1 (P ) < · · · < mr (P ) < · · · are the non-gaps of X at P . Then (A) jr (P ) = mr (P ) = q + 2q0 + 1 for any P ∈ X (Fq ), and there exists P1 ∈ X (Fq2 ) such that mr−1 (P1 ) = q + 2q0 ; (B) ji (P ) = mr (P ) − mr−i (P ) for any P ∈ X (Fq ) and i = 0, . . . , r; (C) D is simple and base-point-free; (D) 2q0 and q are D-orders, and so r ≥ 3;

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(E) ǫr = νr−1 = q; (F) m1 (P ) = q for any P ∈ X (Fq ). From (A), (C), and Proposition 9.83 (i), for any P ∈ X (Fq ), jr−1 (P ) = jr (P ) − m1 (P ) = 2q0 + 1, whence (G) 2q0 ≤ ǫr−1 ≤ 2q0 + 1 . L EMMA 10.97 The order ǫr−1 = 2q0 . Proof. Suppose on the contrary that ǫr−1 > 2q0 . Then ǫr−2 = 2q0 ,

ǫr−1 = 2q0 + 1.

By Corollary 8.58, νr−2 ≤ jr−1 (P ) − j1 (P ) ≤ 2q0 = ǫr−2 , and thus the Frobenius orders of D are ǫ0 , ǫ1 , . . . , ǫr−2 , and ǫr . By Proposition 8.56 (i), for P ∈ X (Fq ), Pr vP (S) ≥ i=1 (ji (P ) − νi−1 ) ≥ (r − 1)j1 (P ) + 1 + 2q0 ≥ r + 2q0 . (10.67) Hence

Pr deg S = ( i=0 νi )(2g − 2) + (q + r)(q + 2q0 + 1) ≥ (r + 2q0 )|X (Fq )|.

Since 2g − 2 = (2q0 − 2)(q + 2q0 + 1) and |X (Fq )| = (q − 2q0 + 1)(q + 2q0 + 1), Pr−2 Pr−2 i=1 νi = i=1 ǫi ≥ (r − 1)q0 . Since ǫi + ǫj ≤ ǫi+j for i + j ≤ r by Exercise 9 in Chapter 7, it follows that Pr−2 (r − 1)2q0 = (r − 1)ǫr−2 ≥ 2 i=0 ǫi ≥ 2(r − 1)q0 .

Hence ǫi + ǫr−2−i = ǫr−2 for i = 0, . . . , r − 2. In particular, ǫr−3 = 2q0 − 1 and, by Lemma 7.62, ǫi = i for i = 0, 1, . . . , r − 3. Then r = 2q0 + 2. Now Castelnuovo’s Bound, see Theorem 7.112, applied to D gives the following: 2g = 2q0 (q − 1) ≤

Since r = 2q0 + 2, this implies that

q + 2q0 − (r − 1)/2)2 . r−1

2q0 (q − 1) < 21 (q + q0 )2 q0 = q0 q + 12 q + 21 q0 , a contradiction. C OROLLARY 10.98 There exists P1 ∈ X (Fq ) such that  1 for i = 1, j1 (P1 ) = νi−1 + 1 for i = 2, . . . , r − 1.

2

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Proof. Taking (10.67) into account, it suffices to show the existence of a point P1 ∈ X (Fq2 ) for which vP1 (S) = r + 2q0 . So, suppose that vP (S) ≥ r + 2q0 + 1 for every P ∈ X (Fq ). Then (8.14) implies that Pr−1 i=0 νi ≥ q + rq0 + 1 . Therefore

Pr−1 i=0

ǫi ≥ rq0 + 2 ,

since ǫ1 = 1, νr−1 = q and νi ≤ ǫi+1 . Then, from Exercise 9 in Chapter 7, rǫr−1 ≥ 2rq0 + 4. But this means that ǫr−1 > 2rq0 , which contradicts Lemma 10.97. 2 L EMMA 10.99

(i) The Frobenius order ν1 > ǫ1 = 1;

(ii) the order ǫ2 = 2e . Proof. (i) Suppose on the contrary that ν1 = 1. Then, from Corollary 10.98, there exists a point P1 ∈ X (Fq ) such that j1 (P1 ) = 1, j2 (P1 ) = 2. Therefore H(P1 ) ⊂ H = hq, q + 2q0 − 1, q + 2q0 , q + 2q0 + 1i ,

by (A), (B) and (F). In particular, g = q0 (q − 1) ≤ g˜, where g˜ counts the nonnegative integers which do not belong to H. On the other hand, g > g˜. To show this, it suffices to prove the following: g˜ = g − 41 q02 .

(10.68)

2q0 −1 The key idea is to observe that Λ = ∪i=1 Λi is a complete system of residues modulo q, where

Λi = {iq + i(2q0 − 1) + j | j = 0, . . . , 2i} if 1 ≤ i ≤ q0 − 1; Λq0 = {q0 q + q − q0 + j | j = 0, . . . , q0 − 1};

Λq0 +1 = {(q0 + 1)q + 1 + j | j = 0, . . . , q0 − 1} for 2 ≤ i ≤ 21 q0 ; Λq0 +i = {(q0 + i)q + (2i − 3)q0 + i − 1 + j | j = 0, . . . , q0 − 2i + 1} ∪{(q0 + i)q + (2i − 2)q0 + i + j | j = 0, . . . q0 − 1}

for 1 ≤ i ≤ 12 q0 − 1; Λ3q0 /2+i = {(3q0 /2 + i)q + (q0 /2 + i − 1)(2q0 − 1) + q0 + 2i − 1 + j | j = 0, . . . , q0 − 2i − 1}.

Also, d ∈ Λ, d ∈ H and d − q 6∈ H for each d ∈ Λ. Hence g˜ can be calculated by summing the coefficients of q in this list. The result is as follows: P 0 −1 P 0 /2 g˜ = qi=1 i(2i + 1) + q02 + (q0 + 1)q0 + qi=2 (q0 + i)(2q0 − 2i + 2) Pq0 /2−1 + i=1 (3q0 /2 + i)(q0 − 2i) 2 = q0 (q − 1) − q0 /4 . (ii) This follows from (i) and Lemma 7.62.

2

Now, let P0 = P1 be an Fq -rational point satisfying Corollary 10.98, and let mi = mi (P1 ).

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Lemma 10.99 (i) implies that νi = ǫi+1 for i = 1, . . . , r − 1. Therefore, from (A), (B) and Corollary 10.98, mi = 2q0 + q − ǫr−i for i = 1, . . . r − 2, mr−1 = 2q0 + q, mr = 1 + 2q0 + q.

(10.69)

Let x, y2 , . . . , yr ∈ Fq (X ) be such that div(x)∞ = m1 P1 ,

div(yi )∞ = mi P1 for i = 2, . . . , r.

Note that x is a separable variable of K(X ). Since ν1 > 1, Proposition 8.42 shows that the matrix   1 xq y2q ... yrq  1 x y2 ... yr  0 1 Dx y 2 . . . Dx y r has rank 2. Therefore

yiq − yi = Dx yi (xq − x) for i = 2, . . . , r. L EMMA 10.100

(10.70)

(i) The divisor (2g − 2)P is canonical for any P ∈ X (Fq2 ).

(ii) If m ∈ H(P1 ) satisfies m < q + 2q0 , then m ≤ q + q0 .

(iii) There exists gi ∈ Fq (X ) such that Dx yi = giǫ2 for i = 2, . . . , r, and qmi − q 2 P1 . ǫ2

div(gi )∞ =

Proof. (i) It may be assumed that P = P1 from the identity (10.66) and the fact that 2g − 2 = (2q0 − 2)(q + 2q0 + 1). When i = r in (10.70), ordP1 (dx) = 2g − 2 and the result follows since ordQ (dx) ≥ 0 for Q 6= P1 . (ii) From (10.69), q, q + 2q0 , q + 2q0 + 1 ∈ H(P1 ). Then (2q0 − 2)q + q − 4q0 + j,

j = 0, . . . , q0 − 2,

are also non-gaps at P1 . Since (i) implies that H(P ) is symmetric, it follows that q + q0 + 1 + j,

j = 0, . . . , q0 − 2,

are gaps at P1 , and this completes the proof. (iii) Put fi = Dx yi . Then

Dx(j) yi = (xq − x)Dx(j) fi + Dx(j−1) fi , for 1 ≤ j < q, by Lemmas 5.72 and 5.80. the matrix  1 x y2  0 1 Dx y2 (j) 0 0 Dx y2

(j)

Then, Dx fi = 0 for 1 ≤ j < ǫ2 , since  ... yr . . . Dx y r  (j) . . . Dx y r

has rank 2 for each j in 2 ≤ j < ǫ2 . Consequently, as ǫ2 is a power of 2 by Lemma 10.99 (ii), by Lemma 5.84, there exists gi ∈ K(X ) such that fi = giǫ2 . Since fi ∈ Fq (X ), also gi ∈ Fq (X ). Finally, from the proof of (i), x − x(P ) is a local parameter at P if P 6= P1 . Then, by the choice of the elements yi , the function gi 2 has no pole other than P1 . From (10.70), ordP1 (gi ) = −(qmi − q 2 )/ǫ2 .

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L EMMA 10.101 The dimension r = 4 and the order ǫ2 = q0 . Proof. From Theorem 10.47, r ≥ 3. Suppose that r = 3. Then ǫ2 = 2q0 , n1 = q, n2 = q + 2q0 , n3 = q + 2q0 + 1, and hence ordP1 g2 = −q for the same g2 as in Lemma 10.100 (iii). Therefore, after a change over Fq of the coordinate system of PG(3, K), the case i = 2 of (10.70) is as follows: y2q − y2 = x2q0 (xq − x).

Now, the element z = y2q0 − xq0 +1 satisfies z q − z = xq0 (xq − x). Therefore q0 + q is a non-gap at P1 . This contradiction implies that r 6= 3. Let r ≥ 4 and 2 ≤ i ≤ r. By Lemma 10.100 (iii), qni − q 2 ∈ H(P1 ). ǫ2

Since qmi − q 2 ≥ mni−1 ≥ q, ǫ2 from (10.69), 2q0 ≥ ǫ2 + ǫr−i for i = 2, . . . , r − 2. In particular, ǫ2 ≤ q0 . On the other hand, by Lemma 10.100 (ii), mr−2 ≤ q + q0 and hence, by (10.69), ǫ2 ≥ q0 ; that is, ǫ2 = q0 . Note that ǫ2 = q0 implies ǫN −2 ≤ q0 . Since n2 ≤ q + q0 by Lemma 10.100) (ii), it follows from (10.69) that ǫr−2 ≥ q0 . Therefore ǫr−2 = q0 = ǫ2 and hence r = 4. 2 T HEOREM 10.102 If q = 2q02 , q0 = 2s and s ≥ 1, and X is an irreducible non-singular Fq -rational curve of genus g such that g = q0 (q − 1),

|X (Fq )| = q 2 + 1 ,

(10.71)

then X is birationally equivalent over Fq to the DLS curve. Proof. Let P1 ∈ X (Fq ) be as above. From (10.70), Lemma 10.100 (iii) and Lemma 10.101, y2q − y2 = g2q0 (xq − x) , where g2 has no pole other than P1 . Also, from (10.69), m2 = q0 + q and so ordP1 g2 = −q by Lemma 10.100 (iii). Thus g2 = ax + b with a, b ∈ Fq , a 6= 0. After a change over Fq of the coordinate system of PG(4, K), the result follows. 2 R EMARK 10.103 The DLS curve X is also investigated in Section 12.2. From the proof of Theorem 10.102, some further properties of X related to its Frobenius linear series D = |(q + 2q0 + 1)P0 |, P0 ∈ X (Fq ), can be deduced. 4 , the curve X has a complete, simple, base-point-free, (i) Since D = gq+2q 0 +1 Fq -invariant linear series with orders 0, 1, q0 , 2q0 , q and with Frobenius orders

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0, q0 , 2q0 , q. If K(X ) is given by K(x, y) with x2q0 (xq + x) = y q + y as in Section 12.2, and u(x, y) = x2q0 +1 + y 2q0 , then D consists of all divisors Ac = div(c0 + c1 x + c2 y + c3 u(x, y) + c4 (xu(x, y) + y 2 )) + (q + 2q0 + 1)P0

with c = (c0 , c1 , c2 , c3 , c4 ) ∈ PG(4, K). Therefore the birational transformation π : F 7→ X , where F = v(X 2q0 (X q + X) − (Y q + Y )), is given by the coordinate functions x0 = 1, x1 = x, x2 = y, x3 = u(x, y), x4 = xu(x, y) + y 2 . From this representation of X it can be verified that X is a non-singular curve of PG(4, K). To do this, note that the points of X coming from the points of F other than Y∞ = (0, 0, 1) are non-singular since Y∞ = (0, 0, 1) is the unique infinite point of F and all other points are non-singular; see Theorem 12.13. As Y∞ is the centre of a unique branch of F, the corresponding point P = (0, 0, 0, 0, 1) of X is the only candidate to be a singular point of X . But, if P were singular, then every automorphism of X would fix P and the automorphism group Sz(q) of X could not act transitively on the set of all Fq -rational point of X , contradicting Theorem 12.13 (iv). (ii) There exists P1 ∈ X (Fq ) whose (D, P1 )-orders are j0 (P1 ) = 0, j1 (P1 ) = 1, j2 (P1 ) = q0 +1, j3 (P1 ) = 2q0 +1, j4 (P1 ) = q+2q0 +1. They are also the (D, P )-orders for each P ∈ X (Fq ). To see this, note that deg S = (3q0 + q)(2g − 2) + (q + 4)(q + 2q0 + 1) = (4 + 2q0 )|X (Fq )|. From (10.67), vP (S) =

P4

i=1

(ji (P ) − νi−1 ) = 4 + 2q0 .

Now, using Corollary 8.58, it follows that ji (P ) = ji (P1 ) for all i. (iii) From (ii) and (B) it follows that H(P ) contains the semigroup H = hq, q + q0 , q + 2q0 , q + 2q0 + 1i,

whenever P ∈ X (Fq ). Actually, H(P ) = H since the number of non-negative integers not in H is g = q0 (q − 1). This can be shown as in the proof of Lemma 10.99 (i). (iv) From (i) and (ii), P deg R = 4i=0 ǫi (2g − 2) + 5(q + 2q0 + 1) = (2q0 + 3)|X (Fq )| , and vP (R) = 2q0 + 3 for P ∈ X (Fq ). Therefore the D-Weierstrass points of X are precisely the Fq -rational points of X . In particular, the (D, P )-orders for P 6∈ X (Fq ) are 0, 1, q0 , 2q0 , q. (v) Regarding the order sequence of the canonical curve of X , from (iv) and Lemma 10.100 (i), it follows that every integer m = a + q0 b + 2q0 c + qd with

a + b + c + d ≤ 2q0 − 2

is an order of the canonical curve at every non-Fq -rational point. (vi) From Proposition 9.83, the canonical curve of X is non-classical as g ≥ q.

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10.11 EXERCISES 1. Over F7 , compare the Serre and St¨ohr–Voloch Bounds for a plane quartic curve. Show that the curve F = v(F ) over F7 , with F = X 4 + Y 4 + Z 4 + 3(X 2 Y 2 + X 2 Z 2 + Y 2 Z 2 ),

has 20 rational points, namely, (±1, ±3, 1), (±3, ±1, 1), (±3, ±2, 1), (±2, ±3, 1), (±2, ±2, 1). Deduce that F is optimal; that is, N7 (3) = 20. 2. If q = 3t ≥ 27 and X is an Fq2 -maximal curve over of genus 61 q(q − 3) with dim D = 4, show that X is birationally equivalent over Fq2 to the plane curve T3′′ = v(T3′′ ), with T3′′ = Y q/3 + Y q/9 + . . . + Y 3 + Y + cX q+1 ,

where c ∈ Fq with cq−1 = −1. 3. Show that there is a maximal curve over Fq2 whose genus is Halphen’s number c1 (q + 1, 4) = ⌊ 81 (q 2 − 2q + 5)⌋. 4. Let X be a non-singular Fq2 -rational plane curve and let L1 be the linear series cut out on X by lines. Show that (a) L1 is both non-classical and Frobenius non-classical if and only X is projectively equivalent over Fq2 to the Hermitian curve; (b) if q = p or q = p2 , then L1 is classical. 5. Let d be a positive integer such that q ≡ −1 (mod d2 ). Show that the curve F = v(F ), with 2

2

F = Y (q+1)/d + X (d+1)(q+1)/d + X (q+1)/d , is an Fq2 -maximal curve of genus 1 + (q + 1)(q − d2 − 1)/(2d2 ). 6. Let d be a divisor of q + 1. Use the Natural Embedding Theorem 10.22 to show that the Fermat curve Dm = v(F ), with m = (q + 1)/d and F = X m + Y m + 1,

is Fq2 -maximal. It has genus g = 21 (m − 1)(m − 2) and the number of its Fq2 -rational points is q 2 + 1 + (m − 1)(m − 2)q. 7. With d be a divisor of q + 1 and m = (q + 1)/d, show that the Artin–Schreier curve Am = v(F ), with F = Y q + Y − X m,

is Fq2 -maximal. It has genus g = 21 (m − 1)(q − 1), and the number of its Fq2 -rational points is mq 2 − (m − 1)q + 1. 8. Prove Theorem 10.47 without using the Natural Embedding Theorem 10.22.

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9. Let X be a Fq2 -maximal curve and P an ordinary point, that is, an ordinary point of X . Show that, if q + 1 ∈ H(P ), then g = q − N1 + 1. 10. Let q be a prime power of and let d > 2, i > 0 be integers such that di divides q + 1. Prove that there exists a sequence of curves X0 = H → X1 → . . . → Xi , such that each covering Xj−1 → Xj , j = 1, . . . , i, is unramified, where H is the Hermitian curve. 11. For q ≡ 1 (mod 3), let G = v(G) with

G = Y q − Y X 2(q−1)/3 + ωX (q−1)/3 ,

where ω q+1 = −1. Show that the genus of G is g = 61 (q 2 − q). 12. Find all F64 -maximal curves of genus 3 up to isomorphism. 13. Let X be an Fq2 -maximal curve of genus g. Show that, if X is hyperelliptic, then g ≤ 12 q. 14. Show that no F64 -maximal curve has genus 8. 15. Show that the curves F1 = v(X 4 + X + Y 3 ),

F2 = v(X 4 + X 2 + X + Y 3 ) both have 113 rational points over F64 , and so are maximal but not isomorphic over F64 . 16. Let F be an irreducible curve of genus g defined over F2 . Show that, if Ni = 2i + 1 for every 1 ≤ i ≤ g, then F is a F22g -maximal curve. 10.12 NOTES Surveys on maximal curves are found in papers by Garcia, Stichtenoth and van der Geer: [149], [141], [142], [472]. See also Fuhrmann [131]. Minimal curves have not received much attention so far, but fundamental results are treated in the survey paper by Viana and Rodriguez [488]; see also Lauter [299]. Theorem 10.2 was originally stated by Serre; see Lachaud [286]. An alternative proof of the Fq2 -maximality of the curve in Example 10.3 is due to Ebin, Hor, Jatz and Shchogolev [113]. Maximal curves of Kummer type (10.2) over Fq2 , with f (X) ∈ Fq [X] additive and separable, were studied by Garcia, Kawakita and Miura [145], Moisio [332], ¨ and Ozbudak [354]. In particular, Kq+1 is covered over Fq2 by the Hermitian curve; see [145]. Maximal Artin–Schreier curves over Fp2m of type n

F = v(Y p

+1

a

− Y + cX p

+1

+ L(X)),

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P(2m/n)−1

ni

where n | a, a | m, and L(X) = bi X p , were investigated by i=0 ¨ Coulter [95], C¸akc¸ak and Ozbudak [62], van der Geer and van der Vlugt [474], and for L = 0 by Wolfmann [506]. The construction of Fq2 -maximal curves not covered over Fq2 by the Hermitian curve Hq or, alternatively, the possibility of a proof of their non-existence is still an open problem. It should be noted that the apparently easier question in this context, namely, the existence of an Fq2 -maximal curve other than the quotient curves of Hq has been resolved. An example of such a curve for q = 27 is described in Example 10.4, and is due to Garcia and Stichtenoth. An unpublished result by Rains and Zieve is the existence of another example for q = 27; it was shown that, for q0 = 1, the DLR curve F = v(f (X, Y ) with f (X, Y ) as in (12.34) is not a quotient of H27 . The most likely candidate for a further example seems to be the DLS curve for q = 8. Regarding Example 10.4, the case ℓ = 2 and the first statement in Theorem 10.5 are due to Serre. For the extension to any ℓ and the proof of the second statement in Theorem 10.4, see Garcia and Stichtenoth [154]. The problem of whether the second statement holds true for any ℓ is still open. In Table 10.2 the sources and characterisations are as follows: 1. g = 21 q(q − 1), 2. 3. 4. 5, 7. 8. 9. 10. 11. 12. 13, 15.

[378], [223];

g = 14 (q − 1)2 , [132]; g = 14 q(q − 2), [5]; g = 16 (q 2 − q + 4), [92]; 6. g = 16 (q 2 − q), [158]; g = 61 (q 2 − q − 2), [91], [158]; g = 16 (q − 1)(q − 2), [282]; g = 16 q(q − 3), [91], [5]; g = 81 (q 2 − 2q + 5), [282]; g = 81 (q − 1)2 , [282]; g = 18 q(q − 2), [282]; 14. g = 81 (q − 1)(q − 3), [282]; g = 18 q(q − 4), [3].

The minimality of m in the preamble to Lemma 10.16 follows from a result due to Kaji [255, Prop. 1]; see also [163, Prop. 2], General properties of quadrics in PG(3, q) can be found in [213, Chapter 16]. For detailed studies of Hermitian varieties, see [397], [213, Chapter 19], and [224, Chapter 23]. Section 10.2 is based on the survey paper [134]. The proof of Theorem 10.7 comes from [281]. A slight modification of the proof of Proposition 10.11 shows that each P ∈ Supp D\X (Fq ) is a Weierstrass point of X ; see [159].

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Related to Proposition 10.12, Garcia and Viana [160] showed that the Weierstrass points of the Fq2 -maximal curves, F = v(Y q + Y − X m ) with m | q + 1 and m > 2, are exactly the Fq2 -rational points of F. Theorem 10.47 is due to Stichtenoth and Xing [509]. For a proof of this theorem independent of the the St¨ohr–Voloch approach, see Fuhrmann [131]. For Remark 10.49, see Garcia, Stichtenoth and Xing [158, Remark 6.1]. The results in Section 10.3, in particular, the Natural Embedding Theorem 10.22 and Theorem 10.31, come from [281]. Sections 10.4, 10.5 and 10.6 follow [282]. The proof of Theorem 10.43 given in [132] is independent of the Natural Embedding Theorem 10.22 but it uses Galois theory. An alternative proof may be carried out by adapting the arguments in the proof of Theorem 10.43 and using results on Hermitian varieties and quadrics in permutable position. For the lengthy calculation omitted from Theorem 10.45, see [5]. For Theorem 10.58, see [88]. Sections 10.7 and 10.8 are based on [13]. Theorem 10.102 and its proof come from [134]. For the examples of maximal curves in Remark 10.64, see the following: (i) [478, Prop. 5.2(ii)], [158, Theorem 3.3]; (ii) [92, Proposition 3.3(3)]; (iii) [92, Proposition 3.3(3)(1)], [158, Example 5.10]. For the quintic in Remark 10.68 (iii), see [406, Section 4]. For Proposition 10.69, see [13]. For the result on arithmetic progressions in Lemma 10.77, see [180]. For Remark 10.81, see [65, Lemmes 3.3, 3.6] and [66]. For generalised Hurwitz curves, see [47, Section 4], [66], [45, Example 4.5], [13]. The Weierstrass semigroup H(P1 ) in Lemma 10.91 is calculated for k = n − 1, and (n, k) = (5, 2) in [47]. For Remark 10.94, see [47, Remarque 4]. The proofs of Theorems 10.95 and 10.96 are found in [171]. For a generalisation of these theorems to m for which d = (q + 1)/m is not prime, see [172]. Section 10.10 follows [134]. Theorem 10.2 implies that the quotient curves of an Fq2 -maximal curve are also Fq2 -maximal. This makes it possible to obtain several infinite families of Fq2 maximal curves, especially from the Hermitian curve; see Section 12.3, and [4], [5], [3], [6], [91], [92], [158], [151], [171], [172]. Others can be obtained from the DLS and DLR curves; see Sections 12.2, 12.4, as well as [174] and [60]. However, Theorem 10.2 fails for optimal curves in general. But the curves attaining the Serre Bound 9.27 for a non-square q are optimal curves of this type, such as the DLS and DLR curves over Fq . Sporadic examples are the Klein quartic over F8 , as in Remark 9.29, and the curve v(Y 5 + Y 4 − X 23 ) over F211 ; see Section 12.5. See also Remark 9.29. The value of Nq (g) was found by Waterhouse for g = 1 and by Serre for g = 2 and for g = 3 when q ≤ 25. Also, Nq (g) is known for a number of sporadic pairs (g, q); see the survey in [242, Chapter 2]. See also van der Geer and van der Vlugt

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457

[481], where tables of known values of Nq (g) are maintained. See also Howe and Lauter [237]. For Remark 10.68, see [401]. For an explicit description of the action of Sz(q) on the DLS curve X , see [174]. Also, X (Fq ) is an ovoid of the non-singular quadric v(X0 X4 − X1 X3 − X22 ) of PG(4, q); see [134]. For other results on the orders of the canonical curve of X , see [147, Section 4]. The non-classicality of the canonical curve is also shown in [147]. For the curve in Exercise 1, see [456]. Exercise 16 is related to quasi-Hermitian curves F = v(Y n + bY + cX m ); see [327] and [371].

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Chapter Eleven Automorphisms of an algebraic curve The concept of a symmetry, or automorphism in modern terminology, plays a prominent role in all branches of geometry. The saying, the larger its automorphism group the richer its geometry, is especially appropriate for algebraic curves. For instance, if there exists an automorphism of an irreducible curve Γ, then another irreducible curve, the quotient curve, can be constructed, and it inherits important properties of Γ. Also, certain families of curves have characterisations obtained from the structure and action of their automorphism groups. As in several concepts introduced previously, an automorphism of an algebraic curve is a birational invariant. In the function field K(Γ), the corresponding term is a K-automorphism, that is, a field automorphism of K(Γ) which fixes every element of K. So, K-automorphisms are mostly investigated by using algebraic rather than geometric tools. However, since Γ is birationally equivalent to an irreducible non-singular curve X embedded in a projective space PG(r, K) such that every K-automorphism of K(Γ) can be represented as a linear collineation of PG(r, K) preserving X , so a K-automorphism can be viewed as a truly geometric concept. In this spirit, AutK (Σ) is the K-automorphism group of X . More generally, the term, K-automorphism group of F, is adopted for any curve F birationally equivalent to Γ, that is, for every model of K(Γ). Since the K-automorphisms of Γ form a group, their study greatly benefits from group theory. A fundamental result is that the K-automorphism groups of an irreducible curve is finite, apart from rational and elliptic curves. So, finite group theory is relevant in this context, which is a further great advantage as it is in the finite case that group theory has been most developed. So, algebraic geometry and group theory combine to make it possible to find an upper bound for the order of AutK (Σ) depending only on the genus g of Σ. The classical bound is 84(g − 1). In positive characteristic, the result is achieved in Section 11.12 by showing that |AutK (Σ)| < 8g 3 , apart from four exceptions. As throughout the book, Σ denotes a function field of transcendence degree 1, that is, the function field of an irreducible curve F. Further, AutK (Σ) stands for the group of all K-automorphisms of Σ. For the special case that F is defined over Fq , the concept of Fq -rationality for K-automorphisms is introduced. Let Fq (F) be the Fq -rational function field of F. A K-automorphism ω of F is Fq -rational if ω preserves Fq (F); that is, the birational transformation x′ = ω(x), y ′ = ω(y) of Σ is Fq -rational. Such elements form a subgroup, the Fq -rational K-automorphism group of F, of AutK (Σ).

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11.1 THE ACTION OF K-AUTOMORPHISMS ON PLACES The first result is a characterisation of K-automorphisms in terms of birational transformations. L EMMA 11.1 Let α be a K-automorphism of Σ. If (F; (x, y)) is any model of Σ, then the birational transformation ω given by the equations x′ = α(ξ),

y ′ = α(y)

preserves the curve F.

Proof. Since α ∈ AutK (Σ), from f (x, y) = 0 and α(f (x, y)) = f (x′ , y ′ ) it follows that f (x′ , y ′ ) = 0. Hence ω preserves F. 2 L EMMA 11.2 Let (F; (x, y)) be a model of Σ. If a birational transformation ω of Σ, defined by (x, y) 7→ (x′ , y ′ ),

preserves the curve F, then there is a K-automorphism α of Σ such that x′ = α(x) and y ′ = α(y). Proof. Since ω preserves F, the point P ′ = (x′ , y ′ ) is another generic point of F. Now, let x′ = α(x), y ′ = α(y), and extend this definition to any element ξ ∈ Σ in a natural way; namely, α(ξ) =

u(x′ , y ′ ) v(x′ , y ′ )

when

ξ=

u(x, y) , v(x, y)

with u(X, Y ), v(X, Y ) ∈ K[X, Y ] and f (X, Y ) ∤ v(X, Y ). Note that the expression of ξ as a rational function of x, y is not unique; however, α(ξ) does not depend on the form of the expression. To verify this, let u(x, y) u1 (x, y) = . v(x, y) v1 (x, y) Then u(X, Y )v1 (X, Y ) − u1 (X, Y )v(X, Y ) is a polynomial that vanishes at the generic point P = (x, y) of F. By the definition of generic point, this polynomial is divisible by f (X, Y ). Since f (x′ , y ′ ) = 0, this yields the result: u1 (x′ , y ′ ) u(x′ , y ′ ) = . ′ ′ v(x , y ) v1 (x′ , y ′ ) So, α is correctly defined on Σ, and it is indeed a K-automorphism of Σ.

2

It is intuitively apparent that K-automorphisms of Σ are an important tool in investigating places, divisors, and linear series by means of the action of AutK (Σ) on such objects. In this context, the fixed field ΣG consisting of all elements in Σ which are fixed by every element in G plays an important role.

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A geometric view of this is also possible, since there is a one-to-one correspondence between places of Σ and points of any non-singular model X of Σ, which gives rise to a natural action of AutK (Σ) on the points of X . In some circumstances, such an action is linear in the sense that it arises from a linear collineation group of the projective space in which X is embedded. The first step is to define the action of K-automorphisms on places. Let α ∈ AutK (Σ) and let σ be a primitive representation of a place P of Σ. Then the image σα of σ under α is defined by the equation (σα)(ξ) = σ(α(ξ)),

ξ ∈ Σ.

As σα is a primitive place representation, it defines a place of Σ: this place is independent of the choice of σ, since equivalent primitive place representations have equivalent images. Such a place is called the image of P by α, and denoted by P α . Once the action of α on places has been P defined, this extends to Div(Σ) in a natural way. Let D ∈ Div(Σ) with D = nP P; then P D α = nP P α . In particular, α preserves the set of all effective divisors of Σ. For any α-invariant subfield Σ′ = K(ξ, η) of Σ, it is appropriate to keep the notation α to denote the induced K-automorphism of Σ′ . Then α has an action, which may be trivial, on the set of all places PΣ′ of Σ′ . It follows that, if P ∈ PΣ and P ′ ∈ PΣ′ are places such that P lies over P ′ , then P α lies over P ′α . First, some elementary properties of K-automorphisms of Σ are collected. L EMMA 11.3 Let α ∈ AutK (Σ) and P ∈ PΣ . Then, for any z ∈ Σ, ordP z = ordP α (α−1 (z)).

(11.1)

Proof. Let σ be a primitive representation of P. Then ordP z = ordt σ(z) for z ∈ Σ. Also, ordt σ(α(α−1 (z))) = ordt (σα)α−1 (z) = ordP α (α−1 (z)). This proves the assertion. 2 Note that (11.1) can also be written in the following way: ordP (α(z)) = ordP α (z).

(11.2)

Lemma 11.3 has some useful corollaries. L EMMA 11.4 If α ∈ AutK (Σ) and z ∈ Σ, then div(z)α = div(α−1 (z)); more particularly, (div(z)0 )α = div(α−1 (z))0 , (div(z)∞ )α = div(α−1 (z))∞ . (11.3) P the summation is over all places Proof. First, div(z)0 = P ordP (z)P, whereP P ∈ PΣ with ordP (z) > 0. Then (div(z)0 )α = P ordP (z)P α . By (11.1), P P α −1 α (z))P α . P ordP (z)P = P ordP (α By (11.1), ordP (z) > 0 if and only if ordP α (α−1 (z)) > 0. Hence P −1 (z))P α = div(α−1 (z)), P ordP α (α

which shows the second assertion. The equation for the divisor of poles is proved similarly. The first assertion is consequence of the second and third assertions. 2

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L EMMA 11.5 If A ≡ B with A, B ∈ Div(Σ), then Aα ≡ B α for α ∈ AutK (Σ). Proof. A − B = div(z) for some z ∈ Σ. By (11.1), (A − B)α = div(α−1 (z)). Since (A − B)α = Aα − B α , the result follows. 2 A similar argument may be used to prove the following two results. L EMMA 11.6 Every K-automorphism of Σ preserves the set of all Weierstrass points of Σ. L EMMA 11.7 Let F be a subfield of Σ of transcendence degree 1 such that the field extension Σ/F is separable. If α ∈ AutK (Σ) and F ′ = α(F ), then D(Σ/F )α = D(Σ/F ′ ),

where D(Σ/F ) and D(Σ/F ′ ) are the different divisors of Σ/F and Σ/F ′ . L EMMA 11.8 If α ∈ AutK (Σ) and div(z)α = div(z) for every non-zero element z ∈ Σ, then α is the identity automorphism of Σ. Proof. As α is a K-automorphism, it suffices to prove that α(z) = z for any z in Σ\K. Since α−1 also satisfies the hypothesis, so, from Lemma 11.4, div(z) = div(α(z)), whence div(z/α(z)) = 0. By Theorem 6.4, α(z) = cz for a non-zero constant c ∈ K. Replacing z by z + 1, there is c′ ∈ K such that α(z + 1) = c′ (z + 1). Thus α(z) + α(1) = c′ z + c′ . These equations show that cz + 1 = c′ z + c′ , whence (c − c′ )z = c′ − 1. Since z ∈ Σ\K, this is only possible when c − c′ = 0 and c′ = 1. Hence c = 1, and the result follows. 2 L EMMA 11.9 If α ∈ AutK (Σ) and P α = P for every place of Σ, then α is the identity automorphism of Σ. Proof. From the hypothesis, Dα = D for every D ∈ Div(Σ). In particular, the hypothesis in the preceding lemma is satisfied. Hence the assertion is just a corollary of that lemma. 2 In fact, Lemma 11.9 still holds if it is only required that α fixes infinitely many places of PΣ . To show this, the residue map resP associated to any place P ∈ PΣ , as in Example 7.130, is needed. L EMMA 11.10 If α ∈ AutK (Σ) and P ∈ PΣ , then resP z = resP α α−1 (z) for any z ∈ Σ. Proof. Let σ be a primitive representation of P. If z ∈ Σ, then σ(z) = σ(αα−1 (z)) = (σα)(α−1 (z)),

and hence resP z = res(σ(z)) = res(σα(α−1 (z)) = resP α α−1 (z).

2

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T HEOREM 11.11 If α ∈ AutK (Σ) and P α = P for infinitely many places P of Σ, then α is the identity automorphism of Σ. Proof. Choose an α-invariant place P of Σ. Lemma 11.10 applied to α−1 shows that resP α(z) = resP z holds for all z ∈ Σ. If neither z nor α(z) has a pole at P, the additive property of resP yields that resP α(z) − resP z = resP (α(z) − z), giving resP (α(z) − z) = 0. Now, assume that α has infinitely many fixed places, and remove those places that are poles for either z or α(z). The remaining fixed places of α are still infinite, but none of them is a pole of any of the elements z, α(z) and α(z) − z. For each such place P, the equation resP (α(z) − z) = 0 implies that P is a zero of α(z) − z. By Theorem 5.33, there exists c ∈ K such that α(z) − z = c. Since α is a K-automorphism, so α(z) − z = 0. 2 The number of fixed places that a K-automorphism can have is an important topic to which the whole Section 11.11 is devoted. Meanwhile, the following result is required. L EMMA 11.12 Every non-trivial K-automorphism of Σ has at most 2g + 2 fixed places. Proof. Let α be a non-trivial K-automorphism of Σ. By Theorem 11.11, α has a finite number of fixed places. Take g + 1 distinct places P1 , . . . , Pg+1 of Σ such that the linear series |D| with D = P1 + · · · + Pg+1 has no fixed point and that D and its image Dα share no place. By Theorem 6.82, Σ contains an element z 6∈ K such that div(z)∞ = D. Now, consider w = z − α(z). Since z and α(z) have different poles, w 6= 0; more precisely, w has exactly 2g + 2 poles, because each pole of w is either a pole of α(z) or a pole of z. By Theorem 5.35, α has exactly 2g + 2 zeros. On the other hand, every fixed place of α is a zero of w. Hence the number of such places is at most 2g + 2. 2 L EMMA 11.13 Let m be the smallest non-gap of Σ at a place P, and suppose that ξ ∈ Σ has a pole of order m at P and is regular elsewhere. If P α = P for an element α ∈ AutK (Σ), then α(ξ) = cξ + d with c, d ∈ K, c 6= 0. Proof. Choose a primitive representation σ of P. Then

σ(ξ) = c1 t−m + c2 t−m+1 + · · · ,

with c1 6= 0. By Lemma 11.3,

σ(α(ξ)) = d1 t−m + d2 t−m+1 + · · · ,

with d1 6= 0. Let η = d1 ξ − c1 α(ξ). Then

σ(η) = e2 tk + · · · , with k ≥ −m + 1.

The minimality of m implies that k ≥ 0. This shows that P is not a pole of η. Since ξ, and by Lemma 11.3 also α(ξ), is regular outside P, the same holds for η. Hence η has no poles at all. By Theorem 5.33, η ∈ K, whence the result follows. 2

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T HEOREM 11.14 Let Σ = K(ξ) be rational. (i) The group AutK (Σ) ∼ = PGL(2, K), and AutK (Σ) acts on the set PΣ of all places of Σ as PGL(2, K) naturally on PG(1, K). (ii) (a) α ∈ AutK (Σ) if and only if α(ξ) is bilinear; that is, α(ξ) : ξ 7→

aξ + b , cξ + d

ad − bc 6= 0, a, b, c, d ∈ K;

(b) the identity of AutK (Σ) is the only K-automorphism in AutK (Σ) fixing at least three places of X ; (c) every K-automorphism of AutK (Σ) fixes a place P ∈ PΣ ;

(d) if p ∤ ord α, then α has exactly two fixed places; (e) if p | ord α, then

(1) ord α = p; (2) α has exactly one fixed place; (3) when the fixed place is a pole of ξ, the map is α(ξ) = ξ + b with b in K\{0}.

Proof. The projective line ℓ = v(Y ) is a non-singular model of Σ = K(x) where P = (x, 0) is a generic point of ℓ. For every a, b, c, d ∈ K such that ad − bc 6= 0, the rational transformation ω : x′ = (ax + b)/cx + d) of K(x) is birational, and preserves ℓ. By Lemma 11.2, ω defines a K-automorphism α of K(x) such that α : ξ 7→

aξ + b , cξ + d

ad − bc 6= 0,

a, b, c, d ∈ K.

(11.4)

These constitute a subgroup L of AutK (Σ) isomorphic to the projective linear group PGL(2, K). Also, L acts on the points of ℓ, that is on the places of K(x), as PGL(2, K) acts on PG(1, K) in its natural representation. Now, let β be any K-automorphism in AutK (Σ). Let P∞ denote the unique place centred at the infinite point of ℓ. Since PGL(2, K) is transitive, there exists δ ∈ L such that the K-automorphism βδ of Σ fixes P∞ . Since Σ = K(ξ) and since div(ξ)∞ = P∞ , so 1 is the smallest non-gap at P∞ . Thus Lemma 11.13 applies, showing that βδ ∈ L. But this shows that β itself belongs to L. This completes the proof of (ii)(a) and (ii)(b). To show (ii)(c), let α be given as (11.4). If either c 6= 0, or c = 0 but a 6= d, then the equation aX + b = X(cX + d) has a solution u in K, and the unique place Pu centred at u is fixed by α. Otherwise, α(ξ) = ξ + b and hence α fixes P∞ . Conversely, if α fixes P and its order is divisible by p, then α(ξ) = ξ + b with b ∈ K\{0}. In particular, ord α = p, which proves (ii)(e). To show (ii)(d), a little more is needed. Since PGL(2, K) is transitive on ℓ, so assume that P∞ is fixed by α. Then α(ξ) = aξ + b with a, b ∈ K and a 6= 0. Since p ∤ ord α, so a 6= 1. Therefore α fixes not only P∞ but also the place centred at the point P = (u, 0) with u = −b/(a − 1). 2 R EMARK 11.15 All finite subgroups of the group PGL(2, K) are known; see Theorem 11.91.

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Let P be any place of Σ. Choose a primitive place representation σ of P. Assume that α ∈ AutK (Σ) fixes P. Then σα is a primitive place representation σ ′ of P, and λσ = σ ′ for a K-automorphism λ of K[[t]]. Here, λ depends on α, and this is stressed by writing λα and saying that λα is the companion K-automorphism of α in Aut(K[[t]]). For any K-automorphism group of Σ which fixes P, this leads to the introduction of the following map: ϕ : G → Aut(K[[t]]),

α 7→ λα .

(11.5)

L EMMA 11.16 The mapping ϕ is an epimorphism. Proof. Define the image ΛG = {λα | α ∈ G} of ϕ to be the companion group of G. Let α, β ∈ G. Then λαβ σ = σ(βα) = (σβ)α = λα (σβ) = λα (λβ σ) = (λα λβ )σ,

whence λαβ = λα λβ . Let α ∈ ker ϕ. Then σ = σα. As σ is an isomorphism, α is the the identity in AutK (Σ). This completes the proof. 2 L EMMA 11.17 Let α be a finite K-automorphism of Σ that fixes a place P. Then, α has order pr s with r ≥ 0 and p ∤ s if and only if λα = ut + · · · , where s is the smallest integer satisfying us = 1. Proof. Let G = hαi. The elements β ∈ G with λβ = t + · · · form a subgroup H of G. To show that H is a p-group, take a prime divisor r of |H|, and an element β ∈ H of order r. If λβ = t + vti + · · · , then λβ r = t + rvti + · · · , which gives r = p. On the other hand, every element β ∈ G whose order is a power of p is in H. This follows from the fact that, if λβ = ut + · · · , then λβ n = un t + · · · , while m up = 1 implies that u = 1. Since G is cyclic, G = H × N with N a cyclic group whose order s is prime to p. If N = hδi, then λδ = ut + · · · and s is the smallest positive integer such that us = 1. From this, the assertion follows. 2 It may be noted that ΛG depends on the choice of σ, even though only formally. In fact, if σ is replaced by another primitive place representation σ ¯ of P, then the ¯ G are conjugate in Aut(K[[t]]). To show this, write resulting groups ΛG and Λ ¯α (¯ τσ = σ ¯ with τ ∈ Aut(K[[t]]). Then τ σα = σ ¯ α; that is, τ λα σ = λ σ ), which ¯ α τ )σ. Since σ ∈ Aut(K[[t]]) ¯ α τ σ. Hence λα σ = (τ −1 λ implies that τ λα σ = λ and α is independent of τ , the result follows.

11.2 LINEAR SERIES AND AUTOMORPHISMS The following theorem shows how K-automorphisms of Σ act on linear series. T HEOREM 11.18 Let α ∈ AutK (Σ) and let P L = {div( ci xi ) + B | c = (c0 , . . . , cr ) ∈ PG(r, K)}, α

B ∈ Div(Σ), (11.6)

be a linear series of Σ. Then the divisors D with D ranging over L are the divisors of a linear series Lα of Σ : P Lα = {div( ci α(xi )) + B α | c = (c0 , . . . , cr ) ∈ PG(r, K)}.

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Note that deg L = deg Lα and dim L = dim Lα . Also, if both Lα = L and B = B, then α can be viewed as an element of EndK (E), where E is the Riemann–Roch space, that is, the (r + 1)-dimensional vector space over K with basis {x0 , . . . , xr }. By Theorem 6.72 (i), the canonical series of Σ is the only effective linear series of Σ of order 2g − 2 and dimension g − 1. This gives the following result. α

L EMMA 11.19 Every K-automorphism of Σ preserves the canonical series of Σ. Similarly, the following result holds. L EMMA 11.20 If a K-automorphism of Σ fixes a place P, then α preserves the complete linear series |kP| for every non-negative integer k. If Σ is the function field of a hyperelliptic curve, then every K-automorphism of Σ preserves g21 , the unique complete linear series of dimension 1 and order 2 of Σ. By Theorems 7.94 and 7.100, the set consisting of all places P of Σ such that 2P ∈ g21 is finite. This has the following consequence. L EMMA 11.21 For a hyperelliptic curve F, the K-automorphism group of F preserves a finite set of places. From a geometric point of view, it is important to have a linear representation of α in the projective space in which a suitable irreducible model of Σ is embedded. This can be done by using the following result. L EMMA 11.22 For a function field Σ, let (a) (11.6) be a simple, base-point-free linear series of Σ; (b) Y be the irreducible curve of PG(r, K) arising from L; (c) C(P) ∈ Y denote the centre of the corresponding branch of Y for any place P ∈ PΣ ; (d) α ∈ AutK (Σ) preserve L. Then there is a projectivity T of PG(r, K) such that (i) C(P α ) = T(C(P)); (ii) T preserves Y; (iii) if Y is non-singular, T acts on the points of Y as α on PΣ . Proof. The series L may be taken to be normalised. Then B ∈ L by Theorem 6.22. P Hence there is a non-zero element z = ci xi such that B α = div(z) + B. From this, for i = 0, . . . , r, P α(xi )z = aij xj ,

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with aij ∈ K, and det(aij ) 6= 0. Let P be a place of Σ. If σ is a primitive representation of P, it follows that P (σα)(xi )σ(z) = aij σ(xj ). (11.7)

Let P = (u0 , . . . , ur ) and P ′ = (u′0 , . . . , u′r ) denote the centres of the branches arising from P and P α , let ord σ(xi ) ≥ 0 for all indices i = 0, P. . . , r, and suppose that ord σ(xi ) = 0 for some index i. Now, from (11.7), u′i = aij uj . This shows that P ′ = T(P ), where T is the projectivity of PG(r, K) associated to the matrix T = (tij ). 2 A corollary of Lemmas 11.19 and 11.22 is the following. T HEOREM 11.23 Let F be an irreducible curve which is not rational, elliptic or hyperelliptic. Then the K-automorphism group of F acts on the points of the canonical curve of K(F) as a linear collineation of PG(g − 1, K). Theorem 11.23 makes possible the use of geometric methods in investigating automorphisms. Important results in this direction are the following theorems that extend to all characteristics p in Section 11.7. T HEOREM 11.24 Let F be an irreducible curve which is not rational, elliptic or hyperelliptic. If either p = 0 or p > 2g − 2, then every K-automorphism of F is of finite order.

Proof. From Theorem 7.64, the set of all Weierstrass points of Σ = K(F) is finite. By Lemma 11.6, it suffices to prove the finiteness of any K-automorphism α fixing every Weierstrass point. By Corollary 7.61, the canonical curve Γ of Σ is classical. Let P ∈ Γ be a Weierstrass point. The hyperplanes H through P with intersection multiplicity I(P, H ∩ Γ) at least jg−2 constitute a pencil through a common subspace Π of dimension g − 3. Then, Γ is preserved by α. It may be that Π contains some more points from Γ, and let k + 1, with k ≥ 0, denote the number of distinct points of Γ lying in Π. Then every hyperplane in the pencil has at most 2g − 2 − (g − 2) − k = g − k points outside Π. On the other hand, the lower bound in (7.15) shows the existence of at least 2g + 2 Weierstrass points. Therefore at least three, but finitely many, distinct hyperplanes in the pencil contain a Weierstrass point not lying in Π. Therefore a power β of α preserves at least three hyperplanes in the pencil. Since, in the dual space of PG(g − 1, K), the pencil corresponds to a line, it follows from Theorem 11.14 (ii)(b) that β preserves every hyperplane through Π. If H is such a hyperplane, then a sufficiently large power of β fixes every common point of H and Γ. Repeating this argument for another hyperplanes through Π, it turns out that, if α is infinite, then a power of α has more than 2g + 2 fixed points in Γ. Therefore, to prove the assertion, it suffices to show that no non-trivial K-automorphism can have more than 2g + 2 fixed points in Γ. To do this, let δ be a non-trivial K-automorphism, and take an ordinary point Q ∈ Γ such that Qδ 6= Q. Note that such a point exists by Theorem 11.11. Since Γ is classical, g + 1 is a non-gap at Q, and hence there is a function x ∈ Σ such that div(x)∞ = (g + 1)Q,

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where Q is the place arising from Q. Letting y = x − δ(x), then div(y)∞ = (q + 1)Q + (g + 1)Qδ .

Hence y has 2g + 2 zeros. On the other hand, every fixed place of δ is a zero of y. This proves the assertion and completes the proof of the theorem. 2 Theorem 11.24 extends to the hyperelliptic case provided that either p = 0 or p is large enough with respect to g. L EMMA 11.25 If p = 0 and g ≥ 2, then every K-automorphism of an irreducible curve F of genus g is finite. Proof. Let W be a canonical divisor of Σ = K(F). Then the tri-canonical series L = |3W | of Σ is a non-special linear series of order n = 3(2g − 2) > 2g and dimension r = 5g − 6. By Theorem 7.40, the associated irreducible curve Γ in PG(r, K) is non-singular. As p = 0, so Γ is a classical curve. By Corollary 7.61, vP (R) ≤ 21 g(g + 1) for every P ∈ PΣ ,P and the number of L-Weierstrass points, each counted according to its weight, is vP (R) = 25(g − 1)2 g. Thus Γ has at least 25(g − 1)2 g 1 2 g(g + 1) L-Weierstrass points. For g ≥ 2, this number is larger than 2g + 2. Since every K-automorphism α of Σ preserves the set W of all L-Weierstrass points, Lemma 11.12 shows that α cannot fix W element-wise. Hence α acts faithfully on W, and α can be viewed as a permutation in the symmetric group on W, which is finite. Therefore ord α is finite. 2 In the elliptic case, Theorem 11.24 holds for K-automorphisms fixing a place. This is shown here for p 6= 2, 3. For the general case, see Section 11.4. L EMMA 11.26 Let p 6= 2, 3. If F is an elliptic curve, then every K-automorphism F fixing a place is finite, and its order divides 4 or 6. Proof. Since g = 1, the smallest non-gap at P is equal to 2. Choose ξ ∈ K(F) with div(ξ)∞ = 2P. By Lemma 11.13, α(ξ) = aξ + b with a, b ∈ K and a 6= 0. Since p 6= 2, 3, there exists η ∈ Σ such that η 2 = 4ξ 3 − g2 ξ − g3 .

(11.8)

In other words, Σ has a model (F; (ξ, η)) with a generic point P = (ξ, η) which satisfies (11.8). Then η has a pole of order 3 in P. Hence, there exists c ∈ K such that α(η) − cη has a pole of order at most 2. Therefore α(η) − cη = dξ + e with d, e ∈ K. Hence, α(η) is a K-linear combination of ξ, η and 1. Applying α to (11.8), this yields that (cη + dξ + e)2 = 4(aξ + b)3 − g2 (ξ + b) − g3 , which can also be written as c2 ξ 2 + d2 η 2 + e2 + 2ceξ + 2deη + 2cdξη = 4(a3 ξ 3 + 3a2 bξ 2 + 3ab2 ξ + b3 ) − g2 (aξ + b) − g3 .

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By Lemma 11.1, this differs from (11.8) only by a constant factor. Comparing coefficients of ξη, η, ξ 2 shows that d 6= 0 and that c = e = b = 0. Hence α(ξ) = aξ,

α(η) = dη,

with d2 = a3 = 1.

If g3 = 0, g2 6= 0 then either a = 1, d = ±1 or a = −1, d = ±j with j 2 = −1, and hence ord α ∈ {1, 2, 4}. If g2 = 0, g3 6= 0, then a = ±1, d3 = 1, and hence ord α ∈ {1, 2, 3, 6}. Otherwise, a = 1, d = ±1, and hence ord α ∈ {1, 2}. 2 11.3 AUTOMORPHISM GROUPS OF PLANE CURVES The K-automorphism groups of rational, elliptic, hyperelliptic, and many other irreducible curves are known; see Section 11.10 and Chapter 12. On the other hand, every finite group is isomorphic to the K-automorphism group of some irreducible curve. For any irreducible curve F of genus g which is not rational, elliptic, or hyperelliptic, Theorem 11.19 shows that its K-automorphism group is isomorphic to a subgroup of PGL(g, K) which preserves the canonical curve Γ of K(F) embedded in PG(g − 1, K). This provides information about K-automorphisms, especially when the dimension g − 1 is low. In fact, Σ = K(F) may have other, possibly singular, models in PG(r, K) such that AutK (Σ) is a subgroup of PG(r + 1, K). If this is the case for an irreducible plane curve C, a plane model of Σ, the structure and the action of AutK (Σ) can be examined by using the classification of all subgroups of PGL(3, K). However, finding out whether a given function field Σ has such a plane model may be very involved and may depend on special features of Σ. An important, general result in this direction is that this problem can be solved for those function fields that have a non-singular plane model. The elementary proof depends on two lemmas of independent interest. L EMMA 11.27 Let {P1 , . . . , Pn } be a set of n > 3 distinct points in the plane. If there are at least 21 (n − 2)(n − 3) linearly independent curves C n−3 of order n − 3 passing through all the points P1 , . . . , Pn , then these points are collinear. Proof. The result is true for n = 4; so proceed by induction on n. Let n > 4. On a line ℓ through one of these points, say P1 , disjoint from the remaining points P2 , . . . , Pn , choose n − 3 distinct points Q1 , . . . , Qn−3 which are also distinct from P1 . The number of linearly independent curves C n−3 of order n − 3 passing all the points P1 , . . . , Pn , Q1 , . . . , Qn−3 is at least 1 2 (n

− 2)(n − 3) − n − 3 = 12 (n − 3)(n − 4).

Such curves are reducible as each of them contains n − 2 points from ℓ, namely the points P1 , Q1 , . . . , Qn−3 . It follows that they split into ℓ and curves C n−4 of order n − 4 passing through all the points P2 , . . . , Pn . The number of such linearly independent curves is at least 21 (n − 3)(n − 4). By induction, P2 , . . . , Pn are collinear. Therefore any n − 1 of the points P1 , . . . , Pn are collinear. This can only happen when all these points lie on a line. 2

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L EMMA 11.28 Let C n be a non-singular plane curve of order n > 3. The linear series cut out on C n by lines is the unique linear series gn2 of C n . Proof. Choose a divisor D = P1 + · · · + Pn in gn2 such that the places Pi are centred at distinct points, say P1 , . . . , Pn . Since C n is non-singular, its genus is g = 12 (n − 1)(n − 2). From Theorem 6.61 applied to gn2 , at least i = g − n − 2 linearly independent curves C n−2 of order n − 2 pass through P1 , . . . , Pn . By Lemma 11.27, these points are collinear, and the assertion follows. 2 So the following theorem is a corollary of Lemmas 11.28 and 11.22. T HEOREM 11.29 The K-automorphism group of any non-singular plane curve X of order n > 3 is linear; that is, it is isomorphic to the subgroup of PGL(3, K) that fixes X . It should be noted that Theorem 11.29 holds true for n = 2 but it fails for both n = 1 and n = 3; see Theorem 11.14 and the proof of Theorem 11.94. An application of Theorem 11.29 is the following result. P ROPOSITION 11.30 If K is the algebraic closure of Fq2 with q > 2, then the K-automorphism group of the Hermitian curve Hq = v(X q + X + Y q+1 ) is the projective unitary group PGU(3, q). Proof. Since Hq is a non-singular plane curve, Theorem 11.29 implies that its Kautomorphism group G consists of all projectivities fixing Hq . A straightforward computation shows that every α ∈ G is defined over Fq2 ; that is, a matrix associated to α has all entries from Fq2 . Therefore G preserves the set Hq (Fq2 ) of all Fq2 -rational points of Hq . By Example A.9, Hq (Fq2 ) coincides with the set U of all self-conjugate points of a unitary polarity of PG(2, q 2 ), and PGU(3, q) is a subgroup of G. It remains to show that every α ∈ G belongs to PGU(3, q). As PGU(3, q) acts 2-transitively on U, with the notation used in Example A.9 it may be assumed that α fixes both the points X∞ and O. After a change of coordinates, U consists of Y∞ plus all points U = (1, u, cuq+1 + m) as u ranges over Fq2 and m ranges over M . By another computation, the hypothesis that α preserves U implies that α is associated to a matrix  q+1  t 0 0  0 t 0  0 0 1

for a non-zero element t in Fq2 . Therefore α ∈ PGU(3, q).

2

T HEOREM 11.31 If p ∤ n and n 6= q + 1, then the K-automorphism group G of the Fermat curve Fn = v(X n + Y n + 1) has order 6n2 and has a normal abelian subgroup N of order n2 such that G/N ∼ = S3 .

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Proof. For any two n-th roots of unity, λ, µ ∈ K, the projectivity (X0 , X1 , X2 ) 7→ (X0 , λX1 , µX2 )

preserves Fn . These projectivities form an abelian group N of order n2 fixing each vertex of the fundamental triangle. Both the projectivities (X0 , X1 , X2 ) 7→ (X1 , X2 , X0 ),

(X0 , X1 , X2 ) 7→ (X0 , X2 , X1 )

also preserve Fn and permute the vertices of the fundamental triangle. The projective group H generated by all these projectivities has order 6n2 , contains N as a normal subgroup, and induces S3 on the vertices of the fundamental triangle. It remains to shows that H coincides with G. Since Fn is a non-singular plane curve, Theorem 11.29 implies that G is a subgroup of PGL(3, q). A straightforward computation shows that, if α ∈ PGL(3, q) preserves Fn , then α also preserves the fundamental triangle, except for n = q + 1. Hence α ∈ H, and the result follows. 2 11.4 A BOUND ON THE ORDER OF A K-AUTOMORPHISM A major result on automorphisms of algebraic curves is that, apart from rational curves, every K-automorphism fixing a place is finite. In characteristic zero, this follows from Lemmas 11.25 and 11.26. For p > 0, the proof is much more involved and requires some lemmas of independent interest on the invariant subfields of Kautomorphisms. In this section it is assumed that p > 0 although almost all results, but not all proofs, remain valid in zero characteristic. L EMMA 11.32 If α ∈ AutK (Σ) fixes every element of a subfield F of Σ of transcendence degree 1, then α is finite and its order does not exceed [Σ : F ]. Proof. Let n = [Σ : F ]. As the field extension Σ/F is algebraic, it is also simple. There exists z ∈ Σ such that Σ = F (z). Let f (X) = X n + . . . + an−1 X + an be the minimal polynomial of z over F . Since α fixes every coefficient ai of P (X), it follows that α(z) is also a root of f (X). So, the set S = {αi (z) | i ≥ 0} has cardinality at most n. Since αi (z) = αj (z) if and only if αi−j (z) = z and since Σ = F (z), this last condition can only occur when αi−j is the identity automorphism of Σ. Hence, ord α ≤ |S| ≤ n. 2 L EMMA 11.33 Assume that Σ is non-rational and that α ∈ AutK (Σ) preserves a rational subfield K(ξ) of Σ. If p ∤ ρ, where ρ = [Σ : K(ξ)], then α is finite and ord α ≤ 2p(g + 1)ρ2 . Proof. By Lemma 11.32, the idea is to find an element η ∈ Σ\K which is fixed by α or a power of α. By Theorem 11.14 (ii) and Lemma 11.13, it may be assumed that, after a Klinear change of ξ, either α(ξ) = ξ + v or α(ξ) = cξ for some v, c ∈ K, and c 6= 0. In the former case, as p 6= 0, so αp (ξ) = ξ, and hence ord α cannot exceed pρ, by Lemma 11.32.

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Henceforth, let α(ξ) = c ξ, where c ∈ K\{0}. By Theorem A.5, there exists η ∈ Σ such that Σ = K(ξ, η). Let F = v(f (X, Y )) be the irreducible plane curve with generic point P = (ξ, η). The projection of F onto the X-axis, that is, the rational transformation τ:

x′ = ξ,

y′ = 0

is not birational, as Σ is not rational. Let D = D(Σ/K(ξ)) be the associated different divisor. Note that, if ξ is a separating variable of Σ, then Hurwitz’s Theorem 7.27 implies that deg D = 2g − 2 + 2ρ,

(11.9)

α

where ρ = [Σ : K(ξ)]. By Lemma 11.7, D = D since α preserves K(ξ). Also, α viewed as a K-automorphism of K(ξ) preserves the set of places Q ∈ PK(ξ) which lie under the ramified places of PK(X ) . First, consider the case where div(ξ) = ρP1 − ρP2 . As in the proof of Theorem 5.33, after replacing η by c0 (ξ)η with c0 (ξ) ∈ K(ξ), it may be assumed that f (X, Y ) = Y ρ + g(X, Y ),

where g(X, Y ) ∈ K[X, Y ] has degree in Y less than ρ. Since fY (X, Y ) = ρY ρ−1 + gY (X, Y )

does not vanish, the hypothesis p ∤ ρ ensures that ξ is indeed a separating variable of Σ. Let σ1 and σ2 be primitive place representations of P1 and P2 . Then σ1 (ξ) = ctρ + · · · ,

c ∈ K\{0},

and the contribution of P1 in D is ρ − 1 since p ∤ ρ. Similarly, σ2 (ξ) = ct−ρ + · · · ,

c ∈ K\{0},

and the contribution of P2 in D is again ρ − 1. Thus deg D = 2(ρ − 1) + m, with m ≥ 0. Comparison with (11.9) gives m = 2g. This yields that, apart from P1 , P2 , at least one but at most 2g places contribute to D. Therefore a power β = αℓ , with ℓ ≤ 2g, of α fixes a place P3 different from both P1 , P2 . If Q1 , Q2 , Q3 are the places of K(ξ) under P1 , P2 , P3 , then β fixes them. By Theorem 11.14 (ii)(b), it follows that β fixes every place Q ∈ PK(ξ) . By Lemma 11.9, the subfield Σβ = {z ∈ Σ | β(z) = z} contains K(ξ). Hence ord β ≤ ρ, yielding that ord α ≤ 2gρ. Therefore it may be assumed that either div(ξ)0 or div(ξ)∞ , say the latter, contains two different places P1 and P2 . Note that P1β = P1 for a power β = αℓ with ℓ ≤ ρ. Put ordP1 (ξ) = −k1 and ordP2 (ξ) = −k2 , where k1 , k2 are positive integers. Let n1 be the smallest non-gap of Σ at P1 . Then there exists ζ ∈ Σ such that div(ζ)∞ = n1 P1 . Since ξ has at least two poles, ζ 6= kξ for every k ∈ K. After changing ζ to uζ + v, with suitable u, v ∈ K, Lemma 11.13 allows that β(ζ) = bζ, b ∈ K\{0}. P Let f1 (X, Y ) = aij X i Y j be an irreducible polynomial with coefficients in K such that f1 (ξ, ζ) = 0. From β ∈ AutK (Σ), it follows that f1 (β(ξ), β(ζ)) = 0. Since β(ξ) = cℓ ξ, β(ζ) = bζ, this implies that f1 (cl X, bY ) = ef1 (X, Y ),

e ∈ K\{0}.

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Now, choose two distinct pairs (i, j) and (m, n) with n > j such that both aij and amn are distinct from zero. Then cℓi bj = e and cℓm bn = e. Thus η = ξ i−m ζ j−n is fixed by β. It must be shown that η 6∈ K. Assume by contradiction that ζ n−j = kξ i−m with k ∈ K. Then div(ξ i−m )∞ = div(ζ n−j )∞ , but div(ξ i−m )∞ = (i − m)k1 P1 + (i − m)k2 P2 + · · · , div(ζ n−j )∞ = (n − j)n1 P1 .

Hence, (i − m)k2 = 0, But then i = m whence n − j = 0, a contradiction. Finally, it is shown that [Σ : K(η)] ≤ 2ρ(g + 1). Since the degree of f1 (X, Y ) in X is at most [Σ : K(ζ)] = n1 ≤ g + 1, while in Y it is at most [Σ : K(ξ)] = ρ, so |i − m| ≤ g + 1 and |j − n| ≤ ρ. Then [Σ : K(ξ i−m )] = [Σ : K(ξ)] · [K(ξ) : K(ξ i−m )] ≤ ρ(g + 1), [Σ : K(ζ j−n )] = [Σ : K(ζ)] · [K(ζ) : K(ζ j−n )] ≤ (g + 1)ρ.

i−m j−n Since div ξ∞ + div ζ∞  div η, the assertion follows. From Lemma 11.32, ord β ≤ 2ρ(g+1). Therefore ord α ≤ 2pρ2 (g+1), and Lemma 11.33 is completely proved. 2

Now, the finiteness of every α ∈ AutK (Σ) fixing a place P of Σ is shown under the hypothesis that g ≥ 1. By the Weierstrass Gap Theorem 6.89, there are exactly g gaps of Σ at P, and each of them is at most 2g. In particular, 2g + 1 is a non-gap of Σ at P. By Corollary 6.75, |(2g + 1)P| is a simple, base-point-free, complete linear series. Its dimension r is g + 1 ≥ 2, by the Riemann–Roch Theorem 6.61. Let m0 = 0 < m1 < · · · < mr denote the first r+1 = g +2 non-gaps of Σ at P. In particular, mr−1 = 2g, mr = 2g + 1. Choose x0 = 1, x1 , . . . , xr ∈ Σ such that div(xi )∞ = mi , for i = 0, 1, . . . , r. Since x0 , x1 , . . . , xr are linearly independent over K, the linear series, P L = {div( ci xi ) + (2g + 1)P | c = (c0 , . . . , cr ) ∈ PG(r, K)}, coincides with |(2g + 1)P|. By Lemma 11.20, α preserves |(2g + 1)P|. By Lemma 11.3, α preserves each subseries, P Lj = {div( ri=j ci xi ) + (2g + 1)P | c = (c0 , . . . , cr ) ∈ PG(r, K)}.

Consider the Riemann–Roch space L((2g + 1)P), that is, the (r + 1)-dimensional vector space E over K with basis {x0 , . . . , xr }. Then α ∈ End(Ej ), and Pj α(xj ) = i=0 aji xi

for j = 0, . . . , r. In terms of E, the matrix A associated to α is lower-triangular:   a0 0 . . . 0  ..   ∗ a1 .  . A=   .. ..  . . 0  ∗ ... ∗ ar

T HEOREM 11.34 Every K-automorphism of a non-rational irreducible curve that fixes a place is finite and its order is bounded above by 2p(g + 1)(2g + 1)2 .

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Proof. Let ξi and ξj be eigenvectors of α associated to the eigenvalues ai and aj . It is shown first that, if ai 6= aj , then ordP ξi 6= ordP ξj . Since P cannot be a pole of ξi ξj−1 and its inverse ξi−1 ξj , it may be assumed that P is not a pole of ξi ξj−1 . Then, for the residue map resP associated to P, −1 −1 resP ξi ξj−1 = resP α(ξi ξj−1 ) = resP ai a−1 = ai a−1 j ξi ξj j resP ξi ξj ;

−1 see Example 7.130. As ai a−1 = 0. Therefore P is j 6= 1, this shows that resP ξi ξj −1 a zero of ξi ξj , and the assertion is proved. Now suppose that all eigenvalues are different from one other. Then, for every non-gap mi of Σ at P with 0 ≤ i ≤ r = g + 1, there is an eigenvector ξi of α such that ordP ξi = mi . The subfields K(ξi ) are left invariant by α. Here

[Σ : K(ξr−1 )] = 2g,

and [Σ : K(ξr )] = 2g + 1.

Since p cannot divide two consecutive integers, Lemma 11.33 applies. This immediately implies that ord α ≤ 2p(g + 1)(2g + 1)2 . Next, assume that two eigenvalues, say ai and aj , coincide. Then the Jordan form of α is either diagonal or contains a non-trivial block. So there are independent elements ξ1 , ξ2 ∈ E, such that α(ξ1 ) = ai ξ1 and α(ξ2 ) = aj (λξ1 + ξ2 ) with λ ∈ K. Put z = ξ2 ξ1−1 . Then α(z) = z + λ, and the subfield K(z) is left invariant by α. Since p 6= 0, this implies that αp (z) = z. Also, ρ = [Σ : K(z)] is at most 2(2g + 1) since both div(ξ1 )∞ and div(ξ2 )∞ have degree less than or equal to 2g + 1. From Lemma 11.32, ord α ≤ 2p(g + 1). This completes the proof. 2

11.5 AUTOMORPHISM GROUPS AND THEIR FIXED FIELDS For any subgroup G of AutK (Σ), the set ΣG = {z ∈ Σ | σ(z) = z for all σ ∈ G} is a subfield of Σ, the fixed subfield of G. The most important result on such subfields states that the field extension Σ/ΣG is finite if and only if G is finite; also, if this is the case, then [Σ : ΣG ] = |G|. Actually, as shown later, this occurs for any g > 1, AutK (Σ) being infinite only for rational and elliptic function fields. For a finite subgroup G of AutK (Σ), this yields that Σ/ΣG is a finite Galois extension with Galois group Gal(Σ/ΣG ) isomorphic to G. Geometrically speaking, if Γ is a model of Σ, the irreducible curve whose function field is ΣG is the quotient curve of Γ with respect to G and denoted by Γ/G. So, in the study of quotient curves using function field theory, the principal technical tool is Galois theory. Here, the main goal is to prove that AutK (Σ) is finite for g > 1; this is achieved after some preliminary results established in Section 11.7. In this section, some properties of G that depend on ΣG are discussed. First, a straightforward generalisation of Lemma 11.35 to any subgroup G of the Kautomorphism group AutK (Σ) is stated.

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L EMMA 11.35 Let G be a subgroup of AutK (Σ) which fixes every element of a subfield F of Σ of transcendence degree 1. Then G is finite and |G| ≤ [Σ : F ]. For a finite subgroup of AutK (Σ), the fundamental result on ΣG is the following. T HEOREM 11.36 Let G be a finite subgroup of AutK (Σ). Then Σ/ΣG is a finite Galois extension, and G ∼ = Gal(Σ/ΣG ). Proof. Let ξ ∈ Σ\K, and let {α1 , . . . , αr } be a maximal set of elements of G such that α1 (ξ), . . . , αr (ξ) are distinct. If β ∈ G, then βα1 (ξ), . . . , βαr (ξ) is just a permutation Qr of α1 (ξ), . . . , αr (ξ); otherwise, the latter set is not maximal. Let f (X) = i=1 (X − αi (ξ)). Since the coefficients of a monic polynomial are symmetric polynomials of its roots, every coefficient of f (X) is in ΣG . Further, f (X) is separable. Therefore every element ξ ∈ Σ is a root of a separable polynomial f (X) of degree at most n with coefficients in ΣG . Also, f (X) splits into linear factors in Σ. Thus, the field extension Σ/ΣG is normal and separable, and hence a Galois extension. By Lemma 11.35, it remains to show that [Σ : ΣG ] ≥ n. Let G = {1, α1 , . . . , αn−1 }. For every 1 ≤ i ≤ n − 1, let Fi be the subfield of Σ fixed by αi . Choose an element η ∈ Σ such that no Fi contains η. Then η, α1 (η), . . . , αn−1 (η) are distinct and hence the above polynomial f (X) has degree n. Also, the notation F is used for ΣG , and an element ζ ∈ Σ is chosen such that [F (ζ) : F ] is as large as possible, say m. Then m ≥ n. To prove that [Σ : F ] ≥ n, it must be shown that F (ζ) = Σ. If this is not true, then there exists an element δ ∈ Σ such that δ 6∈ F (ζ). By Theorem A.5, there is an element ω ∈ F (ζ, δ) such that F (ω) = F (ζ, δ). But, from the tower F ⊂ F (ζ) ⊂ F (ζ, δ), it follows that [F (ζ, δ) : F ] > m, whence [F (ω) : F ] > m; this contradicts the choice of m. 2 L EMMA 11.37 Let G be a finite subgroup of AutK (Σ). Then two places P1 and P2 of Σ lie over the same place P ′ of ΣG if and only if there is a K-automorphism α ∈ G such that P2 = P1α . Proof. Let τ1 , τ2 ∈ Σ be primitive representations of the places P1 , P2 . Suppose first that P2 = P1α . Then τ1 α = λτ2 with λ a K-automorphism of K [[t]] . Since α fixes ΣG element-wise, it follows that τ1 (u′ ) = (λτ2 )(u′ ) for every u′ ∈ ΣG . This shows that the place of ΣG represented by τ1 coincides with that represented by τ2 . If P ′ is such a place, both places P1 and P2 are over P ′ . Conversely, suppose that P1 and P2 lie over the same place P ′ of ΣG . There is a K-automorphism λ of K[[t]] such that τ1 (u′ ) = (λτ2 )(u′ ) for every u′ ∈ ΣG . Hence, (τ1−1 λτ2 )(u′ ) = u′ for each u′ ∈ ΣG . This shows that β = τ1−1 λτ2 is a K-automorphism of Σ which fixes the subfield ΣG element-wise. Hence β ∈ G, and α = β −1 takes P1 to P2 . 2 As a by-product, there is also the following result. L EMMA 11.38 Let G be a finite subgroup of AutK (Σ). If P1 and P2 are two places of Σ lying over the same place P ′ of ΣG , then P1 and P2 have the same ramification index ρ.

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The next step is to investigate subgroups G of AutK (Σ) by means of properties depending on the action of G on PΣ . Now, the definition of a group acting on a set is particularised. D EFINITION 11.39 For a group G, (i) the orbit of a place P under G is the set

P G = {P α | α ∈ G} ;

(ii) the stabiliser of P in G is the subgroup

GP = {α ∈ G | P α = P} ;

(iii) an orbit P G is either long or short according as GP is trivial or not. R EMARK 11.40 For G finite, the orbit P G is short if and only if |P G | < |G|. With this terminology, Lemma 11.37 can be reworded as follows. L EMMA 11.41 Let G be a finite subgroup of AutK (Σ). Then two places of Σ lie over the same place of ΣG if and only if they are in the same orbit under the action of G. In other words, there is a one-to-one correspondence between places of ΣG and G-orbits of places of Σ. It may be noted that the number of the short orbits under a finite subgroup G of AutK (Σ) is finite. If this were not true, then the stabiliser GP for infinitely many P ∈ PΣ would contain some non-trivial K-automorphisms. Since G is finite, this could only occur when at least one non-trivial K-automorphism in G fixes an infinite number of places of Σ, a contradiction to Lemma 11.9. A numerical result on short orbits is given in the following theorem. T HEOREM 11.42 Let P be a place of Σ lying over a place P ′ of ΣG . If n = |G| and m = |GP |, then the number of distinct places lying over P ′ is n/m, and the ramification index of each of them is eP = m. Proof. If P = P1 , . . . , Ps are the places of Σ over P ′ , then s = n/m. In fact,   |G| = |GP | · |P G | = |GP | · Σ : ΣG ,   whence s = Σ : ΣG = |G|/|GP | = n/m. Finally, from Lemma 11.38, it follows that ρ1 = · · · = ρs = m. 2 To end this section, a result applied in later sections is given. L EMMA 11.43 Let α be a non-trivial K-automorphism of Σ. If α fixes an element ξ ∈ Σ and a place P of Σ, then ξ is not a uniformising element of Σ at P.

Proof. By (7.1), ordP ξ = eP ordP ′ ξ, where P ′ is the place of Σα under P, and eP is the ramification index of P with respect to the extension Σ/Σα . Since α fixes P, so eP = ord α, whence ordP ξ 6= 1. 2

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11.6 THE STABILISER OF A PLACE In this section, the genus g of Σ is assumed to be positive. L EMMA 11.44 Let z ∈ Σ be a uniformising element of Σ at P. (i) For any α ∈ GP , there is just one cα ∈ K such that ordP (α(z) − cα z) > 1. (ii)

(a) cα is independent of the choice of the uniformising element; (b) the mapping ϕ: G→K α 7→ cα ,

(11.10)

is a group homomorphism with kernel N ; ∼ Zm with gcd(m, p) = 1; (c) the group G/N = (d) if p = 0, then N = {1};

(e) if p > 0, then N is a nilpotent group consisting of all elements in GP whose order is a power of p, and G = N ⋊ H, with H a subgroup of G whose order is prime to p.

Proof. Choose a primitive place representation σ : Σ 7→ K((t)) of P such that σ(z) = t. By (11.3), σ(α(z)) = kt + · · · . Thus ordP (σ(α(z)) − cσ(z)) > 1 if and only if c = k. It is straightforward to check that c is independent of the choice of σ. For another uniformising element z ′ at P, there exists u ∈ K\{0} such that σ(z ′ ) = ut + · · · . Hence ordP (α(z ′ ) − uα(z)) = ordP (z ′ − uz) > 1.

(11.11)

Then ordP (α(z ′ ) − kz ′ ) = ordP ((α(z ′ ) − uα(z)) + u(α(z) − kz) + k(uz − z ′ )), which is greater than 1 by (11.11). Also, ordP (α(z) − kz) > 1. So, (ii)(a) holds. Since k depends on both P and α, but is independent of the choice of the place representation σ of P, so ϕ is well defined. Let α, β ∈ G. Then ordP ((βα)(z) − cβ cα z) = ordP ((βα)(z) − cβ α(z)) + cβ (α(z) − cα z)) ≥ min{ordP ((βα)(z) − cβ α(z)), ordP (cβ (α(z) − cα z))}

= min{ordP (β(α(z)) − cβ α(z)), ordP (cβ (α(z) − cα z))}.

Since both weights are greater than 1, so ordP ((βα)(z) − cβ cα (z)) > 1. By the uniqueness, shown previously, cβα = cβ cα which shows that ϕ is indeed a homomorphism. Hence, G/ ker ϕ is isomorphic to a subgroup of the multiplicative group of K. By Theorem 11.34, there is an integer M which is larger than the order of each element in G. This holds true for any factor group of G, and in particular for G/ ker ϕ. On the other hand, every finite multiplicative subgroup of K is the group of m-th roots of unity in K for a suitable integer m prime to p.

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As H ranges over the finite subgroups of G/ ker ϕ, only a finite number of distinct images ϕ(H) are obtained, as |H|, and hence |ϕ(H)|, is bounded by M . So, all such subgroups ϕ(H) generate a multiplicative cyclic subgroup Γ of K whose order is prime to p and bounded above by the product of all integers smaller than M. Since Γ ∼ = G/ ker ϕ, so (ii) (c) follows. To prove the last two parts, it is first shown that every eigenvalue of α is a power of cα . With the notation in the P proof of Theorem 11.34, every eigenvalue aj can be extracted from α(xj ) = ri=0 aji xi by taking the first coefficient aj = ajj . Since ordP xj = −mj , so xj z mj +1 is a uniformising element of Σ at P. Write σ(xj ) = bj t−mj + . . . , with bj ∈ K ∗ . Then σ(α(xj z mj +1 )) − cα xj z mj +1 ) = σ(α(xj )σ(α(z mj +1 )) − cα σ(xj )σ(z mj +1 ) j +1 = (aj bj cm t + · · · ) − (cα bj t + · · · ), α −m

whence aj = cα j ; this proves the assertion. In particular, if α ∈ ker ϕ, then every eigenvalue of α is equal to 1. Hence, ker ϕ consists of all those α ∈ G for which the associated upper-triangular matrix A has the form   1 0 ... 0  ..   ∗ 1 .  . A=  .  ..  .. . 0  ∗ ... ∗ 1

However, if p = 0, such an α cannot have finite order unless A is the identity matrix, and hence α is the identity automorphism, see Lemmas 11.25 and 11.26 This proves (ii) (d). On the other hand, if p is positive, ker ϕ is a nilpotent group, and any non-trivial element in ker ϕ has order a power of p. In fact, ker ϕ is isomorphic to a group of matrices of the above form. This completes the proof of (ii) (e). 2 L EMMA 11.45 Let p > 0 and g > 0. Assume that H is a non-trivial abelian subgroup of the stabiliser of P satisfying the following properties: (a) the order of each element in H\{1} is a power of p; (b) the fixed field of any non-trivial finite subgroup of H is rational.

Then H is a cyclic group of order p or p2 . Proof. The result holds trivially when H has order p. Let U be a subgroup of H of order p generated by α ∈ H. Then ΣU = Σα is a rational subfield K(ξ) of Σ. From Theorem 11.36, [Σ : K(ξ)] = p. An element in K(ξ) with a pole of multiplicity p at P may be found by a suitable substitution ξ → (aξ + b)/(cξ + d), ad − bc 6= 0, a, b, c, d ∈ K. Actually, P is the unique pole of ξ by (7.1) and Theorem 11.42. So, there is ξ ∈ Σ with ΣU = K(ξ), div(ξ)∞ = pP, and [Σ : ΣU ] = p. Now, to prove the assertion, assume by contradiction that there is an element τ of order p such that τ ∈ H but τ 6∈ U . Since H is abelian, τ and α generate an

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elementary abelian subgroup V of H of order p2 . By Lemma 11.32, ΣV 6= ΣU , showing that τ does not fix all elements in ΣU . On the other hand, ατ = τ α implies that τ preserves ΣU . Since P τ = P, from Theorem 11.14 (ii)(e) it follows that τ (ξ) = ξ + a with a ∈ K\{0}. Replacing ξ by ξ/a gives the relations α(ξ) = ξ,

τ (ξ) = ξ + 1,

div(ξ)∞ = pP.

(11.12)

Interchanging the roles of α and τ in the above argument shows the existence of η ∈ Σ with the following properties: α(η) = η + 1,

τ (η) = η,

div(η)∞ = pP.

(11.13)

Since [Σ : K(ξ, η)] · [K(ξ, η) : K(ξ)] = [Σ : K(ξ)] = p, and K(ξ, η) 6= K(ξ), so Σ = K(ξ, η). Then the curve F with generic point P = (ξ, η) has genus g, and both α and τ act on the plane as translations. In particular, the centre of the branch of F associated to P lies on the line at infinity. To find an equation f (X, Y ) = 0 of F, deduce from the first part of (11.13) that η p − η ∈ K(ξ). Since both η p − η and ξ have no pole centred at an affine point, it follows that η p − η = f1 (ξ) with f1 (X) ∈ K[X]. As div(η p − η)∞ = p2 P while div(ξ)∞ = p, so deg f1 (X) = p. Also, since (η − i)p − (η − i) = η p − η, so f1 (ξ − i) = f1 (ξ)

for every i = 0, 1, . . . , p−1. Given a root a of f1 (X), it follows that the roots of the polynomial f2 (X) = f1 (X − a) are 0, 1, . . . , p − 1. Hence f2 (X) = c(X p − X). Thus η p − η = c(ξ p − ξ) + d, p

p

with c, d ∈ K.

(11.14)

Therefore F = v(d−cX +Y +cX −Y ). This shows that the point E = (0, 1, e), with ep = c, is a (p − 1)-fold point of F. But then F is a rational curve by Exercise 6 in Chapter 1, contradicting the hypothesis g > 0. 2 L EMMA 11.46 Let p > 0 and g > 0. Assume that H is a non-trivial abelian subgroup of the stabiliser of P. If H has property (a) of Lemma 11.45, then H is finite and its order does not exceed p2 (g − 1). Proof. From Hurwitz’s Theorem 7.27, 2(g − 1) = n(2g ′ − 2) + deg(Σ/ΣU ),

where U is a subgroup of H of order n, and g ′ is the genus of ΣU . Further, deg(Σ/ΣU ) > n − 1, because P completely ramifies in the extension Σ/ΣU . For g ′ > 0, it follows that n ≤ 2(g − 1)/(2g ′ − 1) < 2g.

Therefore, either the fixed field of every non-trivial finite subgroup of H has genus 0 or there is a subgroup U of H of order n < 2g such that ΣU has genus g ′ ≥ 1. In the former case, Lemma 11.45 applies and the assertion follows. In the latter case, replace U by a finite maximal subgroup V of H containing U . Both properties hold true for V ; that is, |V | < 2g and ΣV has genus g ′ > 0. Now, it is shown that Lemma 11.45 applies to ΣV in so far as the factor group H = H/V is regarded as a K-automorphism group of ΣV , and the place under

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P is taken as the fixed place of H. Since both commutativity and property (a) are inherited by H, it is only necessary to deal with property (a). Under the natural homomorphism H 7→ H, any subgroup W of H corresponds to a subgroup W of H containing V . By the maximality of V , the genus of ΣW is 0. Then ΣW viewed as a subfield of ΣV has genus 0, showing that H has property (a) of Lemma 11.45. Consequently, the order of H is at most p2 and hence the order of H itself does not exceed p2 (2g − 1). 2 At this point an elementary result from group theory is needed. L EMMA 11.47 Let G be a finite or infinite group of order at least n, containing its centre Z of order p, such that G = G/Z is an elementary abelian p-group. Then √ G contains an abelian subgroup of order at least pn. Lemma 11.44 is improved by showing the finiteness of N . L EMMA 11.48 Let p > 0 and g > 0. Then N is finite and its order does not exceed p2 (g + 1)(2g − 1)2 . Proof. Choose an element x1 ∈ Σ such that div(x1 )∞ = m1 P, where m1 denotes the smallest non-gap of Σ at P. The subfield K(x1 ) of Σ is invariant under any K-automorphism fixing P. From Theorem 11.14 (ii)(e), if α ∈ N , then α(x1 ) = x1 + aα , with aα ∈ K. Hence the map α 7→ aα is a homomorphism of N into the additive group of K; let N1 be its kernel. If N1 is trivial, then |N | ≤ p2 (2g − 1), from Lemma 11.46. So, for the rest of the proof, N1 may be assumed non-trivial. Then the factor group N/N1 is an elementary abelian p-group, possibly trivial. Also, since any α ∈ N1 fixes K(x1 ) element-wise, the order of N1 is at most m1 = [Σ : K(x1 )] by Lemma 11.32. As N is nilpotent, N1 has a subgroup N2 of index p such that N2 is a normal subgroup of N and N1 /N2 is contained in the centre of the factor group N/N2 . By Theorem 11.36, p · [Σ : ΣN2 ] = p · |N2 | = |N1 | ≤ [Σ : K(x1 )].

As [Σ : K(x1 )] = [Σ : ΣN2 ] · [ΣN2 : K(x1 )], it follows that p ≤ [ΣN2 : K(x1 )]. From this, ΣN2 has positive genus g ′ , for, otherwise, ΣN2 = K(ξ) with ξ ∈ Σ and hence div (ξ)∞ = m′ P with m′ < m. Now, as the factor group N/N2 can be regarded as a group of K-automorphisms of ΣN2 , from Lemma 11.46 it follows that any abelian subgroup of N/N2 has order at most p2 (2g ′ − 1). On the other hand, for Z = N1 /N2 , the quotient group N/N2 has the structure described in Lemma 11.47. Therefore, if n′ is chosen such that the order of N/N2 ′ is √ not′ less than n , then N/N √2 contains an abelian subgroup whose order is at least pn . It follows then that pn′ ≤ p2 (2g ′ − 1). Hence, N/N2 has order at most p3 (2g ′ − 1)2 , and the order of N does not exceed p3 (2g ′ − 1)2 m1 p−1 = p2 m1 (2g ′ − 1)2 .

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As m1 ≤ g + 1 and g ′ ≤ g, the order |N | ≤ p2 (g + 1)(2g − 1)2 .

2

To sum up, a complete description of the structure of the stabiliser of a place P of Σ is obtained. T HEOREM 11.49 Let Σ be a function field of genus g > 0. If P is a place of Σ, then the subgroup of AutK (Σ) fixing P is finite and its structure is described as follows. (i) If p = 0, then GP is a cyclic group of order at most 8(2g + 1)3 . (ii)

(a) If p > 0, then a p-Sylow subgroup N of GP is a normal subgroup with |N | ≤ p2 (g + 1)(2g − 1)2 ,

(11.15)

and the quotient group GP /N is a cyclic group with |GP /N | ≤ 2p(2g + 1)2 (g + 1).

(11.16)

(b) GP contains a cyclic subgroup H ∼ = GP /N such that GP = N ⋊ H. All such subgroups H are conjugate in GP . (c) |GP | has an upper bound depending only on p and g. Both (11.15) and (11.16) are approximate bounds. As is shown below, (11.16) can be significantly improved to 4g + 2, and (11.15) can be sharpened at least to |N | ≤ 4pg 2 /(p − 1)2 . 11.7 FINITENESS OF THE K-AUTOMORPHISM GROUP By Theorems 11.14 and 11.94, the K-automorphism group of any rational or elliptic curve is infinite, and these are the only irreducible curves with this property. This important result has already been discussed before. Now, a proof is given. T HEOREM 11.50 The K-automorphism group of an irreducible curve F of genus g ≥ 2 is finite. Proof. Let Σ = K(F). By Theorem 11.49, it suffices to show that AutK (Σ) preserves a finite set of places of Σ. In fact, if an AutK (Σ)-invariant set of places has size at most m, and the stabiliser of a place in the set has order n, then mn is an upper bound on AutK (Σ). If Σ is hyperelliptic, Lemma 11.21 provides such a set of places of size at most 2g + 2. In the non-hyperelliptic case, the canonical series is used. By Lemma 11.23, AutK (Σ) is a linear collineation group of PG(g − 1, K) which preserves the canonical curve Γ of Σ. In particular, AutK (Σ) preserves the set of all Weierstrass points of Γ. This set is finite as its size does not exceed  P g−1 i=0 ǫi (2g − 2) + g(2g − 2), by (7.13).

2

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R EMARK 11.51 Let F be an irreducible, curve of genus g > 2. Assume that F is not hyperelliptic, and consider its canonical curve K. Then the order sequence of K is ǫ0 = 0 < ǫ1 = 1 < · · · < ǫg−1 ≤ 2g − 2, whence the number of its Weierstrass points is at most (g − 1)(2g − 1)(2g − 2). Hence, |AutK (Σ)| < 4g 3 |GP | for P a Weierstrass point. Thus, an upper bound for |AutK (Σ)| follows from the upper bound for |AutK (Σ)P | given in Theorem 11.49. An important consequence of Theorem 11.50 is that, if Σ has genus g ≥ 2, then Hurwitz’s Theorem 7.27 applies to every extension Σ/ΣG such that ΣG is the fixed field of a non-trivial subgroup G of AutK (Σ). Using the geometric term of quotient curve, the following result is obtained. C OROLLARY 11.52 Let F be an irreducible curve of genus g > 1. For a nontrivial K-automorphism group G of F, let g ′ be the genus of the quotient curve F/G. Then 2g − 2 = |G|(2g ′ − 2) + d,

(11.17)

where d = deg D(Σ/ΣG ) with Σ = K(F) and ΣG = K(F/G). R EMARK 11.53 Corollary 11.52 holds true for rational and elliptic curves when G is finite. Although the aim of the proof of Theorem 11.50 was only to prove the finiteness of AutK (Σ), an upper bound for the order of AutK (Σ) in terms of p and g can be extracted from it; see Remark 11.51. A further effort shows that such an upper bound can be improved significantly. In this direction, the first step is Theorem 11.56, which makes evident the importance of short orbits in investigating curves with large K-automorphism groups. Two types of short orbits are distinguished. D EFINITION 11.54 A short orbit of a K-automorphism group G is tame or nontame (wild) according as the order of GP for one and hence every place P in the orbit is prime or not to the characteristic p of K. R EMARK 11.55 If p = 0, then every short orbit is tame. T HEOREM 11.56 Let F be an irreducible curve of genus g ≥ 2. (i) If G is a K-automorphism group of F, then Hurwitz’s upper bound |G| ≤ 84(g − 1) holds in general with exceptions occurring only in positive characteristic. (ii) In terms of the function field Σ of F, exceptions can only occur when p > 0, the fixed field ΣG is rational, and G has at most three short orbits, as follows: (a) exactly three short orbits, two tame and one non-tame, with p ≥ 3;

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(b) exactly two short orbits, both non-tame; (c) only one short orbit which is non-tame; (d) exactly two short orbits, one tame and one non-tame. Proof. The proof uses Corollary 11.52 following the original idea of Hurwitz. By Theorem 11.42, equation (11.17) may be rewritten in the form 2g − 2 = |G| · (2g ′ − 2 + d′ ),

(11.18)

where d′P =

dP dP = , eP |GP |

and d′ =

P

d′P ;

here, the summation is only over a set of representatives of places in PΣ , exactly one from each short orbit. So, it is necessary to investigate the possibilities for |G| according to the number r of short orbits of |G| on PΣ . Now, take (7.7) into account; this gives dP ≥ eP − 1, with equality holding if and only if either p = 0 or eP is prime to p. Therefore, if dP > 0, then d′P ≥ 21 . Also, if d > 0, then d′ ≥ 12 . If g ′ ≥ 2, then |G| ≤ g − 1. For g ′ = 1, it follows that d′ > 0 since g ≥ 2, and hence |G| ≤ 4(g − 1). So, assume that g ′ = 0; that is, ΣG is rational. Then ′

2g − 2 = |G| · (d′ − 2).

(11.19)

In particular, d > 2. Therefore G has some, say r ≥ 1, short orbits on PΣ . Take representatives, Q1 , . . . , Qr , from each short orbit, and let d′i = d′Qi . After a change of indices, it may be assumed that d′i ≤ d′j for i ≤ j. (I) When r ≥ 5, then d′ ≥ 52 , and hence |G| ≤ 4(g − 1). (II) When r = 4, then d′ > 2 and d′i > 21 for at least one place P. As d′i > implies d′i ≥ 23 , so d′ − 2 ≥ 16 , whence |G| ≤ 12(g − 1).

1 2

(III) When r = 3, then again use d′ − 2 > 0. If d′1 = 32 then d′3 ≥ 43 and hence |G| ≤ 24(g −1). If d′1 = 12 , d′2 ≥ 43 , then |G| ≤ 40(g −1). Also, d′1 = 12 and d′2 = 23 imply that d′3 ≥ 76 , whence |G| ≤ 84(g −1). Finally, if d′1 = d′2 = 12 , then d′3 > 1, which can only occur when p > 2 and eQ1 = eQ2 = 2 while eQ3 is divisible by p. This leads to (a). (IV) When r = 2, then d′ = d′1 + d′2 > 2. This can only occur when either d′1 , or d′2 , or both are greater than 1. Hence, one of (b) and (d) occurs. (V) When r = 1, then d′ = d′1 > 2, and case (c) occurs.

2

For p 6= 0, there are so many exceptions to Hurwitz’s bound that their classification appears out of range. However, those with |G| greater than g 3 seem not too numerous, at least for g large enough, and their determination might be possible. So far, this has been done for |G| ≥ 8g 3 , and the classification is stated in Theorem 11.127; it requires some deeper preliminary results that are proved in the next two sections.

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11.8 TAME AUTOMORPHISM GROUPS Let G be a non-trivial K-automorphism group of an irreducible curve F. In this and the next section, the important equation (11.17) linking the genus g of the curve F to the genus g ′ of its quotient curve F ′ = F/G is given a more explicit form. The idea is to examine the action of G on the set of places of the function field Σ of F. Here, the tame case is considered; that is, it is supposed that the orders of the stabilisers are all prime to p. T HEOREM 11.57 Let G be a finite K-automorphism group of an irreducible curve F. If |GP | is prime to p for every place P of Σ = K(F), then Ps 2g − 2 = n(2g ′ − 2) + i=1 (n − li ), (11.20) where n = |G|, and l1 , . . . , ls denote the sizes of the short orbits of G.

Proof. Choose a place P of Σ, and let P ′ be a place of ΣG lying under P. By Theorem 11.42, the ramification index ρ is equal to m, with m = |GP |. Since by hypothesis p ∤ n, by the remark made after the proof of Hurwitz’s Theorem 7.27, ordP dξ = (ρ − 1) + ordP ′ dξ. Hence D(Σ/ΣG ) =

P

(m − 1) P,

(11.21) P with P rangingPover all places of Σ. Therefore deg D(Σ/Σ ) = (m − 1). P To evaluate (m−1), consider the finite sub-sum (mi −1) restricted to those places P of Σ which lie over a given place Pi′ of ΣG . By Theorem 11.42, there are precisely n/mi distinct places over Pi′ , each of them contributing mi − 1. Hence, from every place Pi′ , a contribution n(mi − 1)/mi is obtained. Therefore P P P (m − 1) = (n/mi )(mi − 1) = (n − n/mi ). As ℓi = n/mi , so

deg(Σ/ΣG ) = From Corollary 11.52,

P

(m − 1) =

Ps

i=1

G

(n − li ).

  2g − 2 = Σ : ΣG · (2g ′ − 2) + deg D(Σ/ΣG ) P = n(2g ′ − 2) + si=1 (n − li ).

2

E XAMPLE 11.58 Over F13 , let

F = v(X06 + X16 + X26 + 3X02 X12 X22 ), F ′ = v(X03 + X13 + X23 + 3X0 X1 X2 ).

Then F → F ′ by the map (x0 , x1 , x2 ) 7→ (x20 , x21 , x22 ), and ∼ Z2 × Z2 . G = {(X0 , X1 , X2 ) 7→ (±X0 , ±X1 , X2 )} =

Then n = 4, g ′ = 1, g = 10, s = 9, li = 2 for all i; so (11.20) is satisfied for these values. The short orbits consist of the pairs of points of F mapping to an inflexion of F ′ ; see Table 11.1.

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Table 11.1 A sextic curve 4-covers a cubic

(±1, ±2, 1) (±2, ±1, 1) (±2, ±5, 1) (±4, ±6, 1) (±6, ±4, 1) (±6, ±6, 1)

7→ 7→ 7→ 7 → 7 → 7 →

(1, 4, 1) (±5, ±2, 1) → 7 (−1, 4, 1) (4, 1, 1) (±3, ±5, 1) → 7 (−4, −1, 1) (4, −1, 1) (±5, ±3, 1) 7→ (−1, −4, 1) (3, −3, 1) = (−1, 1, 4) (−3, 3, 1) = (1, −1, 4) (−3, −3, 1) = (1, 1, 4)

(0, ±2, 1) (±2, 0, 1) (±2, 1, 0) (0, ±6, 1) (±6, 0, 1) (±6, 1, 0)

7→ 7→ 7→ 7→ 7 → 7 →

(0, 4, 1) (4, 0, 1) (4, 1, 0) (0, −3, 1) = (0, 1, 4) (−3, 0, 1) = (1, 0, 4) (−3, 1, 0) = (1, 4, 0)

(0, ±5, 1) (±5, 0, 1) (±5, 1, 0)

7→ (0, −1, 1) 7→ (−1, 0, 1) 7→ (−1, 1, 0)

In the special case that G has prime order, Theorem 11.57 reads as follows. T HEOREM 11.59 Let α be a finite non-trivial K-automorphism of an irreducible curve whose order is prime to p. If ρ(α) is the number of fixed places of α and n = ord α, then 2g − 2 = n(2g ′ − 2) + ρ(α)(n − 1).

(11.22)

For tame K-automorphism groups, a significant improvement on the bound (ii) in Theorem 11.49 is the following result. T HEOREM 11.60 Let F be an irreducible curve of genus g > 0, and let GP be a K-automorphism group of F fixing a place P. If the order n of GP is prime to p, then n ≤ 4g + 2.

(11.23)

Proof. The place P itself constitutes a short orbit of size 1. With ℓs = 1 in (11.20), Ps−1 2g − 1 = n(2g ′ − 1) + n i=1 (1 − ℓi /n), (11.24) where ℓ1 , . . . , ℓs−1 denote the sizes of the remaining short orbits of GP . If g ′ > 0, the assertion follows. So, now assume that g ′ = 0. Then (11.24) becomes the following: Ps−1 2g − 1 = i=1 (n − ℓi ) − n. (11.25)

Since ℓi divides n, so n − ℓi ≥ 21 n. Hence, if s ≥ 4, then 2g − 1 ≥ 23 n − n, whence n ≤ 2(2g − 1) ≤ 4g + 2. The same argument works for s = 3 as long as ℓ1 + ℓ2 ≤ 21 n. Further, neither s = 2 nor s = 1 can actually occur. So it remains to investigate the case that 2g−1 = n−(ℓ1 +ℓ2 ) with ℓ1 +ℓ2 > 21 n. Up to an index change, it may be assumed that ℓ1 ≥ ℓ2 . Then the possibilities for (ℓ1 , ℓ2 ) are the following: ( 13 n, 51 n),

( 31 n, 14 n),

( 31 n, 13 n),

( 21 n, ≤ 13 n).

(11.26)

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Write G = GP for brevity, and denote by Hi the subgroup of G which fixes one, and hence every, place in the short orbit of size ℓi for i = 1, 2. Let H be the subgroup of G generated by H1 and H2 . Since G is cyclic by Theorem 11.49, so |H| = n/gcd(ℓ1 , ℓ2 ). It will be shown that G = H. By way of contradiction, ΣG is assumed to be a proper subfield of ΣH . Then H [Σ : ΣG ] = |G|/|H|. Let P ′ be the place of ΣH lying under P in the extension Σ/ΣH . Then P ′ completely ramifies in ΣH /ΣG . From the definition of H, it can be deduced that no further place of ΣH ramifies in ΣH /ΣG . To do this, note that, for any place Q from the orbit of length ℓ1 , GQ = HQ and |G| |GQ | ℓ1 ℓ1 |QG | = · H = H = H . |H| |HQ | |Q | |Q | |Q |

Then the same is true for ℓ2 . If Q is a place not in a short orbit, then GQ is the identity subgroup, and hence GQ = HQ . Therefore |G|/|H| = |QG |/|QH | for every place Q = 6 P of Σ. This proves the assertion. Corollary 11.52 gives the equation 2g − 2 = −2m′ + m′ − 1,

where g is the genus of ΣH , and m′ = |G|/|H|. But this can only occur for m′ = 1, giving the desired result that H = G. Hence ℓ1 and ℓ2 are prime to each other. From (11.26), the following cases can occur: (ℓ1 , ℓ2 ) = ( 13 n, 15 n), (ℓ1 , ℓ2 ) = ( 31 n, 14 n), (ℓ1 , ℓ2 ) = ( 12 n, k1 n), k odd,

|GP | = 15, |GP | = 12,

|GP | = 2k,

g = 4; g = 3; g = 21 (k − 1).

In each of these cases, n ≤ 4g + 2. This completes the proof.

2

R EMARK 11.61 If the hypothesis that p ∤ n is dropped, then the proof of Theorem 11.57 only provides a lower bound for 2g − 2; that is, Ps 2g − 2 ≥ n(2g ′ − 2) + i=1 (n − ℓi ). (11.27) However, for a p-group G of order n, if g is replaced by the p-rank γ, a relationship similar to (11.20) holds.

T HEOREM 11.62 (Deuring–Shafarevich) Let G be a K-automorphism group of an irreducible curve F whose order n is a power of p. Then Ps γ − 1 = n(γ ′ − 1) + i=1 (n − ℓi ), (11.28) where γ and γ ′ are the p-ranks of F and its quotient curve F/G, while ℓ1 , . . . , ℓs are the sizes of the short orbits of G on the places of Σ.

If G has no short orbits, the possibilities for the structure of G is described in the following theorem.

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T HEOREM 11.63 Let G be a K-automorphism group of an irreducible curve F whose order n is a power of p. If G has no short orbits, then G is isomorphic to a quotient group of the free group with γ generators. Since G has no short orbits means that the extension Σ/ΣG is unramified, the question of determining the maximum number of unramified independent extensions of a given degree ph arises. Interestingly, the answer for h = 1 gives a characterisation of the p-rank of Σ. T HEOREM 11.64 The p-rank of Σ is equal to the maximum number of cyclic unramified independent extensions of degree p of Σ. Theorem 11.63 may be reworded in terms of such an extension.

11.9 NON-TAME AUTOMORPHISM GROUPS As pointed out in Remark 11.61, the exact formula (11.20) depending only upon the action of G on places does not hold true for non-tame subgroups. However, a similar, although more complicated, formula, relying on more concepts and arguments exists for non-tame subgroups; this is the Hilbert Different Formula stated in Theorem 11.70. The goal in this section is to give a proof of this. Further, for non-tame K-automorphism groups, a significant improvement to Theorem 11.49 (ii) is obtained by using Theorem 11.70. Given a place P of Σ, choose a primitive representation τ of P together with a uniformising element x ∈ Σ at P, and assume that τ (x) = t. By Lemma 11.43, α(x) 6= x for any non-trivial K-automorphism α ∈ GP . Also, n = |GP | divides ordt τ (ξ) for any ξ ∈ ΣGP ; see (7.1) and Theorem 11.36. More precisely, there exists s(t) ∈ K[[t]] such that τ (ξ) = σ(s(t)) with σ ∈ K((s)). Such an element s(t) is independent of the choice of ξ, and σ is a primitive representation of the place P ′ of ΣGP lying under P; see Section 7.2. L EMMA 11.65 The set {1, x, x2 , . . . , xn−1 } is a basis for Σ/ΣGP .

Proof. By Theorem 11.36, it suffices to prove that 1, x, x2 , . . . , xn−1 are linearly independent over ΣGP . Suppose, on the contrary, that ξ0 + ξ1 x + · · · + ξn−1 xn−1 = 0,

where ξ0 , ξ1 , . . . , ξn−1 ∈ ΣGP but not all ξi = 0. Then

τ (ξ0 ) + τ (ξ1 )τ (x) + · · · + τ (ξn−1 )(τ (x))n−1 = 0,

showing that two or more of the n terms on the left-hand side have the same order. This cannot actually occur by the following argument. If 0 ≤ i ≤ n − 1, then ordt τ (ξi )τ (xi ) = ordt τ (ξi ) + i ordt τ (x) = ki n + i

for an integer ki ; the same holds for every j with 0 ≤ j ≤ n − 1. This implies that ordt τ (ξi )τ (xi ) = ordt τ (ξj )τ (xj ), and hence n(kj − ki ) = i − j. But this is impossible since 0 ≤ i, j < n and i 6= j. 2

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L EMMA 11.66 There exist ξ0 , . . . , ξn−1 ∈ ΣGP satisfying

ξ0 + ξ1 x + · · · + ξn−1 xn−1 + xn = 0,

(11.29)

with ordt τ (ξ0 ) = n, and ordt τ (ξi ) ≥ n for i = 1, . . . n − 1.

Proof. By Lemma 11.65, there exist η0 , . . . , ηn ∈ ΣGP , ηn 6= 0, satisfying η0 + η1 x + · · · + ηn−1 xn−1 + ηn xn = 0.

(11.30)

It follows that τ (η0 ) + τ (η1 )τ (x) + · · · + τ (ηn )(τ (x))n = 0. By the argument used in the proof of Lemma 11.65, the equation ordt (τ (ηi )τ (xi )) = ordt (τ (ηj )τ (xj )) implies that n(kj − ki ) = i − j. Here, this is only possible for i = n, j = 0 and ordt τ (η0 ) = n + ordt τ (ηn ). It also follows that ordt τ (ηi ) ≥ n + ordt τ (ηn ). Let ξi = ηi /ηn ; then ordt τ (ξ0 ) = n, while ordt τ (ξi ) ≥ n for 1 ≤ i ≤ n − 1. Dividing both sides in equation (11.30) by ηn , the result follows. 2 For i = 0, . . . , n − 1, let τ (ξi ) = σi where σi ∈ K((s)) and s ∈ K[[t]] with ordt s = n. By Lemma 11.66, ords σ0 = 1 and ords σi ≥ 1 for i = 1, . . . , n − 1. From equation (11.29), σ0 + σ1 t + · · · + σn−1 tn−1 + tn = 0,

(11.31)

Derivation of (11.31) gives the equation   dσ0 dσ1 dσn−1 n−1 ds + t+ ··· + t ds ds ds dt

+σ1 + 2σ2 t + · · · + (n − 1)σn−1 tn−2 + ntn−1 = 0.

Define the polynomial Φ(T ) ∈ K(s)[T ]:

Φ(T ) = σ0 + σ1 T + · · · + σn−1 T n−1 + T n .

Then



dσ1 dσn−1 n−1 dσ0 + t+ ···+ t ds ds ds

L EMMA 11.67

Φ(T ) =

Q



α∈GP (T

ds dΦ(T ) =− . dt dT T =t

− (τ α)(x)).

(11.32) (11.33)

Proof. From (11.31), Φ(t) = 0. To find the other roots of Φ(T ), first note in (11.29) that α(ξi ) = ξi for every α ∈ GP . From (11.29), it follows that ξ0 + ξ1 α(x) + · · · + ξn−1 α(x)n−1 + α(x)n = 0.

Let δ(t) = (τ α)(x); then, since α(x) is also a uniformising element of Σ, so ordt δ(t) = 1. Applying τ to the last equation shows that σ0 + σ1 δ(t) + · · · + σn−1 δ(t)n−1 + δ(t)n = 0.

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As α ranges over GP , the number of distinct δ(t) obtained is n. This depends on Lemma 11.43 taking into account that τ is a K-isomorphism from Σ to K((t)). Since Φ(T ) has degree n, it follows that the roots of Φ(T ) in Σ are the n distinct elements δ(t) = (τ α)(x) with α ranging over GP . 2 From (11.33),

Q dΦ(T ) = α∈G∗ (t − (τ α)(x)), P dT T =t

(11.34)

with α ranging over the set G∗P of non-trivial K-automorphisms in GP . From (11.32) and (11.34),   Q dσ1 dσn−1 n−1 ds dσ0 + t + ···+ t = − α∈G∗ (t − (τ α)(x)). P ds ds ds dt

Since ords σ0 = 0 and ords δi ≥ 1 for i = 1, . . . , n − 1, this gives the equation ds P ordt (11.35) = α∈G∗ ordt ((τ α)(x) − t). P dt Therefore the following result is obtained. T HEOREM 11.68 For a subgroup G of AutK (Σ) and a place P of Σ, let G∗P be the set of all non-trivial elements of GP . Then the different D(Σ/ΣG ) of the separable extension Σ/ΣG is given by the formula P  P (11.36) D(Σ/ΣG ) = P∈PΣ α∈G∗ ordP (α(x) − x) P P

for any uniformising element x ∈ Σ at P. (i)

For i = 0, 1, . . ., define GP to be the set of all elements α ∈ GP such that For α, β ∈

ordP (α(x) − x)) ≥ i + 1.

(i) GP ,

ordP (βα(x) − x) = ordP (βα(x) − α(x) + α(x) − x)

≥ min{ordP (β(α(x)) − α(x)), ordP (α(x) − x)}.

Since α(x) is still a uniformising element at P, so

ordP (β(α(x)) − α(x)) ≥ i + 1.

Hence ordP (βα(x) − x) ≥ i + 1. This leads to the following definition. (i)

D EFINITION 11.69 The i-th ramification group GP at P is (i)

GP = {α | ordP (α(x) − x) ≥ i + 1, α ∈ GP },

where x ∈ Σ is a uniformising element of Σ at P.

From Theorem 11.68, the following important result is deduced. T HEOREM 11.70 (Hilbert Different Formula) P P (i) D(Σ/ΣG ) = P∈PΣ dP P, with dP = i≥0 (|GP | − 1).

(11.37)

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E XAMPLE 11.71 As in Example 4.33, Hq = v(Y q + Y − X q+1 ) is the Hermitian curve; it is non-singular and contains the origin. If Σ = K(x, y) with the relation y q + y = xq+1 , then Σ = K(H). For any a, b ∈ Fq2 with bq + b = aq+1 , the birational transformation x + ay y α(a,b) : x′ = q , y′ = q a x + by + 1 a x + by + 1 defines a K-automorphism of Σ. Such a K-automorphism fixes the place P arising from the unique branch of Hq centred at the origin. Further, these automorphisms form a subgroup G of AutK (Σ) of order q 3 , and those with a = 0 form a subgroup M of order q. Since the tangent to H at the origin is v(Y ), so x is a uniformising element of Σ at P, and −aq x2 + ay − bxy . 1 + aq x + by From Example 4.33, ordP x = 1, ordP y = q + 1. Hence  2 for a 6= 0, ordP (α(a,b) (x) − x) = q + 2 for a = 0. α(a,b) (x) − x =

(0)

(q+2)

(q+1)

(2)

(1)

= M , but GP Therefore GP = GP = G and GP = . . . GP From this, dP = 2(q 3 − 1) + (q − 1)q = 2q 3 + q 2 − q − 2.

is trivial.

Theorem 11.70 and Hurwitz’s Theorem 7.27, see also Corollary 11.52, have the following corollary. T HEOREM 11.72 For a finite K-automorphism group G of an irreducible curve F, let F/G be the corresponding quotient curve; equivalently, in terms of function fields, Σ = K(F) and ΣG = K(F/G). Then P P (i) (11.38) 2g − 2 = |G|(2g ′ − 2) + P∈PΣ i≥0 (|GP | − 1), where g and g ′ are the genera of F and F/G.

E XAMPLE 11.73 With the notation as in Example 11.71, no non-trivial element in G fixes a place of Σ other than P. Further, ΣG is rational. Now, (11.38) reads, 2g − 2 = −2q 3 + 2q 3 + q 2 − q − 2, verifying that Hq has genus g = 21 q(q − 1). (1)

(0)

The subgroup chain GP ≥ GP ≥ . . . is described in the following theorem. T HEOREM 11.74

(0)

(i) GP is the stabiliser of P.

(1)

(ii) GP is the unique maximal normal p-subgroup of GP . (i)

(i)

(i+1)

(iii) For i ≥ 1, the group GP is normal in GP and the group GP /GP elementary abelian p-group. (0)

Proof. Statement (ii) follows from the definition of GP and Lemma 11.44. To prove (iii), let i ≥ 1. From (11.34), (τ α)(x) − t = cα ti + a(t),

a(t) ∈ K[[t]], ordt a(t) > i,

is an

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for every K-automorphism α ∈ GP . The mapping (i)

φ : GP → (K, +), α 7→ cα

(11.39) (i) GP

and is a group homomorphism. To show this, let α, β ∈ a(x) = α(x) − x − cα xi , b(x) = β(x) − x − cβ xi . Then both ordP b(x) and ordP c(x) are greater than i. Now, (αβ)(x) = α(β(x)) = α(x + cβ xi + b(x)) = α(x) + cβ α(x)i + α(b(x)) = x + cα xi + a(x) + cβ (x + cα xi + a(x))i + α(b(x)) = x + (cα + cβ )xi + d(x). Since ordP d(x) > i, the assertion follows. (i+1) On the other hand, ker φ consists of all K-automorphisms belonging to GP . (i+1) (i) (i) (i+1) is is a normal subgroup of GP , and the factor group GP /GP Hence GP isomorphic to an additive subgroup of K. Part (iii) now follows from the fact that finite additive subgroups of K are abelian groups in which every non-trivial element has order p. 2 (1)

(0)

In the subgroup chain, GP = GP > GP > · · · , consecutive non-trivial subgroups may coincide. Sometimes, repetitions in the chain are deleted so that the Hilbert Different Formula (11.37) only involves distinct ramification groups. To do this, let v−1 = 0 < v0 < v1 < · · · be the strictly increasing sequence of all non-negative integers which are of the form ordP (α(x) − x) for some α ∈ GP . Then vi is the i-th ramification number of GP . Note that (i) v0 = min{ordP (α(x) − x) | α ∈ GP }, vi = min{ordP (α(x) − x) | α ∈ UP }, (0)

where UP = GP and

(i+1)

UP

(i)

= {α ∈ UP | ordP (α(x) − x) ≥ vi + 1}. (1)

(1)

Also, v0 = 1 if and only if GP > GP ; that is, GP = GP ⋊ H with H a non-trivial tame subgroup. (1) (0) The subgroup chain GP = UP > UP > · · · is strictly decreasing, and P (i) (11.40) dP = i≥0 (vi − vi−1 )(|UP | − 1). Therefore (11.38) can also be written as follows: P P (i) (11.41) 2g − 2 = |G|(2g ′ − 2) + P∈PΣ i≥0 (vi − vi−1 )(|UP | − 1). (i)

(i+1)

if and only if vi = ordP (α(ξ) − ξ) for a A useful fact is that α ∈ UP \UP uniformising element of Σ. Some further properties of the ramification subgroups may be proved by using companion K-automorphisms, introduced in Section 11.1. This essentially depends on the fact that, in terms of the companion K-automorphism λα of α, the equation, ordP (α(x) − x) = k, means that λα (t) = t + ctk + · · · , c 6= 0.

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L EMMA 11.75 Let α ∈ GP and β ∈ GP , k ≥ 1. (k+1)

(1)

(i) If α 6∈ GP , then the commutator [α, β] = αβα−1 β −1 belongs to GP (1) (k+1) and only if either αk ∈ GP or β ∈ GP . (l)

(k+l+1)

(ii) If α ∈ GP with l ≥ 1, then the commutator [α, β] belongs to GP (k+1)

(iii) Let ord α = pr s with p ∤ s. If β 6∈ GP

if

.

and αβ = βα, then s divides k. (1)

(iv) Let k be an integer not divisible by |GP |/|GP |. If GP is abelian, then (k+1) (k) . GP = GP (k)

(k+1)

are all congruent modulo p; in (v) The integers k ≥ 1 satisfying GP 6= GP terms of ramification indices, vi ≡ vj (mod p), for all i, j ≥ 1. (k)

Proof. Since GP is a normal subgroup of GP , the commutator γ = [α, β] belongs (k) to GP . Write P λα (t) = ut + i≥i0 ui ti , with ui0 6= 0, P λβ (t) = t + j≥j0 wj tj , with wj0 6= 0, P λγ (t) = t + m≥m0 vm tm .

By Lemma 11.17, ord α = pr s, where s is the smallest positive integer satisfying (k) us = 1; the hypothesis that β ∈ GP with k ≥ 1 means that k ≤ j0 − 1. Also, it may be that λγ (t) = t, and this occurs if and only if αβ = βα; otherwise, γ is a (m −1) non-trivial K-automorphism belonging to ∈ GP 0 . Then P P P λβ λα (t) = ut + i≥i0 ui ti + j≥j0 wj (ut + i≥i0 ui ti )j , P P P λγ λα λβ (t) = ut + u j≥j0 wj tj + i≥i0 ui (t + j≥j0 wj tj )i P P P P + m≥m0 vm [ut + u j≥j0 wj tj + i≥i0 (t + j≥j0 wj tj )i ]m .

If u 6= 1, then βα = γαβ implies that

0 = wj0 (uj0 − u)tj0 + vm0 um0 tm0 + ctr + · · · ,

where r ≥ j0 + 1. Hence, two cases can occur: (a) uj0 −1 = 1 and either γ = 1, or m0 ≥ j0 + 1; (b) uj0 −1 6= 1 and j0 = m0 . In the former case, αj0 −1 has order a power of p, while in the latter case γ 6= 1. Now, a straightforward argument proves (i). If u = 1, then αβ = γβα implies m0 ≥ i0 + j0 − 1. Hence m0 − 1 ≥ (i0 − 1) + (j0 − 1) + 1, whence (ii) follows.

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Let u 6= 1. If αβ = βα, then λγ = t, and hence uj0 −1 = 1. Thus αj0 −1 ∈ GP . (k+1) , then k = j0 − 1, and hence s | k, as This implies that s | (j0 − 1). If β 6∈ GP stated in (iii). (1) Since GP is the semidirect product of its normal subgroup GP by a cyclic (1) subgroup of order s = |GP |/|GP |, some α ∈ GP has order s. If k does not divide (k+1) s, then αk 6= 1. By (i), β ∈ GP . This proves (iv). (l) Choose an element α from the last non-trivial ramification group, say GP . Then (k) (l+1) = 1. Since β ∈ GP with k ≥ 1, from (ii) αβ = βα follows. To show GP that i0 − j0 is divisible by p, assume that p does not divide both i0 and j0 . Then, αβ = βα implies that 0 = (j0 − i0 )uj0 wj0 tj0 −1+i0 + · · · .

Therefore j0 ≡ i0 (mod p). On the other hand, β 6∈ Gk+1 implies that k = j0 − 1; P similarly, l = i0 − 1. Hence, k ≡ l (mod p). From this, (v) follows. 2 The following two results are corollaries of Lemma 11.75 (i). L EMMA 11.76 Let A be an abelian subgroup of AutK (Σ) that fixes m ≥ 1 places. (i) If P is one of the fixed places and g ′ is the genus of ΣA , then   |A| (1) 2g − 2 ≥ |A|(2g ′ − 2) + m |A| − 1 + (1) (|AP | − 1) . |AP |

(11.42)

(ii) If G is non-tame, then 2g − 2 ≥ |A|(2g ′ + 23 m − 2) − m.

(11.43)

(1) AP

Proof. For a fixed place P ∈ PΣ of A, let A = AP = ⋊ H with p ∤ |H|; see Theorem 11.49. Since A is abelian, from Lemma 11.75 (i), it follows that (h) (1) AP = · · · = AP with h = |H|. Now, (11.42) follows from Theorem 11.72 (1) (1) (1) applied to A. If AP is non-trivial, then |AP | − 1 ≥ 21 |AP |. Hence (11.42) implies (11.43). 2 L EMMA 11.77 Let G be an abelian non-tame subgroup of AutK (Σ) fixing a place (1) P. If G = GP ⋊ H with p ∤ |H|, then (1)

(1)

2g ≥ 2g ′ |GP | + (|H| − 1)(|GP | − 1). (1)

(11.44) (h)

Proof. Since G is abelian, from (i) of Lemma 11.75, GP = · · · = GP where (1) 2 h = |H|. Then (11.44) follows from Theorem 11.72 applied to GP . From Theorem 11.49, if g > 0, then GP is finite and its order has an upper bound depending only on p and g. In the notation of this section, Theorem 11.49 (1) states that |GP | ≤ p2 (g + 1)(2g − 1)2 . The aim is to improve this bound. T HEOREM 11.78 Let F be an irreducible curve of positive genus g, and let GP be a K-automorphism group of F fixing a place P. Then 4p 2 (1) |GP | ≤ g . p−1

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More precisely, if Fi is the fixed subfield of GP , then one of the following cases occurs: (1)

(i) F1 is not rational, and |GP | ≤ g; (1)

(ii) F1 is rational, GP has a short orbit on PΣ other than {P}, and p (1) |GP | ≤ g; p−1 (1)

(iii) F1 and F2 are rational, {P} is the unique short orbit of GP on PΣ , and (2)

(1)

|GP | ≤

4|GP |

(2) (|GP |

−

1)2

g2 ≤

4p g2. (p − 1)2

(1)

Proof. From Theorem 11.72 applied to GP , (1)

2g − 2 = |GP |(2g1 − 2) + d,

(11.45)

where g1 is the genus of F1 and d = deg D(Σ/F1 ). Since P completely ramifies (1) (1) (1) in Σ/F1 , so eP = |GP |. Since |GP | is a power of p, hence P |GP | is equal to the order of the first ramification group. Write D(Σ/F1 ) = dP P. By the Hilbert Different Formula (11.37), (1)

dP ≥ 2|GP | − 2.

(11.46)

When g1 > 0, (11.45) and (11.46) imply that (1)

2g − 2 ≥ d ≥ dP ≥ 2|GP | − 2, (1)

whence |GP | ≤ g. From now on, assume that g1 = 0. Then (11.45) becomes the following: (1)

2g − 2 = −2|GP | + d.

(11.47)

In case (ii), there is one more place Q with the above properties. As usual, eQ is its ramification index, and dQ is its weight in the different divisor D(Σ/F1 ). In (1) particular, eQ ≤ |GP |. Since eQ is a power of p and F1 is rational, again from the Hilbert Different Formula (11.37), dQ ≥ 2eQ − 2. Taking (11.46) into account, it follows that (1)

(1)

2g − 2 ≥ −2|GP | + dP + (1)

|GP | dQ eQ

(1)

(1)

≥ −2|GP | + 2|GP | − 2 + 2|GP | hence g≥ which is the desired result.

p − 1 (1) eQ − 1 (1) |GP | ≥ |GP |, eQ p

eQ − 1 ; eQ

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Now, consider the case that F1 is rational and P is the unique ramified place. First, it is shown that F2 is rational as well. Put d1 = deg D(Σ/F1 ),

d∗ = deg D(F2 /F1 ).

d2 = deg D(Σ/F2 ),

From (7.8), (2)

d1 = d2 + [Σ : F2 ]d∗ = d2 + |GP |d∗ .

Also, d1 d2

= =

(11.48)

P (i) (1) 2(|GP | − 1) + i≥2 (|GP | − 1), P (2) 2(|GP | − 1) + i≥2 (|T (i) | − 1),

(11.49)

where T (i) denotes the i-th ramification group of Σ/F2 at P. As F2 is the fixed (i) (2) field of GP , the groups T (i) and GP coincide for i ≥ 2. Hence (1)

(2)

d1 − 2(|GP | − 1) = d2 − 2(|GP | − 1),

which, together with (11.48), gives the relation (1)

d∗ = 2

|GP |

− 2. (2) |GP | Then Theorem 11.72 shows that F2 has genus 0; that is, F2 is rational. (2) (k) Let k denote the smallest integer such that k ≥ 3 and GP 6= GP . All ramifi(2) (k) cation groups, and thus both GP and GP , are normal subgroups of GP . Hence, (k) (2) ′ there is a normal subgroup G containing GP , which is a subgroup of GP of index p. The next step is to compute the genus g ′ of the fixed field F ′ of G′ . Let d′ = deg D(Σ/F ′ ),

d = deg D(F ′ /F2 ).

As before, the following equations hold: (2)

Also,

d2 = d′ + p−1 |GP | d,

(2)

d2 = k(|GP | − 1) + (p)

d′ = k(p−1 |GP | − 1) + (2)

(k)

P

P

(i)

i≥k

(|GP | − 1).

(i)

i≥k

(|GP | − 1),

as G′ lies between GP and GP . It follows that d = k(p − 1). Now, Theorem 11.72 shows that ′

g ′ = 12 (k − 2)(p − 1). ′

(11.50) ′

Since k ≥ 3, so F is not rational. Let P the place of F lying under P in the (1) extension Σ/F ′ , and let GP ′ be the stabiliser of P ′ in the K-automorphism group (1) (1) of F ′ . As G′ is a normal subgroup of GP , the factor group GP /G′ can be viewed (2) (1) as a subgroup of GP ′ . Since [GP : G′ ] = p, so [F ′ : F2 ] = p. Also, P ′ is the only ′ place which ramifies in F |F2 . Choose x ∈ F2 having P ′ as a pole. Then F ′ /F2 is an Artin–Schreier extension, and Theorem 12.5 ensures the existence of y ∈ F ′ such that y p − y = B(x) with B(X) ∈ K[X]. From Theorem 12.7, 4p 2 (1) g′ . |GP ′ | ≤ (p − 1)2

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(1)

Since the quotient group GP /G′ is a subgroup of GP ′ , from (11.50), (1)

|GP | ≤ |G′ |

(2)

4|GP | ′ 2 4p 2 (2) g′ = g = |GP |(k − 2)2 . 2 (p − 1) (p − 1)2

(11.51)

(2)

From Theorem 11.72 applied to GP , (2)

(2)

2g − 2 ≥ −2|GP | + k(|GP | − 1), whence k−2≤

2g (2) |GP |

−1

.

By substituting this in (11.51) the final result is obtained: (1)

|GP | ≤

(2)

4|GP |

(2) (|GP |

−

1)2

g2. 2

The next objective is to obtain sharper estimates on the orders of abelian K-automorphism groups of Σ. T HEOREM 11.79 Let G be an abelian K-automorphism group of an irreducible curve F of genus g ≥ 2. Then  4g + 4 for p 6= 2, |G| ≤ 4g + 2 for p = 2. Proof. Put Σ = K(F). If Σ/ΣG does not ramify, then |G| ≤ g − 1 by Theorem 11.72. Another case in which the proof is a direct consequence of Theorem 11.72 when |G| = p, and ΣG is rational. In fact, if k denotes the number of non-trivial ramification groups of G at P, then 2g −2 = −2p+k(p−1) follows from Theorem 11.72. This implies that k ≥ 3. whence |G| ≤ 2g + 1 < 4g + 2. Let P be a place of Σ which ramifies in Σ/ΣG . Since G is abelian, GP fixes every place in the orbit of P under G. Let A = GP and a = |A|. Then |G|/a is the size of the orbit of P under G. Let g ′ denote the genus of ΣA . From Theorem 11.72 applied to A, 2g − 2 ≥ a(2g ′ − 2) + |G|(a − 1)/a ≥ a(2g ′ − 2) + 12 |G|.

If g ′ ≥ 1, this gives |G| ≤ 4g − 4. Therefore ΣA is assumed to be rational. Then Theorem 11.72 applied to A shows that 2g − 2 ≥ −2a + |G|(a − 1)/a,

(11.52)

and the inequality is strict when A is non-tame. Suppose that a = 2. Then (11.52) implies that |G| ≤ 4g + 4 giving the assertion for p > 2. If p = 2, then A is non-tame, and hence |G| < 4g + 4. To prove the assertion for p = 2, the possibility of |G| = 4g + 3 must be ruled out. In such a case, G is tame, and the strict inequality occurs in (11.52). From Theorem 11.57, some place Q outside the G-orbit of P must be ramified in Σ/ΣG . In other words, GQ contains a non-trivial element α. Since α has order at least 3, this implies

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that (11.52) remains valid when 2 is subtracted from the right-hand side. But then |G| ≤ 4g + 2, a contradiction. Therefore (11.79) holds for a = 2. Suppose that a = 3. From (11.52), |G| ≤ 3g + 6 which implies |G| ≤ 4g + 2 for g > 3. The preceding argument can be used to show that if p = 2 then |G| = 14 is impossible. Therefore (11.79) holds for a = 3. From now on, let a > 3. Theorem 11.79 holds if the size ℓ of the G-orbit of P is at least 4. In fact, from (11.52), 2g + 2 ≥ 21 |G| + 12 ℓ(a − 1) − 2a + 4 ≥ 12 |G| + ( 12 a − 1)(ℓ − 4),

(11.53)

which implies that |G| ≤ 4g + 4 for ℓ ≥ 4 (and |G| ≤ 4g + 2 for ℓ ≥ 5). The possibility of 4g + 3 ≤ |G| ≤ 4g + 4 for p = 2 can be ruled out by the argument previously used for a = 2. Let ℓ1 , . . . , ℓs be the sizes of the short G-orbits. From the above result, ℓi ≤ 3. Suppose that A is tame. Since ΣG is rational, Theorem 11.57 applied to G gives the relation 2g + 1 = −2|G| + s|G| − (ℓ1 + · · · + ℓs ) + 3.

In particular, s ≥ 3. If ℓi = 3 for some i = 1, then |G| ≥ 12, and

2g + 1 ≥ 21 |G| + (s − 25 )|G| − 3s + 3 ≥ 12 |G|,

whence |G| ≤ 4g + 2. Otherwise, ℓ1 = · · · = ℓs = 2, |G| ≥ 8, and, in this case, 2g + 1 = −2|G| + s|G| − 2s + 3 = 21 |G| + (s − 52 )|G| − 2s + 3 > 21 |G|,

which gives the result. Suppose that A is non-tame. If a short G-orbit has length 3, then A has at least three fixed places. Now, for m = 3, (11.43) gives the inequality 2g + 1 ≥ 52 |A| = 56 |G|,

whence |G| ≤ 51 (12g + 6) < 4g + 2. The same argument with m = 2 shows that, if G has a short orbit of length 2, then 2g ≥ |A| = 12 |G|, whence |G| ≤ 4g. Also, if G fixes at least two places, then (11.43) applied to G gives |G| ≤ 2g. (1) It remains to consider the case that G = A = GP , that GP is non-trivial and that the identity is the only K-automorphism in GP that fixes a place distinct from (1) P. Write GP = GP ⋊ H with p ∤ |H|. Suppose that H is non-trivial. From Lemma 11.77, (1)

(1)

2g ≥ 21 (|H| − 1)(|GP | − 1).

Since |H| = |GP | = 2 is not possible, so

(|H| − 1)(|GP | − 1) ≥ 12 |H||GP | − 1

Therefore |G| = |GP | = |GP ||H| ≤ 4g + 2. (1) Suppose that G = GP = GP . A slightly sharper bound is shown, namely that |G| ≤ 4g. By Theorem 11.99, this assertion holds for g = 2, but assume on the contrary that it does not extend to every g > 2. Let g be denote the minimum (1) value of such exceptional genera g. Then there is an abelian subgroup G = GP of AutK (Σ) that fixes no place distinct from P and that its order |G| is at least 4g + 1.

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Since |G| = 6 p, it has a non-trivial subgroup M . If ΣM has genus g ′ ≥ 2, Theorem 11.72 applied to M implies that g ≥ g ′ |M |. Therefore 4g + 1 ≥ 4g ′ |M | + 1,

whence |G| ≥ 4g ′ |M | + 1. Since |G|/|M | is an integer, it follows that that ¯ = G/M can be viewed as a K-automorphism |G|/|M | ≥ 4g ′ + 1. Note that G group of ΣM whose genus g ′ is smaller than g. From what has been shown so far, ¯=G ¯ P ′ , where P ′ is the place of ΣM lying under P in the extension Σ/ΣM , and G ¯ fixes no place of ΣM distinct from P ′ . But this contradicts the minimal choice G of g. If ΣM is elliptic, Theorem 11.72 applied to M implies that g ≥ |M |. Since ¯ = G/M can be viewed as a K-automorphism group of ΣM , Theorem 11.94 G shows that either p = 2 and |G|/|M | = 2, 4, or p = 3 and |G/M | = 3. Therefore |G| ≤ 4g for p = 2 and |G| ≤ 3g for p = 3. If ΣM is rational for every non-trivial subgroup M of G, Lemma 11.45 implies that either |G| = p, or |G| = p2 . Therefore |G| = p2 is assumed. If p = 2 then (2) |G| = 4 ≤ 4g + 1. Let p ≥ 3. Since g > 0, Theorem 11.72 implies that GP is (1) (2) not trivial. From this, |G| ≤ 2g − 1 when G = GP = GP . In the case where (2) (1) |GP | = p2 and |GP | = p, Lemma 11.75 (v) gives the equalities (2)

(p+1)

(3)

GP = GP = · · · = GP

.

Therefore Theorem 11.72 gives 2g ≥ p(p − 1), whence 4g > p2 = |G|.

2

The following lemmas are related to the Hilbert Different Formula (11.37) and Theorem 11.72; they are used in later sections. L EMMA 11.80 Let D(Σ/ΣG ) =

P

dP P. Then (1)

dP ≥ |GP | + |GP | − 2. (1)

L EMMA 11.81 Suppose that a subgroup M of GP contains no non-trivial ele(2) ment from the second ramification group GP . If H ≤ GP is in the normaliser of M and |H| is prime to p, then |H| ≤ |M | − 1. Proof. Let h ∈ H commute with a non-trivial element of M . By Theorem 11.75 (i), this can only occur when h = 1. So, |M | > |H|. 2 L EMMA 11.82 Let G = GP be a K-automorphism group of an irreducible curve F that fixes a place P, and let ν = dP − |GP | with dP = dP (Σ/ΣGP ) and (i) (i+1) qi = [UP : UP ]. Then, with vi the i-th ramification number of G at P, (i) ν = (µ + 1)|H| − 1 with µ a non-negative integer; (ii) |H| | (vi − 1)(qi − 1); (iii) qi | (vi+1 − vi )2 , for every i ≥ 1.

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Proof. (i) Since dP (Σ/ΣH ) = |H|−1, and dP (Σ/ΣGP ) = |H||GP |+v, equation (7.8) reads as follows: (1)

|H| |GP | + ν = |H| − 1 + |H| dP ′ (ΣH /ΣGP ), (1)

(1)

whence (i) follows for µ = dP ′ (ΣH /ΣGP ) − |GP |. Since [ΣH : ΣGP ] = |GP | (1) and p divides |GP , from the remark after the Hurwitz Theorem 7.27 it follows that (1) dP ′ (ΣH /ΣGP ) ≥ |GP |, whence µ ≥ 0. (i) (ii) For i ≥ 1, put Ui = UP ⋊ H. Applying (i) to Ui gives that vi ≡ −1 (mod |H|).

Hence, |H| divides vi − vi+1 . This, together with the equations (i)

(i+1)

vi − vi+1 = vi (|UP | − |UP

(i)

(i+1)

|) − |UP | + |UP

(i+1)

| = |UP

|(vi − 1)(qi − 1)

proves (ii). (i) (iii) It may assumed that i = 1 after replacing GP by UP ⋊ H. Also, take (2) (2) |UP | = p , after replacing Σ by ΣN , where N is a normal subgroup of UP of index p. (1) First, consider the case that GP is not abelian. Note that a non-abelian group (1) has non–trivial commutator subgroup. Hence, the commutator subgroup C of GP (2) (1) is non–trivial. By Theorem 11.74 (iii), the quotient group GP /UP is abelian, and (2) (2) hence UP contains C. Since UP has no non-trivial subgroup, this implies that (2) (2) (1) (2) C = UP . Also, Z(GP ) ∩ UP is non-trivial, as UP is a normal subgroup of (1) (1) (1) (1) GP ; see Theorem 11.74 (ii). Thus Z(GP ) contains C, and hence GP /Z(GP ) is an elementary abelian p–group. From this, (1)

(1)

[GP : Z(GP )] = p2m ,

(1)

and, if A is a maximal abelian normal subgroup of GP , then (1)

(1)

Z(GP ) ≤ A and |A| = pm |Z(GP )|. (2)

Since every non-trivial element in A/GP has order p, the possible orders of elements in A are 1, p, p2 . Therefore the Main Theorem for Finite Abelian Groups implies that A is a direct product of cyclic groups A = A1 × hτ1 i × · · · × hτm+n−2 i,

where |A1 | = p2 , (2)

(1)

Z(GP ) = pn ,

(1)

p = 1, τ1p = · · · = τm+n−2

and A1 contains UP . In particular, |GP | = p2m+n and q1 = p2m+n−1 . (2) Let A2 = hτ1 i × · · · × hτm+n−2 i and B = A2 UP . Then |A2 | = pm+n−2 and m+n−1 |B| = p . Also, A2 ⊳ B ⊳ A. Now, the contribution of P to d = deg(D(Σ/ΣB ) is calculated in two different (2) ways by applying (7.8) to the towers Σ ⊃ ΣUP ⊃ ΣB and Σ ⊃ ΣA2 ⊃ ΣB . This gives the equation v1 (pm+n−2 − 1)p + v2 (p − 1) = v¯2 (p − 1)pm+n−2 + v1 (pm+n−2 − 1),

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where v¯2 is the ramification number in the extension ΣA2 /ΣB . It follows that v2 = v¯2 pm+n−2 − v1 (pm+n−2 − 1). In particular, v¯2 > v1 . Since the group A/A2 is abelian and v1 is also the ramification number in the extension ΣA2 /ΣA , Lemma 11.75 (v) implies that v¯2 ≡ v1 (mod p); that is, v¯2 = v1 + kp for a positive integer k. Therefore v2 = v1 pm+n−2 + kpm+n−1 − v1 pm+n−2 + v1 = v1 + kpm+n−1 ,

whence v2 − v1 = kpm+n−1 . This implies that q1 divides (v2 − v1 )2 . (1) If GP is abelian, then the Main Theorem for Finite Abelian Groups applies (1) (1) directly to GP . Therefore GP contains an elementary abelian subgroup A such that (2)

(2)

A ∩ UP = {1} and |A| = |UP |/p2 . Using the above argument, the sharper assertion that q1 | (v2 − v1 ) follows. (i)

2 (i+1)

L EMMA 11.83 Let z ∈ Σ be an element such that p ∤ ordP z. If α ∈ UP \UP then α(z) = z + z ′ with z ′ 6= 0, and vi = 1 − ordP z + ordP z ′ .

,

Proof. There exists c ∈ K\{0} such that z ′ = α(z) − cz with ordP z ′ > a and m a = ordP z. Equivalently, α(z) = cz +z ′ . If ord α = pm , then cp = 1. Therefore c = 1. Note that z ′ 6= 0, otherwise z ∈ Σα which contradicts p | a by (7.1). Now, let x ∈ Σα with ordP ′ x = 1 where P ′ is the place of Σα lying under P. By (7.1), b = ordP x = pm and hence gcd(a, b) = 1. Further, choose integers n1 , n2 such that n1 a+n2 b = 1. Now, p | b implies that p ∤ n1 and that w = z n1 xn2 is a uniformising element at P. Therefore vi = ordP (α(w) − w)

= ordP ((z + z ′ )n1 xn2 − z n1 xn2 ) = ordP (n1 z n1 −1 z ′ xn2 )

= (n1 − 1)a + n2 b + ordP z ′ = 1 − ordP z + ordP z ′ .

2

The following result is a refinement of Theorem 11.78. T HEOREM 11.84 If γ is the p-rank of Σ, then every p-subgroup G of AutK (Σ) has the following properties. (i) If γ ≥ 2, then |G| ≤ κp (γ − 1) with κp = p/(p − 2) for p ≥ 3, and κ2 = 4. (ii) If γ = 1 and p ≥ 3, then G is a cyclic group and |G| divides γ − 1; also, G has no fixed place. (iii) If γ = 1 and p = 2, then |G| ≤ 4(g − 1). (iv) If γ = 0, then |G| ≤ max{g, 4pg 2 /(p − 1)2 }.

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Proof. Let Σ′ = ΣG , n = |G| and λ = (γ ′ − 1)/n where γ ′ denotes the p-rank of Σ′ . If G has s short orbits with lengths ℓ1 , . . . , ℓs , then (11.28) reads as follows: Ps λ = γ ′ − 1 + i=1 (1 − ℓi /n). (11.54) Since G is a p-group, n/ℓi ≥ p for i = 1, . . . , s. Assume that γ ≥ 2; that is, λ > 0. ′ −1 Then (i) can be reworded as λ ≥ κ−1 p . If γ ≥ 2, (11.54) shows that λ ≥ 1 > κp . ′ Also, it shows that if γ = 1 then s ≥ 1, and hence λ ≥ 1 − ℓi /n ≥ 1 − p > κ−1 p .

If γ ′ = 0 and p ≥ 3, then s ≥ 2, and hence

λ ≥ −1 + 2(1 − 1/p) = κ−1 p .

If γ ′ = 0 and p = 2, two cases occur: either there are at least two short orbits, one of length ≥ 4, or there at least three short orbits. From (11.54), λ ≥ 41 in the former case and λ ≥ 12 in the latter. Thus (i) is established. Assume next that γ = 1; that is, λ = 0. From (11.54), either γ ′ = 0, or γ ′ = 1. If γ ′ = 1, no short orbit exists, and G must be cyclic by Theorem 11.63. Further, from Theorem 11.72, 2g − 2 = |G|(2g ′ − 2), which shows that |G| divides g − 1. In particular, |G| ≤ g − 1. The case g ′ = 0 cannot occur when p ≥ 3, since 2 3

≤ 1 − 1/p ≤ 1 − ℓ/n < 1

P for each short orbit of length ℓ. Hence, if p ≥ 3, then (1 − ℓi /n) 6= 1. This proves (ii). If γ ′ = 0 and p = 2, there are exactly two short orbits both of length 21 |G|, and (11.18) reads as follows: P (i) (1) 2g − 2 = |G| (2g ′ − 2) + 12 |G| [2(|GP − 1) + i≥2 (|GP − 1)] P (i) (1) + 21 |G| [2(|GQ − 1) + i≥2 |GQ − 1)], (11.55)

where P and Q come from the two distinct short orbits. Since |GP | = |GQ | = 2, (1) (1) this gives |G| ≤ g − 1 when either g ′ ≥ 1, or g ′ = 0 and |GP | > 2 or |GQ | > 2. (1)

(1)

In the remaining case, g ′ = 0 and |GP | = |GQ | = 2. As g ≥ 2, in (11.55) either (2)

(1)

|GP | ≤ 2 or |GQ | ≥ 2. Therefore (11.55) becomes the equation 2g − 2 ≥ −2|G| + 2|G| + 21 |G|,

whence (iii) follows. Finally, assume that γ = 0; that is, λ < 0. Then γ ′ = 0 from (11.54). If there were at least two short orbits, the right-hand side in (11.54) would be at least −1 + 2(1 + 1/p) ≥ 0, contradicting that λ < 0. So, there is just one short orbit and, in this case, λ = −ℓ/n. But then ℓ = 1; that is, G fixes a place P. From Theorem 11.78, part (iv) follows. 2 R EMARK 11.85 Theorem 11.84 shows that the order of a K-automorphism group of an irreducible curve, whose order is a power of p, is bounded above by a linear polynomial in g except when the p-rank γ is zero. Some of the exceptions are characterised in Theorem 11.140.

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T HEOREM 11.86 Let g ≥ 2. (i) If γ = 1 and p ≥ 3, then AutK (Σ) is tame. (ii) If p ≥ 5 and 2 ≤ γ ≤ p − 2, then |AutK (Σ)| is not divisible by p. Proof. Part (i) follows from Theorem 11.84 (ii). Also, Theorem 11.84 (i) shows that γ ≤ p − 2, whence |G| ≤ p(p − 3)/(p − 2) < p. Therefore |G| = 1 since G has order a power of p. This proves that AutK (Σ)| has no p-subgroups. 2 D EFINITION 11.87 For p 6= 0, an irreducible curve F is ordinary if the p-rank γ of K(F) is equal to the genus g of F. T HEOREM 11.88 For any ordinary curve F of genus g ≥ 2, the K-automorphism group of F has order at most 84(g − 1)g. E XAMPLE 11.89 An example of an ordinary curve is F = v(F ) with n

F (X, Y ) = (X q − X)(Y q − Y ) − 1,

where q = p2 . It has genus g = (q − 1)2 and its K-automorphism group has order at least 2q 2 (q − 1). An interesting property of F, valid for n = 0, is established in Theorem 11.93 (i). (1)

T HEOREM 11.90 Let p = 2. If GP contains a cyclic characteristic subgroup U, (1) then GP has cyclic subgroup V such that [GP : V ] = [GP : U ]. (1)

Proof. By Theorem 11.44, GP = GP ⋊ H, where H is a cyclic group of odd (1) order, and GP is a normal 2-subgroup of GP . From the latter assertion, since U is characteristic, it follows that U is a normal subgroup of GP . Therefore the set V = U H is a subgroup of GP . More precisely, V = U ⋊ H, and U is a cyclic Sylow 2-subgroup of V . Therefore V = O(V ) ⋊ U . So, both U and O(V ) are normal subgroups, and V = O(V ) × U . Since both O(V ) and U are cyclic, V itself is cyclic. Also (1)

(1)

(1)

|GP | = |GP ||H| = [GP : U ]|U ||H| = [GP : U ]|V |, whence the assertion follows.

2

11.10 K-AUTOMORPHISM GROUPS OF PARTICULAR CURVES This section describes the groups of curves of low genus, namely rational and elliptic curves, as well as those of hyperelliptic and Artin–Schreier curves. Theorem 11.14 states that Aut(K(x)) ∼ = PGL(2, K). The finite subgroups of PGL(2, K) can be determined by the techniques used in Section 11.9, although a deeper result, namely Theorem A.16, is also required. Here the two main results are stated without proofs.

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T HEOREM 11.91 Let G be a non-trivial finite group of K-automorphisms of a rational function field K(x). Let s be the number of short orbits of G on the set of all places of K(x), and let ℓ1 , . . . , ℓs be their lengths. Then G is a group of one of the following types: (i) the cyclic group Zn of order n, with p ∤ n, s = 2, ℓ1 = ℓ2 = 1; (ii) an elementary abelian p-group, with s = 1, ℓ1 = 1; (iii) the dihedral group Dn of order 2n, with p ∤ n, s = 2, ℓ1 = 2, ℓ2 = n, or p 6= 2, s = 3, ℓ1 = ℓ2 = n; (iv) the alternating group A4 , with p 6= 2, 3, and s = 3, ℓ1 = 6, ℓ2 = ℓ3 = 4; (v) the symmetric group S4 , with p 6= 2, 3, s = 3, ℓ1 = 12, ℓ2 = 8, ℓ3 = 6; (vi) the alternating group A5 , with p = 3, s = 2, ℓ1 = 10, ℓ2 = 12, or p 6= 2, 3, 5, s = 3, ℓ1 = 30, ℓ2 = 20, ℓ3 = 12; (vii) the semidirect product of an elementary abelian p-group of order q with a cyclic group of order n, with n | (q − 1), s = 2, ℓ1 = 1, ℓ2 = q; (viii) PSL(2, q), with p 6= 2, q = pm , s = 2, ℓ1 = q(q − 1), ℓ2 = q + 1; (ix) PGL(2, q), with q = pm , s = 2, ℓ1 = q(q − 1), ℓ2 = q + 1. T HEOREM 11.92 Let K = Fq and let G be the subgroup of Aut(K(x)) preserving the set of all Fq -rational places of K(x). If G ∼ = PGL(2, q), then the extension K(x)/K(x)G has the following properties. (i) Every Fq -rational P place wildly ramifies; G0 (P ) is the semidirect product of an elementary abelian p-group of order q by a cyclic group of order q − 1; G1 (P ) is an elementary abelian p-group of order q; G2 (P ) is trivial; all Fq -rational places form a single orbit of size q + 1 under G. (ii) Every Fq2 -rational place which is not Fq -rational tamely ramifies; G0 (P ) is a cyclic group of order q + 1; such places form a single orbit of size q 2 − q under G. (iii) No other place of K(x) ramifies. The main result on Artin–Schreier curves in terms of the associated function field is the following. T HEOREM 11.93 Let Σ/K(x) be an Artin–Schreier extension such that the genus of Σ is greater than 1. Then the K-automorphism group G of Σ/K(x) is an extension of a cyclic group of order p by a finite group of K-automorphisms of K(x) except in the following cases for p > 2 : (i) Σ = K(x, y), (xp − x)(y p − y) = c with c ∈ K, and G is the semidirect product of an elementary abelian p-group of order p2 with Dp−1 ;

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(ii) Σ = K(x, y), y 3 − y = ν/(x(x − 1)) where ν 2 = 2, and G is an extension of a cyclic group of order 2 by S4 ; (iii) Σ = K(x, y), y p − y = xm with m | (p + 1), m < p + 1, and G is an extension of Zm by PGL(2, p); (iv) Σ = K(x, y), y p − y = xp+1 ; that is, Σ is the Hermitian function field over Fp2 , and G ∼ = PGU(3, p). The K-automorphism groups of elliptic curves are known. Here the following result, which is an extension of Lemma 11.26, is shown. T HEOREM 11.94 he automorphism group of an elliptic curve F has the following properties. (i) The K-automorphism group of F is infinite, and it acts on the places as a transitive permutation group. (ii) For every place P, the stabiliser of P is finite. More precisely, if G is a non-trivial K-automorphism group of F, then  when p 6= 2, 3,  2, 4, 6 2, 4, 6, 12 when p = 3, (11.56) |GP | =  2, 4, 6, 8, 12, 24 when p = 2. (1)

(1)

(iii) For p = 2, the stabiliser GP is cyclic when |GP | ≤ 4, and it is the quater(1) nion group Q8 when |GP | = 8. Proof. Let p 6= 2, 3. By Theorem 7.89, F is taken to be the plane cubic curve v(Y 2 − X 3 − uX − v),

with 4u3 + 27v 2 6= 0. Let Σ = K(F), and let P∞ be the place of Σ = K(F) arising from the branch of F with centre at Y∞ . For another place P of Σ, let P = (a, b) be the centre of the corresponding branch of F. It is straightforward to show that the birational transformation ω(a, b), given by the equations 2  y−b ′ y−b − x − a, y ′ = (x − a) + b x′ = x−a x−a

extends to a K-automorphism of Σ; see Lemma 11.2. Let Q be the place of Σ corresponding to the branch centred at the the point Q = (a, −b). Then ω(a, b) sends Q to P∞ ; hence ω(a, b)−1 sends P∞ to Q. Therefore the K-automorphism group generated by ω(a, b) acts transitively on the set of all places of Σ. The structure of the stabiliser of a place P can be deduced from Lemma 11.26. In fact, Lemma 11.26 shows that AutK (Σ)P contains no p-element, and hence it is tame. By Theorem 11.49, AutK (Σ)P is cyclic. Therefore Lemma 11.26 shows that the possibilities for |AutK (Σ)P | are 2, 3, 4, 6. Now, let p = 3. By Theorem 7.89, Σ = K(F), where F = v(Y 2 − X 3 − uX 2 − vX + w),

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with v 3 6= u3 w. To show (i), the previous argument still works, the only change being in the definition of ω(a, b) where −u is added in x′ . Also, the stabiliser G of P∞ contains ω(a, b). If G is tame, then G is a cyclic group of order 2 or 4 by Theorem 11.60. If G is non-tame, then G has a normal subgroup N of order 3, and either |G| = 6 or |G| = 12. Finally, let p = 2. Theorem 7.89 gives two possibilities. If F is supersingular, then G = v(Y 2 + uY + X 3 + vX + w),

with u 6= 0, is a non-singular model of Σ = K(F). It is possible to use the previous argument, provided that ω(a, b) is defined as follows: 2  y−b ′ y+b ′ + x + a, y ′ = (x + a) + b + u. x = x+a x−a The stabiliser GP∞ is non-tame as it contains the involutory K-automorphism ω0 :

x′ = x,

y ′ = y + u.

By Theorem 11.60, a Sylow 2-subgroup of GP∞ has order at most 8. From (11.20), GP∞ has no element of order 5. Hence, by Theorem 11.60, 3 is the only possibility for the order of a non-trivial K-automorphism of Σ of odd order. Hence |GP∞ | is an even divisor of 24. From the proof of Theorem 12.7, GP∞ consists of linear transformations of type ω′ :

x′ = x + d,

y ′ = y + ax + b,

with a, b, d ∈ K. Then ω ′ is involutory if and only if ad = 0. As ω ∈ GP∞ , the condition ad = 0 implies that a = d = 0 and b = u. Therefore ω0 is the unique (1) (1) involutory automorphism in GP∞ . In particular, if |GP∞ | ≤ 4, then GP∞ is cyclic. (1)

(1)

(2)

If |GP∞ | = 8, Theorem 11.72 applied to GP∞ implies that |GP∞ | = 2. From (1)

(2)

(1)

this, the group GP∞ /GP∞ is elementary abelian of order 4. Therefore GP∞ is not cyclic, and hence is a quaternion group of order 8. Finally, if F is not supersingular, then G = v(Y 2 + XY + X 3 + uX 2 + v),

with v 6= 0, is a non-singular model of Σ = K(F). Again, the previous argument can be used with 2  y+b y+b ′ y+b ′ ω(a, b) : x = + + x + a + u, y ′ = (x + a) + x′ + b, x+a x+a x+a ω0 : x′ = x, y ′ = y + x, to obtain the results.

2

R EMARK 11.95 Theorem 11.94 implies that, for every place P of an elliptic curve,   6 when p 6= 2, 3, 12 when p = 3, (11.57) |GP | ≤  24 when p = 2.

In each case, this upper bound is achieved by some elliptic curves. For p 6= 2, 3, see the proof of Lemma 11.26. Here, a few examples are given for p = 2, 3.

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E XAMPLE 11.96 Let p = 3. Let F = v(Y 2 − X 3 + X). If δ is a primitive fourth root of unity, then both birational transformations x′ = −x + 1,

ω: ω′ :

x′ = x + 1,

y ′ = δy, y′ = y

extend to K-automorphisms of Σ whose orders are 4 and 3. The group generated by ω and ω ′ has order 12. E XAMPLE 11.97 Let p = 2, let F = v(Y 2 + Y + X 3 ), and choose a cube root of unity ǫ in Σ. Then the birational transformation ω(i, k) :

x′ = x + ǫi ,

y ′ = y + ǫ2i x + ǫk ,

with i = 0, 1, 2 and k = 1, 2, extends to a K-automorphism of Σ. These, together with the K-automorphism ω:

x′ = x,

y′ = y + 1

and the identity form a quaternion group Q8 of order 8. Also, the birational transformation ω′ :

x′ = ǫx,

y′ = y

extends to a K-automorphism of Σ of order 3; this and Q8 generate a group of order 24. Now, the K-automorphism groups of hyperelliptic curves are investigated. By Theorem 11.50, such a group is finite. As in Section 7.10, two cases are distinguished. First, suppose that K has zero or odd characteristic. By Theorem 7.94, F is an irreducible plane curve v(Y 2 − f (X)), where f (X) ∈ K[X] is monic, has degree 2g + 1, and its roots α0 , . . . , α2g are distinct. Let Σ = K(x, y) with y 2 = f (x) be the function field Σ of F. The Weierstrass points are the places associated to the branches of F centred at the non-singular points Ui = (αi , 0) for each i in {0, . . . , 2g}, together with the place associated to the unique branch centred at Y∞ . As in Theorem 7.95, aX + b w(X) = , with ad − bc 6= 0, cX + d is a rational function which maps the set {α0 , . . . , α2g , ∞} onto itself. Such rational functions constitute a subgroup W of PGL(2, K). Hence, if w ∈ W fixes three distinct elements from the set {α0 , . . . , α2g , ∞}, then w(X) = X. Since g ≥ 2, this implies that W is a finite group. From the proof of Theorem 7.95, the K-automorphisms α of F are as follows: (i)

α(x) = ax + b, with aX + b ∈ W , α(y) = ǫy with ǫ2 = a2g+1 ;

(ii)

(11.58)

aX + b ax + b , with ∈ W, x − α0 X − α0 Q y , with ǫ2 = (b + aα0 ) 2g α(y) = ǫ i=1 (α0 − αi ). (11.59) g+1 (x − α0 )

α(x) =

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Hence, |AutK (Σ)| = 2|W |; more particularly, the hyperelliptic involution of F, that is, the birational transformation (x, y) 7→ (x, −y), extends to a K-automorphism ω of F such that ω ∈ Z(AutK (Σ)) and W ∼ = AutK (Σ)/hωi. In even characteristic, let F = v(Y 2 + h(X)Y + g(X)),

where deg h(X) ≤ g + 1 and deg g(X) = 2g + 1. Then the function field of F is Σ = K(x, y), with y 2 + h(x)y + g(x) = 0. If n = deg h(X) ≥ 1, let α1 , . . . , αn denote the roots of h(X), each counted with multiplicity. As before, W is the subgroup of PGL(2, K) that preserves the multi-set {∞, α1 , . . . , αn }. The Weierstrass points are the places arising from the √ branches centred at the non-singular points Ui = (αi , αi ), with i = 1, . . . n, together with the place associated to Y∞ . From the proof of Theorem 7.101, the K-automorphisms α of F are as follows: (i)

(ii)

h(w(x)) y, h(x) g(w(x)) g(x) + ; with w(X) ∈ W and v(x)2 + v(x) = h(x)2 h(w(x))2 h(w(x)) α(x) = w(x), α(y) = h(w(x)) + y, h(x) h(w(x))2 g(x) + g(w(x)) = 0. with w(X) ∈ W and h(x)2

α(x) = w(x),

α(y) = v(x)h(w(x)) +

The centre Z(AutK (Σ)) of AutK (Σ) is non-trivial as the hyperelliptic involution of F, that is, the birational transformation (x, y) 7→ (x, y + 1) extends to a Kautomorphism ω of F lying in Z(AutK (Σ)). Therefore W ∼ = AutK (Σ)/hωi and |AutK (Σ)| = 2|W |. If h(X) is constant, then AutK (Σ) fixes the unique Weierstrass point. Therefore the following result is obtained. T HEOREM 11.98 Let F be a hyperelliptic curve. Then (i) its hyperelliptic involution ω is in the centre of the K-automorphism group G of F; (ii) the quotient group G/hωi is isomorphic to a subgroup W of PGL(2, K) that preserves the set or, for p = 2, the multi-set consisting of all Weierstrass points of F; (iii) if F has at least three Weierstrass points, the identity is the only K-automorphism of F that fixes each Weierstrass point. By Remark 11.15, all possibilities for W are known. Now, some particular classes of hyperelliptic curves are considered. The following proposition is a special case of a more general result that is established later; see Theorem 11.138. P ROPOSITION 11.99 Let F be an irreducible hyperelliptic curve of genus 2.

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(i) Assume that either p 6= 2, or p = 2 and F has more than one Weierstrass point, and let G be a K-automorphism group of F. Then  1 for p = 0 and p ≥ 7,    5 for p = 5, (1) |GP | ≤ 3 for p = 3,    8 for p = 2.

(ii) If F has only one Weierstrass point and G is an abelian p-group, then the order |G| ≤ 8. (iii) If G is a soluble group, then |G| ≤ 48. Proof. By Theorem 11.78, (i) holds for p 6= 2, 3. To prove (i) for p = 3, Theorem 11.98 is relevant, as Σ has six Weierstrass points. If G has a subgroup of order at least 9, then some non-trivial elements of H fixes three Weierstrass points, in contradiction to Theorem 11.98 (iii). To prove (i) for p = 2, let ω ∈ G; Theorem 11.98 and Remark 11.15 are used. If Σ has three Weierstrass points, then AutK (Σ)/hωi is a subgroup of S3 . In particular, the order of a Sylow 2-subgroup S2 of G is at most 4. Similarly, |G/hωi| ≤ 2, and hence |S2 | ≤ 4 when Σ has two Weierstrass points. To prove (ii), let p = 2. The essential tools are Theorem 11.94 and Lemma 11.46. Let M be a non-trivial subgroup of G. Suppose first that ΣM is elliptic. Since M fixes the unique Weierstrass point P, Theorem 11.72 applied to M implies that |M | = 2. Therefore G/M is an abelian K-automorphism group of an elliptic curve. Theorem 11.94 (II) implies that |G/M | ≤ 4, whence it follows that |G| ≤ 8. Finally, in the case that the fixed field of every non-trivial subgroup of G is rational, Lemma 11.46 implies that |G| ≤ 4. In proving (iii), Theorem 11.98 and Remark 11.15 play a role since they, together with the solubility condition, imply that, if Σ has more than two Weierstrass points, then |G|/hωi| ≤ 24. Therefore (iii) holds for p 6= 2, To complete the proof for p = 2 it remains to examine the case that Σ has exactly two Weierstrass points, say P1 and P2 . Then, G has a subgroup U of index 2 fixing ¯ = U/hωi is a subgroup of both P1 and P2 . Since G contains ω, the group U PGL(2, K) fixing two objects, namely the places P1′ and P2′ of Σω lying under ¯ is a cyclic group of P1 and P2 in the extension Σ/Σω . From Theorem 11.14, U ¯ | ≤ 4g + 2 = 10. Therefore |G| ≤ 18, and (iii) odd order. By Theorem 11.60, |U follows. 2 E XAMPLE 11.100 Let p > 2. For a power q of p, the curve F = v(Y 2 − g(X))

with g(X) = X q − X is a special case of Example 5.59. So F has genus 21 (q − 1). The field Fq is viewed as a simple extension of the prime field Fp of K, and Fq is assumed to be a subfield of K. The distinct roots of g(X) are elements in Fq . Hence the group W consists of all rational functions w(X) =

aX + b , cX + d

with a, b, c, d ∈ Fq , ad − bc 6= 0.

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Therefore W ∼ = PGL(2, q) and W acts on Fq ∪ {∞} as PGL(2, q) acts naturally on PG(1, q). If ω : x′ = x, y ′ = −y is the central involution in AutK (Σ), then AutK (Σ)/hωi ∼ = PGL(2, q). It may be noted that if q = 5 then g = 2 and |AutK (Σ)| = 240. This shows that the solubility condition in Proposition 11.99 cannot be relaxed. The following result is a characterisation of the curve in Example 11.100. P ROPOSITION 11.101 Let p > 2 and q = ph = 2g + 1. If W ∼ = PSL(2, q) in Theorem 11.98, then F is birationally equivalent to the curve in Example 11.100. Proof. The notation is the same as in the proof of Theorem 11.98. Take α0 = 0, α1 = 1 and f (X) monic, after a suitable substitution of type X ′ = uX + v, with u, v ∈ K and u 6= 0. Since W ∼ = PSL(2, q), the polynomial f (X) is invariant under the substitution X 7→ cX for every 21 (q − 1)-st root of unity c ∈ K; it should be noted that c ∈ Fq . Since α1 = 1, and {c α1 , . . . , c α2g } = {α1 , . . . , α2g }, this shows that every c with c(q−1)/2 = 1 is a root of f (X). Also, since W ∼ = PSL(2, q), so f (X) is invariant under the substitution X 7→ X + 1. Choose a 12 (q − 1)-st root of unity u ∈ K in such a way that v = u + 1 is not also such a root of unity. Then the elements cv with c ranging over all 21 (q − 1)-st roots of unity provide the remaining 12 (q − 1) non-zero roots of f (X). Hence the roots of f (X) are precisely the elements of Fq . Thus, f (X) = X q − X. 2 E XAMPLE 11.102 Let p = 2, F = v(Y 2 + Y + X 5 ) and Σ = K(F). The field F16 is viewed as a subfield of K and as an extension of the prime field F2 of K. Since h(X) = 1, the place P∞ of Σ arising from the unique branch of F centred at Y∞ is the only Weierstrass point. Therefore AutK (Σ) fixes P∞ . A straightforward computation shows the following results. For every b, c ∈ F16 with c2 + c + b5 = 0, the birational transformation αb,c (x) = x + b,

αb,c (y) = b8 x2 + b4 x + c + y

of Σ is a K-automorphism fixing P∞ . For b = c = 0, the identity is obtained; otherwise the order of αb,c is either 2 or 4 according as b10 + b5 is 0 or 1. The set {αb,c | b, c ∈ F16 ; c2 + c + b5 = 0} is a group U of order 32 containing 11 elements of order 2 and 20 elements of order 4. Such a group is uniquely determined up to isomorphism, and hence it is (1) isomorphic to Z4 × D4 . Since U fixes P∞ , Theorem 11.78 implies that GP = U . 5 Further, for every d ∈ F16 with d = 1, the birational transformation αd (x) = dx,

αd (y) = y

is a K-automorphism of Σ fixing P∞ . These form a subgroup C5 of AutK (Σ) of order 5. By Theorem 11.60, C5 is a maximal subgroup of odd order in the stabiliser of P. Therefore AutK (Σ) = U ⋊ C5 ; so, AutK (Σ) ∼ = (Z4 ⋊ D4 ) ⋊ Z5 , and |AutK (Σ)| = 160. This shows that Proposition 11.99 can fail when AutK (Σ) has a fixed place.

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E XAMPLE 11.103 Let p > 2 and a ∈ Fq \{0}. The hyperelliptic curve G0 = v(X(Y p − Y ) − (aX 2 + 1))

of genus p − 1 has the following properties:

(i) the Fq -rational K-automorphism group of G0 is the direct product of the hyperelliptic involution by a dihedral group Dp of order 2p; (ii) the fixed places of Dp are Fq -rational. Conversely, up to birational equivalence over Fq , the curve G0 is the unique hyperelliptic curve defined over Fq of genus g = p − 1 which satisfies both (i) and (ii). If Dp is replaced by the cyclic group Zp , similar properties and characterisations hold for hyperelliptic curves Gb of genus p − 1 with Gb = v(X(Y p − Y ) − (aX 2 + bX + 1)),

(11.60)

where a, b ∈ Fq , a 6= 0 and TFq /Fp (b) 6= 0.

E XAMPLE 11.104 Let Fn,m = v(X n +Y m +1) with n > m. Assume that p does not divide n and m, and that Fn,m has genus g > 1. Let G be the K-automorphism group of Fn,m . Then G has a normal cyclic subgroup N order m, and one of the following holds: (i) G/N is a cyclic group of order 2n and m ∤ n; (ii) G/N is a dihedral group of order 2n and m | n but n 6= q + 1;

(iii) G/N ∼ = PGL(2, q) with m | n and n = q + 1. 11.11 FIXED PLACES OF AUTOMORPHISMS

Some simplification in the statements of the results in this section is made by using the following notation. For a non-trivial K-automorphism α, the integer ρ(α) denotes the number of fixed places of α in the action of α on the set PΣ of all places of Σ. More generally, for a non-trivial subgroup G of AutK (Σ), let ρ(G) be the number of places of Σ fixed by every K-automorphism in G. If G has prime order and G = hαi, then ρ(G) = ρ(α), but this is not always true for subgroups G of composite order. L EMMA 11.105 Let α ∈ AutK (Σ) be a non-trivial K-automorphism such that ρ(α) > 2n, where n = [Σ : K(z)] for a non-constant element z of Σ. Then α(z) = z, and n is divisible by ord α. If, in addition, n is prime, then ord α is also a prime and Σα is a rational function field. Proof. Let P be a place of Σ and take a primitive place representation σ of α. If σ(z) = cti + · · · and P α = P, then there is a K-automorphism λ of K[[t]] such that σ(α(z)) = cλ(t)i + · · · = cti + · · · .

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Then ordP (α(z) − z) > ordP z provided that α(z) − z 6∈ K and P = P α . Now, let h = α(z) − z, and assume that h 6∈ K. Then every pole of h is either a pole of z or a pole of its image α(z). Since n = [Σ : K(α(z))], this implies that [Σ : K(h)] ≤ 2n. Using this observation, this can be improved to [Σ : K(h)] ≤ 2n − r, where r is the number of poles of z fixed by α. In particular, r ≤ 2n. From the above observation it also follows that each fixed place of α that is not a pole of z is a zero of h. Thus h has ρ(α) − r > 2n − r zeros. But this is in contradiction with Corollary 5.35; hence h ∈ K. More precisely, h = 0; that is, α(z) = z, because α fixes some places which are not poles of z. To show the other two assertions, choose a place P fixed by α, and let P ′ be a place lying under P in the extension Σ/Σα . Then P completely ramifies. Hence [Σ : Σα ] = ord α since eP = ord α by Theorem 11.42. From the equation, [Σ : K(z)] = [Σ : Σα ] · [Σα : K(z)], n is the product of ord α and [Σα : K(z)]. If n is prime, this gives n = ord α and Σα = K(z). 2 L EMMA 11.106 Let α ∈ AutK (Σ) be a non-trivial K-automorphism of order n, where Σ has genus g. Then ρ(α) ≤ 2 +

2g . n−1

(11.61)

Proof. Since ρ(α) ≤ ρ(αj ) for j ≥ 1, Remark 11.61 applied to G = hαi gives that 2g − 2 ≥ n(2g ′ − 2) + ρ(α)(n − 1), where g ′ is the genus of the function field Σα . Since g ′ ≥ 0, the result follows. 2 L EMMA 11.107 If the order n of a K-automorphism α is a prime different from p, then ρ(α) = 2 +

2g − 2g ′ n , n−1

(11.62)

where g and g ′ are the genera of Σ and Σα . In this case, equality holds in (11.61) if and only if g ′ = 0. Proof. If n is prime, then ρ(α) = ρ(αj ) for j ≥ 1. Then, the proof of Lemma 11.106 gives the result. 2 T HEOREM 11.108 If a prime number n other than p is the order of a K-automorphism of Σ, then one of the following holds: (i) n ≤ g + 1; (ii) n = 2g − 1 and g > 2; (iii) n = 2g + 1.

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Proof. Using the formula (11.27) several times, an upper bound for n can be obtained. If g ′ ≥ 2, then 2g − 2 ≥ 2n + ρ(α)(n − 1) ≥ 2n; hence n ≤ g − 1, yielding that n < g + 1. Again, by (11.22), g ′ = 1 and ρ(α) = 0 cannot occur simultaneously. If g ′ = 1 and ρ(α) = 1, then n = 2g − 1. If g ′ = 1 and ρ(α) ≥ 2, then 2g − 2 ≥ 2(n − 1), and hence n ≤ g, yielding n < g + 1. Similarly, by (11.22), g ′ = 0 and ρ(α) ≤ 2 cannot occur simultaneously. If g ′ = 0 and ρ(α) = 3, then n = 2g + 1. If g ′ = 0 and ρ(α) ≥ 4, then 2g − 2 ≥ −2n + 4(n − 1), which implies that n ≤ g + 1. This completes the proof. 2 R EMARK 11.109 In the extremal case that n = 2g + 1, all possibilities are known. Let n be as in Lemma 11.108. Then n = 2g + 1 if and only if Σ is the function field of the curve F = v(F (X, Y )), where F is one of the following: (i) F (X, Y ) = Y m−r (Y − 1)r − X n , with 1 ≤ r < m ≤ g + 1, n 6= p;

(ii) F (X, Y ) = Y 2 − X n + X with n = p. Theorem 11.108 is a slight improvement on Theorem 11.79 for groups of prime order. The following result is another refinement for particular cyclic groups. T HEOREM 11.110 Let F be a non-singular plane curve of degree d defined over Fq , and suppose that its K-automorphism group contains a Singer group, that is, a cyclic subgroup G of PGL(3, q) of order q 2 + q + 1 acting transitively on the points of PG(2, q). Then one of the following holds: (i)

(a) d = q + 2; (b) F is projectively equivalent over Fq3 to the curve G = v(X q+1 Y + Y q+1 + X); (c) AutK (F) is the normaliser of G in PGL(3, q);

(ii) d ≥ q 2 + q + 1. Theorem 11.56 shows that large K-automorphism groups may have only one short orbit. To improve Theorem 11.116, the following lemma is used. L EMMA 11.111 Let G be a K-automorphism group of Σ with only one short orbit. If Σ has genus g ≥ 2, then the size of the short orbit divides 2g − 2.

Proof. Let g ′ be the genus of ΣG . If the stabiliser GP of a place P in the short orbit o is tame, then Theorem 11.57 reads as follows: 2g − 2 = |G|(2g ′ − 2) + |G| − |o|. As |G| = |GP | |o|, the result follows. In the non-tame case, this equation may fail. However, in the Hilbert Different Formula (11.37), dP = dQ for P, Q ∈ o and dQ = 0 for Q 6∈ o. Hence |o| divides d, and the result follows as in the tame case. 2 The next two theorems are classical results linking Weierstrass points and fixed places of K-automorphisms. For their validity in positive characteristic, some hypotheses are needed.

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T HEOREM 11.112 Let F be an ordinary curve. If (a) p does not divide the order of a non-trivial K-automorphism α of F, (b) α has more than four fixed places, then every fixed place of α is a Weierstrass point. Proof. It suffices to deal with the case that n = ord α is prime, since a suitable power αj of α has prime order, and then α may be replaced by αj , as every fixed place of α is fixed by αj . Let P be a fixed place of α, and take a place P ′ of Σα under P. By the Weierstrass Gap Theorem 6.89, there exists z ∈ Σα such that div(z)∞ = m′ P ′ with m′ ≤ g ′ + 1 where g ′ is the genus of Σα . Theorem 11.36, together with (7.1), shows that ordP z = n · ordP ′ z = n · m′ ≤ (g ′ + 1)n. On the other hand, Theorem 11.72 applied to the subgroup G generated by α, together with the hypothesis ρ(α) > 4, gives the inequalities 2g − 2 ≥ n(2g ′ − 2) + (n − 1)ρ(α) > n(2g ′ − 2) + 4(n − 1), showing that g + 1 > n(g ′ + 1). Hence ordP z < g + 1. Since F is ordinary, P is a Weierstrass point. 2 T HEOREM 11.113 Let G be a tame K–automorphism group of Σ. Suppose that α, β ∈ G\{1} satisfy the following conditions: (a) g > n2 g ′ + (n − 1)2, where n = ord α is a prime, and g ′ is the genus of Σα ; (b) ρ(β) > 2n(g ′ + 1). Then the following hold: (i) each fixed place of α is a fixed by β, and ord β ≤ n; (ii) if ord β = n, then hαi = hβi; (iii) if ord β = n, then hαi is a normal subgroup of G; (iv) if ord β = n = 2, then α ∈ Z(G). Proof. Let P1 , . . . , Pρ(α) denote the fixed places of α. From the proof of Theorem 11.112, for each j there is a non-constant zj ∈ Σα such that div(zj )∞ = rj Pj with rj ≤ n(g ′ + 1). From Lemma 11.105, zj = β(zj ) for each j, and thus Pj = Pjβ . So the first part of (i) holds. Also, ρ(β) ≥ ρ(α). Rewrite g > n2 g ′ + (n − 1)2 in the form g+n−1 g − ng ′ > . n−1 n

(11.63)

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Now (g + n − 1) 2(g − ng ′ ) >2+2 n−1 n 2g (n − 1) 2g =2 + +2 >2+ . n n n This, together with (11.61), gives ord β ≤ n, completing the proof of (i). Both α and β are in the stabiliser of P1 . Since G is tame, GP1 is a cyclic group, by Theorem 11.49. So, ord α = ord β yields (ii). Next, suppose δ ∈ G is conjugate to β under G. Then ρ(β) ≥ ρ(α) = 2 +

ρ(δ) = ρ(β) > 2n(g ′ + 1).

Thus, by (ii), hδi = hβi showing that hβi is a normal subgroup of G. This and (ii) imply (iii). Finally, (iv) is a consequence of (iii). 2 T HEOREM 11.114 Let G be a subgroup of AutK (Σ) of order n which has a partition with components G1 , . . . , Gk , with ni = |Gi | for i = 1, . . . k, and let g, g ′ , gi′ be the genera of Σ, ΣG , ΣGi for i = 1, . . . , k. Then Pk (k − 1)g + ng ′ = i=1 ni gi′ .

Proof. Let G0 = G and n0 = n. Let Di = D(Σ/ΣGi ) denote the different divisor P of the extension Σ/ΣGi , for i = 0, 1, . . . , k. Then D0 = ki=1 Di . From Corollary 11.52, 2g − 2 = ni (2gi′ − 2) + deg Di ,

for i = 0, 1, . . . , k. Summing over i = 1, . . . k and subtracting for i = 0, since P deg D0 = k1=1 deg Di , Pk (k − 1)(2g − 2) = i=1 ni (2gi′ − 2) − n(2g ′ − 2). Hence

Pk Pk (k − 1)g = i=1 ni gi′ − ng ′ − n + (k − 1) − i=1 ni . Pk Since n = i=1 ni − (k − 1), the result follows.

2

R EMARK 11.115 Three important examples of finite groups with a partition are PGL(2, q), PSL(2, q), Sz(q).

11.12 LARGE AUTOMORPHISM GROUPS OF FUNCTION FIELDS In this section, Theorem 11.56 is refined. T HEOREM 11.116 In Theorem 11.56, the upper bounds can be sharpened for the cases (a), (b), (c) as follows: (a′ ) |G| < 24g 2 ;

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(b′ ) |G| < 16g 2 ; (c′ ) |G| < 8g 3 . Proof. The notation used in the proof of Theorem 11.56 is maintained. Assume first that (a) occurs. Then d′1 = d′2 = 12 , d′3 = dQ3 /eQ3 , and hence d′ − 2 =

dQ3 − eQ3 . eQ3

(11.64)

(1)

Write eQ3 = e = e1 e′Q3 with e1 = |GQ3 | ≥ p and p ∤ e′Q . In other words, eQ3 = |GQ3 |,

(1)

e1 = |GQ3 |,

e′Q3 = |H|,

(1)

where GQ3 = GQ3 ⋊ H. By Lemma 11.80, dQ3 ≥ e + e1 − 2. Since p ≥ 3, from (11.64),     1 1 1 2 2 e − 1 + e1 − 1 − e ′ = ′ ≥ ′ . ≥ ′ 1− 1− d −2≥ e eQ3 e1 eQ3 p 3eQ3 From (11.19) and (11.23), |G| ≤ 3e′Q3 (2g − 2) ≤ 3(4g + 2)(2g − 2) < 24g 2. Now, assume that (b) occurs. There are two short orbits, both non-tame. As above, for p ≥ 3, d′ − 2 =

dQ1 − eQ1 1 2 dQ2 − eQ2 1 + ≥ ′ + ′ ≥ , eQ1 eQ2 3eQ1 3eQ2 3(4g + 2)

whence |G| ≤ 3(2g + 1)(2g − 2) < 12g 2 . For p = 2, a further argument is required. Since |GQ1 | = eQ1 , the Hilbert Different Formula (11.37) implies that P (1) (i) dQ1 = |GQ1 | − 1 + |GQ1 | − 1 + i≥2 (|GQ1 | − 1) P (i) (1) = eQ1 + GQ1 − 2 + i≥2 (|GQ1 | − 1).

Since the same holds for Q2 , so P2 dQi − eQi d′ − 2 = i=1 eQi  P P2 1  (1) (i) |GQj | − 2 + i≥2 (|GQj | − 1) . = j=1 eQj Three cases (I), (II), (III) are investigated separately. (1) (1) (I) For |GQ1 | ≥ 4 and, similarly for |GQ2 | ≥ 4, (11.65) shows that ! (1) (1) (1) |GQ1 | |GQ1 | − 2 |G | 2 ′ 1 − (1) = d −2≥ ≥ Q1 . eQ1 eQ1 2 eQ1 |G | Q1

Then (11.23) together with (11.19) gives the bound |G| ≤ 2(4g + 2)(2g − 2) < 16g 2 .

(11.65)

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(1)

(2)

(2)

(II) The two equations |GQ1 | = |GQ2 | = 2 and |GQ1 | = |GQ2 | = 1 cannot hold simultaneously because of (11.65) and d′ − 2 > 0. (1) (1) (2) (2) (III) If |GQ1 | = |GQ2 | = 2, and either |GQ1 | = 2 or |GQ2 | = 2, from (11.65), ′

d −2≥

(1)

1 eQ1

|G | = Q1 , 2 eQ1

whence again |G| < 16g 2 , by (11.23) and (11.19). This completes the proof of (b′ ). To investigate (c), two cases are distinguished according as the unique short orbit consists of just one place or more. (1) In the former case, G = GP . If F1 = ΣGP is not rational, then |G| ≤ g(4g + 2) by Theorems 11.60 and 11.78. So, assume that F1 is rational. It is shown that (1) G = GP . Assume on the contrary that G contains a tame subgroup H whose order is prime to p. Choose a non-trivial K-automorphism α in H. Since α is in (1) the normaliser of GP , it induces a K-automorphism α′ of F1 . By Theorem 11.14 ′ (ii)(d), α has at least two fixed places, and hence one of them lies under a place Q (1) distinct from P. So the group generated by α together with GP has a non-trivial stabiliser at Q. Then the orbit of such a place Q under the action of G is short. But then G has at least two short orbits, a contradiction. Hence, G = GP . Now, from (1) Theorem 11.78, |G| = |GP | ≤ 8g 2 . In the latter case, G does not fix P. Let o denote the G-orbit of P, that is, the unique short orbit of G. Then |o| > 1, and deg D(Σ/ΣG ) = |o| deg D(Σ/ΣGP ). Since |G| = |GP ||o|, this gives the equation 2g − 2 = |o|(deg D(Σ/ΣGP ) − 2|GP |).

(11.66)

(1)

Therefore |o| ≤ 2g − 2. If F1 is not rational, then |GP | ≤ g, by Theorem 11.78 (i). This together with Theorem 11.60 give |GP | ≤ g(4g + 2), whence |G| ≤ g(2g − 2)(4g + 2) < 8g 3 .

Assume that F1 is rational. Then (1)

(1)

(1)

deg D(Σ/ΣGP ) = |GP | − |GP | + deg D(Σ/ΣGP ) = |GP | + |GP | + 2g − 2. Thus (11.66) reads: (1)

2g − 2 = |o|(2g − 2 + |GP | − |GP |). (1)

(1)

If |o| = 2g − 2, then 2g − 2 + |GP | − |GP | = 1 with GP = GP ⋊ H, whence (1)

2g − 2 = (|H| − 1)|GP | + 1.

(1)

This implies that H is non-trivial. Also, 2g − 2 > 21 |H||GP | = 12 |GP |, whence |G| = |o||GP | < 2(2g − 2)2 < 8g 3 . (1)

So, let |o| ≤ g − 1. If F1 is rational and GP has a short orbit disjoint from P, then Theorem 11.78 (ii) again gives the desired upper bound, |G| < 8g 3 . Otherwise, o (1) (1) contains a long orbit of GP , and |o| ≥ 1 + |GP |. As |o| ≤ g − 1, this shows

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that |GP | < g. Using the above argument, it follows that |G| < 4g 3 , and this completes the proof of (c′ ). 2 (1)

The methods in this study of GP may also be used to investigate case (d) of Theorem 11.56. But the upper bound on |G| which may be shown in this way is only about g 5 . Therefore a substantial improvement is needed. This requires a careful analysis using deeper group-theoretical results. The final result is stated in Theorem 11.127. In Theorem 11.56 (d), G has two short orbits, one tame and one non-tame. (1) Choose a place P in the non-tame orbit. By Lemma 11.44 (e), GP = GP ⋊H with (1) (2) |H| prime to p. As before, write F1 for ΣGP , and F2 for ΣGP . The following five cases are treated separately. (iv.1) F1 is not rational. (iv.2) F1 is rational, and there is a place R distinct from P such that the stabiliser of R in GP has order pt with t ≥ 1. (iv.3) F1 is rational, and there is a place R distinct from P such that the stabiliser of R in GP has order hpt with t ≥ 1, h > 1 and h prime to p. (1)

(iv.4) F1 is rational, no non-trivial element of GP fixes a place distinct from P, and there is a place R distinct from P but lying in the orbit of P in G such that the stabiliser of R in GP is trivial. (1)

(iv.5) F1 is rational, no non-trivial element of GP fixes a place distinct from P, and, for every place R distinct from P but lying in the orbit of P in G, the stabiliser of R in GP is non-trivial. Before starting with (iv.1), another closed formula for the order of G is deduced. Choose a place Q from the tame orbit of G. Then dQ = eQ − 1 = |GQ | − 1. From Theorem 11.72, (1)

|G| = 2(g − 1)

|GP | |H| |GQ | , N

(11.67)

where G

N = 2g1 |GP | |GQ | + dP |GQ | − |GP | |GQ | − |GP |.

(11.68)

If Σ is rational, which is the relevant case, then N = |GQ |(dP − |GP |) − |GP |. (1) Case (iv.1). If F1 is not rational, then its genus g1 ≥ 1. Since P is fixed by GP (1) and |GP | is a power of p, (1)

dP ≥ 2(eP − 1) = 2(|GP | − 1). (1)

Hence, by Theorem 11.72 applied to GP , (1)

(1)

2g − 2 ≥ |GP |(2g1 − 2) + 2(|GP | − 1).

(1) (1) This implies that |GP | ≤ g/g1 . As the group H ∼ = GP /GP can be viewed as a K-automorphism group of F1 , Theorem 11.60 shows that |H| ≤ 4g1 + 2. The

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same theorem also shows that |GQ | ≤ 4g + 2 as Q belongs to the tame orbit. Also, N ≥ 6, since (1)

N = |GP |2g1 (|GQ | − 1) + |GQ |(dP − |GP |) + (2g1 − 1)|GP | and dP ≥ |GP | ≥ 2. By (11.67),

|G| ≤ 62 (g − 1)(4g + 2)(4g1 + 2)g/g1

< 68 g(g − 1)(g + 1)(4 + 2/g1 ) < 8g 3 .

(11.69)

Case (iv.2). Here, F1 is rational, and the orbit of R under H has size |H|. Let (1) M be the subgroup of GP fixing R, that is M = GP,R . Then (1)

d(Σ/F1 ) ≥ 2(|GP | − 1) + By Theorem 11.72, it follows that

(1)

|GP | (2|M | − 2) |H|. |M |

(1)

g ≥ |GP | (1 − 1/|M |) |H| ≥

1 2

(1)

|GP | |H| = 21 |GP |.

If |GQ | ≤ 2g + 2, this and (11.67) imply that

|G| ≤ 2(g − 1)(2g + 2)2g/N ≤ 8(g 3 − g) < 8g 3 .

If |GQ | > 2g + 2, then a sharper bound is obtained from (11.67), namely, |G| < (2g − 2)2g < 4g 3 .

This depends on the fact that |GQ | < N when |GQ | > 2g + 2. That |GQ | > 2g + 2 implies |GQ | < N is shown in the following way. Since |GP |/|GQ | ≤ 2g/(2g + 2) < 1, N > |GQ |(dP − |GP | − 1). Therefore it may be assumed that dP − |GP | = 1. On the other hand, (11.38) can be re-written in the form P (i) dP − |GP | − 1 = −2 + i≥1 (|GP | − 1).

Therefore

(1)

N (3) (2) (1) > |GP | − 3 + |GP | − 1 + |GP | − 1. |GQ |

If |GP | ≥ 4, in particular for p ≥ 5, the assertion follows. (1) To show that this holds true for |GP | ≤ 3, a little more is needed. For |GP | ≤ 3, (11.67) implies that 3

|G| ≤ 3(2g − 2)(4g + 2),

which is smaller than 8g for g ≥ 2. So this case can be dismissed. (1) (1) Let |GP | ≥ 4 and |GP | ≤ 3. If |GP | = 3 and |GP | = 6, then N = |GQ | − 6. From (11.67), |G| = 12(g − 1)

|GQ | < 24(g − 1) < 8g 3 ; |GQ | − 6

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(1)

so this case can also be dismissed. If either |GP | = 3 and |GP | > 6, or |GP | = 2 and |GP | > 4, then (1)

(1)

|H| > |GP | − 1 with GP = GP ⋊ H.

(1)

Since GP is a normal subgroup of GP , this implies the existence of a non-trivial (2) (1) (1) element α ∈ H in the centraliser of GP . From Theorem 11.75 (i), GP = GP (3) (2) (1) when p = 3, and GP = GP = GP when p = 2. Therefore |GQ | < N . This completes the proof for (iv.2). Case (iv.3). The stabiliser of R in GP contains a non-trivial subgroup T of GP,R of prime order m with p 6= m. By Sylow’s theorem, a subgroup H1 conjugate to H in GP contains T . Now, consider the action of T on F1 or, equivalently, on the set (1) of all orbits of GP . By Theorem 11.14 (ii)(d), T preserves exactly two such orbits: one is the trivial orbit consisting of P and the other is the orbit o(R) containing R. Note that either o(R) = {R} or |o(R)| is a power of p. Since H1 is cyclic and its order is prime to p, a generator of H1 has exactly two (1) invariant orbits of GP , again by Theorem 11.14 (iv). Therefore {P} and o(R) are the invariant orbits of H1 as well. Since o(R) is either trivial or its size is a power of p, so H1 fixes a place in o(R). Since H and H ′ are conjugate in GP , this implies that H fixes a place R′ in o(R). After replacing this place by R, it may be assumed that H fixes R. From the above discussion, a non-trivial element of H may fix some other places of Σ, but (1) such places must be in o(R). So, if some non-trivial element of GP fixes a place distinct from those in P ∪ o(R), then (iv.2) occurs. Therefore it is assumed that this does not happen, since |G| < 8g 3 has already been shown in Case (iv.2). In particular, if o(R) = {R}, then P and R are the only fixed places of any non(1) trivial element in H, and GP fixes both P and R; but no non-trivial element of (1) (1) GP fixes another place of Σ. Since GP = GP ⋊ H, this extends to GP . (1) (1) Let M = GP ∩ GR . Then M is non-trivial p-group. A first consequence is the following result. (1)

L EMMA 11.117 Assume that (iv.3) holds. If o(R) is larger than R, then |GP | is divisible by p2 . (1)

(1)

(1)

(1)

Proof. If |GP | = p, then GP = M, and hence GP = GR . In other words, (1) GP fixes R. 2 (1)

Now, upper bounds on |H| and |GP |, depending only on g, are given. L EMMA 11.118 If (iv.3) holds, then |GP | ≤ 32 (2g + 1)g. Proof. By Theorem 11.72 applied to GP , and Lemma 11.82 (i), 2g−2 ≥ −2|GP |+dP +|o(R)| dR = v−|GP |+|o(R)| dR ≥ |H|−1+|o(R)| dR . From Lemma 11.82 (i), applied this time to GP ∩ GR = M ⋊ H, it follows that dR ≥ |M ||H| + |H| − 1. Therefore 2g − 2 ≥ |H| − 1 − |GP + |o(R)|(|M ||H| + |H| − 1) ≥ (|o(R)| + 1)(|H| − 1),

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whence |H| ≤



for |o(R)| ≥ 2, for o(R) = {R}.

(2g + 1)/3 g

For |o(R)| ≥ 2, this gives |H| ≤ (2g + 1)/3. (1) Also, by Theorem 11.72 applied to GP and Lemma 11.80, (1)

(1)

2g − 2 ≥ −2|GP | + (2|GP | − 2) + |o(R)|(2|M | − 2), whence (1)

g ≥ |o(R)|(|M | − 1) ≥ 12 |o(R)||M | = 12 |GP |. If |o(R)| ≥ 2, the result follows. (1) Therefore let o(R) = {R}. Then Theorem 11.72 applied to GP gives the inequality, (1)

(1)

(1)

2g − 2 ≥ −2 |GP | + 2(2 |GP | − 2) = 2|GP | − 4, (1)

(1)

whence g + 1 ≥ |GP |. Thus, |GP | = |GP ||H| ≤ g(g + 1) < 23 (2g + 1)g.

2

3

The desired upper bound |G| < 8g can be deduced from Lemma 11.118 if the size of the G-orbit of P does not exceed 6(g − 1). The next two lemmas show (2) that this hypothesis is fulfilled in many cases, for instance, when GP is trivial, apart from a few exceptions described in the following lemma. However, such exceptional curves do not provide counter-examples for the bound, |G| < 8g 3 . (2)

L EMMA 11.119 Assume that (iv.3) holds. If the second ramification group GP is trivial and if the orbit of P under G has size r ≥ 6(g − 1), then |G| < 8g 3 . (2)

Proof. Since |GP | = 1, (1)

(1)

N = |GQ |(|GP | − 2) − |GP ||H|. (1)

Thus N > 0 implies that |GQ | = |H|+ǫ with ǫ > 0. So N = ǫ(|GP |−2)−2|H|, which, together with (1)

|GP | − 2 = v = (µ + 1)|H| − 1, given in Lemma 11.82 (i), shows the following equations for µ ≥ 1 and ǫ ≥ 2:

|GQ | |H| + ǫ |H| + ǫ |H| + 2 r = = ≤ ≤ ≤ 2. 2g − 2 N |H|[(µ + 1)ǫ − 2] − ǫ |H|(2ǫ − 2) − ǫ 2|H| − 2

If ǫ = 1, then µ ≥ 2 and

r |GQ | |H| + 1 |H| + 1 = = ≤ ≤ 3. 2g − 2 N |H|(µ − 1) − 1 |H| − 1 (1)

If µ = 0, then ǫ ≥ 3 and |H| = |GP | − 1. For |H| ≥ 6, |GQ | |H| + ǫ r = = ≤ 3, 2g − 2 N |H|(ǫ − 2) − ǫ

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Table 11.2 Order of automorphism group

|GP |

|H|

|GQ |

r 2g − 2

|G|

o(R) 6= {R}

o(R) = {R}

5

4

7

7

280(g − 1)

impossible

g=4

4

3

7

7/2

84(g − 1)

g≥3

g=3

3

2

7

7

84(g − 1)

impossible

g=2

3

2

8

4

48(g − 1)

impossible

g=2

(1)

(1)

while, for |H| = |GP | − 1 < 6,

2 ≤ |H| ≤ 4. In the last case, r ≤ 6(g − 1) provided that either |GQ | < 7 and 3 ≤ |H| ≤ 4, or |GQ | > 8 and |H| = 2. Otherwise, by Lemmas 11.117 and 11.118, one of the cases in Table 11.2 occurs. In the second and fourth cases, |G| < 8g 3 . In the third case, Σ is hyperelliptic, and since no non-soluble group has order 84, Proposition 11.99 shows that again |G| < 8g 3 . Some more is needed to rule out the first case. Here, |G| = 840 = 23 · 3 · 5 · 7, and the orbit o(Q) of Q, that is, the tame orbit of G, has size 120. The key idea is to prove that GQ must have another fixed place in o(Q). Let T be minimal normal subgroup of G. If T is an elementary abelian group of prime power order wk , then wk ∈ {23 , 3, 5, 7}. If |T | < 7 = |GQ |, then a non-trivial element α ∈ T is contained in the centraliser of GQ . This implies that Qα is also fixed by GQ . Since α 6∈ GQ , it follows that Q = 6 Qα , and the assertion is proved. If |T | = 7, then GQ ∼ = T , and hence GQ is a normal subgroup of G. Then G preserves the set of all fixed places of GQ , showing that GQ fixes each place in o(Q). If |T | = 8, then T is a Sylow 2-subgroup of G. Since T is a normal subgroup of G, H is a subgroup of T . Therefore T contains an element of order 4, and hence T has at most 5 involutions. So at least one involution of T is contained in the centraliser of GQ . As before, this implies that GQ has at least two fixed places. Otherwise, T is either a non-abelian simple group or the direct product of isomorphic non-abelian simple groups. The latter possibility does not occur in this situation, since no factor of |G| distinct from 2 is a power of an integer with exponent greater than 1. Also there are only two simple groups whose order divides 840, namely A5 and PSL(2, 7). If T ∼ = A5 , then GQ is contained in the centraliser of T in G. This follows from the fact that 7 is not a factor of 120, and that |Aut(A5 )| = |PGL(2, 5)|. Therefore GQ must have at least two fixed places in o(Q). As 7 is not a factor of 120 and |Aut(PSL(2, 7))| = |PGL(2, 7)|, a similar argument may be used for the case T ∼ = PSL(2, 7) showing that GQ coincides with

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one of the eight Sylow 7-subgroups of T . Then an element of order 3 is in the normaliser of GQ . As before, this implies that GQ cannot have just one fixed place. Once it has been shown that GQ fixes not only Q but some more places in o(Q), it follows that GQ must have at least 7 fixed places. In fact, 120 minus the number of fixed places of GQ must be divisible by 7. By Theorem 11.72 applied to GQ , 2g − 2 ≥ 7(2g ′ − 2) + 7 · 6 ≥ 42 − 14 = 28. From this, g ≥ 15, a contradiction.

2

L EMMA 11.120 Assume that (iv.3) holds. Then |G| < 8g 3 provided that at least one of the following conditions is satisfied: (1)

(a) v < |GP |, where v = dP − |GP |; (b) |GQ | > |H|; (1)

(c) (µ + 1)|GQ | = 6 |GP | + 1, where (µ + 1)|H| = v + 1, as in Lemma 11.82; (1)

(d) v ≥ 2|GP | − 3. Proof. By Lemma 11.118, it suffices to show that r ≤ 6(g −1) if some of the above conditions are satisfied. By (11.67), the bound to be established is the following: r |GQ | |GQ | = = ≤ 3. 2g − 2 N |GQ |(dP − |GP |) − |GP |

By the Hilbert Different Formula (11.37) and Lemma 11.75 (iv), Case (a) can only occur in one of two circumstances: (1)

v1 = 3,

|GP | = 2;

(1)

v1 = 2,

|GP | = 1.

(i) v = |GP | − 1, (ii) v = |GP | − 2,

(2)

(2)

(1)

In case (i), Lemma 11.82 (i) implies that |GP | = (µ + 1)|H|, a contradiction, (1) as both gcd(|H|, |GP |) = 1 and |H| > 1. In case (ii), either |G| < 8g 3 or r < 6(q − 1); see Lemma 11.119. (1) Assume that (b) holds. From (a), take v ≥ |GP |. Put |GQ | = |H| + ǫ with ǫ > 0. Then |GQ | |H| + ǫ |H| + 1 |H| + ǫ r = = ≤ . ≤ 0< (1) 2g − 2 N vǫ v |H|(v − |G |) + vǫ P

On the other hand, by Lemma 11.82 (i), |H| divides (v + 1). Thus |H| ≤ 3v, and hence r ≤ 6(g − 1). Now assume that (c) holds. From (b), take |H| ≥ |GQ |. From Lemma 11.82 (i), 0
|o(R)|v ′ ≥ 12 |H|(|H| − 1) ≥ 21 |H||GQ |. From the proof of Lemma 11.118, (1)

(1)

|G| ≤ 2(g − 1)|H| |GP | |GQ | ≤ 4(g − 1)2 |GP | ≤ 4(g − 1)2 g < 8g 3 . Suppose 2|o(R)| < v + 1. Then (11.71) gives the inequality (1)

(1)

|GP | + v ≥ (3|o(R)| − 1)|M | + |M | + v ′ ≥ 3|GP |. Therefore (d) in Lemma 11.120 occurs. By Lemma 11.118, this shows that (1)

|G| = r|H| |GP | ≤ g(g + 1)6(g − 1) < 8g 3 .

If o(R) = {R}, the bound |G| < 8g 3 can be established by showing that (b), (c) or (d) of Lemma 11.120 holds. To do this, suppose (b) does not hold. By Lemma (2) 11.119, take GP to be non-trivial; that is, v1 ≥ 2. Actually, it may be assumed that v1 = 2, as v1 ≥ 3 implies that case (d) of Lemma 11.120 holds. (2)

L EMMA 11.122 Assume that (iv.3) holds with o(R) = {R}. If GP is not trivial, (2) (2) (1) then g2 = q1 − 1 where g2 is the genus of F2 = ΣGP and q1 = |GP |/|GP |. (1)

Proof. No non-trivial element of GP fixes a place distinct from P and R. Also, GP = GR , as some element of G sends P to R; this was shown in the proof of (1) (2) Lemma 11.121. From (11.17) applied to GP and GP , P (1) (1) (i) 2g − 2 = −2|GP | + 2[2(|GP | − 1) + i≥2 (|GP | − 1)], P (i) (2) (2) 2g − 2 = (2g2 − 2)|GP | + 2[2|GP | − 1) + i≥2 (|GP | − 1)].

Subtracting the two equations gives the result.

2

Let P be the place of F2 lying under P in the covering Σ → F2 . Also, let P¯ be the place of F1 lying under P in the covering Σ → F1 . Then P¯ is the place lying ¯ under P ′ in the covering F2 → F1 , and similarly for R, R′ and R. ¯ Since F1 is rational, there exists y ∈ F1 such that div y = R − P¯ in F1 . As F1 is a subfield of F2 , so y may be viewed as an element of F1 . From (7.1), div y = q1 R′ − q1 P ′ in F2 . Let m be the smallest non-gap at P ′ , and choose x ∈ F2 such that div(x)∞ = rP ′ in F2 . (1) (2) If m < q1 , then x 6∈ F1 by (7.1). Hence there exists α ∈ GP \GP which induces a non-trivial K-automorphism α′ of F2 such that α′ (x) 6= x. From Lemma 11.13, α′ (x) = x + u with u ∈ K and u 6= 0. Since R′ is a zero of x, this implies that α′ does not fix R′ , a contradiction. Therefore m = q1 . ′

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From the Weierstrass Gap Theorem 6.6 and Lemma 11.122, m + 1 = q1 + 1 is also a non-gap. Choose z ∈ F2 such that div z = kR′ + D − (q1 + 1)P ′ ,

where k ≥ 0 and D is an effective divisor of F2 of degree q1 + 1 − k. Since z 6∈ F1 , (1) (2) there exists α ∈ GP \GP that induces a non-trivial K-automorphism α′ of F2 ′ such that α (z) 6= z. Since q1 + 1 is the second smallest non-gap at R′ , it follows that α′ (z) = z + uy + v with u, v ∈ K. Since α′ also fixes R′ , so v = 0 and u 6= 0. From Lemma 11.83, 2 = v1 = q1 + 1 − k, showing that deg D = 2. (2) Now take a generator h of H. Then h is in the normaliser of GP , and hence h may be viewed as a K-automorphism of F2 of order |H| such that h(z) = uz + vy + w with u, v, w ∈ K and u 6= 0. Since h fixes both P ′ and R′ , so v = w = 0 up to a substitution of z by z + u′ y with u′ ∈ K. Thus h(z) = uz, whence it follows that div h(z) = div z and hence h preserves D. Since deg D = 2, so h2 fixes both places in D. Therefore h2 as a K-automorphism of F1 fixes at least four places, ¯ R ¯ and the places lying under those of D in the covering F2 → F1 . namely, P, From Theorem 11.14 (b), ord h = 2, and hence |H| = 2. Since case (b) of Lemma 11.120 does not hold by hypothesis, |H| = 2 implies (1) that |GQ | = 2. Hence N = 2(v − |GP |). But then case (d) of Lemma 11.120 holds, as N 6= 0. Therefore |G| < 8g 3 holds true for case (iv.3) when o(R) = {R}. Case (iv.4). Here, F1 is rational and no non-trivial element in GP fixes R. The latter hypothesis means that the orbit o(R) of R under GP is long. Let o′ (R) denote the orbit of R under G. Then |o′ (R)| · |GR | = |G|. Since P and R lie in the same orbit of G, so GR ∼ = GP . Also, o(R) is contained in o′ (R). From (11.67), |GP | ≤

|GQ | |G| = 2(g − 1) · ≤ 2(g − 1)|GQ |. |GP | N

Now, a lower bound on N , as in (11.68), is given. As

N = dP |GQ | − |GP ||GQ | − |GP | ≥ dP |GQ | − |GP ||GQ | − 2(g − 1)|GQ |, so N ≥ |GQ |(dP − |GP | − 2(g − 1))

(11.73)

From Theorem 11.72 applied to GP , (1)

(1)

2(g − 1) = −2|GP | + (dP − |GP |(|H| − 1)) = dP − |GP | − |GP |. Hence (1)

dP − |GP | − 2(g − 1) = |GP |.

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Taking (11.73) into account, N ≥ |GP ||GQ |. Then (11.67), together with Theorem 11.60, gives the desired result: (1)

|G| ≤ 2(g − 1) ·

|GP ||H||GQ | (1)

|GP ||GQ |

≤ 8(g + 1)(g − 1) < 8g 3 .

Case (iv.5). The type of argument used for (iv.4) only gives an estimate that |G| < cg 5 in this case. So, further investigation is needed, which also requires a few technical results, such as the inverse formula of (11.67): (1)

2g − 2 =

|G| (|GP | − |GP | |GQ |) ; |GQ |(|G| − |GP |)

(11.74)

this holds whenever |GP ∩ GQ | = 1. First, the possibility that the unique non-tame orbit consists of a single place P (1) is considered. From Theorem 11.72 applied to GP , P (i) 2g − 2 = −2 + i≥2 (|GP | − 1). Arguing as in the proof of (11.72), it follows that 2g ≥ |H|. By Theorem 11.78, (2)

4|GP |

(1)

|G| = |H||GP | ≤ 2g

(2) (|GP |

−

1)2

g2,

(2)

which is greater than or equal to 8g 3 only if p = 2, |GP | = 2, and F2 is rational. Theorem 12.5 implies that Σ = K(x, y) with y 2 − y = B(x), where B(X) is in K[X] and has odd degree m. Now Theorem 12.7 provides all the necessary information: pn = 2,

m − 1 = 2g,

|H| | m.

From Theorems 12.8 and 12.9, |G| ≥ 8g 3 only occurs when m − 1 = 2k ≥ 4, (1) |H| = m and |GP | = 2(m − 1)2 . Therefore |G| < 8g 3 apart from just one exception occurring for every k > 1 and p = 2, namely, the hyperelliptic curve k

v(Y 2 + Y + X 2

+1

),

which has genus 2k−1 . In fact, if P is the unique place centred at Y∞ = (0, 0, 1), then GP has order 22k+1 (2k + 1), which is greater than 8g 3 = 23k . The possibility that the non-tame orbit is non-trivial is now investigated. Permutation group theory plays an important role, as in the next result. L EMMA 11.123 If case (iv.5) occurs, then G acts on its non-tame short orbit o as a 2-transitive permutation group. In particular, o has size q + 1 with q = pt , and ¯ induced by G on o are as follows: the possibilities for the permutation group G ¯∼ (I) G = PSL(2, q) or PGL(2, q); ¯∼ (II) G = PSU(3, n) or PGU(3, n), with q = n3 ; ¯∼ (III) G = Sz(n), with p = 2, n = 2n20 , n0 = 2k , with k odd, and q = n2 ;

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¯∼ (IV) G = Ree(n) with p = 3, n = 3n20 , n0 = 3k , and q = n3 ; ¯ is soluble, and the size of o is a prime (V) a minimal normal subgroup of G power. (1)

Proof. For a place P ∈ o, let o0 = {P}, o1 , . . . ok denote the orbits of GP conSk tained in o; then, o = i=0 oi . To prove that G acts 2-transitively on o, it must be shown that k = 1. For any i with 1 ≤ i ≤ k, take a place R ∈ oi . In case (iv.5), R is fixed by an element α ∈ GP whose order m is a prime different from p, and hence divides |H|. By Sylow’s theorem, there is a subgroup H ′ conjugate to H in GP which contains α; here, α preserves ok . As noted previously, F1 being rational implies that α fixes (1) at most two orbits of GP . Therefore o0 and oi are the orbits preserved by α. As (1) H ′ is abelian and it fixes o0 , so the orbits o0 and oi are also the only orbits of GP (1) ′ ′ which are fixed by H . Since GP = GP ⋊ H , this implies that the whole group GP fixes oi . As i can be any integer between 1 and k, it follows that GP fixes each of the orbits o0 , o1 , . . . , ok . Therefore, either k = 1 or GP fixes at least three orbits (1) of GP . The latter case cannot actually occur, by Theorem 11.14 applied to F1 . (1) Also, the size of o is of the form q + 1 with q = |GP |; in particular, q is a power of p. ¯ denote the 2-transitive permutation group induced by G on o. The 2Let G ¯ is cyclic, as the subgroup of G fixing two distinct places is point stabiliser of G cyclic when (iv.5) occurs; see Theorem 11.49. Two-transitivity together with this special behaviour of the 2-point stabiliser is enough to determine completely both ¯ Theorem A.17 states indeed that, up to the abstract structure and the action of G. ¯ is one of the groups in the list, with G ¯ acting in each of the first isomorphism, G four cases in its natural 2-transitive permutation representation. 2 A consequence of Lemma 11.123 is the following result. L EMMA 11.124 In case (iv.5), GP and GQ have trivial intersection. Proof. Let α ∈ GP ∩ GQ be non-trivial. Then p ∤ ord α, and hence α ∈ H. This shows that α fixes not only P but another place in o, say R. Since Q 6∈ o, this shows that α has at least three fixed places. These places are in three different (1) orbits of GP . Since F1 is rational, Theorem 11.14 implies that α fixes every orbit (1) of GP , a contradiction. 2 Now, the cases in Lemma 11.123 are investigated further. Let M be the kernel ¯ = G/M . Since M of the above permutation representation of G over o. Then G ¯ may also be viewed as an automorphism is a normal subgroup of G, the group G group of ΣM . When M is not trivial, then it influences the size of G. T HEOREM 11.125 Assume that case (iv.5) occurs. If the subgroup M fixing all places in o is non-trivial, then the following assertions hold. (i) If ΣM is rational, then

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(a) the genus

g = 12 (q − 1)(|M | − 1);

(11.75)

(b) |G| ≥ 8g 3 only occurs when Σ = K(x, y) with y q − y = x2 , and ¯ = G/M ∼ q = pk is odd, g = 1 (q − 1), |M | = 2, G = PGL(2, q). 2

(ii) If Σ

M

is elliptic, then g =

1 2 (q

+ 1)(|M | − 1) + 1, and |G| ≤ 8g 3 .

(iii) If ΣM has genus g¯ ≥ 2, then g = g¯|M | + 12 (q − 1)(|M | − 1), and ¯ 1 ¯ |G| |G| |G| < 3 < 3. 3 2 g g¯ |M | g¯

Proof. Since F1 is rational, M has no fixed place outside o, by Theorem 11.14 (iv). From Theorem 11.72 applied to M , 2g − 2 ≥ |M |(2¯ g − 2) + (q + 1)(|M | − 1).

(11.76)

For g¯ ≤ 1, this shows the first statement in both (i) and (ii). If g¯ > 1, write (11.76) as follows: g = g¯|M | + 21 (q − 1)(|M | − 1).

Since |M | > 1 and q > 1, it follows that g > g¯|M |. This together with the relation ¯ |G| = |G||M | proves (iii). To prove (i)(b) and (ii)(b), take a non-trivial element µ in M . From Theorem (1) (1) 11.49, µ is in the normaliser of GP . Also, µ fixes two orbits of GP , namely {P} (1) and o\{P}. Since F1 is rational, they are all the orbits of GP fixed by µ. This implies that µ fixes no place outside o. Hence µ 6∈ GQ , showing that M and GQ ¯ Q |. So, (11.74) reads as follows: have trivial intersection; that is, |GQ | = |G 2g − 2 =

¯ Q )| ¯ G ¯ P ||M | − |G ¯ (1) ||G |G|(| P ¯ ¯ ¯ |GQ |(|G| − |GP |)

(11.77)

¯ For the proof of (i)(b), let g¯ = 0. Then no non-trivial K-automorphism in G fixes three distinct places of ΣM in o. From the list in Theorem 11.123, the groups of type (II) and (IV) do not comply with this requirement. ¯∼ If G = PSL(2, q) as in Lemma 11.123 (I), then ¯ = 1 (q 3 − q), |G| 2

¯ P | = 1 (q 2 − q), |G 2

(1)

|GP | = q.

From (11.77), ¯Q| (q − 1)|M | − 2|G . ¯Q| |G ¯ Q | = 1 (q + 1). This together with 2g = (q − 1)(|M | − 1) gives |G 2 1 3 3 Now, let |G| ≥ 8g . Since |G| = 2 (q − q)|M |, from the above computation, 2g − 2 = 21 (q + 1)

(|M | − 1)3 q(q + 1) > 2 . (q − 1)2 |M |

(11.78)

Since g > 1, (11.78) implies that |M | = 2. Hence, g = 12 (q − 1). In particular, p > 2. Since |M | = [Σ : ΣM ] = 2 by Theorem 11.36, Σ is hyperelliptic. From ¯ = PSL(2, q). Proposition 11.101, (ii)(b) follows for G

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¯ = PGL(2, q). If G ¯∼ The above argument also works for G = Sz(n) as in Lemma 11.123 (III), then (1)

¯ = (n2 + 1)n2 (n − 1), |G ¯ P | = n2 (n − 1), |G ¯ | = n2 . |G| P From (11.77), 2g − 2 = (n2 + 1)

¯Q| (n − 1)|M | − |G . ¯Q| |G

Comparison with (i)(a) shows that |GQ | = n−1+2/(n+1) which is impossible for ¯∼ n > 1. Therefore the case G = Sz(n) cannot actually occur. A similar computation ¯ = Ree(n) is also impossible. shows that G ¯ regular on o, then T is an elementary If T is a minimal normal subgroup of G abelian w-group acting on o as a sharply transitive permutation group. Hence |o| = |T | = wk for a prime w 6= p and k ≥ 1. Thus wk = q + 1.

Therefore either q = 8, wk = 9 or k = 1, q = 2r . Up to an isomorphism, there is only one 2-transitive permutation group of prime degree w, namely, GL(1, w). Hence, when k = 1, Lemma 11.124 implies that |GQ | = w, whence 2g − 2 = |M | − w by (11.77). Comparison with (i)(a) shows that |M | − w + 2 is equal to (w − 2)(|T | − 1), whence ¯ = 6, |G| = 6|M |. w = 3, q = 2, g = 1 (|M | − 1), |G| 2

Therefore |G| ≥ 8g 3 only occurs when |M | = 3. But then g = 1 contradicting the hypothesis. In the other case, wk = 9. From Lemma 11.124, either |GQ | = 9 or 3. In the ¯ P | = 40. But this is imposformer case, (i)(a) together with (11.77) implies that |G ¯ has order 48 and hence contains sible, because GL(2, 3), the normaliser of T in G, no element of order 5. If |GQ | = 3, then (i)(a) and (11.77) cannot simultaneously hold. ¯ P | divides 24. This is consistent Finally, let g¯ = 1. From Theorem 11.94, |G ¯ on o only if |o| is one of 3, 4, 5, 7, 9, 13, 25. Since with the 2-transitive action of G ¯ and G ¯ is a K-automorphism group of ΣM , Lemmas 11.60 and |o| divides |G| 11.60 together with (11.59) also imply that the possibilities for |o| are actually four, namely, 3, 4, 5, 9. The last one has already been ruled out. ¯ = 6, as the 2-transitivity of G ¯ on o implies If |o| = 3, then q = p = 2 and |G| 3 ∼ ¯ that G = S3 . Hence |G| = 6|M | and g = 2 (|T | − 1) + 1. Therefore |G| < 8g 3 . If |o| = 4, then q = p = 3, and g = 2(|T | − 1) + 1. From the 2-transitive action ¯ on o, either |G| ¯ = 12 or |G| ¯ = 24. In both cases, |G| < 8g 3 . of G If |o| = 5, then q = 4, and g = 25 (|T | − 1) + 1. Since a 2-transitive group on ¯ = 20. Hence, 5 letters with a regular normal subgroup is sharply 2-transitive, |G| 3 again, |G| < 8g . 2 T HEOREM 11.126 If |G| ≥ 8g 3 , then (iv.5) only occurs in the following cases: k

(I) p = 2, the field Σ = K(x, y) with y 2 + y = x2 +1 and genus g = 2k−1 , where G fixes a place P, and |G| = 22k+1 (2k + 1);

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(II) p > 2, the field Σ = K(x, y) with y 2 = xq − x and genus g = 12 (q − 1), with G/M ∼ = PGL(2, q), where q = pr and |M | = 2; = PSL(2, q) or G/M ∼ (III) p ≥ 2, the field Σ is the Hermitian function field K(x, y) with y q + y = xq+1 and genus g = 21 (q 2 − q), with G ∼ = PSU(3, q) or G ∼ = PGU(3, q); (IV) p = 2, q0 = 2r , q = 2q02 , the field Σ = K(x, y) with xq0 (xq + x) = y q + y and genus g = q0 (q − 1), with G ∼ = Sz(q). Proof. If the non-tame orbit consists of a single place, then (I) holds. This was the first result in case (iv.5). Therefore the non-tame orbit o is assumed non-trivial. The few possibilities for ¯ induced by G on o are described in Lemma the 2-transitive permutation group G 11.123. First, the faithful actions are investigated. In doing so, Lemma 11.123 together with classification theorems of subgroups of the groups in that lemma are the main ingredients. Also, by Lemma 11.124, no non-trivial element in GQ fixes a place in o. In particular, GQ , which is cyclic by Theorem 11.49 (ii)(b), contains no element of order 2 or p. Take G ∼ = PGL(2, q). Then |G| = q 3 − q,

|GP | = q 2 − q,

From (11.74), 2g − 2 = (q + 1) If |G| ≥ 8g 3 then 1


(q 3 + 1)(q 2 − 1) − (q 3 − 1). d |GQ |

whence |GQ | > q + 1. Further, the order of GQ divides (q 3 + 1)(q + 1)(q 2 − 1). Therefore GQ must be contained in one of the subgroups (ii), (iii), (iv) in Theorem A.10. Since |GQ | > q + 1, (ii) and (iii) cannot occur; hence |GQ | =

q2 − q + 1 td

for an odd integer t, and 2g = (q + 1)(q 2 − 1)t − (q 3 − 1) = (q − 1)(t(q + 1)2 − (q 2 + q + 1).

As |G| ≥ 8g 3 , this implies that t = 1 and 2g = q 2 − q. From Theorem 12.25 case (III) follows. The same argument works for G ∼ = PGU(3, q). In the case that G ∼ = Sz(q), q0 = 2r and q = 2q02 , |G| = (q 2 + 1)q 2 (q − 1),

|GP | = q 3 − q,

(1)

|GP | = q 2 .

From (11.74), 2g =

(q + 2q0 + 1)(q − 2q0 + 1)(q − 1) − q 2 + 1. |GQ |

By Lemma 11.124 and Theorem A.12, |GQ | must be prime to q − 1. Hence, |GQ | divides either q + 2q0 + 1 or q − 2q0 + 1. If |GQ | = (q + 2q0 + 1)/t, then 2g = (t − 1)(q 2 − 1) − 2tq0 (q − 1),

and hence t > 1. On the other hand, |G| < 8g 3 when t > 1. If |GQ | = (q − 2q0 + 1)/t, then 2g = (t − 1)(q 2 − 1) + 2tq0 (q − 1).

If, in addition, |G| ≥ 8g 3 , then t = 1 and hence g = q0 (q − 1). From Theorem 12.14, case (IV) follows. Now assume that G ∼ = Ree(q), q = 3q02 and q0 = 3h . Then |G| = (q 3 + 1)q 3 (q − 1), From (11.74), 2g = (q − 1)



|GP | = q 3 (q − 1),

(1)

|GP | = q 3 .

 (q + 3q0 + 1)(q − 3q0 + 1)(q + 1) − (q 2 + q + 1) . |GQ |

From Theorem A.14, no prime divisor of |GQ | divides q − 1.

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Suppose that |GQ | = (q + 1)/t. Then

2g = (q − 1)((t − 1)(q 2 + 1) − q);

hence t ≥ 2 and 2g ≥ (q − 1)(q 2 − q + 1). Thus |G| < 8g 3 . Suppose that |GQ | = (q + 3q0 + 1)/t. Then

2g = (q − 1)((t − 1)q 2 − t(3q0 q − 2q + 3q0 − 1) − q − 1);

hence t ≥ 2 and

2g ≥ (q − 1)(q 2 − 3(2qq0 − q + 2q0 ) + 1).

Thus |G| < 8g 3 . Suppose that |GQ | = (q − 3q0 + 1)/t. Then

2g = (q − 1)((t − 1)q 2 + t(3q0 q + 2q + 3q0 + 1) − q − 1);

hence t ≥ 2 and

2g ≥ (q − 1)(q 2 + 3(2q0 q + q + 2q0 ) + 1).

Again, |G| < 8g 3 . Now assume that a minimal normal subgroup T of G is regular on o. Then the size |o| = q + 1 is a power of a prime. It is shown in the proof of Theorem 11.125 that this is only possible when q = 8 or q + 1 = w is a prime number. In both cases, G is sharply 2-transitive on o, and hence |G| = q(q + 1). Therefore, either |GQ | = q + 1 or |GQ | ∈ {3, 9} and q = 8. In the former case, q + 1 ≤ 4g + 2 by Theorem 11.60, and hence |G| ≤ (4g + 1)(4g + 2) < 8g 3 for g > 2. This holds true for g = 2 as |G| = 6 and 8g 3 = 64. In the latter case, |G| = 72 and |G| < 8g 3 for g > 2. Since G is soluble, |G| = 72 and g = 2 do not simultaneously occur by Theorem 11.99 (iii). ¯ of G on o Finally, the possibility of a 2-transitive permutation representation G 3 with non-trivial kernel M is considered. Suppose that |G| ≥ 8g . Theorem 11.125 states that, if ΣM is rational, then (I) occurs while no example exists when ΣM ¯ > 8¯ is elliptic. Also, if the genus g¯ of ΣM is at least 2, then |G| g 3 . Hence case ¯ ¯ acts (iv.5) occurs, where G is viewed as a K-automorphism group of ΣM . As G faithfully on its 2-transitive non-tame orbit, from the first part of this proof there ¯ But each can be ruled out as follows. are three possibilities for G. ∼ ¯ If G = PSU(3, q), then 2¯ g = q 2 − q,

2g = (q 2 − q)|M | + (q 3 − 1)(|M | − 1) ¯ by Theorem 11.125 (iii). Since |G| = |G||M |, it follows that

|G| = (q 3 + 1)q 3 (q 2 − 1)|M | < (q 3 + 1)q 3 (q 2 − 1)( 12 |M |)3 (q − 1)3 (q 2 + 3q + 1)3 , ¯∼ which is smaller than 8g 3 . The same argument works for G = PGU(3, q). ¯ If G = Sz(q), then Thus

2¯ g = 2q0 (q − 1),

2g = 2q0 (q − 1)|M | + (q 2 + 1)(|M | − 1).

¯ |G| = |G||M | = (q 2 + 1)q 2 (q − 1)|M |

< ( 21 |M |)3 (q 2 + 4q0 q − 4q0 + 1)3 < 8g 3 .

In geometric terms, the following result is obtained.

2

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T HEOREM 11.127 Let F be an irreducible curve of genus g ≥ 2. If |G| ≥ 8g 3 , then F is birationally equivalent to one of the following: k

(I) the hyperelliptic curve v(Y 2 + Y + X 2 +1 ) with p = 2, g = 2k−1 ; also, G fixes a place P and |G| = 22k+1 (2k + 1);

(II) the hyperelliptic curve v(Y 2 − (X q − X)) with p > 2, g = 21 (q − 1); also, G/M ∼ = PSL(2, q) or G/M ∼ = PGL(2, q), where q = pr and |M | = 2; (III) the Hermitian curve v(Y q + Y − X q+1 ) with p ≥ 2, g = also, G ∼ = PSU(3, q) or G ∼ = PGU(3, q);

1 2 2 (q

− q);

(IV) the DLS curve v(X q0 (X q + X) − (Y q + Y )) with p = 2, q0 = 2r , q = 2q02 , g = q0 (q − 1); also, G ∼ = Sz(q). 11.13 K-AUTOMORPHISM GROUPS FIXING A PLACE In this section, K-automorphism groups of Σ whose order is divisible by p and in which every element of order p has exactly one fixed place (11.80) are investigated. R EMARK 11.128 A suitable power β of any non-trivial element α of a p-group has order p. Also, every fixed place of α is fixed by β as well. Therefore condition (11.80) is satisfied by a subgroup G of AutK (Σ) if and only if every p-element in G has exactly one fixed place. Also, since every p-subgroup has non-trivial centre, the non-trivial elements of a Sylow p-subgroup Sp of a K-automorphism G of AutK (Σ) satisfying (11.80) have the same fixed place. Condition (11.80) occurs in several circumstances, for instance when G has property (iv.5) in the proof of Theorem 11.116, or when G is the K-automorphism group of certain curves v(A(X) − B(Y ) = 0) investigated in Section 12.1. Another such circumstance is described in the following two lemmas. L EMMA 11.129 If Σ has p-rank 0, then (11.80) holds in AutK (Σ). Proof. Let α ∈ AutK (Σ) have order p. Applying the Deuring–Shafarevich formula (11.28) to the group generated by α gives −1 = p(γ ′ − 1) + m(p − 1), where γ ′ is the p-rank of Σα and m is the number of fixed places of α. This is only possible when γ ′ = 0 and m = 1. 2 R EMARK 11.130 The converse of Lemma 11.129 is not true, a counter-example being the curve F of 2-rank equal to 4 given in Exercise 10 in Chapter 9. To show this, let Σ = K(x, y), y 6 + y 5 + y 4 + y 3 + y 2 + y + 1 = x3 (y 2 + y), be the function field of F. Then the birational transformation x′ = x, y ′ = y + 1 is a K-automorphism of Σ which has only one fixed place, namely that arising from the unique branch of F tangent to the infinite line.

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L EMMA 11.131 Assume that AutK (Σ) contains a p-subgroup G of order pr . If ΣG has p-rank zero, and (11.80) holds, then Σ has p-rank zero. Proof. Let P be the fixed place of G. Since γ ′ = 0, (11.28) applied to G gives γ − 1 = −|G| + |G| − 1, whence γ = 0. 2 R EMARK 11.132 Exercises 9 and 10 in Chapter 9 show that the hypothesis on the p-rank of ΣG is essential. Suppose that a subgroup G of AutK (Σ) has a Sylow p-subgroup with property (11.80). Then (11.80) is satisfied by all Sylow p-subgroups of G. Suppose further that G has no normal Sylow p-subgroup. Let Sp and Sp′ two distinct Sylow psubgroups of G. By Remark 11.128, both Sp and Sp′ each have exactly one fixed place, say P and P ′ . Also, Sp is the unique Sylow p-subgroup of the stabiliser of (1) 6 P ′ ; for, if P under G, and GP = GP ⋊ H; see Theorem 11.49. Further, P = ′ ′ P = P , then Sp and Sp are two distinct Sylow p-subgroups in GP . However, (1)

this is impossible as GP is a normal subgroup of GP . Therefore any two distinct Sylow p-subgroups in G have trivial intersection. A subgroup H of a group G is a trivial intersection set if, for every g ∈ G, either H = g −1 Hg or H ∩ g −1 Hg = {1}. A Sylow p-subgroup T of G is a trivial intersection set in G if and only if that T meets every other Sylow p-subgroup of G trivially. Hence Lemma 11.129 has the following corollary.

T HEOREM 11.133 If Σ has p-rank 0, then every Sylow p-subgroup Sp in AutK (Σ) is a trivial intersection set. A classical result on trivial intersection sets is the following. T HEOREM 11.134 (Burnside) If a Sylow p-subgroup Sp of a finite soluble group is a trivial intersection set, then either Sp is normal or cyclic, or p = 2 and S2 is a generalised quaternion group. Finite groups whose Sylow 2-subgroups are trivial intersection sets have been classified. This has been refined to groups containing a subgroup of even order which intersects each of its distinct conjugates trivially. T HEOREM 11.135 (Hering) Let Q be a subgroup of a finite group G with trivial normaliser intersection; that is, the normaliser NG (Q) of Q in G has the following two properties: (i) Q ∩ x−1 Qx = {1} for all x ∈ G\NG (Q); (ii) NG (Q) 6= G. If |Q| is even and S is the normal closure of Q in G, then S = O(S) ⋊ Q, the semidirect product of O(S) by Q, with Q a Frobenius complement, unless S isomorphic to one of the groups PSL(2, n), Sz(n), PSU(3, n), SU(3, n), (11.81) ¯ be the permutation group induced by G on the set where n is a power of 2. Let G Ω of all conjugates of Q under G. Then

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(iii) CG (S) is the kernel of this representation; (iv) S is transitive on Ω; (v) |Ω| is odd; (vi) in the exceptional cases, (a) |∆| = n + 1, n2 + 1, n3 + 1, n3 + 1; ¯ = G/CG (S) contains a normal subgroup isomorphic to one of the (b) G linear groups in (11.81); ¯ is isomorphic to an automorphism group of S containing S. (c) G Hering’s result does not extend to subgroups of odd order with trivial normal intersection. A major result which can be viewed as a generalisation of Theorem 11.134 is the following. T HEOREM 11.136 Let Sd , with d > 11, be a Sylow d-subgroup of a finite group G that is a trivial intersection set but not a normal subgroup. Then Sd is cyclic if and only if G has no composition factors isomorphic to either PSL(2, dn ) with n > 1 or PSU(3, dm ) with m ≥ 1. Now, some consequence of the above results are stated. Two cases are distinguished according as p = 2 or p > 2. When p = 2, if (11.80) is satisfied by a subgroup G of AutK (Σ), then Theorem 11.135 applies to G with Q a 2-subgroup of a Sylow 2-subgroup S2 of G. In fact, (i) is true, while (ii) holds provided that G is larger than GP , where P is the fixed place of S2 . When GP = G, the structure of G is fully described in Theorem 11.49. Otherwise, G has no fixed place, and Hering’s Theorem determines the abstract structure of the normal subgroup S of G generated by all elements whose order is a power of 2. Note that, since S2 has only one fixed point, NG (S2 ) = GP holds. If S = O(S) ⋊ S2 , then S2 is a Frobenius complement, and hence it has a unique involutory element. Note that it is not claimed that S is a Frobenius group. In the exceptional cases, if CG (S) is the centraliser of S in G, then G/CG (S) is the K-automorphism group of a group listed in Hering’s Theorem. Therefore the following result is obtained. T HEOREM 11.137 Let p = 2 and assume that (11.80) holds. If S is the subgroup of AutK (Σ)(X ) of even order generated by all elements whose order is a power of 2, then one of the following alternatives holds. (i) S fixes no place and either (a) or (b) is satistfied: (a) S is isomorphic to one of the groups, PSL(2, n), PSU(3, n), SU(3, n), Sz(n), with n = 2r ;

(11.82)

(b) S = O(S) ⋊ S2 with S2 a 2-Sylow subgroup of G; here, S2 is either a cyclic group or a generalised quaternion group.

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(ii) Aut(X ) fixes a place, and AutK (Σ)(X ) = S ⋊ H, with H a subgroup of odd order. Using Theorem 11.135, the action of S on the set Ω of all places of Σ fixed by some involution or, equivalently, on the set of all involutions in G can be investigated. Assume that Aut(X ) fixes no place, and let S¯ be the permutation group induced by S on Ω. If M is the kernel of the permutation representation of S, then M is the centre of Aut(X ), the group S¯ = S/M , and S¯2 = S2 M/M is a 2-Sylow ¯ Here, S¯ is transitive on Ω; also, S¯2 fixes P and acts on Ω\{P } as a subgroup of S. semiregular permutation group. Further, M = CG (S). If S is isomorphic to one of the groups in (11.82), then S¯ is 2-transitive on Ω, and |Ω| = n + 1, n2 + 1, n3 + 1 according as S¯ ∼ = PSL(2, n), Sz(n), PSU(3, n). Theorem 11.134 is the main ingredient in the proof of the following result. T HEOREM 11.138 Let G be a soluble K-automorphism group of Σ satisfying condition (11.80). Assume that G fixes no place, and that  2 p when p > 2, |G| is divisible by (11.83) 16 when p = 2. If Σ has genus g ≥ 2, then |G| ≤ 24g(g − 1). Proof. For g = 2, Theorem 11.138 follows from Theorem 11.99. Let N be a minimal normal subgroup of G. Since G is soluble, N is an elementary abelian group of order dr with a prime d. If d = p, then (11.80) implies that N fixes a unique place. But then G itself must fix P, contradicting one of the hypotheses. Therefore d 6= p. In particular, the Sylow p-subgroups of G and G/N are isomorphic, and hence Theorem 11.134 applies to G/N , while (11.83) remains valid for G/N . ¯ = G/N , viewed as a K-automorphism group of ΣN , This suggests that G ¯ also satisfies (11.80). should be investigated. The aim is to show that G ′ The place P lying below P in the extension Σ/ΣN is fixed by a Sylow p¯ Since G/N can be viewed as a K-automorphism group of ΣN , subgroup of G. this rules out the possibility that ΣN is elliptic, because (11.83) is inconsistent with Theorem 11.94. Similarly, ΣN cannot be rational. In fact, if ΣN were rational, then no p-subgroup of G/N and hence of G would be either cyclic or a quaternion group, by Theorem 11.14. Therefore ΣN has genus g¯ ≥ 2. Assume that Theorem 11.138 does not hold, and choose a minimal counter-example with genus g as small as possible. Then |G| > 24g(g − 1), but Theorem 11.138 holds for all genera g ′ with 2 ≤ g ′ < g. Theorem 11.72 applied to N gives 2g − 2 ≥ |N |(2¯ g − 2), whence |G| > 12g(2g − 2) > |N | 12g(2¯ g − 2). Therefore |G/N | > 24¯ g(¯ g − 1). By the divisibility condition (11.83) on |G|, the ¯ = G/N is not a prime. Hence G ¯ is a soluble group. Since d 6= p, so order of G ¯ satisfies (11.83). On the other hand, G ¯ can be regarded as a K-automorphism G group of ΣN .

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¯ also satisfies condition (11.80), let α ¯ of order To show that G ¯ be any element of G p. Choose an element α in G whose image is α ¯ under the natural homomorphism G → G/N . By (11.80), Σ has a unique place P fixed by α. Then, α ¯ fixes the place P¯ lying under P in the extension Σ/ΣN . Assume on the contrary that α ¯ fixes ¯ Then the set of places of Σ lying over Q ¯ is preserved another place of ΣN , say Q. by α. The number of such places is prime to p since their number divides |N |. But then α must fix one of these places, contradicting (11.80). ¯ fixes a place of ΣN . Equivalently, an Since G is a minimal counter-example, G orbit of N is preserved by G. Such an orbit consists of places of Σ fixed by pelements of G.Since Sp has only one fixed place, so |Sp | ≤ |N | − 1. This together with (11.83) implies that |N | ≥ 10. As g − 1 ≥ |N |(¯ g − 1), it follows that g ≥ 11. ¯ with H ¯ a cyclic group ¯ = |G ¯ P | = |G ¯ (1) | |H|, From Theorem 11.49 (ii)(b), |G| P ¯ ≤ 4¯ of order prime to p. By Theorem 11.60, |H| g + 2. Let S¯p be the Sylow ¯ ¯ then S¯p = G ¯ (1) p-subgroup of G; ¯ . Since p ∤ |N |, so Sp is isomorphic to Sp . P ¯ (1) g + 4. Therefore Assume that Sp is cyclic. From Theorem 11.79, |G ¯ | ≤ 4¯ P

¯ ≤ (4¯ |G| g + 4)(4¯ g + 2),

whence |G| ≤ (4¯ g +4)(4¯ g +2)|N |. Since g −1 ≥ |N |(¯ g −1), g¯ ≥ 2 and |N | ≥ 10, it follows that |G| ≤ 16(¯ g + 1)(¯ g + 21 )|N | ≤ ≤

g¯ + 12 16 g¯ + 1 · (¯ g − 1)|N | · (¯ g − 1)|N | 10 g¯ − 1 g¯ − 1

8 2¯ g 2 + 3¯ g+1 (g − 1)2 ≤ 12(g − 1)2 < 24g(g − 1). · 5 2(¯ g − 1)2

But then G is not a counter-example. If Sp is not cyclic, Theorem 11.134 implies that p = 2 and that S2 is a quaternion group. Since a quaternion group has a cyclic subgroup of index 2, Theorem 11.79 ¯ (1) | ≤ 2(4¯ g + 4). Now the argument above can be used to show that shows that |G P 1 2 |G| ≤ 12(g − 1) , whence |G| ≤ 24(g − 1)2 < 24g(g − 1). Again, G is not a 2 counter-example. 2 Finally, an alternative proof for p = 2 which does not require the hypothesis (11.83) is given. T HEOREM 11.139 Let p = 2, g ≥ 2, and assume that (11.80) holds. If G is a soluble K-automorphism group of Σ fixing no place of Σ, then |G| ≤ 24g 2 . Proof. Since 84(g − 1) < 24g 2 , the group G may be supposed to be one of the exceptions in Theorem 11.56. Since G has no fixed place, from Theorem 11.137, S = O(S) ⋊ S2 , and O(S) acts transitively on the set o consisting of all places fixed by involutory elements in G. In particular, o is the unique non-tame orbit of (1) G. Let P ∈ o. By assumption (11.80), GP is a Sylow 2-subgroup S2 of G. From |O(S)| = |o||O(S)P |, |G| =

|GP ||O(S)| . |O(S)P |

(11.84)

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In particular, o has odd size. As O(S) is transitive on o and NG (S2 ) = GP , each element in G is the product of an element fixing P and an element from O(S), that is, G = GP O(S). Also, as O(S) is a characteristic subgroup of S and S ⊳ G, ¯ = G/O(S) is a factor group which can be viewed as a so O(S) ⊳ G. Hence G K-automorphism group of ΣO(S) . ¯ is Assume that ΣO(S) is rational. By Theorem 11.14, every 2-subgroup of G ∼ elementary abelian. On the other hand, as S2 = S2 O(S)/O(S), from Theorem 11.135 the factor group S2 O(S)/O(S) has only one involutory element. Thus |S2 O(S)/O(S)| = 2, whence |S2 | = 2. This implies that O(S) = O(G) and G = O(G) ⋊ S2 . Hence |G| = 2|O(G)| ≤ 168(g − 1) ≤ 24g 2 for g ≥ 6. To show that this holds true for 2 ≤ g ≤ 5, note that, if O(G) has more than three short orbits, then |G| ≤ 12(g − 1) < 24g 2 , as observed in the proof of Theorem 11.56 (I). Since S2 has exactly one fixed place, it preserves only one orbit of O(G). Hence the number of short orbits of O(G) is odd, and thus it is one or three. Since p = 2, in the latter case, |G| ≤ 84(g − 1), as in the proof of Theorem 11.56 (III). Hence |G| < 24g 2 . If o is the unique short orbit of O(G), then Lemma 11.111 implies that |o| = 3, g = 4. Hence O(G) acts on o as a group of order 3. Let H be the subgroup of O(G) fixing every place in o. Then |O(G)| = 3|H| and, from Theorem 11.57, |H|(2˜ g + 1) = 9, where g˜ is the genus of ΣH , a contradiction. Assume that ΣO(S) is elliptic. Then O(S) has at least one short orbit o1 . Since |O(S)| − |o1 | > 12 |O(S)|, Corollary 11.52 implies that 2g − 2 ≥ |o1 | > 12 |O(S)|, whence |O(S)| < 4(g − 1). From Theorem 11.94, |G| < 96(g − 1) ≤ 24g 2 . It remains to consider the case that the genus of ΣO(S) is greater than 1. From Corollary 11.52, |O(S)| ≤ g − 1. Hence, by (11.84), |G| ≤ (g − 1)|GP |.

Assume that F1 = ΣS2 has genus g ′ > 1. Then |S2 |(g ′ − 1) < g − 1 by the same Corollary 11.52. Since GP /S2 is a tame automorphism group of F1 fixing the place of F1 lying under P, from Theorem 11.60, |GP /S2 | ≤ 4g ′ + 2. Hence |GP |
1 for every R ∈ o. Then, from the proof of Theorem ¯ and o comprises P 11.123, G acts on o as a 2-transitive permutation group G, together with all places in the unique non-trivial orbit of S2 . The latter assertion

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¯ is soluble, implies that |o| = |S2 | + 1 = q + 1, q = 2r . On the other hand, as G k k |o| = w with an odd prime w. From q + 1 = w , either k = 1, or k = 2, q = 8, w = 3. Let M be the kernel of the permutation representation of G on o. ¯ and this possibility is investigated It may be that M is trivial, that is G = G, first. Assume that k = 1. Then G = GL(1, w) and G is sharply 2-transitive on o. Let N be the normal subgroup of G of order w. If ΣN has genus g ′ ≥ 2, then 2g − 2 ≥ w(2g ′ − 2) ≥ 2w, and hence |G| = |o|(|o| − 1) = w(w − 1) ≤ (g − 1)(g − 2) < 24g 2 . If ΣN is elliptic, then |S2 | is either 2, or 4, or 8 by Theorem 11.94. The last case cannot occur because 9 is not a prime number. If |S2 | = 2, then w = 3, and hence |G| = 6, while |S2 | = 4 occurs when w = 5 and hence |G| = 20. If ΣN is rational, then S2 , viewed as a K-automorphism group of ΣN , must be an elementary abelian group by Theorem 11.14. On the other hand, S2 is the 1-point stabiliser of GL(1, w) which is cyclic. It follows that |S2 | = 2. Therefore |o| = 3, whence |G| = 6. If k = 2, w = 3, q = 8, then G has a normal subgroup N of order 9, and S2 is the quaternion group. Since G is a subgroup of the normaliser of N , it follows that GP is a subgroup of GL(2, 3) whose order is divisible by eight but not by sixteen. Since |GL(2, 3)| = 48, either |GP | = 8, or |GP | = 24. In the former case, |G| = 72 and hence |G| < 24g 2 . In the latter case, |G| = 216, which is greater than 24g 2 only for g = 2. But this does not actually occur, as the assertion holds for g = 2. Suppose M is non-trivial. By Theorem 11.49, M is cyclic of odd order, and ¯ = G/M can be viewed as a K-automorphism group of ΣM . The Sylow 2G ¯ is isomorphic to S2 . Now, it is shown that G ¯ has subgroup S¯2 = S2 M/M of G property (11.80). If this did not hold, then some non-trivial K-automorphism α ¯ of ΣM in S¯2 would fix not only the place P ′ under P but at least one more place Q′ . Equivalently, some non-trivial element α ∈ S2 would fix P and the orbit θ of M consisting of all places Q of Σ lying over Q′ . Since θ has odd size, then α would fix at least one place in θ, contradicting (11.80). ¯ ≤ 24¯ From above, if the genus g¯ of ΣM is at least 2, then |G| g 2 ; in consequence, |G| ≤ 24|M |¯ g 2 . Since |o| ≥ 3, from Theorem 11.57, 2g − 2 ≥ |M |(2¯ g − 2) + 3(|M | − 1) ≥ 2|M |¯ g − 2. This shows that |G| ≤ 24g¯ g < 24g 2 . If ΣM is elliptic, then S2 as a subgroup of AutK (ΣM ) fixes a place and hence has order at most 8, by Theorem 11.94, whence |G| = |S2 ||Ω||H| = |S2 |(|S2 | + 1)|H| ≤ 72(4g + 2). For g ≥ 4, this gives the desired result. If 2 ≤ g ≤ 3, then 2g − 2 ≥ |Ω|(|M | − 1) ≥ |Ω| = |S2 | + 1, a contradiction. Finally, if ΣM is rational, then |S2 | ≤ 2, and hence |G| = |S2 ||Ω||H| = |S2 |(|S2 | + 1)|H| ≤ 6(4g + 2). Therefore |G| < 6(4g + 2) < 24g 2 by Theorem 11.60. 2

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11.14 LARGE p-SUBGROUPS FIXING A PLACE T HEOREM 11.140 Let F be a curve of genus g ≥ 2 with function field Σ, and let G be a K-automorphism group of Σ such that Σ has a place P satisfying the condition, (1)

|GP | > 2g + 1.

(11.85)

Then one of the following occurs: (i) G = GP ; (ii) one of the cases (III), (IV) in Theorem 11.127 holds; (iii) (V) p = 3, g = n0 (n − 1), with n = 3n20 , n0 = 3r , and F is the DLR curve with 2

F = v(Y n −(1+(X n −X)n−1 )Y n +(X n −X)n−1 Y −X n (X n −X)n+3n0 ); here, G ∼ = Ree(n).

(1)

Proof. By Theorem 11.78, it may be assumed that the fixed subfield F1 of GP is rational. (1) Let Ω be the set of all places R of Σ with non-trivial first ramification group GR . Since P ∈ Ω and G 6= GP , so Ω contains at least two places. From (XXVIII) of Section A.5, Ω is a full G-orbit, and hence Ω is the only non-tame G-orbit. If there are at least two more short G-orbits, say Ω1 and Ω2 , then, from Theorem 11.72, (1)

(2)

2g − 2 ≥ −2|G| + (|GP | + |GP | + |GP | − 3) |Ω| +(|GQ1 | − 1) |Ω1 | + (|GQ2 | − 1) |Ω2 |, where Qi ∈ Ωi for i = 1, 2. Note that (|GQ1 | − 1)|Ω1 | + (|GQ2 | − 1)|Ω2 | ≥ |G|

since |GQi | − 1 ≥ 21 |GQi |, and |G| = |GQi | |Ωi |. Also, |G| = |GP | |Ω|, and (2) |GP | > 1 by Theorem 11.78. Therefore (1)

2g − 2 ≥ (|GP | − 1) |Ω|. (1)

Since |Ω| ≥ 2, this implies that g ≥ |GP |. If ΣG has genus g ′ ≥ 2, then Theorem 11.72 gives |G| ≤ g − 1. If ΣG is elliptic, then (1)

2g − 2 ≥ 2(|GP | − 1 + |GP | − 1) ≥ 2|GP |, whence |G| ≤ g − 1. From what has been shown so far, it may be assumed that ΣG is rational and that either Theorem 11.56 (iii) holds, or case (iv.4) or (iv.5) in Section 11.11 occurs. In case (iii), Ω is the unique short G-orbit. By Theorem 11.72 applied to G, 2g − 2 = −|G| + deg D(Σ/ΣG ) = |Ω| (deg D(Σ/ΣGP ) − 2|GP |).

(11.86)

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Therefore |Ω| ≤ 2g − 2. On the other hand, |Ω| > g − 1 as |Ω| ≥ |GP | + 1 is inconsistent with (11.85) for |Ω| ≤ g − 1. Since F1 is rational, (1)

deg D(Σ/ΣGP ) = |GP | − |GP | + deg D(Σ/F1 ) (1)

Thus (11.86) reads as follows:

= |GP | + |GP | + 2g − 2. (1)

2(g − 1) = |Ω|(2g − 2) + |GP | − |GP |). (1)

(1)

Therefore |Ω| = 2g − 2. Here, 2g − 2 + |GP | − |GP | = 1. Since GP = GP ⋊ H, (1) hence 2g − 2 = (|H| − 1)|GP | + 1. This and (11.85) imply that |H| = 2. (1) Therefore p 6= 2 and |GP | = 2g − 3. Since |Ω| = 2g − 2 and GP only ramifies (1) at P, this implies that GP acts on Ω\{P} as a transitive permutation group. Hence (1) |Ω| = q + 1 with q = |GP |. Since Ω is a G-orbit, it follows that G induces on Ω ¯ whose one-point stabiliser has order either q or a 2-transitive permutation group G ¯ ¯ 2q according as |G| = 2|G| or G = G. ¯ If |G| = 2|G|, the subgroup H is the kernel of the permutation representation of G on Ω; that is, H fixes every place in Ω. In particular, H is a normal subgroup of ¯ can be viewed as a K-automorphism group of ΣH . Let P ′ be the G. Therefore G (1) (1) H ¯ (1)′ . Also, the place of Σ lying under P. Since 2 ∤ |GP | it follows that GP ∼ =G P H ¯ places of Σ lying under the places in Ω form the unique short G-orbit. Therefore ¯ From what was shown before, this implies that case (iii) occurs for ΣH and G. ¯ ¯ ¯ ¯ ¯ P ′ has GP ′ = GP ′ ⋊ H with |H| = 2 But this is impossible as the stabiliser G order q. ¯ two cases are distinguished according as G is soluble or not. If G = G, In the former case, from (XXVI) of Section A.4, q + 1 = dk with d prime. Since |Ω| = 2g − 2 is even, so d = 2. From Theorem A.18, GP is a subgroup of the 1-point stabiliser of AΓL(1, q + 1), and hence |GP | divides kq. On the other hand, q + 1 = 2k can only occur when k and q are both primes. Since |GP | = 2q, this (1) implies that k = 2. Hence q = g = 3; that is, p = 2 and |GP | = g = 3. Suppose that G is not soluble. If G has a soluble regular normal subgroup M , then G/M is not soluble. On the other hand, |GP | = 2q, and hence |G/M | = 2q. But the latter is not possible for a non-soluble group. From Theorem A.17, |Ω| = 6 (1) and G ∼ = PSL(2, 5), as Ree(3) with q = 27 does not occur. Therefore |GP | = 5 and g = 4. Thus (11.85) does not hold in case (iii). In case (iv), G has not only a short non-tame orbit Ω but also a short tame orbit ∆. Let P ∈ Ω and Q ∈ ∆. If (iv.4) occurs, from the proof of Theorem 11.116, |GP | ≤ 2(g − 1)|GQ |/N (1) and N ≥ |GP ||GQ |, whence (1)

|GP ||GP | ≤ 2(g − 1).

Since |GP | ≥ 2, this implies that |GP | ≤ g − 1 contradicting (11.85). Assume that case (iv.5) holds. From the proof of Theorem 11.116, the possible ¯ on Ω are those in Lemma 11.123. First the case when this action actions of G = G is faithful is considered.

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If G ∼ = PGL(2, q), then (1)

|G| = q 3 − q, |GP | = q 2 − q, |GP | = q. (2)

From Theorem 11.116, the second ramification group GP of GP is non-trivial. Since GP has a unique conjugacy class of elements of order p, (1)

(2)

(k+1)

(k)

GP = GP = . . . = GP , |GP

| = 1.

Since F1 is rational, from the Hilbert Different Formula (11.37), 2g = (q − 1)(k − 1). But then (11.85) does not hold. If G ∼ = PSL(2, q), then (1)

|G| = 21 (q 3 − q), |GP | = 12 (q 2 − q), |GP | = q. The above argument showing that 2g=(q-1)(k-1) still holds, as GP has two conju(1) gacy classes of elements of order p, and each of them generates GP . Again, this contradicts (11.85). If G ∼ = PSU(3, n) with q = n3 , then (1)

|G| = (n3 + 1)n3 (n2 − 1)/µ, |GP | = n3 (n2 − 1)/µ, |GP | = n3 , where µ = gcd(3, n + 1). From the proof of Theorem 11.126, there is a divisor t of (n2 − n + 1)/µ such that 2g = (n − 1)(t(n + 1)2 − (n2 + n + 1)),

|GQ | = (n2 − n + 1)/(tµ).

Since m is odd, this and (11.85) imply that t = 1. Then g = 12 n(n − 1), and hence, 8g 3 < |G|. Therefore (III) in Theorem 11.126 holds. The same argument works for G ∼ = PGU(3, n). 2 If G ∼ Sz(n) with q = n and n = 2n20 for a power n0 ≥ 2 of 2, then = (1)

|G| = (n2 + 1)n(n − 1), |GP | = n2 (n − 1), |GP | = n2 . From the proof of Theorem 11.126, there is an odd integer t such that either (A) or (B) holds, where (A) (B)

2g = (t − 1)(n2 − 1) − 2tn0 (n − 1),

2g = (t − 1)(n2 − 1) + 2tn0 (n − 1),

|GQ | = (n + 2n0 + 1)/t;

|GQ | = (n − 2n0 + 1)/t.

In the former case, t must be at least 3. But then (11.85) does not hold. In the latter case, (11.85) implies that t = 1. Then g = n0 (n − 1), and hence 8g 3 < |G|. Therefore (IV) in Theorem 11.126 holds. If G ∼ = Ree(n) with q = n3 and n = 3n20 for a power of 3, then |G| = (n3 + 1)n3 (n − 1),

|GP | = n3 (n − 1),

(1)

|GP | = n3 .

As was shown in the proof of Theorem 11.126, three cases may occur. If |GQ | = (n + 1)/t, then 2g = (n − 1)((t − 1)(n2 + 1) − n), with t ≥ 2. Let t = 2. Then |GQ | = 21 (n + 1). Since 12 (n + 1) is even, GQ must contain

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an involution. But this is impossible by Theorem A.15 and Lemma 11.124. When t > 2, (11.85) does not hold. If |GQ | = (n + 3n0 + 1)/t, then 2g = (n − 1)((t − 1)n2 − t(3n0 n − 2n + 3n0 − 1) − n − 1).

Hence t ≥ 2. Since t is odd and t 6= 3, t must be at least 5. But then (11.85) does not hold. If |GQ | = (n − 3n0 + 1)/t, then 2g = (n − 1)((t − 1)n2 + t(3n0 n + 2n + 3n0 + 1) − n − 1).

When t = 1, this becomes 2g = 3n0 (n − 1)(n + n0 + 1). From Theorem 12.31, case (b) follows. Otherwise, t ≥ 5. But then (11.85) does not hold. ¯ of G It remains to show the impossibility of the permutation representation G on Ω having non-trivial kernel. Such a kernel M is a cyclic normal subgroup of G whose order is relatively prime to p. Note that no place outside Ω is fixed by a non-trivial element in M , by Lemma 11.124. Let g˜ be the genus of ΣM . From Theorem 11.57 applied to M , 2g − 2 = |M |(2g˜ g − 2) + (|M | − 1)(q + 1). Therefore 2g ≥ (|M | − 1)(q − 1). But this is inconsistent with (11.85).

2

11.15 NOTES The proof of the finiteness of the K-automorphism group of a complex irreducible curve of genus g ≥ 2 was a major achievement of the 19th century, due to Schwartz, Klein, Noether, Weierstrass, Poincar´e, and Hurwitz. A detailed account of the group AutK (Σ) in zero characteristic is found in [11, 127]; see also [306], [284]. The classical result was extended to curves over any field by Schmid [383]. The proof given in Sections 11.4 and 11.6 is based on [249, 250, 251]; see also [372]. An alternative proof using Jacobian varieties is found in [376]. For the result that every finite group is isomorphic to the K-automorphism group of some irreducible curve, see [319], [379], [426]. Related to this is Abhyankar’s project to find families of ‘nice’ polynomials whose Galois groups are linear groups over Fq . For a survey, see [10]; for symplectic groups, see also [9]. In connection with Section 11.2, the uniqueness of certain linear series on curves is important. So far, this has been investigated in zero characteristic; see [82]. Lemmas 11.27, 11.28 and Theorem 11.29 are due to Segre [393]. Theorem 11.29 can be generalised to show that any birational transformation between two plane non-singular curves of the same degree n ≥ 4 is induced by a projectivity of PG(2, K); see [73]. For Theorem 11.31, see [303] and [414]. For the proof of Lemma 11.47, see [249, Lemma 5]. Related to Section 11.5, several good textbooks on the topic of quotient curves using function field theory are available in the literature, such as [428] and [335]. Theorems 11.56 11.60 11.78 are due to Stichtenoth [423].

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The proof of Theorem 11.56 follows [244]. The plane sextic F in Example 11.58 arises in the study of arcs in PG(2, 13); see Example 13.46. Since F over F13 achieves the St¨ohr–Voloch Bound of 54 for a plane curve of this degree over F13 , so F is a candidate to be an optimal curve for the genus g = 10. For more on this curve, see [170]. The study of plane curves of low degree with large automorphism groups was initiated by Klein and Wiman. The Klein quartic F, regarded as a complex curve, has long been known to attain Hurwitz’s upper bound, the K-automorphism group G of F being isomorphic to PSL(2, 7); see [117]. Tuffery [463, 464] showed that this holds true for p 6= 3, 7. In both exceptional characteristics, |G| = 6048 and G contains a subgroup isomorphic to PSL(2, 7). The Wiman sextic F = v(10X 3 Y 3 + 9(X 5 + Y 5 ) − 45X 2 Y 2 − 135XY + 27),

is defined for p 6= 2, 3, 5 and its K-automorphism group G of order 360 is isomorphic to the alternating group A6 . In particular, F possesses the maximum number of K-automorphisms of all curves of genus 10, and F is the unique non-singular plane sextic, up to projectivity, whose K-automorphism group contains a subgroup isomorphic to A6 : this was shown by Kaneta, Marcugini and Pambianco [256]. Theorem 11.62 was stated by Deuring for s ≥ 1 and then by Shafarevich for s = 0. Later Rosen and, independently, Sullivan gave proofs for special cases. Subrao [435] proved the theorem in full generality. Madan [318] revised Deuring’s original proof fixing some errors and giving a proof for every s using Deuring’s approach. A short proof valid for n = p and K = Fq is found in [375]; see also [369]. Theorem 11.63 is due to Witt [505] and independently to Shafarevich [409]. The group-theoretic result required in Lemma 11.82 is [239, Theorem 13.7] in (1) (2) which the hypothesis that Z(GP ) is cyclic is unnecessary, since GP has order p. Related to Section 11.9, a detailed account of ramification groups is found in [404, Chapter IV]; see also [428]. Theorems 11.79, 11.84, 11.86 and 11.88 are due to Nakajima, see [342], [341] and the survey paper [343]. Bounds on the order of automorphism groups in terms of the p-rank of the curve are also found in [465]. Example 11.89 comes from [435]. For another example of an ordinary curve with large K-automorphism group, see [341]. Theorem 11.79 for complex curves goes back to Wiman; a proof valid for p = 0 is found in [194]. For a tame cyclic group G with at least two fixed places, Theorem 11.79 can be sharpened to |G| ≤ 4g; see [441]. Also, if G is cyclic of order at least 2g + 1 and p = 0, then either F is a quotient curve of a Fermat curve or F is hyperelliptic and |G| = 2g + 2. This was shown by Irokawa and Sasaki [248]. For improvements to Theorem 11.79, also valid in positive characteristic but under some further conditions, see [458]. Proofs of Theorems 11.91, 11.92, and 11.93, are found in [319]; see also [279]. For further results on the K-automorphism group of an Artin–Schreier curve h

v(Y p − Y − f (X)),

with f (X) ∈ K[X], see Section 12.1.

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There is a large literature on automorphism groups of complex hyperelliptic curves, whose study was initiated by Clebsch, Hurwitz, and Bolza in the late nineteenth century; see [11] and [127]. Many of the results hold true in zero characteristic, see for instance [59], but this does not happen in positive characteristic when non-tame automorphisms come into play. Automorphism groups of curves of genus 2 of PG(2, K) with p = 2 were investigated by Ancochea [14] in 1943. As mentioned in Section 8.10, important applications in cryptography have stimulated research on hyperelliptic curves over finite fields, especially of genus 2, with particular attention to their automorphism groups; see [274]. Recent results improving Proposition 11.99 are found in a series of papers by Cardona, Duursma, Gonzalez, Gutierrez, Kiyavash, Lario, Quer, Rio, Shaska, and V¨olklein; see [111], [67], [69], [413]. A relevant result is that, when p 6= 2, the possible K-automorphism groups of curves of genus 2 are the following: Z2 , Z10 , Z2 × Z2 , D8 , D12 , Z3 ⋊ D8 , GL(2, 3), and an extension of S5 by the hyperelliptic involution. The classification of curves of genus 2 by their K-automorphism groups is being investigated. For p 6= 2, the curves of genus 2 whose K-automorphism group G is isomorphic to Z3 ⋊ D8 , GL(2, 3), or Z10 are determined. Such curves are birationally equivalent to the irreducible plane curves v(Y 2 − f (X)), with  6 G∼ = Z3 ⋊ D12 ,  X −1 X5 − X G ∼ f (X) = = GL(2, 3),  5 X −1 G∼ = Z10 .

Results on K-automorphisms of hyperelliptic curves of genus 3 are found in [184]. In [516] and [411], hyperelliptic curves whose hyperelliptic involution is the only K-automorphism are considered. In [185] and [411] hyperelliptic curves with a K-automorphism group isomorphic to Z2 × Z2 are considered. Duma’s paper [108] is a survey of results contained in the unpublished thesis of Brandt, who gave a complete classification of K-automorphism groups of Kummer extensions of the rational field K(x), by exhibiting explicit equations for the curves to which they correspond. The main ingredients in Brandt’s analysis are a refinement of Dickson’s classification and the computation of all central extensions of cyclic groups with these groups. Example 11.103 comes from [473]. Example 11.104 is due to Kontogeorgis [278]; see also [334]. Section 11.11 comes from [11]. For the result in Remark 11.109, see [228]. For a classification of finite groups with partitions, see [438]. The curve in case (ii) has large K-automorphism group; see Theorem 11.127(II). Theorem 11.110 is due to Cossidente and Siciliano [93, 94]. The set of Fq rational points of the curve G has interesting configurational properties; this was pointed out by Pellikaan [361]. Theorem 11.112 was generalised by Garcia in [138, 139] to higher-order Weierstrass points; such a generalisation in zero characteristic was previously given by Farkas and Kra [127]. The number of fixed places of an involutory K-automorphism was investigated by Towse [459] and, independently, in [457] using the earlier

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545

work of Garcia [136] and Kato [258, 259] on double coverings and Weierstrass points. Levcovitz [305] gave a bound on the number of fixed points of an automorphism α acting on a non-singular model X embedded in PG(r, K). Ideas from the St¨ohrVoloch approach, as in Section 8.5, are used to count the number of points P ∈ X for which the m-osculating space to X at P meets the (r − m − 1)-osculating space to X at P α . The result is an inequality involving the number of fixed points of α, the order sequence of X and certain orders arising from the counting argument. Section 11.12 is based on [206] and [423]. Theorem 11.127, relying on previous results of Stichtenoth, is due to Henn. As a consequence, Hurwitz’s upper bound, |G| ≤ 84(g − 1), remains valid in positive characteristic p provided that p is large with respect to the genus, namely, p > g +1. This was first pointed out by Roquette in [373], who also showed that, if p = g+1 and |G| exceeds Hurwitz’s upper bound, then (II) of Theorem 11.127 holds. Sing’s bound [417] valid for p ≤ 2g + 1,    2g 4pg 2 4pg 2 +1 , · +1 |G| ≤ p−1 p−1 (p − 1)2 is weaker than the bound in Theorem 11.127. For a proof of Theorem 11.134, due to Burnside, see [179]. In finite group theory, trivial intersection sets play an important role. Suzuki [440] classified all finite groups whose Sylow 2-subgroups are trivial intersection sets. Hering [209] refined and extended Suzuki’s classification to finite groups containing a subgroup of even order which intersects each of its distinct conjugates trivially. Here, Hering’s result Theorem 11.135 is quoted. In the special case that Q is a Sylow 2-subgroup of G, this is essentially Suzuki’s classification. Theorem 11.136 is stated in [515]. For more on the curves given in Theorems 11.127 and 11.140, see Sections 12.2, 12.3 and 12.4. For Huppert’s classification referred to in the proof of Theorem 11.140, see [241, Theorem 7.3] Theorem 11.140 holds true for |GP | > pg/(p − 1) provided that the DLR curve and case (II) from Theorem 11.127 are added to (ii); see [173]. In connection with Example 11.103 and Kloosterman sums, see van der Geer and van der Vlugt [473], where it is shown that the irreducible plane curves v(A(Y ) − R(X)), with A(Y ) ∈ Fq [Y ] and R(X) ∈ Fq (X), are precisely the curves Γ over Fq of genus p − 1 with the three properties listed. If C is an irreducible curve of genus g ≥ 2 covering the projective line PG(1, K) with a cyclic p-covering ramified in exactly one point, Lehr and Matignon [302] showed that p-subgroups of AutK (Σ) providing such coverings are exactly the extensions of a cyclic p-group by an elementary abelian p-group. They also characterised curves C of genus g ≥ 2 and p-subgroups of AutK (Σ) such that 2p/(p − 1) < |G|/g and 4/(p − 1)2 ≤ |G|/g 2 . Automorphism groups of modular curves were investigated by Bending, Camina, and Guralnick [46].

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Chapter Twelve Some families of algebraic curves The aim of this chapter is to provide families of algebraic curves in positive characteristic with properties that a complex algebraic curve cannot have. As mentioned in the earlier chapters, these curves illustrate concepts and results unknown in the classical literature on curves.

12.1 PLANE CURVES GIVEN BY SEPARATED POLYNOMIALS In this section, the curve with C = v(A(Y ) − B(X)),

(12.1)

is investigated in detail under the following conditions: (I) deg C ≥ 4 and gcd(p, deg B(X)) = 1; n

n−1

(II) A(Y ) = an Y p + an−1 Y p

+ · · · + a0 Y,

aj ∈ K, a0 , an 6= 0;

(III) B(X) = bm X m + bm−1 X m−1 + · · · + b1 X + b0 ,

bj ∈ K, bm 6= 0;

(IV) m 6≡ 0 (mod p); (V) n ≥ 1, m ≥ 2. Note that (II) occurs if and only if A(Y + a) = A(Y ) + A(a) for every a ∈ K, that is, the polynomial A(Y ) is additive. The basic properties of C are collected in the following lemma. L EMMA 12.1 The curve C is an irreducible plane curve with at most one singular point. (i) If |m − pn | = 1, then C is non-singular. (ii)

(a) If m > pn + 1, then P∞ = (0, 0, 1) is an (m − pn )-fold point of C.

(b) If pn > m + 1, then P∞ = (0, 1, 0) is a (pn − m)-fold point of C.

(c) In both cases, P∞ is the centre of only one branch of C; also, P∞ is the unique infinite point of C.

(iii) If Σ = K(x, y), with A(y) = B(x), is the function field of C, and P∞ is the place associated to the branch centred at P∞ , then (a) div(dx) = ((pn − 1)(m − 1) − 2)P∞ ;

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(b) X has genus g = 21 (pn − 1)(m − 1);

(c) a translation (x, y) 7→ (x, y + a) preserves C if and only if A(a) = 0;

(d) these translations form an elementary abelian group of order pn , and AutK (Σ) contains an elementary abelian p-group G of order pn that fixes a unique place P∞ and acts transitively on the zeros of x; (e) the sequence of ramification groups of G at P∞ is (1)

(2)

(m)

G = GP∞ = GP∞ = · · · = GP∞ ,

(m+1)

GP∞

= 1;

(f) {P∞ } is the unique short orbit of G, and

div(Σ/ΣG ) = (pn − 1)(m + 1)P∞ ;

(g) ΣG is rational, and Σ has p-rank zero. Proof. Put F (X, Y ) = A(Y ) − B(X). Since ∂F/∂Y = a0 6= 0, no affine point is a singular point of C. The case m > pn is worked out in detail; the arguments can be adapted to the case pn ≥ m. First, P∞ = (0, 0, 1) is the unique point of C at infinity and P∞ is, in fact, an (m − pn )-fold point, with a unique tangent line v(X0 ). It is now shown that P∞ is the centre of only one branch of C. To do this, P∞ is taken to the origin by interchanging X0 and X2 . Then C = v(A(Y ) − C(X, Y )), where C(X, Y ) = bm X m + bm−1 X m−1 Y + · · · + b1 XY m−1 + b0 Y m .

Let x(t) = uti + · · · , y(t) = vtj + · · · , with u, v 6= 0, i < j, be a primitive representation of a branch of C centred at O. Since A(y(t)) = C(x(t), y(t)), it follows that j(m − pn ) = im. As gcd(m, pn ) = 1, this leaves only one case, namely i = m − pn , j = m. Since O is an (m − pn )-fold point, the assertion follows. As a consequence, C is irreducible. Going back to the original frame,  m−pn  ut + ··· 1 (x(t), y(t)) = , vtm + · · · vtm + · · ·  n  = t−p (u1 + · · · ), t−m (v1 + · · · ) is a primitive representation of the unique branch of C centred P∞ . As ordt x(t) is divisible by p, the calculation of ordt (dx(t)/dt) is not simple. To overcome the difficulty, take derivatives in A(y(t)) = B(x(t)). This gives the equation
dx dy = (m − 1)bm x(t)m−1 + · · · + b1 . a0 dt dt Since ordt y(t) is not divisible by p and ordt x(t) < 0, so −(m + 1) = −pn (m − 1) + ordt (dx/dt).

Therefore ordt (dx/dt) = (pn − 1)(m − 1) − 2. By assumption, C has degree at least 4 and this implies that (pn − 1)(m − 1) > 2. In particular, x is a separable

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variable of Σ. Also, as no vertical line is tangent to C, so ordγ (dx(t)/dt) = 0 for every branch γ of C centred at an affine point. Since C has degree m, so (iii)(b) follows from Definition 5.55. Let P∞ be the place associated with the branch γ. So (iii) (a) holds. The roots of the polynomial A(Y ) are the elements of an additive subgroup A of K of order pn . Also, for every a ∈ A, the linear collineation (x, y) 7→ (x, y + a) preserves C; that is, αa : (x, y) 7→ (x, y + a) is a K-automorphism of the function field Σ = K(x, y) of C which fixes the place P∞ . Let G be the group consisting of these αa with a ranging over A. For a 6= 0, the automorphism αa fixes no more place of Σ showing (iii) (c) and the first part of (f). To prove (e), take P∞ again to the origin by interchanging the coordinates X0 and X2 . Then Σ is assumed to be K(x, y) with A(x) = C(x, y) and, for every a ∈ A,   x y . αa : (x, y) 7→ , 1 + ay 1 + ay A uniformising element at O can be found by using the following argument. Since gcd(m, pn ) = 1, there exist integers k, ℓ such that k(m − pn ) + ℓm = 1. Let n ξ = xk y ℓ . As shown before, x(t) = utm−p +· · · , y(t) = vtm +· · · is a primitive representation of the unique branch centred at O. If O is the corresponding place of Σ, then n

ordO ξ = ordt (tk(m−p

k ℓ

)+ℓm

+ · · · ) = 1,

showing that ξ = x y is a uniformising element of Σ at O. As gcd(pn , m) = 1), for every a ∈ A, a 6= 0,   x(t)k y(t)ℓ k ℓ ordt(α(ξ(t)) − ξ(t)) = ordt − x(t) y(t) (1 + ay(t))ℓ+k = ordt (aty(t) + · · · ) = m + 1,

whence (e) follows. Hilbert’s Different Formula (11.37), together with (e), gives the second part of (f). From this, by Theorem 11.72, the first part in (g) follows. Finally, Lemma 11.131 shows the last assertion. 2 L EMMA 12.2

(i) For every k > 1,

P |kP∞ | = div( cij xi y j ) + kP∞ ,

summed over all i, j ≥ 0 satisfying both j ≤ pn − 1 and ipn + jm ≤ k. P g−1 g2g−2 = div( cij xi y j ) + (2g − 2)P∞ , (ii) summed over all i, j ≥ 0 satisfying ipn + jm ≤ (pn − 1)(m − 1) − 2.

Proof. With the notation in the previous proof, ordt (x(t)i y(t)j ) = −(ipn + jm). This shows that xi y j has a pole at P∞ of order at most k when ipn + jm ≤ k. Since every place distinct from P∞ arises from a branch of C centred at an affine i j point, no more poles; hence div(xi y j ) + kP ∈ |kP|. This implies that Px y ihas j div( cij x y ) + kP ∈ |kP| provided that ipn + jm ≤ k when cij 6= 0.

(12.2)

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Those xi y j satisfying not only (12.2) but also j < pn provide a linearly independent set over K. Every element in Σ has a representation a(x, y)/b(x) where a(X, Y ) ∈ K[X, Y ] and b(X) ∈ K[X]. As C = v(A(Y ) − B(X)), with deg A(Y ) = pn , it may be assumed that degY a(X, Y ) < pn . Now, suppose that   a(x, y) div ≻ kP. (12.3) b(x) Then a(x, y)/b(x) has no pole at an affine point. To show that b(X) is a constant polynomial, assume on the contrary that u ∈ K be a root of b(X). Then every point P = (u, v) of C is a zero of a(x, y) ∈ Σ; that is, every root of the polynomial A(Y ) − B(u) is a root of the polynomial a(u, Y ). The former polynomial has pn distinct roots since dA(y)/dY = 1, but the latter one has fewer roots, as its degree does not exceed pn − 1. This contradiction proves the assertion. So, itP may be assumed that b(X) = 1. Let a(X, Y ) = cij X i Y j . If some (i, j) satisfies (12.2), replace a(X, Y ) by i j a(X, Y ) − cij X Y . In doing so, ipn + jm > k with cij 6= 0. To show that this leads to a contradiction, let a primitive representation of the unique branch centred at P∞ be  n  (x(t), y(t)) = t−p (u1 + · · · ), t−m (v1 + · · · ) . Therefore, if cij 6= 0, then ordt (x(t)i y(t)j ) = −(ipn + jm). As k < ipn + jm for cij 6= 0, it turns out that either ordt a(x(t), y(t)) < −k, or there are two terms in a(X, Y ), say cij X i Y j and c¯ı,¯ X ¯ı Y ¯ such that ordt (x(t)i y(t)j ) = ordt (x(t)¯ı y(t)¯). In the former case, the contradiction is with (12.3) while, in the latter case, (¯ı − i)pn = (¯  − j)m,

which gives j ≡ ¯ (mod pn ). Since degY a(X, Y ) < pn this implies that j = ¯ and hence i = ¯ı, a contradiction. The second assertion is a corollary to the first when applied to k = 2g − 2, as (2g − 2)P∞ is the canonical series on X . 2 From Lemma 12.2, it also means that the Weierstrass semigroup of Σ at P is generated by pn and m. Hence the following result holds. L EMMA 12.3 There are exactly 21 (pn −1)(m−1) gaps, every other positive integer being a linear combination of m and pn with positive integer coefficients. A characterisation of the curve C of (12.1) is given in the following theorem. T HEOREM 12.4 Let F be an irreducible plane curve such that a place P of its function field Σ = K(F) satisfies the following conditions: (a) Σ has a K-automorphism group N of order pn whose non-trivial elements fix P but no other place of Σ;

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(b) ΣN is rational; (c) Σ contains an element y such that div(y)∞ = mP with gcd(m, p) = 1; (d) Σ has genus g ≥ 21 (pn − 1)(m − 1). Then F is birationally equivalent to the irreducible plane curve C of (12.1) with A(Y ), B(X) satisfying (I)–(V). Proof. Choose x ∈ Σ such that ΣN = K(x). Let σ be a primitive representation of the place P. If σ(x) = ci ti + · · · with ci 6= 0, i ≥ 0, replace x by x−1 or (x − c0 )−1 depending on whether i > 0 or i = 0. Then P may be taken to be a pole of x and is the unique pole of x. If Q were another pole of x, then every place in the orbit of Q under N would be a pole of x as well. So x would have more than pn poles. But this contradicts Theorem 5.34 since pn = |N | = [Σ : ΣN ] = [Σ : K(x)].

Thus div(x)∞ = pn P. From this two facts follow. First, the Weierstrass semigroup at P contains the semigroup generated by pn and m, where m is defined as in (c). From the Weierstrass Gap Theorem 6.89, 2g ≤ (pn − 1)(m − 1). This and (d) imply that g = 12 (pn − 1)(m − 1). Secondly, div(x)∞ = pn P, together with (c), implies that Σ = K(x, y). In fact, [Σ : K(x, y)] divides gcd(m, pn ) = 1, as [Σ : K(x, y)][K(x, y) : K(x)] = [Σ : K(x)] = pn , [Σ : K(x, y)][K(x, y) : K(y)] = [Σ : K(y)] = m. It remains to show that x and y satisfy an equation of type A(y) = B(x), with A(Y ), B(X) as in (I)–(V). To do this, two cases are distinguished. Assume first that m < pn , and choose α ∈ N . Since m is the smallest nongap at P, there exists dα ∈ K such that α(y) = y + dα , by Lemma 11.13. Then dα + dβ = dαβ for any two α, β ∈ N . As N is a group, {dα | α Q ∈ N } is an additive subgroup of K of order pn . Then the polynomial A(Y ) = (Y + dα ) with α ranging of all elements in N , is of type (II). Also, A(y) ∈ K(x), because A(y) is fixed by N . Therefore A(y) = u(x)/w(x), where u(X), w(X) ∈ K[X] are polynomials without common roots. In fact, w(X) must be constant. If c is a root of w(X), replace x by x − c. Let Q be a zero of x; then Q is not a zero of u(x). Also, Q = 6 P because P is a pole of x. Hence Q is a pole of A(y). As A(Y ) ∈ K[Y ], this is only possible when y itself has a pole at Q, contradicting that div(y)∞ = mP. So there exists B(X) ∈ K[X] such that A(y) = B(x). Then div(A(y))∞ = div(B(x))∞ . n

As div(A(y))∞ = p mP, and div(x)∞ = pn P, this implies that deg B(X) = m. So, the proof is complete for m < pn . Now, assume that m > pn , and take a non-trivial K-automorphism α ∈ N . If σ is a primitive representation of P, then σ(y) = ct−m +· · · . If λα : t 7→ t+uti +· · · is the companion automorphism of α, then σ(α(y)) = c(t + uti + · · · )−m + · · · .

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Thus σ(α(y) − y) = dt−j + · · · with j < m. As m > pn , this implies that div(α(y) − y)∞ = ℓ pn P.

(12.4)

Since every non-gap at P smaller than m is a multiple of pn , Pℓ |ℓpn P| = div( i=0 ci xi ) + ℓpn P. Hence there is a polynomial Pα (X) such that α(y) − y = Pα (x),

s = deg Pα (X) ≤ ℓ.

(12.5)

u −v

with −um+vp = 1.

div(α(η) − η)∞ = (m + 1 − spn )P.

(12.6)

Change y to a uniformising element η by putting η = y x Since α(x) = x, so (12.4) becomes the equation

n

In fact, α(η) − η = x−v (α(y)u − y u ) = x−v (uy u−1 Pα (x) + · · · + Pα (x)u ), which implies that ordt σ(α(η) − η) = pn v − m(u − 1) − pn s = m + 1 − pn s. On the other hand, the Hilbert Different formula (11.37) applied to N gives the relation P (i) (m + 1)(pn − 1) = i≥0 (|NP | − 1), (i)

where NP is the i-th ramification group of N at P. From (12.6) and ℓpn < m it (i) (i) follows that NP is trivial for any i > m. Hence N must coincide with NP for 0 ≤ i ≤ m. Again by (12.6), Pα (X) is a constant, say dα ∈ K, for every α ∈ N . Now the proof can be completed in the same way as in the case m < pn . 2 For n = 1, Theorem 12.4 becomes the following. T HEOREM 12.5 Let F be an irreducible plane curve such that its function field Σ = K(F) satisfies the following conditions: (a) Σ has an automorphism α of order p fixing only one place; (b) Σα is rational;

If P is the fixed place of α and m is the smallest pole number at P prime to p, then F is birationally equivalent to the curve C = v(Y p − Y − B(X))

with B(X) ∈ K[X] of degree m. Proof. The background and arguments from the preceding proof are used. Let Σ = K(x, y), Σα = K(x), div(x)∞ = pP, div(y)∞ = mP. If m < p, then α(y) = y + d with d ∈ K. Therefore αi (y) = y + id, for Qp−1 i = 0, 1, . . . , p − 1. Since i=0 αi (y) is fixed by α, Qp−1 y p − dp−1 y = i=0 (y + id) ∈ K(x).

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Replacing y by yd−1 gives y d − y ∈ K(x). This implies that y p − y = B(x) with B(X) ∈ K[X]. In particular, deg B(X) = m. Therefore C has equation of type (12.1) with A(Y ) = Y p − Y . If m > p, then α(y) = y + P (x) with P (X) ∈ K[X] and p · deg P (X) < m. To prove that P (X) is a constant polynomial, suppose on the contrary that P (X) has a root, say a. Replacing x by x − a, it may be assumed without loss of generality that P (X) = XQ(X) with Q(X) ∈ K[X]. This implies that y p − Q(x)p−1 xp−1 y = B(x)

for a polynomial B(X) ∈ K[X], whose degree is m, since ordP y = −m,

ordP x = p,

p · deg Q(X) < m − p.

Also, B(0) = 0 after replacing y by y − u where up = B(0). Therefore the following equations are valid: y p − Q(x)p−1 xp−1 y = am x + · · · + a1 x; α(x) = x;

α(y) = y + xQ(x),

p · deg Q(x) < m − p.

The plane curve G = v(Y p − Q(X)p−1 X p−1 Y − (am X m + · · · + a1 X)) is a birational model of Σ. If the place Q is a zero of x, then Q is a zero of y, as well. Hence the branch of G corresponding to Q is centred at the origin O. Since the set of zeros of x is preserved by α, but none of these is fixed by α, the origin is the centre of at least p distinct branches of G. On the other hand, O is at most a p-fold point of G. Hence the places corresponding to branches centred at O form a single orbit under α, and hence P αi div(x)0 = p−1 i=0 Q . Pp−1 αi As div(y)0 ≻ i=0 Q , it follows that P is the unique pole of y/x. Further, ordP (y/x) = −(m − p). Since gcd(p, m − p) = 1, this contradicts the choice of y, and proves that P (X) is a constant polynomial. Now, the proof finishes in the same way as in the case m < p. 2 R EMARK 12.6 The following two results are refinements of Theorem 12.5. (i) If m − 1 = pr s, r > 0, s > 1, gcd(p, s) = 1, then  r+1 p for p > 2, (1) |GP | ≤ 2r for p = 2. (1)

(ii) If |GP | ≤ pg/(p − 1), then B(X) = XR(X) with R(X) ∈ K[X]. T HEOREM 12.7 Let G be the K-automorphism group of a plane curve C as in (1) (12.1). Let GP∞ = GP∞ ⋊ H with p ∤ |H|. Then (i) |H| divides m(pn − 1);

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4p g2; − 1)2

(1)

(ii) |GP∞ | ≤ pn (m − 1)2 =

(pn

(1)

(iii) |GP∞ | = pn when m 6≡ 1 (mod pn ), and so g 6≡ 0 (mod pn ); (2)

(iv) |GP∞ | = pn when m ≡ 1 (mod pn ), and so g ≡ 0 (mod pn ).

Proof. Let α ∈ GP∞ . By Lemma 11.20, α preserves both |mP∞ | and |pm P∞ |. First, consider the case m < pn . By Lemma 12.2, |mP| = {div(c0 + c1 y) + mP | c = (c0 , c1 ) ∈ PG(1, K)}, Pr |pn P| = {div(cx + i=0 di y i ) + pn P | c = (d0 , . . . , dr , c) ∈ PG(r + 1, K)},

with r the integer defined by rm < pn < (r + 1)m. Hence α(y) = ay + b,

α(x) = cx + P (y),

where P (Y ) ∈ K[Y ] such that m · deg P (Y ) < pn , and a, b, c ∈ K with ac 6= 0. As α(A(y)) = α(B(x)), the polynomial, H(X, Y ) = A(aY + b) − B(cX + P (Y )) n

n

n−1

= an ap Y p + an−1 ap −c

m−1

n−1

Yp

+ · · · + a0 aY + A(b) − bm cm X m

(bm mP (Y ) + bm−1 )X m−1 − · · · − B(P (Y )),

is divisible by A(Y ) − B(X). Also,

deg H(X, Y ) = pn = deg(A(Y ) − B(X)),

which implies that H(X, Y ) coincides with A(Y ) − B(X) up to a constant factor; that is, H(X, Y ) = k(A(Y ) − B(X)) with k ∈ K\{0}. Since n

k = a p = a = cm ,

cm−1 (bm mP (Y ) + bm−1 ) = kbm−1 ,

so P (Y ) is a constant; namely, P (Y ) = (c − 1)bm−1 /(mbm ). Hence α(x) = cx +

(c − 1)bm−1 , mbm

α(y) = ay + b

n

with ap = a = cm .

(12.7)

The map α 7→ c is a homomorphism of GP∞ into the multiplicative group of K, (1) whose kernel is GP∞ as c = 1 implies that a = 1. (1)

(1)

To calculate the order of GP∞ , the number of elements b when α varies in GP∞ needs to be found. These are given by the roots of the polynomial   (c − 1)bm−1 , A(Y ) = kB mbm

whose number is pn . Further, H is isomorphic to a subgroup of the group of all n n m(pn − 1)-st roots of unity, as cm(p −1) = ap −1 = 1. This completes the proof n for m < p . The proof for the case m > pn is similar. Again, by Lemma 12.2, |pn P| = {div(c0 + c1 x) + pn P | c = (c0 , c1 ) ∈ PG(1, K)}, Pr |mP| = {div(cy + i=0 di xi ) + mP | c = (d0 , . . . , dr , c) ∈ PG(r + 1, K)},

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with r defined by rpn < m < (r + 1)pn . Therefore α(x) = cx + d,

α(y) = ay + Q(x),

(12.8)

where Q(X) ∈ K[X] such that pn · deg Q(X) < m, and a, c, d ∈ K with ac 6= 0. The polynomial H(X, Y ) = A(aY + Q(X)) − B(cX + d) n

n

n−1

n−1

= an ap Y p + an−1 ap Y p + · · · + a0 aY + A(Q(X)) m m m−1 −bm c X − c (bm md + bm−1 )X m−1 − · · · − B(d)

is A(Y ) − B(X) up to a constant factor; that is, H(X, Y ) = k(A(Y ) − B(X)) n with k ∈ K\{0}. As before, this gives ap = a = cm . Again, the map α 7→ c is a homomorphism of GP∞ into the multiplicative group of K. Since n

cm(p

−1)

n

= ap

−1

= 1,

it follows that H is isomorphic to a subgroup of the group of all m(pn − 1)-st roots (1) of unity. As m < pn , the kernel of this homomorphism coincides with GP∞ since c = 1 implies that a = 1. (1) To calculate the order of GP∞ , first consider the case m 6≡ 1 (mod pn ). Here, no monomial in A(Q(X)) has degree m − 1. Therefore, from the equation H(X, Y ) = k(A(Y ) − B(X)),

(12.9)

it follows that cm−1 (bm md + bm−1 ) = 0; that is, d = −bm−1 /(mbm ). Then d = 0 after substituting x − bm−1 /(mbm ) for x. As before, (12.9) implies that A(Q(X)) = 0 and this only occurs when Q(X) (1) is constant. More precisely, Q(X) = u with A(u) = 0. Hence α ∈ GP∞ if and only if α(x) = x,

u ∈ K,

α(y) = y + u, (1)

A(u) = 0.

As A(Y ) has pn distinct roots, so |GP∞ | = pn . This completes the proof of (iii). For the case m ≡ 1 (mod pn ), more is needed. Let m = vpk + 1 with k ≥ n (1) and gcd(p, v) = 1. By (12.8), any α ∈ GP∞ has an equation of type α(x) = x + d,

α(y) = y + Q(x),

with k−n

Q(x) = f1 x + · · · + fv xv + · · · + fvpk−n xvp

+ b.

Since 0 = A(α(y)) − B(α(x)) = A(y) + A(Q(x)) − B(x + d)

= A(Q(x)) + B(x) − B(x + d),

it follows that k−n

A(f1 X + · · · + fv X v + · · · + fvpk−n X vp

+ b) + B(X) − B(X + d)

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is the zero polynomial, and hence n

p an fvp k−n − mbm d = 0. (2)

(2)

To calculate the second ramification group GP∞ , note that |GP∞ | ≥ pn by Lemma (2)

12.1 (iii) (f). To see that equality holds, let α ∈ GP∞ , that is, vpk−n

where z = x

ordt σ(α(z) − z) ≥ 3,

/y is a local parameter at P∞ . As k−n

α(z) − z =

y(xp

k−n

+ dp

k−n

k−n

)v − yxvp − Q(x)xvp (y + Q(x))y

,

straightforward computation shows that fvpk−n = 0, whence d = 0. Therefore (2) α ∈ GP∞ if and only if α(x) = x,

α(y) = y + u,

Since A(Y ) has pn distinct roots, so complete.

(2) |GP∞ |

u ∈ K,

A(u) = 0.

= pn . By Theorem 11.78, the proof is 2 (1)

T HEOREM 12.8 If equality holds in (i) of Theorem 12.7, and |GP∞ | > (p − 1)g/p then B(X) = bm X m . Proof. Let α ∈ H. From the proof of Theorem 12.7, α(x) = cx + d,

pn

α(y) = ay + Q(x),

with a = a = c and Q(X) ∈ K[X]. Also, Q(X) is linear for m < pn while its degree is smaller than m/pn for m > pn . Assume that α has order m(pn − 1). n Then β = αp −1 has order m. After replacing x by x + u with a suitable u ∈ K, β(x) = cx,

m

β(y) = y + Q(x),

with cm = 1,

Q(X) ∈ K[X].

(12.10)

The key idea in the proof is to show that Q(X) is the zero polynomial. To do this, it is first shown that β fixes every place arising from a branch centred on the (2)

Y -axis. As, by Theorem 11.78, ΣGP∞ is rational, Lemma 11.14 (ii)(d) implies (2)

that β fixes two places of ΣGP∞ , one of them is the place under P∞ , the other lies under pn distinct places of Σ, say R1 , . . . , Rpn . Since the K-automorphisms in (2) GP∞ act as (x, y) 7→ (x, y + u), (2) GP∞

with u ∈ K and A(u) = 0,

⋊ H is an abelian group. As p does not divide they commute with β. Thus, the order of β, this can only occur when β fixes each of the places R1 , . . . , Rpn . In particular, Q(0) = 0. By β(x) = cx, the centres of the corresponding branches lie on the Y -axis, and they are indeed all the places of Σ centred at an affine point on the Y -axis. Now, by Theorem 11.72 applied to the K-automorphism group generated by β, 2g − 2 = (2g1 − 2)m + (pn + 1)(m − 1),

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where g1 denotes the genus of Σβ . This implies that g1 = 0. Hence Σβ is rational. Let Σβ = K(η). Then [Σ : K(η)] = m is prime to p, and hence Σ = K(x, η). After replacing η by y in the proof of Theorem 12.7, the argument used there shows that A(x) = B(η) with A(X) and B(Y ) as in (12.1). The advantage is that the action of β on Σ has become as simple as possible; that is, β(x) = cx,

β(η) = η.

As β is a K-automorphism of Σ, so B(cx) = B(x). Since B(X) has degree m and c is an element of order m in the multiplicative group of K, this only happens when B(X) = bm X m . 2 T HEOREM 12.9 Suppose that (a) C is an irreducible plane curve of degree ≥ 4 as in (12.1) with n

A(Y ) = Y p + Y,

B(X) = X m ,

gcd(p, m) = 1;

(b) G is the K-automorphism group of C; (c) P∞ is the place arising from the unique branch centred at the unique infinite point of C; (1)

(d) GP∞ = GP∞ ⋊ H with p ∤ |H|. Then (i) |H| = m(pn − 1); (ii)

(1) |GP∞ | =



pn , if m − 1 is not a power of pn , n(2k+1) p , if m − 1 = pnk .

Proof. For a primitive m(pn − 1)-st root of unity, the birational transformation α of Σ = K(C) defined by α(x) = cx,

α(y) = cm y,

is a K-automorphism generating a cyclic group of order m(pn − 1). Theorem 12.7 (i) implies that H = hαi, showing (i). By Theorem 12.7 (iii), part (ii) holds for m 6≡ 1 (mod pn ). So consider the case m ≡ 1 (mod pn ) and s = (m − 1)/pn . As in the proof of Theorem 12.7, if α ∈ GP∞ , then α(x) = x + d,

α(y) = y + Q(x), pn

pn · deg Q(X) < m. m

(12.11)

m

Hence α ∈ GP∞ if and only if Q(x) +Q(x) = (x+d) −x . So the polynomial equation n

Q(X)p + Q(X) = (X + d)m − X m

(12.12)

is to be solved. For d = 0, only the trivial solution exists; that is, Q(X) = c with n cp + c = 0.

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Now let d 6= 0. To find which coefficients of (X + d)m − X m do not vanish, consider the p-adic representation of s: s = si pi + si+1 ps+1 + · · · ,

si 6= 0.

Suppose first that s is not a power of p; then s − pn ≥ 12 s. Put r = m − pi+n . The coefficients or equal to those of m.
in the p-adic representation of r are smaller than r Hence m does not vanish, by Lemma A.6. Therefore X has no zero coefficient r on the right-hand side in (12.12). So, the same holds for the left-hand side. As gcd(r, p) = 1, Q(X) itself contains X r which implies that r ≤ deg Q(X). On the other hand, deg Q(X) ≤ s by pn · deg Q(X) < m. However, this leads to a contradiction, since r = m − pi+n = (s − pi )pm + 1 > 21 spn ≥ s. This shows that d = 0 provided that s is not a power of p. It remains to investigate the case that s = pi ; that is, m = pn+i + 1. This time, n+i

(X + d)m − X m = dX p

n

n+i

+ dp X + dp

.

With Q(X) = q0 + q1 X + · · · + qr X r ,

qr 6= 0,

it follows from (12.12) that n

i+n

q0p + q0 = dp

+1

,

n+i

q1 = dp

+1

q2 = · · · = qpn −1 = 0.

,

In fact, the non-constant terms in Q(X) have degree of the form pjn , as a direct argument shows. Therefore m = pkn + 1. From Theorem 12.7 (ii), (1)

|GPinf ty | ≤ pn p2kn = p(2k+1)n . To finish the proof, it suffices to exhibit p(2k+1)n K-automorphisms of type (12.11). To do this, choose elements d, e ∈ K such that 2kn

dp

+ (−1)k d = 0,

n

n

ep + e = dp

+1

,

(12.13)

and let α(x) = x + d,

(12.14) (k−1)n

kn

α(y) = y + e + dp x − dp

(2k−1)n

+ · · · + (−1)k−1 dp

It is straightforward to verify that α is a K-automorphism of C: n

n

n

kn

(k−1)n

xp

2kn

α(y)p + α(y) = y p + y + ep + e + dp x + (−1)k−1 dp kn

kn+1

.

kn

xp

kn

= xp +1 + dp + dp x + dxkn kn+1 kn = (x + d)p = α(x)p +1 . For each of the pkn possible values of d, there exist exactly pn values of e that satisfy (12.13). This shows that (12.14) provides p(2k+1)n distinct K-automorphisms of C. 2

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Let C again be the curve (12.1), and let Σ = K(C). Also, define the following notation for Weierstrass semigroups: H(P) = the Weierstrass semigroup at P; H 0 (P) = the Weierstrass semigroup at P generated by m and pn ; S 0 = {P | H(P) = H 0 (P)}.

Note that P∞ ∈ S 0 by Lemma 12.3.

T HEOREM 12.10 Let Σ have a place P = 6 P∞ with P ∈ S 0 . Then, up to a linear substitution on X and Y, one of the following holds: (i) B(X) = X m with m < pn , pn ≡ −1 (mod m), and S 0 consists of the pn distinct zeros of x, together with P∞ ; n

n

(ii) A(Y ) = Y p + Y, B(X) = X p +1 , that is, C is the Hermitian curve over Fp2n , and S 0 consists of the places centred at the p3n + 1 rational points of C; n

j

n

(iii) A(Y ) = Y p + aj Y p + · · · + Y, B(X) = X p +1 , with 1 ≤ j ≤ n − 1 and aj 6= 0, and S 0 consists of the pn distinct zeros of x, together with P∞ . Proof. Take the centre of the branch associated with P to the origin P = (0, 0) by a translation. Since C remains of type (12.1), all previous results hold true. Also, ordP x = 1 because P is a simple point of C and the tangent to C is not the vertical line. By Lemma 12.2, the canonical series of Σ is |W |, where W = ((pn − 1)(m − 1) − 2)P∞ .

Hence div(xi y j ) + W ≻ 0

whenever ipn + jm ≤ (pn − 1)(m − 1) − 2. (12.15)

First, take the case m > pn + 1. If pn > 2, then (12.15) implies that n

div xp

−1

+ W ≻ 0.

This is also true for pn = 2, since gcd(m, p) = 1 together with deg C ≥ 4 implies that m ≥ 5. From the beginning of Section 7.6, the Weierstrass gaps are characterised by means of the canonical series as follows: a positive integer i is not in the Weierstrass semigroup H(P) at P if and only if there exists z ∈ Σ such that div(z) + W ≻ 0 and ordP z = i. This characterisation is used several times in this proof. Hence pn does not belong to H(P), which gives a contradiction. Assume that m ≤ pn − 1 with pn > 4. From (12.15), div xi + W ≻ 0 for 0 ≤ i ≤ m − 2, div y j + W ≻ 0 for 0 ≤ j ≤ pn−1 .

Since A(y) = B(x), n

z = an y p − bm xm − bm−1 xm−1

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satisfies div z + W ≻ 0. If bm−1 6= 0, then m < pn implies that ordP z = m − 1, which is a contradiction by the above characterisation of Weierstrass gaps. Hence either B(X) = bm X m or B(X) = bm X m + bm−2 X m−2 + · · · + bk X k with 1 ≤ k ≤ m − 2, bk 6= 0.

The latter case cannot in fact occur. For m = pn − 1, this is a consequence of the relation ordP z = m = pn − 1

and the above characterisation of Weierstrass gaps. Similarly, for m ≤ pn − 2, there exists a rational function η such that div η + W ≻ 0 and ordP η = m − 1. Letting  m−1 y , if k = 1, η= yxm−k+1 , if k ≥ 2,

this again gives a contradiction to the characterisation of Weierstrass gaps. Thus B(X) = bm X m . In particular, I(P, C ∩ ℓ) = m, where P is the centre of the branch associated to P and ℓ = v(Y ). Now, choose positive integers s < m and t such that pn + s ≡ 0 (mod m); so lm = pn + s. Note that div(xy −l ) = P2 + · · · + Pn + sP∞ − (pn + s − 1)P,

where P2 , . . . , Pn are the zeros of x other than P. Hence pn + s − 1 ∈ H(P) and this gives a contradiction for s > 1. If s = 1, then ordP (xy −t ) = −pn ,

ordP (y −1 ) = −m.

Thus H(P) is generated by m and pn . This result holds true for every zero of x, that is, for every place of Σ arising from a branch centred at a common point of C and the line X = 0. This depends on the fact that each translation (x, y) 7→ (x, y + c), for a root c of A(Y ), preserves C. The possibility of having another place with the same Weierstrass semigroup is ruled out by using the invariance of the intersection multiplicity under translations. If such a place exists and it corresponds to a branch centred at the point P = (u, v), the translation (x, y) 7→ (x − u, y − v) takes P to the origin O and the image of C under any translation is a curve C ′ of the same type. Hence what has already been shown for C applies to C ′ . Therefore, if ℓ′ = v(Y ) and ℓ = v(Y − v), then I(O, C ′ ∩ ℓ′ ) = I(P, C ∩ ℓ),

I(O, C ′ ∩ ℓ′ ) = m;

so I(P, C ∩ ℓ) = m. But a straightforward computation shows that this only occurs when u = 0. Let m ≤ pn − 1 with pn ≤ 4. The case pn ≤ 3 cannot occur because deg C ≥ 4. Since gcd(m, p) = 1, if pn = 4 then m = 3, and A(Y ) = a2 Y 4 + a1 Y 2 + a0 Y,

B(X) = b3 X 3 + b2 X 2 + b1 X.

By (12.15), div z + W ≻ 0, where z = a0 y + b1 x. On the other hand, a calculation shows that  2, if a1 a0−1 b21 + b2 6= 0, ordP z = 2 3, if a1 a−1 0 b1 + b2 = 0.

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But again the characterisation of Weierstrass gaps leads to a contradiction since both pn = 4 and m = 3 are in H(P). Let m = pn + 1. Without loss of generality, let a0 = 1. With η = y + b1 x, A(η) − B(x) = A(y) + A(b1 x) − B(x) = A(b1 x), A(η) = B(x) + A(b1 x) = bm xm + · · · + b′2 x2 ,

with b′2 = b2 for p > 2 and b′2 = b2 +a1 b21 for p = 2. After the linear transformation (x, y) 7→ (x, a0 y + b1 x), it is possible to put b1 = 0, or equivalently, to assume that pn + 1 ≥ ordP y ≥ 2.

In fact, ordP y = pn + 1. To show this, suppose on the contrary that ordP y < pn + 1.

If ordP y = pn = m − 1, there is a contradiction to the characterisation of Weierstrass gaps since div y + W ≻ 0 by (12.15). If ordP y < pn , let n

z = yxp

−k−1

,

with k = ordP y.

n

As ordP z = p − 1 and div z + W ≻ 0, once again there is a contradiction to the characterisation of Weierstrass gaps. Therefore B(X) = X m . In other words, the point P = (0, 0) of C has the property that I(P, C ∩ ℓ) = m where ℓ is the tangent line to C at P . By a preceding argument depending on the invariance of the intersection multiplicity under translations, this holds true for any point of C which is the centre of a branch corresponding to a place of Σ with Weierstrass semigroup H 0 (P) n Now, if A(y) = an y p + a0 y, then C is the Hermitian curve over Fp2n . Otherwise, there exists j with 0 < j < n such that n

j

A(y) = an y p + aj y p · · · + a0 y,

aj 6= 0.

Since ordP (y −1 ) = −m, so H(P) = H 0 (P). As before, this holds true for every zero of x, again by the fact that every translation (X, Y ) 7→ (X, Y + c), for a root c of A(Y ), preserves C. It remains to show that no other place of Σ has Weierstrass semigroup H 0 (P). Again, if P = (u, v) is the centre of a branch corresponding to such a place, then I(P, C ∩ ℓ) = m, where ℓ is tangent line to C at P . But another calculation shows that this is not the case as soon as u 6= 0. 2 T HEOREM 12.11 If C is a curve of type (12.1), then the K-automorphism group AutK (Σ) of C fixes the place P∞ except in the following two cases. (i)

n

(a) C = v(Y p + Y − X m ), with m < pn , pn ≡ −1 (mod m);

(b) the group AutK (Σ) contains a cyclic normal subgroup Cm of order m such that AutK (Σ)/Cm ∼ = PGL(2, pn );

(c) Cm fixes each of the pn +1 places with the same Weierstrass semigroup as P∞ ;

(d) AutK (Σ)/Cm acts on the set of such pn + 1 places as PGL(2, pn ).

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(ii)

n

n

(a) C = v(Y p + Y − X p +1 ), the Hermitian curve; (b) AutK (Σ) ∼ = PGU(3, q); (c) AutK (Σ) acts on the set of all places with the same Weierstrass semigroup as P∞ ; (d) AutK (Σ) acts on the set of such places as PGU(3, q) on a Hermitian curve.

Proof. Assume that G = AutK (Σ) does not fix P∞ . The invariance of the Weierstrass semigroup under the action of AutK (Σ) implies that the orbit of P∞ is a subset of S 0 . From this, Theorem 12.10 shows that there are only a few cases to consider. First, let B(X) = X m , m < pn , A(Y ) = Y

pn

pj

+ aj Y

The K-automorphisms

pn ≡ −1

(mod m),

b0 = 0;

+ · · · + Y, 1 ≤ j ≤ n − 1,

(x, y) 7→ (x, y + d),

aj 6= 0.

A(d) = 0

of Σ form a group acting transitively on the set of all places which are zeros of x. If G 6= GP∞ , then the orbit of P∞ consists of all zeros of x together with P∞ , and one of them, say PO , corresponds to a branch centred at the origin O. The Weierstrass semigroup H(PO ) is generated by m and pn . Also, dim |mPO | = 1 and dim |pn PO | = s where s = (pn + 1)/m. Since the tangent to C at O is not the the line v(X), a primitive representation of the unique branch centred at O is (x = t, y(t) = vti + · · · ). Since A(y(t)) = B(x(t)), m y(t) = a−1 O t + ··· ;

so ordPO (y) = m. From the hypothesis that m < pn ,

|mPO | = {div(c0 + c1 y −1 ) + mPO | c = (c0 , c1 ) ∈ PG(1, K)},

|pn PO | = {div(c0 + c1 y −1 + · · · + cs−1 y −(s−1) + cs xy −s ) + mPO | c = (c0 , . . . , cs ) ∈ PG(s, K)},

Assume that α ∈ AutK (Σ) takes P∞ to PO . Then α takes |mP∞ | to |mP|, and |pn P∞ | to |pn P|. Hence α(x) = y −1 (cx + d1 y + · · · + ds y s ),

Here, b 6= 0, c 6= 0. Then

n

α(y) = y −1 (ay + b).

n

j

j

A(α(y)) = A(a) + bp y −p + aj bp y −p + · · · + by −1 ,

A(α(x)) = y −ms (cx + d1 y + · · · + ds y s )m .

On the other hand,

A(α(y)) = α(A(y)) = α(B(x)) = B(α(x)) implies that y

ms

n

A(α(y)) = y p

A(a)y

n

p +1

n

p

+1

B(α(x)). Hence j

n

+ b y + aj b p y p

−pj +1

n

+ · · · + by p

= cm xm + mcm−1 xm−1 (d1 y + · · · + ds y s ) + · · · = cm A(y) + mcm−1 xm−1 (d1 y + · · · + ds y s ) + · · · .

(12.16)

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Since P = (x, y) is a generic point of C, the polynomial n

n

j

n

j

n

H(X, Y ) = A(a)Y p +1 + bp Y + aj bp Y p −p +1 + · · · + bY p (12.17) m m−1 m−1 s −(c A(Y ) + mc X (d1 y + · · · + ds Y ) + · · · ) is divisible by A(Y ) − B(X). Note that degX = A(Y ) − B(X) = m.

degX H(X, Y ) < m,

Hence no term in X can actually occur in H(X, Y ). Therefore n

A(a)Y p

+1

n

j

n

+ b p Y + aj b p Y p

−pj +1

n

+ · · · + bY p = cm A(Y ).

i

Since A(Y ) has only terms of type Y p , either b = 0 or aj = 0, a contradiction. The next case is the following: b(X) = X m , m = pn + 1, b0 = 0, n

j

A(Y ) = Y p + aj Y p + · · · + Y, 1 ≤ i ≤ n − 1, aj 6= 0. This can be ruled out in a similar way, since |pn PO | = {div(c0 + c1 xy −1 ) + pn PO | c = (c0 , c1 ) ∈ PG(1, K)}, |(p + 1)PO | = {div(c0 + c1 xy −1 + c2 y −1 ) + (pn + 1)PO | n

c = (c0 , c1 , c2 ) ∈ PG(2, K)}.

In the case n

C = v(Y p + Y − X m ),

pn ≡ −1 (mod m),

the rational transformation α(x) = xy −s ,

α(y) = y −1 ,

(12.18)

with s = (pn + 1)/m, is a K-automorphism of Σ. In fact, n

xm 1 yp + y 1 A(α(y)) = pn + = pn +1 = sm = B(α(x)). y y y y This shows that G is larger than GP∞ . Also, the orbit of P∞ consists of all zeros of x together with P∞ . To give an explicit permutation representation for G, the zeros of x are identified with the elements of Fpn in the following way. Choose an n element u ∈ K satisfying up −1 = −1, and change (X, Y ) to (X, uY ). Then n

C = v(Y p − Y − X m ).

The places arising from the branch centred at the point Pv = (0, v) of C and the Y -coordinates v of Pv are in one-to-one correspondence; this can be extended to n P∞ by associating the symbol ∞ to it. Note that v p = v; that is, v ∈ Fpn . ¯ be the To prove that G acts on Fpn ∪ {∞} as PGL(2, pn ) on PG(1, pn ), let G permutation group induced by G on Fpn ∪ {∞}. By Lemma 12.1 (iii)(c), every permutation of Fpn ∪ {∞} of type  v + b for v ∈ Fpn , v = v 7→ (12.19) ∞ for v = ∞,

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563

¯ Also, every permutation of Fpn ∪ {∞} of type with b ∈ Fpn is in G.  cv for v ∈ Fpn , v 7→ (12.20) ∞ for v = ∞, ¯ Further, α defined as in (12.18) acts on Fpn ∪ {∞} as with c ∈ Fpn \{0} is in G. the permutation  −1 for v ∈ Fpn ,  v ∞ for v = 0, v 7→ (12.21)  0 for v = ∞. The group generated by the permutations (12.19), (12.20), (12.21) is PGL(2, pn ). ¯ = PGL(2, pn ). This is a consequence of Theorem A.17. In fact, G An elementary proof is also possible, using Theorem 12.7 in the following way. The K-automorphisms (x, y) 7→ (dx, y) with d ranging over the m-th roots of unity form a cyclic group Cm of order m. Also, the K-automorphisms in (12.20) form a cyclic subgroup Cpn −1 . These two groups generate an automorphism group of order m(pn − 1). On the other hand, from Theorem 12.7, |G∞ | ≤ m(pn − 1)pn , which implies that |GP∞ ,PO | ≤ m(pn − 1). Therefore |GP∞ ,PO | = m(pn − 1). Since G acts 2-transitively on the orbit of P∞ , it follows that |G| = m(pn + 1)pn (pn − 1). As every place in this orbit is fixed by Cm , this shows that G/Cm ∼ = PGL(2, pn ). Finally, if n n C = v(Y p + Y − X p +1 ), then C is the Hermitian curve, and (ii) follows from Theorem 11.30. 2

R EMARK 12.12 Let A(Y ) be a separable additive polynomial of degree pn , and B(X), C(X) polynomials with no common roots such that the rational function R(X) = B(X)/C(X) is admissible; that is, no pole of R(x) regarded as an element of the rational function field K(x) has multiplicity divisible by p. If Pr−1 R(X) = f0 (X) + i=1 fi (X)(X − ei )−1 , with fi ∈ K[X] for 0 ≤ i ≤ r − 1, then R(X) is admissible if and only if the degree di of each fi is relatively prime to p. If, also, di ≥ 1 for 0 ≤ i ≤ r − 1, then R(x) has exactly r distinct poles in K(x). In this situation, the curve C = v(C(X)A(Y ) − B(X)) is irreducible and has genus Pr−1 g = 21 (pn − 1)(d0 − 1 + i=1 (di + 1)). Also, if γ is the p-rank of the function field Σ = K(C), then γ = (pn − 1)(r − 1). Therefore g = γ if and only if di = 1 for 0 ≤ i ≤ r − 1; that is, every pole of R(x) is simple. In the spirit of Theorem 12.5, the particular case R(X) = aX + X −1 , where q = p, is characterised, up to an Fq -birational equivalence, as an irreducible hyperelliptic curve F defined over Fq of genus p − 1 whose Fq -automorphism group G has the following two properties:

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(i) G = hωi × D, where ω is the hyperelliptic involution and D is a dihedral group of order 2p; (ii) the set of fixed places of D is exactly F(Fq ). 12.2 CURVES WITH SUZUKI AUTOMORPHISM GROUP Let p = 2, q0 = 2s , with s ≥ 1 and q = 2q02 = 22s+1 . Let F be the irreducible plane curve with

F = v(X 2q0 (X q + X) + (Y q + Y )).

(12.22)

The Deligne–Lusztig curve associated with the Suzuki group or DLS curve, for short, is any algebraic curve birationally isomorphic to F. So, the function field of a DLS curve is K(F) = K(x, y) with x2q0 (xq + x) = y q + y. Note that F is birationally isomorphic to the curve

v(X q0 (X q + X) + (Y q + Y )).

To show this, let Σ = K(x, y) with xq0 (xq + x) = y q + y. If z = x2q0 +1 + y 2q0 , that is, y q = z q0 + xq+q0 , then Σ = K(x, z). Also, z q0 = xq+q0 + y q = xq0 +1 + y, and hence z q = xq+2q0 + y 2q0 . Now, since z q + z = xq+2q0 + y 2q0 + x2q0 +1 + y 2q0 = x2q0 (xq + x), the assertion follows. So, the definition here is equivalent to the usual definition of a DLS curve. T HEOREM 12.13 If F is the curve (12.22), P∞ = (0, 0, 1) and P∞ is the place associated to the branch centred at P∞ , then the following properties hold: (i) P∞ is a q0 -fold point, it is the unique infinite point of F, and is the centre of just one branch of F; (ii) div(dx) = (2q0 (q − 1) − 2)P∞ ; (iii) F has genus g = q0 (q − 1); (iv) the set F(Fq ) of all Fq -rational places of F has size q 2 + 1; (v) the K-automorphism group G of F is Fq -rational and is isomorphic to the Suzuki group Sz(q); (vi) G acts on F(Fq ) as Sz(q) on the Tits ovoid in PG(3, q); (vii) |G| = (q 2 + 1)q 2 (q − 1), and |GP∞ | = q 2 (q − 1); (1)

(2)

(viii) |GP∞ | = q 2 , and GP∞ is an elementary abelian group of order q;

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(2q −1)

(ix) GP∞ = . . . = GP∞0

(2q )

(2q +1)

= GP∞0 and GP∞0

= {1}.

Proof. It is immediate that P∞ is the only singular point of F and that P∞ is a 2q0 -fold point of F. First, it is shown that just one branch of F is centred at P∞ . As a consequence, the irreducibility of F follows. With h = xy + x2q0 +2 + y 2q0 , the function h has a zero at the place O of the field Σ = K(F) arising from the unique branch γ of F centred at the origin O = (1, 0, 0). Actually, h has no more zeros corresponding to branches centred at an affine point. Equivalently, the system x2q0 (xq + x) − (y q + y) = 0, xy + x2q0 +2 + y 2q0 = 0

(12.23) (12.24)

has only one solution over K, namely (x, y) = (0, 0), as follows From these equations, xq0 y q0 + xq+2q0 + y q = xq0 y q0 + x2q0 +1 + y = 0, whence y 2 = x2q0 y 2q0 + x4q0 +2 = x2q0 (xy + x2q0 +2 ) + x4q0 +2 = yx2q0 +1 . This shows that either x = y = 0, or y = x2q0 +1 . If y = x2q0 +1 , from (12.22) xq = x. But then (12.24) gives x = 0, and again x = y = 0. The utility of h depends on the fact that the rational transformation ϕ, given by (x, y) 7→ (y/h, x/h)

(12.25)

is a K-automorphism of Σ. From above, ϕ takes any branch centred at an affine point to a branch centred at an affine point, except for γ. Since ϕ has order 2 and hence coincides with its inverse, it takes γ to a branch centred at P∞ . Let (x′ (t), y ′ (t)) be a primitive representation of γ. Then x′ (t) = t, y ′ (t) = t2q0 +1 + tq+2q0 + · · · . Therefore t2q0 +1 + · · · 1 = t−q = t−q (1 + · · · ), + t2q+2q0 + · · · 1 + tq−1 + · · · t 1 y(t) = q+2q0 +1 = t−q−2q0 = t−q−2q0 (1 + ...). t + t2q+2q0 + · · · 1 + tq−1 + · · · The resulting branch has order 2q0 . Since P∞ is a 2q0 -fold point, it is the unique branch of F centred at P∞ . So, in terms of places, ϕ interchanges P∞ and O. In particular, x(t) =

tq+2q0 +1

ordP∞ x = −q, ordP∞ y = −(q + 2q0 ), ordP∞ h = −(q + 2q0 + 1). To calculate div(dx), begin from the equation x(t)2q0 (x(t)q + x(t)) = y(t)q + y(t). Taking derivatives, dy(t) dx(y) = x(t)2q0 +1 . dt dt

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Since ordt (dy(t)/dt) = (−2q0 − 2), this gives

dx(t) = 2q0 (q − 1) − 2. dt Therefore P∞ contributes weight 2q0 (q − 1) − 2 to div(dx). To prove (ii), it must be shown that ordP dx = 0 for every place P whose corresponding branch is centred at an affine point. By Lemma 5.56, this only requires that no vertical line is tangent to F or, equivalently, that the polynomial Y q + Y + c has q distinct roots. By Definition 5.55, (ii) implies (iii). Further, F(Fq ) comprises P∞ and every place which arises from a branch centred at a point in AG(2, q). This proves (iv). It can be shown that, for any a, c, d ∈ Fq with d 6= 0, the following transformations are K-automorphisms of K(F): ordt

ψa,c : γd :

x 7→ x + a,

x 7→ dx,

y 7→ a2q0 x + y + c; 2q0 +1

y 7→ d

y.

(12.26) (12.27)

Let Ψ = {ψa,c | a, c ∈ Fq }.

It is straightforward to verify that Ψ is a group of order q 2 which preserves F(Fq ). Also, Ψ fixes P∞ and acts on the other places in F(Fq ) as a sharply transitive permutation group. The involutory elements in Ψ together with the identity form an elementary abelian subgroup Ψ′ of order q. Also, H = {γd | d ∈ Fq \{0}} is a cyclic group of order q − 1 which fixes P∞ and preserves F(Fq ). The group generated by Ψ and H has order q 2 (q − 1) and is the semidirect product Ψ ⋊ H. This group together with the K-automorphism ϕ of (12.25) generate a larger group G which acts on F(Fq ) as a 2-transitive permutation group. From Theorem A.17, G∼ = Sz(q), and G acts on C(Fq ) as Sz(q) in its natural 2-transitive representation. Note that G is Fq -rational as it is generated by Fq -rational K-automorphisms. (1) (1) Also, |G| = (q 2 + 1)q 2 (q − 1). Hence Ψ = GP∞ , and GP∞ = GP∞ ⋊ H. Now the ramification groups at P∞ can be determined by using Hilbert’s Different Formula 11.37. As y/h is a uniformising element, it suffices to calculate ordP∞ (u/h + ψa,c (y/h)). By a straightforward calculation, u/h + ψa,c (y/h) =

ay 2 + a2q0 xh + bh + (a2q0 +1 + b)xy + (a2q0 +2 + ab + b2q0 )y , h2 + ahy + (a2q0 +1 + b)hx + (a2q0 +2 + ab + b2q0 )h

which shows that ordP∞ (u/h + ψa,c(y/h)) is either 2 or q0 + 2 according as a 6= 0 (2q +1) (2) 2 or a = 0. Therefore Ψ′ = GP∞ = · · · = GP∞0 . A group-theoretic characterisation of the DLS curve is established in the following theorem.

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T HEOREM 12.14 With q0 = 2s , q = 2q02 , let F be an irreducible curve of genus q0 (q − 1). If a K-automorphism group of F contains a subgroup G isomorphic to Sz(q), then F is the DLS curve. The proof is organised in a series of lemmas and uses some results obtained (i) in Section 11.12. The same notation is employed here, and Fi = ΣGP . Since |G| > 8g 3 , the relevant case is (iv.5). Here, G has one non-tame orbit o of places, and its faithful action on o is the natural 2-transitive permutation representation of Sz(q). For P ∈ o, the following properties hold: (i) |GP | = q 2 (q − 1); (1)

(ii) GP is a group of order q 2 acting on o\{P} as a sharply transitive permutation group; (1) (iii) GP = GP ⋊ H, where H ∼ = Cq−1 ; (2)

(iv) F2 = ΣGP is rational. (2)

L EMMA 12.15 The group GP has order q and consists of all involutory elements (1) (2) in GP together with the identity. In particular, [GP : GP ] = q. Proof. Each involutory element in Sz(q) is the square of an element of order 4 in (2) (1) Sz(q). Since the factor group GP /UP is an elementary abelian 2-group, every (1) (2) (2) involutory element in GP belongs to UP . In particular, |UP | ≥ q. (1) Choose any non-involutory element α ∈ GP , and look at the structure of the (1) 1-point stabiliser of Sz(q). Every non-involutory element in GP can be written as (2) βαβ −1 δ with β ∈ H and δ 2 = 1. Now, assume on the contrary that α is in UP . (2) (2) Since H is contained in the normaliser UP , it follows that βαβ −1 δ is also in UP , (2) (2) (1) (1) contradicting that GP 6= UP . If ρ1 > 2, that is, if GP = GP , then Theorem (1) 11.72 applied to GP would imply that 2q0 (q − 1) − 2 = 2g − 2 ≥ −2q 2 + 3(q 2 − 1) = q 2 − 1,

which is impossible since q = 2q02 . Therefore ρ1 = 2.

2

L EMMA 12.16 ρ2 = 2q0 + 2. Proof. In the 1-point stabiliser of Sz(q), any two involutory elements are conjugate. (2) (i) (i) Thus, if i ≥ 3, then either GP = GP , or GP = {1}. By Theorem 11.72 applied (2) to GP , it follows that 2q0 (q − 1) − 2 = −2q + 2q − 2 + (ρ2 − 2)(q − 1), whence ρ2 = 2q0 + 2. L EMMA 12.17 The smallest non-gap at P is q.

2

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Proof. Let m denote the smallest non-gap at P. Take an element x ∈ Σ for which (1) div(x)∞ = mP. The group GP induces an automorphism group on K(x). The (2) kernel of such a permutation representation contains GP as every non-trivial Kautomorphism of K(x) whose order is a power of p has order p. Hence K(x) ⊂ F2 . Since m = [Σ : K(x)] = [Σ : F2 ][F2 : K(x)], (2) and |GP = [Σ : F2 ] by Theorem 11.36, it (2) every GP -orbit distinct to P has length q.

follows that m ≥ q. On the other hand, Therefore m ≥ q, whence m = q. 2

This result together with Theorem 11.14 also shows the following lemma. L EMMA 12.18 There is an element x ∈ Σ such that div(x)∞ = qP, F2 = K(x), and, for every α ∈ GP ,  (1) (2)   x + a, a ∈ K, a 6= 0, for α ∈ GP \GP , α(x) = (12.28) bx, bq−1 = 1, b 6= 1, for α ∈ H,   (2) x, for α ∈ GP . L EMMA 12.19 The smallest non-gap m at P which is not divisible by q does not exceed q + 2q0 .

Proof. Since F1 is rational, F1 = K(x1 ) with x1 ∈ Σ. After a change of x1 in K(x1 ), let P be a pole of x1 . Apart from the trivial orbit {P}, all orbits of places (1) under GP have maximum size q 2 . Since [Σ : F1 ] = q 2 , so x1 has at most q 2 distinct poles, each counted with multiplicity. Hence P is the unique pole of x1 . (1) Since GP = GP ⋊ H, an automorphism δ ∈ H of order q − 1 preserves K(x1 ). By Theorem 11.14, δ(x1 ) = ax1 with a ∈ K, a 6= 0. Hence δ preserves the set of zeros of x1 . Since the order of δ is prime to q 2 , a power of δ fixes at least one (1) such zero, say R. As F1 is rational, and o is the only GP -orbit preserved by δ, it follows that R ∈ o. Hence o consists of all zeros of x1 ; that is, P div(x1 ) = R∈o\{P} R − q 2 P. Now, take an element β ∈ G which moves P, and put Q = P β . Note that Q ∈ o. Replacing P by Q in the preceding argument gives the equation, P div(y1 ) = R∈o\{Q} R − q 2 Q,

for an element y1 ∈ Σ. Let z1 = x1 /y1 ; then div(z1 ) = q 2 Q − q 2 P. Similarly, there exists z2 ∈ Σ such that div(z2 ) = (2g − 2)Q − (2g − 2)P.

To show this, note the following two properties of the differential dx1 : (1) dx1 can have at most one pole, namely P; (2) the weight of P in div(dx1 ) does not exceed q 2 , by div(x1 )∞ = q 2 P.

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As a consequence, dx1 has at most 2g − 2 + q 2 < 2q 2 zeros. Since x1 ∈ F1 and (1) |GP | = n2 this implies that the zeros of dx1 are q 2 . Also, (div(dx1 ))δ = div(d(δ(x1 ))) = div(d(ax1 )) = div(dx1 );

so δ fixes div(dx1 ). Now, the above arguments can be used to prove that o\{P} is the set of all zeros of dx1 each with the same weight. Again, moving P to Q by an element β ∈ G, it can also be shown that there exists y1 such that div(dx1 ) − div(dy1 ) = (2g − 2)Q − (2g − 2)P. Since z2 = dy1 − dx1 ∈ Σ, it follows that div(z2 ) = (2q0 (q − 1) − 2)Q − (2q0 (q − 1) − 2)P.

Since q 2 + 1 = (q + 2q0 + 1)(q − 2q0 + 1),

gcd(q 2 + 1, 2(q(q0 − 1) − 1)) = q + 2q0 + 1.

Hence div(z3 ) = (q + 2q0 + 1)Q − (q + 2q0 + 1)P for some element z3 ∈ Σ. (1) (2) Finally, if α ∈ GP \GP , then div(α(z3 ) − z3 )∞ = (q + 2q0 )P. This follows from Lemma 11.83 and the fact that ρ1 = 2. 2 Choose an element z ∈ Σ such that div(z)∞ = mP with m as in Lemma 12.19. L EMMA 12.20 There exists a polynomial f (X) ∈ K[X] with deg f = m such that Σ = K(x, z), where z q + z = f (x). Proof. For any element u ∈ F2 , the equation [Σ : K(u)] = [Σ : F2 ][F2 : K(u)] (2) shows that if div(u)∞ = kP then q | k. Hence GP must contain some element δ (2) (1) such that δ(z) 6= z. Choose α ∈ GP \GP such that α2 = δ. Then α(z) 6= z, and the minimality of m implies that ordP (α(z) − z) > −m. As 2q > q + 2q0 , this is only possible when ordP (α(z) − z) = −q. Therefore α(z) = z + cx + d with

c, d ∈ K, c 6= 0.

Hence δ(z) = z + eδ e−1 δ z

with eδ ∈ K\{0}.

(12.29)

Replacing z with ensures that δ(z) = z + 1. Now, take a generator β of H. From Lemma 11.17, either β(z) = ǫ−m z or div(β(z) − ǫ−m z)∞ < mP, where ǫ denotes a primitive (q − 1)-st root of unity. As before, this implies that β(z) = ǫ−m z + dx + e with d, e ∈ K. Replacing z by z + d′ , where d′ = e/(c − 1), gives β(z) = ǫ−m z + dx, and replacing β 2 by β, this becomes β(z) = ǫ−m z. From this, eγ can be calculated for all non-trivial elements γ ∈ facts:

(12.30) (2) GP

using two

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(1) β −k δβ k (z) = z + ǫ−mk eδ ; (2)

(2) any non-trivial element in GP is conjugate to δ by an element in H.

So, eγ = ǫ−mk eδ , showing that eγ 6= 0. If two of these values, say eγ and eg′ , were (2) equal, then δ ′ = γγ ′ would fix x; that is, eδ′ would be zero. Since δ ′ ∈ GP , this −m is impossible. As a consequence, ǫ is a (q − 1)-st of unity; that is, all non-zero (1) (2) elements of a subfield Fq of K are obtained. Also, as each α ∈ GP \GP is a (1) product of elements from H and GP , the coefficients of the equation of α are all (2) in Fq . Therefore GP is defined over Fq . Now, for every γ ∈ GP , γ(z q + z) = (z + eγ )q + z + eγ = z q + z.

Since K(x) = F2 , this implies that z q + z ∈ K(x). Hence z q + z = f (x) with f (X) ∈ K[X] and deg f = m, since both z and x have only one pole, namely P, and ordP (z)∞ = m, ordP (x)∞ = q. 2 Note that m is even by Lemma 11.83. L EMMA 12.21 f (X) = X q+q0 + X q0 +1 . Proof. By (12.30) and m < 2q, applying β to z q + z + f (x) = 0 gives f (X) = uX m + vX m+1−q , q

u 6= 0.

Note that v 6= because an equation of type z + z = xm can be written as √ 0,m/2 q/2 )2 , showing that z = (z + ux √ div(z m/2 + uy m/2 )∞ = 21 mP, (1)

(2)

in contradiction to the definition of m. Now, take α ∈ GP \GP . Then

a, b, c ∈ Fq , ab 6= 0.

m Applying α to z + z + ux + vx gives m 6= 0 but q+i = 0 for q 1 ≤ i ≤ m − q. Hence m − q is a power of 2. Taking Lemma 12.19 into account, it follows that α(x) = x + a, α(z) = z + bx + c, q

m

m+1−q

k

f (X) = X 2 (rX q + sX),

2k ≤ q0 , r, s ∈ K\{0}. k

Also, raq + sa = 0 for every a ∈ Fq . Hence r = s. Further, ca2 = bq , whence r, s ∈ Fq . Replacing z by rz gives the equation k

f (X) = X 2 (X q + X), q

with 2k ≤ q0 .

Applying β ∈ H to z +z+f (X), where β(x) = ηx, β(z) = ǫz with ǫ, η primitive k (q − 1)-st roots of unity, gives ǫ = η 2 +1 . For brevity, put q1 = 2k , q2 = q/q1 , and q3 = q2 /q1 = q/q12 . Note that each of these numbers q1 , q2 , q3 is an integer. Let v = z q2 + xq2 +1 ; then v q + v = z qq2 + x(q2 +1)q + z q2 + xq2 +1 = (z q + z)q2 + x(q2 +1) = [xq1 (xq + x)]q2 + x(q2 +1)q + xq2 +1 = xq (xqq2 + xq2 ) + xqq2 +q + xq2 +1 = xq+q2 + xq2 +1 ,

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and v q1 = z q + xq+q1 = z + y q1 +1 . Also, in succession, (z q + z)xq1 (q3 −1) = (xq + x)xq1 xq1 (q3 −1) , z q xq1 (q3 −1) + zxq1 (q3 −1) = xq+q1 q3 + xq1 q3 +1 , (z q x(q3 −1) + xq2 +q3 )q1 = zxq1 (q3 −1) + xq1 q3 +1 , (z q2 x(q3 −1) + v q2 )q1 = zxq1 (q3 −1) + v. Let w = z q2 x(q3 −1) + v q2 . Since ordP z = −m = −(b + q1 ), w

q1

q1 (q3 −1)

= zx

q2

+v ,

ordP x = −q,

ordP v ≤ −q(q2 − 1),

it follows that ordP w = q2 −1 (−q − q2 − q1 q3 q − q1 q) = −(1 + q1 + q(q3 − 1)). q2

(2)

Also, if δ ∈ GP , then δ(w) = w + eqδ2 xq3 −1 + eδ2 . From Lemmas 12.16 and 11.83, 2 + 2q0 = 1 + 1 + +q2 + q(q3 − 1) − q(q3 − 1) = 2 + q2 ,

whence q0 = 2q2 . Also, q1 = q0 , which gives the result.

2

T HEOREM 12.22 If F ′ = F/G is a tame quotient curve of the DLS curve, then it is one of the following types, where g ′ is the genus of F ′ and r = |G| : (i) with r | (q − 1) and g ′ = q0 (q − 1)/r,

F ′ = v(Y (q−1)/r f (X) − (1 + X q0 )(X q−1 + Y 2(q−1)/r ));

(ii) with r | (q + 2q0 + 1) and g ′ = (q + 2q0 + 1)(q0 − 1)/r + 1,

F ′ = v(Y (q+2q0 +1)/r g(X) − X q+2q0 +1 + Y 2(q+2q0 +1)/r );

(iii) with r | (q − 2q0 + 1) and g ′ = (q − 2q0 + 1)(q0 + 1)/r − 1, F ′ = v(bY (q−2q0 +1)/r h(X)

−(X q0 −1 + X 2q0 −1 )(X q−2q0 +1 + Y 2(q−2q0 +1)/r )).

Here

and b =

Ps−1

i 2i (2q0 +1)−(q0 +1) (1 + X)2 , i=0 X Ps−1 i i g(X) = 1 + i=0 X 2 q0 (1 + X)2 (q0 +1)−q0 + X q/2 , Ps−1 i i h(X) = 1 + i=0 X 2 (2q0 +1)−(q0 +1) (1 + X)2 , λq0 + λq0 −1 + λ−q0 + λ−(q0 −1) with λ ∈ Fq4 of order q −

f (X) = 1 +

2q0 + 1.

Since Sz(q) contains a large number of non-isomorphic subgroups of even order, especially 2-subgroups, a complete classification for non-tame quotient curves is out of reach. T HEOREM 12.23 There exist non-tame quotient curves F ′ = F/G of the DLS curve F given by the following examples, where g ′ is the genus of F ′ and r = |G|.

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(i) With r = q2 and g ′ = q0 (q/q2 − 1), Pq−1 i F ′ = v( i=0 Y q2 − X 2q0 (X q + X)),

where Fq2 is a proper subfield of Fq . The K-automorphisms ψ0,c defined ′ in (12.26) form an elementary abelian group G of order q2 = 2s +1 with ′ (2s + 1) | (2s + 1).

(ii) With r = 4 and g ′ = 41 q0 (q − 2), P2s i F ′ = v( i=0 Y 2 − (f1 (X) + f2 (X) + f3 (X))), where

f1 (X) = f2 (X) = f3 (X) = Here G ∼ = Z4 .

Ps

i=0

P2s

i=0

P2s

P

s j=i

j

X2

i



X2 , P i−s−2

i=s+2

j=0

i

X2 ,

j

X2

2q0

i

X2 .

12.3 CURVES WITH UNITARY AUTOMORPHISM GROUP Let q = ph > 2. In this section, Hq is the irreducible plane curve defined over Fq2 with Hq = v(Y q + Y − X q+1 ).

(12.31)

A Hermitian curve defined over Fq2 is any algebraic curve birationally isomorphic to Hq . In projective coordinates, equivalent forms of (12.31) are the v(Fi ), for i = 1, 2, 3, 4, with the following Fi : (M1)

F1 = X0q+1 + X1q+1 + X2q+1 ;

(M2)

F2 = X2q X0 − X2 X0q + ωX1q+1 , where ω q−1 = −1;

(M3)

F3 = X1 X2q − X1q X2 + ωX0q+1 , where ω q−1 = −1;

(M4)

F4 = X0q X1 + X1q X2 + X2q X0 .

Each of the first three is obtained from (12.31) by a linear substitution defined over Fq2 , but for (M4) this can be done only over Fq3 ; see Remark 8.19. The function field of Hq is K(Hq ) = K(x, y) with y q + y + xq+1 = 0. The relevant properties of Hq are collected in the following theorem. T HEOREM 12.24 The curve Hq is non-singular, with P∞ the place associated to the branch centred at P∞ = (0, 0, 1), and has the following properties: (i) div(dx) = (q + 1)(q − 2)P∞ ; (ii) Hq has genus g = 21 q(q − 1);

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(iii) the set F(Fq2 ) of all Fq2 -rational places of F has size q 3 + 1; (iv) the K-automorphism group AutK (Σ) of Hq is Fq2 -rational and isomorphic to PGU(3, q), acting projectively on Hq (Fq2 ).

Proof. The curve Hq is of type (12.1) with A(Y ) = Y q + Y and with the degree q + 1 of B(X) not divisible by p. So the results obtained in Section 12.1 apply. Parts (i) and (ii) follow from Lemma 12.1, while (iii) is Theorem 12.10 (ii), and (iv) is Proposition 11.30. 2 T HEOREM 12.25 For q = ph > 2, let F be an irreducible algebraic curve of genus 21 q(q − 1). If a K-automorphism group of F contains a subgroup G isomorphic to PSU(3, q), then F is the Hermitian curve. The proof is similar to that of Theorem 12.14. Since |G| > 8g 3 , case (iv.5) of Section 11.12 occurs. From Theorems 11.123 and 11.125, G has just one nontame orbit o of places, and its action on o is faithful and 2-transitive, the same as PSU(3, q) on its permutation representation of the classical unital. For a place P ∈ o, the following properties hold: (i) |GP | = q 3 (q − 1); (1)

(ii) GP is a group of order q 3 acting on o\{P} as a sharply transitive permutation group; (1) (iii) GP = GP ⋊ H, where H ∼ = Cq−1 ; (2)

(iv) F2 = ΣGP is rational by Theorem 11.78. The following lemmas can be proved using the same arguments as in the proofs of the analogous Lemmas 12.15 and 12.17. The necessary changes are as follows: involutory element 7−→ element of order p; n2 − 7 → q3 ; n 7−→ q 2 ;

2g − 2 = 2n0 (n − 1) − 2 7−→ 2g − 2 = q(q − 1) − 2. (2)

L EMMA 12.26 GP has order q and consists of all elements of order p in GP together with the identity. L EMMA 12.27 The smallest non-gap at P is q. Theorem 12.25 now follows from these two lemmas and Theorems 12.4 and 12.11. Since PGU(3, q) is particularly rich in subgroups, the corresponding quotient curves of Hq provide a large family of Fq2 -maximal curves. However, their explicit determination appears to be difficult, especially for non-tame subgroups. For the simplest case, where G has prime order, the following result is a complete description of the quotient curves.

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T HEOREM 12.28 Let Hq /G be the quotient curve of Hq with respect to a subgroup G of PGU(3, q) of prime order d. (i) One of the following holds: (a) d = 2 6= p; (b) d = p; (c) d ≥ 3 and (q 2 − 1)(q 2 − q + 1) ≡ 0 (mod d). (ii) The curve F ′ = Hq /G is birationally Fq2 -equivalent to one the following cases of irreducible plane curves of genus g ′ defined over Fq2 . (I) F ′ = v(Y q + Y − X (q+1)/2 ), with d = 2, p > 2, g ′ = 41 (q − 1)2 . Ph i (II) (a) F ′ = v( i=1 Y q/p + ωX q+1 ), with d = p, q = ph , g ′ = 21 q(q/p − 1), where ω q−1 = −1; Ph i (b) F ′ = v(Y q + Y − ( i=1 X q/p )2 ), h ′ with d = p, q = p , p ≥ 3, g = q(q − 1)/(2p).

(III) F ′ = v(Y q − Y X 2(q−1)/d + ωX (q−1)/d ), with d ≥ 3, q ≡ 1 (mod d), g ′ = q(q − 1)/(2d), where ω q−1 = −1 .

(IV) (a) F ′ = v(Y q + Y − X (q+1)/d ), with d ≥ 3, q ≡ −1 (mod d), g ′ = 21 (q − 1)((q + 1)/d − 1);

(b) Fi′ = v(X i(q+1)/d + X (i+1)(q+1)/d + Y q+1 ), with d ≥ 3, q ≡ −1 (mod d), g ′ = (q + 1)(q − 2)/(2d) + 1, 1 ≤ i ≤ d − 1.

(V) Fi′ = v(S(X q/d , Y 1/d , X 1/d y q/d )), where Q S(X, Y, Z) = (βX + β q Y + Z), with β ranging over all d-th roots of unity, d ≥ 3, (q 2 − q + 1) ≡ 0 (mod d), g ′ = 12 ((q 2 − q + 1)/d − 1).

(iii) In (IV) (b) two such curves, Fi′ and Fj′ , are birationally Fq2 -equivalent if and only if one of the following relations holds modulo d : i ≡ j, ij ≡ 1, ij + i + j ≡ 0 , i + j + 1 ≡ 0, ij + i + 1 ≡ 0, ij + j + 1 ≡ 0 . The number of birationally Fq2 -equivalent classes of curves Fi is ( 1 for d ≡ 2 (mod 3) , 6 (d + 1) n(d) = 1 for d ≡ 1 (mod 3) . 6 (d − 1) + 1

(12.32)

(12.33)

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12.4 CURVES WITH REE AUTOMORPHISM GROUP In this section, p = 3, q = 3q02 , with q0 = 3s , s ≥ 1. Let F = v(F ) be the irreducible plane curve defined over Fq with 2

F (X, Y ) = Y q − [1 + (X q − X)q−1 ]Y q + (X q − X)q−1 Y − X q (X q − X)q+3q0 . (12.34) The Deligne–Lusztig curve associated with the Ree group or DLR curve, for short, is any algebraic curve birationally isomorphic to F. The function field of F is Σ = K(x, y) with F (x, y) = 0. To obtain a simpler description of Σ, let q  q y −y 0 (12.35) − xq0 (xq − x), y1 = xq − x  q q y −y 0 q0 y2 = x (12.36) − x2q0 (xq − x) − y q0 . xq − x Hence

y1q − y1 = xq0 (xq − x), y2q − y2 = xq0 (y1q − y1 ).

(12.37) (12.38)

Also, y = xy13q0 − y23q0 . Therefore Σ = K(x, y1 , y2 ). The relevant properties of F are collected in the following two theorems. T HEOREM 12.29 If F is the curve (12.34), P∞ = (0, 0, 1), and P∞ is the place associated to the branch centred at P∞ , then the following properties hold: (i) P∞ is a q0 -fold point, it is the unique infinite point of F, and is the centre of just one branch of F; (ii) div(dx) = (3q0 (q − 1)(q + q0 + 1) − 2)P∞ ; (iii) F has genus g = 23 q0 (q − 1)(q + q0 + 1); (iv) the set F(Fq ) of all Fq -rational places of F has size q 3 + 1; (v) the K-automorphism group G of F is Fq -rational and is isomorphic to the Ree group Ree(q); (vi) G acts on F(Fq ) as Ree(q) in its unique 2-transitive permutation representation on the Kantor ovoid; (vii) |G| = (q 3 + 1)q 3 (q − 1) and |GP∞ | = q 3 (q − 1). Proof. (ii) The rational subfield K(x) of Σ = K(F) is regarded as the function ′ field of the line ℓ = v(X). Then [Σ : K(x)] = q 2 . Let P∞ be the unique place of K(x) centred at the infinite point of ℓ. By Lemma 7.20, some places of Σ lie over P ′ . If P∞ is one of them, by (7.1), e∞ = −eP , where e∞ denotes the ramification index of P. From (7.3), e∞ ≤ q 2 . On the other hand, from Lemma 12.30 (ii) (I), ordP∞ (w8 ) = −(1 + q0 −1 + 2q −1 + (q0 q)−1 + q −2 )e∞ .

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Since ordP∞ (w8 ) is an integer, so e∞ = q 2 . This together with (7.3) implies that ′ P∞ is the unique place of Σ over P∞ . Also,

ordP∞ x = −q 2 , ordP∞ y1 = −q(q + q0 ), ordP∞ y2 = −q(q + 2q0 ). (12.39)

Now, let P be any place of Σ other than P∞ . Denote by P ′ the place of K(x) lying under P. Let P ′ = (u, 0) be the centre of P ′ . Choose a primitive representation σ of P. For every a, b ∈ Fq , let σ(a,b) be the K-isomorphism from Σ into K((t)) defined by the relations σ(a,b) (x) = σ(x),

σ(a,b) (y1 ) = σ(y1 ) + a,

σ(a,b) (y2 ) = σ(y2 ) + b.

Then σ(a,b) is also a primitive place representation. By (12.37), (12.38), the corresponding place P(a,b) of Σ lies over P ′ . Note that P = P(0,0) . Since [Σ : Fq ] = q 2 , this shows that there are q 2 places of Σ over every place of K(x). Therefore no place of Σ distinct from P∞ ramifies. From (7.1) applied to ξ = x − u, it follows that ordP (x − u) = 1. Hence x − u is a local parameter of Σ at P. For u 6= 0, σ(x) = σ(y1 ) = σ(y2 ) =

u + t, v + uq0 t − (uq − u)tq0 + tq0 +1 + · · · , w + u2q0 t + uq0 (uq − u)tq0 − uq0 tq0 +1 + · · · ,

(12.40)

with v q − v = uq0 (uq − u),

wq − v = u2q0 (uq − u),

(12.41)

while, for u = 0, σ(x) = σ(y1 ) = σ(y2 ) =

t, v + tq0 +1 + · · · , w + t2q0 +1 + · · · ,

(12.42)

with v, w ∈ Fq . From parts (G) and (I) of Lemma 12.30, ζ = w6 /w8 is a local parameter at P∞ . Hence ζ is a separable variable, and it defines a differential dξ for every ξ ∈ Σ. Again from (G) and (I) of Lemma 12.30, dw6 = w43q0 dx,

dw8 = w73q0 dx.

Since 1 = d(ζ) = d



w6 w8



=

w6 dw8 − w8 dw6 w6 w73q0 − w8 w43q0 = dx, w82 w82

it follows that dx =

w82 w6 w73q0

− w8 w43q0

.

From parts (D) and (G) of Lemma 12.30, ordP∞ (w6 w73q0 − w8 w43q0 ) = ordP∞ (w6 w73q0 )

= −3(q0 q 2 + q 2 + 2q0 q + q + q0 ), ordP∞ w8 = −(q 2 + 3q0 q + 2q + 3q0 + 1).

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Hence ordP∞ (dx) = 3q0 q 2 + q 2 − q − 3q0 − 2. By (12.40) and (12.42), ordP (dx) = 0 for every P = 6 P∞ . Therefore (ii) holds. From this and Definition 5.55, (iii) follows. As Σ = K(x, y1 , y2 ), the point Q = (1, x, y1 , y2 ) defines a model of Σ given by an irreducible curve Γ of PG(4, K). Note that Γ is defined over Fq . From (12.37) to (12.42), Γ consists of the point P∞ = (0, 0, 0, 1) together with all points P(u,v,w) = (1, u, v, w) where u, v, w ∈ K satisfy (12.41). The point P∞ is the unique singular point of Γ; more precisely, P∞ is a (q0 q)-fold point, and it is the centre of a unique branch of Γ, namely that arising from the place P∞ of Σ. Also, the Fq -rational places of Γ are those centred at points with coordinates in Fq , that is, either at P∞ or at P(u,v,w) with u, v, w ∈ Fq . This proves (iv). A computation shows that, if a, b, c, d ∈ Fq with a 6= 0, then the birational transformation φ(a,b,c,d) given by x 7→ ax + b, y1 7→ abq0 x + aq0 +1 y1 + c, y2 7→ ab3q0 x − aq0 +1 bq0 y1 + a3q0 +1 y2 + d,

(12.43)

of Σ is a K-automorphism of Σ. Also, φ(1,b,c,d) φ(1,b′ ,c′ ,d′ ) = φ(1,b′′ ,c′′ ,d′′ ) with b′′ = b + b′ ,

c′′ = c + c′ ,

d′′ = b′3q0 b − bq0 c + d′ + d.

These K-automorphisms form a subgroup Φ of AutK (Σ) of order q 3 (q − 1). Also, Φ1 = {φ(1,b,c,d) | b, c, d ∈ Fq }, Φ3 = {φ(1,0,0,d) | d ∈ Fq },

Φ2 = {φ(1,0,c,d) | c, d ∈ Fq }, Ψ = {φ(a,0,0,0) | a ∈ Fq }

are subgroups of Φ; their orders are |Φ1 | = q 3 ,

|Φ2 | = q 2 ,

|Φ3 | = q,

|Ψ| = q − 1,

and Φ2 is the commutator subgroup of Φ1 . In addition, the set of cubes in Φ1 is its centre Φ3 . Also, Φ = Φ1 ⋊ Ψ. A further K-automorphism of Σ is the birational transformation θ as follows: x 7→ w6 /w8 ,

y1 7→ w10 /w8 ,

y2 7→ w9 /w8 .

(12.44)

Also, θ is an involution. As a consequence, a primitive representation τ of the place P∞ is τ (x) = σ(w6 /w8 ),

τ (y1 ) = σ(w10 /w8 ),

τ (y2 ) = σ(w9 /w8 ),

(12.45)

where σ is given by (12.42). These K-automorphisms of Σ all fix F(Fq ). Also, every φa,b,c,d fixes P∞ , and Φ1 acts on the remaining q 3 places in F(Fq ) as a sharply transitive permutation group. The map θ interchanges P∞ and the place P(0,0,0) corresponding to the branch centred at P(0,0,0) . Therefore the group G generated by Φ and θ acts on F(Fq ) as a 2-transitive permutation group.

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To prove (v), the first step is to show that this permutation representation of G (0) on F(Fq ) is faithful. Since |Φ1 | = q 3 and Φ is a subgroup of GP∞ , Theorem (1)

11.78 together with (iii) imply that no non-trivial element of GP has more than one fixed place. Hence the kernel of the representation is a tame subgroup of GP∞ . By Theorem 11.60, this subgroup is cyclic, that is, generated by an element α. From Lemma 11.106, if α were non-trivial, then it would have at most 2g + 2 fixed places while (iii) implies that 2g + 2 < q 3 . This contradiction shows that the kernel is trivial. The possibilities for the structure of G are given by Theorem A.17, and it remains to show that only the case G = Ree(q) can occur. If G had a regular normal subgroup, then q 3 + 1 = 2m for some m. But this equation has no solution, as q is a power of 3. Also, PSL(2, q) has an abelian Sylow 3-subgroup, and the same holds for PGL(2, q). So, the final step in the proof of (v) consists in showing that G cannot contain PSU(3, q). Since |PSU(3, q)| = q 3 (q 3 + 1)(q 2 − 1)/µ with µ = gcd(3, q + 1), it follows from (iii) that |G| > 8g 3 . From Theorem 11.127, Σ is the Hermitian function field of genus 21 q(q − 1). But this is a contradiction, as this value is different from the genus of Σ given by (iii). Finally, G is Fq -rational as it is generated by Fq -rational K-automorphisms. 2 L EMMA 12.30

(i) With x, y1 , y2 ∈ Σ as in (12.35), (12.36),

ordP∞ (y1 ) = −(1 + (3q0 )−1 )e∞ ,

ordP∞ (y2 ) = −(1 + 2(3q0 )−1 )e∞ .

(ii) Define the further functions on Σ : w1 = x3q0 +1 − y13q0 , w2 = xy13q0 +1 − y23q0 , w3 = xy23q0 − w13q0 , w4 = xw23q0 − y1 w13q0 , v = xw3q0 − y2 w1q0 , w5 = y1 w3q0 − y2 w2q0 ,

w6 = v 3q0 − w23q0 + w43q0 , w7 = y1 w2q0 − xw3q0 − w63q0 , w8 = w53q0 − xw73q0 ,

w9 = w2q0 w4 − y1 w6q0 , w10 = y2 w6q0 − w3q0 w4 . Then

(a) w1q0 = xq0 +1 − y1 , w1q − w1 = x3q0 (xq − x), ordP∞ (w1 ) = −(1 + q0 −1 )e∞ ;

(b) w2q0 = xq0 y1 − y2 , w2q − w2 = y13q0 (xq − x), ordP∞ (w2 ) = −(1 + q0 −1 + q −1 )e∞ ;

(c) w3q0 = xq0 y2 − w1 , w3q − w3 = y23q0 (xq − x), ordP∞ (w3 ) = −(1 + q0 −1 + 2q −1 )e∞ ;

(d) w43q0 = x3q0 w2 − y13q0 w1 , w4q − w4 = w2q0 (xq − x) − w1q0 (y1q − y1 ), ordP∞ (w4 ) = −(1 + 2(3q0 )−1 + q −1 )e∞ ; (e) v 3q0 = x3q0 w3 − y23q0 w1 , v q − v = w3q0 (xq − x) − w1q0 (y1q − y1 ), ordP∞ (v) = −(1 + q0 −1 + q −1 )e∞ ;

(f) w53q0 = y13q0 w3 − y23q0 w2 , w5q − w5 = w3q0 (y1q − y1 ) − w2q0 (y2q − y2 ), ordP∞ (w5 ) = −(1 + q0 −1 + q −1 + (3q0 q)−1 )e∞ ;

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(g) w6q0 = −w2 + xq0 w4 , w6q − w6 = w43q0 (xq − x), ordP∞ (w6 ) = −(1 + q0 −1 + 2q −1 + (q0 q)−1 )e∞ ;

(h) w73q0 = y13q0 w2 −x3q0 w3 −w6 , w7q −w7 = w2q0 (y1q −y1 )−w3q0 (xq −x), ordP∞ (w7 ) = −(1 + 2(3q0 )−1 + q −1 + (3q0 q)−1 )e∞ ; (i) w8q0 = w5 + xq0 w7 , w8q − w8 = w73q0 (xq − x), ordP∞ (w8 ) = −(1 + q0 −1 + 2q −1 + (q0 q)−1 + q −2 )e∞ ;

(j) w93q0 = w2 w43q0 − y13q0 w6 , w9q − w9 = w2q0 (w4q − w4 ) − w6q0 (y1q − y1 ), ordP∞ (w7 ) = −(1 + q0 −1 + 2q −1 + (3q0 q)−1 )e∞ ;

3q0 q (k) w10 = y23q0 w6 −w3 w43q0 , w10 −w10 = w6q0 (y2q −y2 )−w3q0 (w4q −w4 ), ordP∞ (w10 ) = −(1 + q0 −1 + 2q −1 + (3q0 q)−1 )e∞ .

Proof. This is by straightforward calculation.

2

A major result on the DLR curve is the following characterisation, analogous to Theorem 12.14. T HEOREM 12.31 Let p = 3, q = 3q02 with q0 = 3s , s ≥ 1. If F is a curve of genus g = 32 q0 (q − 1)(q + q0 + 1) such that the K-automorphism group of F contains a subgroup G isomorphic to Ree(q), then F is the DLR curve. As in the proof of Theorem 12.14, a series of preliminary results are first established. L EMMA 12.32 There is a place P of Σ such that the ramification groups of GP have the following properties: (1)

(1)

(2)

(i) |GP | = q 3 , and ρ1 = 2, that is, GP 6= GP ; (1)

(2)

(ii) GP is the commutator group of GP , is an elementary abelian group of order q 2 , and ρ2 = 3q0 + 2, that is, (2)

(2)

(3q0 +2)

(3)

UP = GP = GP = · · · = GP

;

(1)

(3q +3)

is the centre of GP , is an elementary abelian group of order q, (iii) GP 0 and ρ3 = q + 3q0 + 2, that is, (3)

(3q0 +3)

UP = GP (4)

(q+3q0 +3)

(iv) UP = GP (i)

(3q0 +4)

= GP

(q+3q0 +2)

= · · · = GP

;

= {1}; (3q0 +3)

(v) ΣGP is rational for 0 ≤ i ≤ 3q0 + 2, and ΣGP

has genus 23 q0 (q − 1).

Proof. A Sylow 3-subgroup S3 of G has order q 3 . Since 3g < q 3 and the 3-rank of F is at most g, Theorem 11.84 implies that the 3-rank of F is zero. From Lemma 11.129, S3 fixes a place P of Σ and no non-trivial element in S3 fixes a (1) place distinct from P. In particular, S3 = GP for some place P. Since G has q 3 + 1 Sylow 3-subgroups, the set Ω of all places fixed by some 3-element of G

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has size q 3 + 1, and G acts on Ω as Ree(q) in its natural 2-transitive permutation representation. (1) From Theorem 11.78, ΣGP is rational. This implies that ΣGP also rational. The normaliser of S3 in G coincides with GP , and hence GP = S3 ⋊ H with H∼ = Zq−1 . Theorem 11.72 applied to GP gives (2)

2(g − 1) ≥ q 3 (2g1 − 2) + 2q 3 − 2 + |GP | − 1, (1)

where g1 denotes the genus of ΣGP . Since g < q 3 , this implies that g1 = 0. Hence (1) (2) GP 6= GP , that is, ρ1 = 2. From Lemma 11.82, (i)

(i+1)

(q − 1) | (ρi − 1)(ni − 1).

(12.46) (2)

|. For i = 1, this gives n1 = q; that is, |UP | = q 2 . where ni = |UP |/|UP (1) (2) By Lemma 11.75 (ii), GP contains the commutator subgroup of GP . On the other hand, the commutator subgroup of a 3-subgroup of Sylow of Ree(q) is ele(2) (1) mentary abelian of order q 2 . Therefore GP is the commutator subgroup of GP . (1) (2) The genus g2 of GP is calculated from Theorem 11.72 applied to GP regarded (2) as a K-automorphism group of ΣGP : 2g2 − 2 = −2q 2 + 2(q 2 − 1) + ρ1 (q − 1),

which gives g2 = 0. Further, ρ2 ≡ 2 (mod 3). This follows from Lemma 11.75 (iv) and the fact that ρ1 = 2. The congruence ρ2 ≡ 2 (mod 3) and the equation P (i) 2g = (ρ2 − 2)(q 2 − 1) + i≥3 (ρi − ρi−1 )(|UP | − 1), (12.47) which is actually a consequence of (11.40), provide the bound ρ2 ≤ 3q0 + 2. On the other hand, Lemma 11.82 (iii) shows that q | (ρ2 − 2)2 , whence 3q0 | (ρ2 − 2). Thus ρ2 = 3q0 + 2. From Lemma 11.82 (ii), (q − 1) | (3q0 + 1)(n2 − 1), which (3) implies that n2 = q. Therefore |UP | = q. (1) Since the centraliser of H in GP intersects GP trivially, every normal subgroup of GP contains at least q − 1 non-trivial elements. In particular, the last non-trivial (3) (4) ramification group of GP has order at least q. Therefore |UP | = {1}, and UP (1) coincides with Z(GP ). Now (12.47) reads as follows: 2g = (ρ2 − 2)(q 2 − 1) + (ρ3 − ρ2 )(q − 1).

(12.48)

It follows that ρ3 = q + 3q0 + 2. (3) Finally, the genus g3 of ΣUP can be calculated by applying Theorem 11.72 to (3) (2) UP regarded as a K-automorphism group of ΣUP : The result is that g3 =

2(g3 − 1) = −2q + ρ2 (q − 1).

3 2 q0 (q

− 1).

2

To find generators of Σ with the desired polynomial relations, the first step consists in showing the existence of elements x, y1 ∈ Σ satisfying (12.37) such that

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(3n +3) GP 0

Fq (x, y1 ) is K-isomorphic to Σ = Σ . Next, an element y2 ∈ Σ is found satisfying both Σ = K(x, y1 , y2 ) and (12.38). Finally, it is shown that G preserves Fq (x, y1 , y2 ). Let P∞ denote a place of Σ satisfying the properties listed in Lemma 12.32. The arguments used in the proofs of Lemmas 12.17 12.18 and 12.19 can be adapted to prove analogous results on Σ. The necessary changes are as follows: q 2 7→ q 3 ;

q 7→ q 2 ;

2g − 2 = 2q0 (q − 1) − 2 7→ 2g − 2 = 3q0 (q − 1)(q + q0 + 1) − 2 = (q + 1)(q + 3q0 + 1)(3q0 − 2);

q 2 + 1 = (q + 2q0 + 1)(q − 2q0 + 1) 7→ q 3 + 1 = (q + 1)(q + 3q0 + 1)(q − 3q0 + 1);

q + 2q0 + 1 7→ (q + 1)(q + 3q0 + 1). L EMMA 12.33

(i) The smallest non-gap at P∞ is q 2 .

(ii) There exists x ∈ Σ such that Pq 2 (a) div x = −q 2 P∞ + i=1 Pi ; (2)

(b) K(x) = UP∞ .

L EMMA 12.34 For α ∈ GP∞ ,  (2) (1)  for α ∈ GP∞ \UP∞ ,  x + a, a ∈ K\{0}, α(x) = bx, bq−1 = 1, b 6= 1, for α ∈ H,   (2) x, for α ∈ UP∞ .

(12.49)

L EMMA 12.35 There exists w ∈ Σ such that

div(w)∞ = (q + 1)(q + 3q0 + 1)P∞ . By Lemma 12.35, the smallest non-gap m at P which is not divisible by q does not exceed (q + 1)(q + 3q0 + 1). From Lemma 12.32, ρ1 = 2, ρ2 = 3q0 + 2, ρ3 = q + 3q0 + 3. This together with Lemma 11.82 gives the following result. (3)

L EMMA 12.36 Let w be as in Lemma 12.35. If α is a non-trivial element in UP∞ and w′ = α(w) − w, then div(w′ )∞ = q(q + 3q0 + 1)P∞ .

L EMMA 12.37 Let y1 ∈ Σ\K(x1 ) with div(y1 ) = kP∞ , such that k is minimal and k ≥ 1. (i) If β is a generator of H, then β(y1 ) = bβ y1 for a primitive (q − 1)-st root of unity bβ . (3)

(ii) y1 ∈ ΣUP∞ .

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Proof. (i) If z ∈ Σ\K(x) with div(z) = kP∞ , then, under β, x 7→ bx, for a primitive (q − 1)-st root of unity b ∈ K; z 7→ uβ x + vβ y1 + wβ , uβ , vβ , wβ ∈ K, uβ 6= 0.

(12.50)

A calculation shows that there exist vβ′ , wβ′ ∈ K for which y1 = z + vβ′ x + wβ′ satisfies (i). (ii) Since y1 6∈ K(x1 ), from Lemma 12.36, q 2 < k ≤ (q + 1)(q + 3q0 + 1). For (1) (2) α ∈ GP∞ \GP∞ , x 7→ x + bα , bα ∈ K\{0}; y1 7→ y1 + dα x + eα , dα , eα ∈ K, dα 6= 0, (1)

(12.51) (2)

whence α3 (y1 ) = y1 . Since α3 , with α ranging over GP∞ \GP∞ , produces all

elements of

(1) Z(GP )

=

(3) UP∞ ,

so (ii) follows.

2

From now on, let x and y1 have the properties stated in Lemmas 12.33, 12.34 (2) (3) and 12.37. Choose an element α from UP∞ . Since UP∞ is a normal subgroup of (3)

(2)

(2)

UP∞ , so α induces a K-automorphism of ΣUP∞ fixing every element in ΣUP∞ . By Lemma 12.32 (i), the latter function field is rational. From Lemma 11.14, it follows that α(y1 ) = y1 + aα with aα ∈ K\{0}. Replacing y1 by a−1 α y1 , it may be supposed that α(y1 ) = y1 + 1. (3)

P ROPOSITION 12.38 The field ΣUP = K(x, y1 ), and y1q − y1 = xq0 (xq − x).

Proof. Let αj = β j αβ −j for j = 1, . . . , q − 1. From (i) of the previous lemma, αj (y1 ) = y1 + bj . Since bq−1 = 1, this shows that αj (y1 ) = y1 + bj with bj ∈ Fq . (3) (2) Since |UP∞ /UP∞ | = q, every non-zero element in Fq is obtained exactly once in this way. Putting b0 = 0, this implies that Qq−1 q j=0 (y1 − bj ) = y1 − y1 ∈ Fq (x). (3)

From the definition of x and Lemma 12.37 (ii), K(x, y1 ) is a subfield of ΣUP∞ . On the other hand, (2)

[ΣUP∞ : K(x, y1 )] = q,

(2)

(3)

|UP∞ /UP∞ | = q.

(3)

Therefore K(x, y1 ) = ΣUP∞ . Since y1 is assumed to have no pole other than P∞ , this implies that y1q − y1 = f (x),

f (X) ∈ Fq [X].

Put r = deg f (X); then r = m/q as q ordP∞ (y1 ) = −r eP∞ . Now, since q 2 < m ≤ (q + 1)(q + 3q0 + 1),

it follows that q < r ≤ q + 3q0 + 2. Therefore Pr q < r ≤ q + 3q0 + 2. y1q − y1 = i=0 ai xi ,

(12.52)

Without loss of generality, let ar = 1. Let b and bβ be as in Lemma 12.37. Applying the map β, in which x 7→ bx, y1 7→ bβ y1 , to (12.52) gives bβ ai = bi bi

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for i = 0, 1, . . . , r. Note that bβ = br as ar = 1. Thus, (br − bi )ai = 0 for i = 0, 1, . . . , r. Since ai = 0 for i neither r − q + 1 nor r, (12.52) can be written as follows: y1q − y1 = xr + cxr−q+1 , (1)

r ∈ Fq .

(2)

(12.53)

Choose an element α from GP∞ \GP∞ ; then (12.51) holds. Applying α to (12.53) gives the following: y1q − y1 + dα (xq − x) = (x + eα )r + c(x + eα )r−q+1 .

(12.54)

y1q

From this, c = −1. Now, replace − y1 in (12.54) by xr + cxr−q−1 and use the binomial expansion. Then (12.54) becomes the following:
P r−q i r−1−i xr−q + dα = xr − q + er−1 + r−q−1 . α i=1 i x eα This implies that

r−q i




≡0

(mod 3),

i = 1, . . . , r − q − 1;

that is, r − q is a power of 3. From (12.52), 1 ≤ r − q ≤ 3q0 . From (12.53), the only possibility is r − q = q0 as m is chosen to be minimal. Also, the genus of (3)

K(x, y1 ) must be 32 q0 (q − 1), the genus of ΣUP∞ .

2

Using the arguments in the proof of Lemma 12.37 the following result can be obtained. (3)

L EMMA 12.39 Let y2 ∈ Σ\ΣUP∞ for which div y2 = kP∞ with k ≥ 1 and minimal. Then y2 can be chosen such that β(y2 ) = cβ y2 ,

cβ ∈ Fq \{0}.

In this lemma, it follows from Lemma 12.36 that q(q + q0 ) < k ≤ q(q + 3q0 + 1),

(3) UP∞

as w′ 6∈ Σ . Proof of Theorem 12.31. So far, the following results have been shown. The cyclic group H is generated by b and, for a primitive element b of Fq , the map β acts as follows: x 7→ bx,

while

(1) GP∞

y1 7→ bq0 +1 y1 , 3

y2 7→ bs y2 ,

for 1 ≤ s ≤ q − 1,

(12.55)

consists of q elements α that act as follows:

x 7→ x + bα ,

y1 7→ y1 + bqα x + eα

y2 7→ y2 + fα y1 + gα x + hα ,

(12.56)

where bα , eα , fα , gα , hα ∈ Fq . Note that α ∈ GP∞ if and only if bα = 0. As the (3) (2) (1) cube of any element in GP∞ \GP∞ belongs to UP∞ , the necessary and sufficient (3)

condition for α to be in UP∞ is that it acts as follows: x 7→ x,

y1 7→ y1 ,

y2 7→ y2 + u,

for u ∈ Fq .

Now, the proof proceeds. By (12.57), (3) Q q UP∞ . u∈Fq (y2 − u) = y2 − y2 ∈ Σ

(12.57)

(12.58)

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Since K(x, y1 , y2 ) is a subfield of Σ, this implies that Σ = K(x, y1 , y2 ) where x, y1 , y2 satisfy (12.37),(12.38) and (12.58). The unique involutory element in H is β ′ = β (q−1)/2 . By (12.55), its action is the following: x 7→ −x, y1 7→ y1 , y2 7→ vy2 ; here either v = 1 or v = −1. Using (12.56), a calculation shows that β ′ commutes (1) with ψ ∈ GP∞ if and only if ψ has the following action: x 7→ x, y1 7→ y1 + u for u ∈ Fq , y2 7→ y2 . (2)

(1)

In particular, the centraliser C of ψ in GP∞ has order q and is a subgroup of GP∞ . (3)

(2)

(3)

(2)

Since C ∩ UP∞ = {1}, this implies that GP∞ = C × UP∞ . Hence GP∞ consists of all τ ∈

Therefore

(1) GP∞

acting as follows: x 7→ x, y1 7→ y1 + u,

y2 7→ y2 + w,

for u, w ∈ Fq .

Q

− w) = y2q − y2 ∈ Fq (x). The assumption that y2 has no pole other than from P∞ implies that y2q − y2 = f (x), with f (X) ∈ Fq [X]. Arguing as before, put t = deg f (X). Then t = k/q as q ordP∞ (y1 ) = −t eP∞ . Since q 2 < k ≤ (q + 1)(q + 3q0 + 1), so q < t ≤ q + 3q0 + 2. Therefore Pt (12.59) y2q − y2 = i=0 bi xi , bi ∈ Fq , q < t ≤ q + 3q0 + 2. Without loss of generality, let bt = 1. Also, let b and bβ be as in Lemma 12.37. Applying the map β, with x 7→ bx, y2 7→ bsβ y2 , to (12.52) gives bβ bi = bi bi for i = 0, 1, . . . , t. Note that bβ = bt as br = 1. Thus (br − bi )bi = 0 for i = 0, 1, . . . , t. As bi = 0 for i 6= t − q + 1, t, so (12.59) can be written as follows: y2q − y2 = xt + cxt−q+1 , c ∈ Fq . (12.60) w∈Fq (y2

(1)

(2)

As above, choose an element α from GP∞ \GP∞ ; then (12.51) holds. Applying α to (12.60) gives the equation (12.61) y2q − y2 + dα (y1q − y1 ) + eα (xq − x) = (x + fα )s−q (xq − x). From this, c = −1. In (12.61), replace y2q − y2 by xs + cxs−q+1 and y1q − y1 by xq0 (xq − x), and use the binomial expansion. Then (12.54) becomes the following: Pt−q−1 t−q
i t−1−i . (12.62) xt−q + dα xq0 + eα = xt−q + fαt−1 + i=1 i x eα t−q In particular, eα = fα . Suppose that dα = 0. Then (12.62) implies that
s−q ≡ 0 (mod 3), i = 1, . . . , r − q − 1; i that is, r − q is a power of 3. Since, q0 ≤ s − q ≤ 3q0 + 1, this yields that (3) r − q = 3q0 . However, in this case, y2 ∈ UP∞ , a contradiction. Therefore dα 6= 0. Now, from (12.62), modulo 3,

t−q t−q 6≡ 0, ≡ 0 for i 6= q0 , 1 ≤ t − q − 1, i q0 showing that t − q = sq0 with s not divisible by 3. Since q0 < t − q ≤ 3q0 + 1, and t − q is distinct from both q0 and 3q0 , the desired result, that t − q = 2q0 , follows. 

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SOME FAMILIES OF ALGEBRAIC CURVES

585

12.5 A CURVE ATTAINING THE SERRE BOUND In this section, K = F2 . Consider the curve C = v(X018 X25 + X019 X2 + X123 )

(12.63)

together with its function field Σ = K(x, y), where y 5 + y 4 = x23 .

(12.64)

T HEOREM 12.40 The curve C has the following properties: (i) it is an irreducible plane curve of genus 11 with exactly two singular points, the 18-fold point Y∞ = (0, 0, 1) and the 4-fold point O = (1, 0, 0), each the centre of a unique branch of C; (ii) if P∞ and P0 are the associated places of Σ, then

div(dx) = 26P0 − 6P∞ ;

(12.65)

(iii) with z = y −1 , x0 = z, x1 = xz, x2 = x2 z, 3 5 2 6 2 x3 = x z, x4 = x z , x5 = x z , x6 = x7 z 2 , x7 = x8 z 2 , x8 = x11 z 3 , x9 = x12 z 3 , x10 = x17 z 4 , the canonical series of Σ is P |dx| = {div( 10 i=0 ci xi ) + 26P0 − 6P∞ | c = (c0 , . . . , c11 ) ∈ PG(11, K)}; (12.66) (iv) the K-automorphism group of C contains a cyclic group of order 23. Proof. The arguments and calculations are similar to those in the proof of Lemma 12.1. The point Y∞ is the centre of a unique branch of C with a primitive representation, (x(t) = t−5 + · · · , y(t) = t−23 + · · · ). From this, ordP∞ (dx) = ordt (dx/dt) = −6.

(12.67)

The other singular point of C is the origin O; the unique branch of C centred here has a primitive representation (x(t) = t4 + · · · , y(t) = t23 + · · · ).

From (12.64), y 4 dy = x22 dx. Since dy/dt = t22 + · · · , so

ordP0 (dx) = ordt (dx/dt) = 4 ordt y(t) − 22 ordt x(t) + ordt (dy/dt) = 26. (12.68) Since C has no more singular points, a first consequence is that every point of C is the centre of just one branch of C. Hence C is an irreducible plane curve and, for a power q of 2, a place of Σ is Fq -rational if and only if the corresponding branch of C is centred at a point with coordinates in Fq .

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Also, ordP (dx) 6= 0 only at these two places, P∞ and P0 . Therefore (12.67) and (12.68) give (12.65). From Definition 5.55, 2g − 2 = 26 − 6 = 20, whence g = 11. Let P1 be the place of Σ associated to the unique branch of C whose centre is the point A = (1, 1, 0). Then div x = 4P0 + P1 − 5P∞ ,

div y = 23P0 − 23P∞ .

From this, (12.66) follows. Finally, for a primitive twenty-third root of unity λ ∈ K, the birational transformation ω:

x′ = λx,

y ′ = λy

of Σ is a K-automorphism of order 23.

2

From (12.66), the gap sequences of Σ at the places P∞ , P1 and P0 are the following: P∞ :

(1, 2, 3, 4, 6, 8, 9, 11, 13, 16, 18);

P0 :

(1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 16).

P1 :

(1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 18);

Next, the Weierstrass points of Σ are determined. From (12.66), the canonical curve Γ of Σ is given by the point P = (x0 , x1 , . . . , x11 ) of PG(10, K) with the xi (i) as defined above. The 11×11 Wronskian matrix W = (Dx xj ) with 0 ≤ i, j ≤ 10 in Figure 12.1 can be calculated using the method illustrated in Example 5.81. In it, let fi = y i + y i−1 + · · · + y + 1, h3 = y 8 + y 3 + 1,

g4 = y 4 + y + 1,

h4 = y 8 + y 4 + 1.

From the matrix, det W = x17 (y 3 + y 2 + 1)(y 8 + y 7 + y 6 + y 5 + y 4 + y + 1) ×(y 35 + y 34 + y 32 + y 30 + y 29 + y 28 + y 27 + y 26 + y 19 +y 18 + y 15 + y 11 + y 8 + y 5 + y 4 + y 3 + 1)(y + 1)3 y −76 . Since W is not identically zero, the canonical series is classical; that is, ǫi = i for i = 0, 1, . . . , 10. Further, the zeros of W are exactly the Weierstrass points of Σ; their total number is 3 + (3 + 8 + 35) · 23 = 1061. The weights of the places P∞ , P0 , P1 in the ramification divisor R of Σ are as follows: ordP∞ W = 536,

ordP∞ (dx) = −6,

vP∞ (R) = 536 − 330 − 66 = 140;

eP∞ = −6,

ordP0 W = −1680, ordP0 (dx) = 26, eP0 = 26, vP0 (R) = −1680 + 1430 + 286 = 36; ordP1 W = 86, ordP1 (dx) = 0, vP1 (R) = 86 + 0 + 0 = 86.

eP1 = 0,

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SOME FAMILIES OF ALGEBRAIC CURVES

From (7.13),

587

P

vP (R) = 1320. Hence P1058 i=1 vP (R) = 1320 − (140 + 36 + 86) = 1058.

Since Σ has 1058 Weierstrass points other than P∞ , P0 , P1 , this is only possible when vP (R) = 1 for each of these 1058 places. Also, since the canonical series of Σ is classical, Theorem 7.55 shows that vP (R) = 1 if and only if the order sequence of Γ at the branch point associated to P is (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11). Therefore the gap sequence of Σ at any Weierstrass point P other than P∞ , P0 , P1 is (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12). This shows that each of the three places, P∞ , P0 , P1 , differs from any other place of Σ in its Weierstrass gap sequence. As a consequence, AutK (Σ) has three fixed places. From Theorem 11.72 applied to AutK (Σ), 20 ≥ −2|AutK (Σ)| + 3(|AutK (Σ)| − 1) = |AutK (Σ)| − 3, whence |AutK (Σ)| ≤ 23. By Proposition 12.40 (iv), |AutK (Σ)| = 23. It may be noted that |AutK (Σ)| = 23 = 2g + 1, which agrees with Remark 11.109 (ii) for m = 5, r = 1, n = 23. Calculating the number of solutions of the equation Y 4 (Y +1) = X 23 over F211 , it is found that the number of F211 -rational places on C is exactly 3039. Since √ 211 + 1 + 11⌊2 211 ⌋ = 3039, so Σ attains the Serre Bound. Now Theorem 9.28 can be applied to calculate the L-polynomial of Σ over F211 . These results are summarised in the following theorem. T HEOREM 12.41

(i) The number of Weierstrass points of Σ is 1061.

(ii) |AutK (Σ)| = 23. (iii) The number of F211 -rational points of C attains the Serre Bound, 3039. (iv) The L-polynomial of C over F211 is (1 + 90t + 2048t2 )11 .

12.6 NOTES Section 12.1 is based on Stichtenoth [424]. The properties of the curve C in Remark 12.12 are due to Sullivan [436]. The characterisation of the particular case is due to van deer Geer and van der Vlugt [473]. The Weierstrass points of curves of type (12.1) were investigated by Garcia and Voloch [160] in the following cases:

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CHAPTER 12

(i) A(Y ) = Y q + Y, B(X) = X m , m | q + 1, m > 2; (ii) A(Y ) = Y q + Y, B(X) = X q

i

+1

, i ≥ 1;

(iii) A(Y ) = Y 4 + Y 2 + Y, B(X) = X 7 + λX, λ ∈ K, p = 2. For a proof of Remark 12.6, see Lehr and Matignon [302]. Section 12.2 is based on [189], [134], [174]; see also Section 10.10 and [358]. For Theorems 12.22 and 12.23, see [174]. The explicit determination of quotient curves of Hq appears to be difficult, especially for non-tame subgroups; Garcia, Stichtenoth, and Xing [158] calculated the genera of many such quotient curves. For the simplest case, where G has prime order, the complete description of the quotient curves in Theorem 12.28 comes from [92]. Using the method developed by Garcia, Stichtenoth, and Xing [158] for the Her¨ mitian curve, C¸akc¸ak and Ozbudak determined many quotient curves of the DLR curve and their genera. Their papers [60], [61] also contains numerous results on the arithmetic and automorphism groups of these curves. Section 12.4 is based on [359] and [188]; see also [187]. Section 12.5 is based on [264]. A similar example for a curve of degree 12 is found in [263]. For more curves which are quotients of Fermat curves, see [485].

2

x2 y

x y

x2 y

x3 y2

x6 y2

x7 y2

x8 y2

x11 y3

x22 y6

1 y2

(y+1)x y2

x2 y2

x4 y2

0

x8 y2

0

x10 y4

x21 y7

(y+1)x22 y7

y 2 +y+1 y3

x y3

(y 2 +1)x3 y4

x4 y4

x5 y4

(y 2 +1)x6 y4

(y+1)x10 y4

x20 y8

21

(y+1)x y8

2

(y +1)x y8

27

1 y4

2

(y +1)x y4

2

x12 y3 (y+1)x y4

x17 y4 11

(y+1)x10 y4

0

x4 y4

0

x8 y4

(y+1)x y4

9

x16 y4

0 0

g4 x19 y 10

(y 4 +f2 )x20 y 10

(y 4 +y 3 +y+1)x21 y 10

f3 x22 y 10

(y 4 +y 2 +1)x y6

x2 y6

x3 y6

(y 2 +1)x4 y6

(y 4 +1)x7 y8

x8 y8

(y 4 +1)x13 y8

x18 y6

(y+1)x19 y 10

(y 2 +1)x20 y8

f3 x21 y 10

(y 4 +y 2 +1) y6

0

x2 y6

0

(y 4 +1)x6 y8

0

(y 4 +1)x12 y8

g4 x17 y 12

(y 8 +g4 )x18 y 12

(y 6 +f4 )x10 y 12

f3 x20 y 12

(y 5 +g4 )x22 y 12

1 y8

x y8

(y 2 +1)x2 y8

(y 4 +1)x5 y8

0

0

x16 y8

(y+1)x17 y 12

(y 2 +1)x18 y8

f3 x19 y 12

(y 5 +g4 )x21 y 12

0

1 y8

0

(y 4 +1)x4 y8

0

0

(y 4 +f2 )x15 y 15

(y 8 +y 7 +f2 )x16 y 15

(y 8 +y 7 +f4 )x17 y 15

(y 9 +y 8 +f4 )x18 y 15

(y 6 +1)f3 x20 y 16

(y 8 +1)x21 y 16 (y+1)

(y 8 +1)x22 y 16 (y+1)

h4 y 12

h3 x3 y 11

(y 8 +y 7 +1)x4 y 11

(y 4 +1)x9 y 12

(y 8 +1)x14 y 16

(y 9 +y 8 +f3 )x15 y 16

x16 y 16

(y 9 +y 8 +f5 )x17 y 16

(y 6 +1)f3 x19 y 16

0

(y 8 +1)x21 y 16 (y+1)

0

h4 x2 y 12

(y 8 +1)(y+1)x3 y 12

(y 4 +1)x8 y 12

(y 9 +1)x13 y 16

(y+1)x14 y8

f3 x15 y 16

f2 x16 y 15

(y 5 +g4 )x18 y 12

(y 8 +1)x19 y 16 (y+1)

(y 8 +1)x20 y 16 (y+1)

0

h3 x y 11

(y 8 +1)(y+1)x2 y 11

0

Figure 12.1 The Wronskian for the curve C

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

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6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 W =6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

1 y

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Chapter Thirteen Applications: codes and arcs Algebraic curves over a finite field are the basic structure in the theory of algebraicgeometry codes, which combines algebraic geometry and error-correcting codes. Here, only a brief exposition of the main construction, due to Goppa, and a few illustrative examples are presented in Section 13.1. Coding theory is also connected with algebraic curves via finite geometry, since complete arcs in PG(r, q) are the geometric counterpart of MDS (Maximum Distance Separable) codes, which are linear codes correcting the greatest number of errors with respect to their parameters. In the other sections, an account of this subject, especially on the Main Conjecture on MDS codes, is given. The key definitions using the terminology of linear algebra are the following. D EFINITION 13.1 (i) Let V (n, q) be the vector space on (Fq )n with the usual addition and scalar multiplication: (x1 , . . . , xn ) + (y1 , . . . , yn ) = (x1 + y1 , . . . , xn + yn ), t(x1 , . . . , xn ) = (tx1 , . . . , txn ), for t, xi , yi ∈ Fq . (ii) An [n, k, d]q code C is a k-dimensional subspace of V (n, q), such that the number of non-zero co-ordinates of any non-zero element of C is at least d, with precisely d for some element of C. (iii) A generator matrix for C is a k × n matrix whose rows form a basis for C. 13.1 ALGEBRAIC-GEOMETRY CODES In this section, K is the algebraic closure of Fq , and F is an algebraic curve defined over Fq . Assume that F has some Fq -rational points, that is, the set F(Fq ) is not empty. For an ordered set of distinct Fq -rational places P1 , . . . , Pn of K(F) associated to the Fq -rational points P1 , . . . , Pn , let D = P1 + · · · + Pn be the associated Fq -rational divisor. Further, for Q1 , . . . , Qs not-necessarilydistinct Fq -rational places of K(F), let Ps E = j=1 mj Qj ,

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APPLICATIONS: CODES AND ARCS

591

P

with mj ≥ 0 and mj = m. Suppose that the associated Fq -rational divisor E is disjoint from D; that is, Supp(D) ∩ Supp(E) = ∅. As in Section 6.4, L(E) is the Riemann–Roch space of E. Then the evaluation map θ at P is θ : L(E) −→ V (n, q), given by f 7−→ (f (P1 ), . . . , f (Pn )). Now, im θ = C = C(D, E) = (F, D, E)L is an algebraic-geometry code. If f1 , . . . , fk is a basis for L(E), then a generator matrix for C is   f1 (P1 ) . . . f1 (Pn )   .. .. G= . . . fk (P1 ) . . . fk (Pn )

T HEOREM 13.2 If n > m > 2g − 2, then (i) k = m − g + 1; (ii) d ≥ n − m.

Proof. (i) This is precisely part (iii) of Theorem 6.61. (ii) If f ∈ L(E) and w(θ(f )) = d, then n − d places Pi1 , . . . , Pin−d are zeros of f and their sum defines the divisor D′ , where deg D′ = n − d and D′ ≺ D. So div(f ) ≻ D′ − E, whence deg div(f ) ≥ deg D′ − deg E;

that is, 0 ≥ n − d − m.

2

C OROLLARY 13.3 With R = k/n, δ = d/n, (i) n − k + 1 − g ≤ d ≤ n − k + 1; (ii) R + δ ≥ 1 − (g − 1)/n. Proof. The upper bound in (i) is the Singleton bound for any linear code. The rest follows from the theorem. 2 Let C ⊥ , the dual of C(D, E), be an [n, k ⊥ , d⊥ ] code. It turns out that C ⊥ is also an algebraic-geometry code. C OROLLARY 13.4 If n > m > 2g − 2, then (i) k ⊥ = n − m + g − 1;

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(ii) d⊥ ≥ m − 2g + 2; (iii) n − k ⊥ + 1 − g ≤ d⊥ ≤ n − k ⊥ + 1. E XAMPLE 13.5 Consider the case that the curve F is the line v(X2 ), and write Fq = {t1 , . . . , tq }. Then the Fq -rational points P , the centres of the Fq -rational places P of F, are Pti = (1, ti , 0), i = 1, . . . , q, and P∞ = (0, 1, 0).

In particular, P0 = (1, 0, 0). Let E = mP∞ . For a generic point Q = (1, x, 0) of F, the divisor div xr = rP0 − rP∞ and L(E) contains the functions 1, x, . . . , xm ,

(13.1)

but not the function xm+1 . Also, D = Pt1 + · · · + Ptq . So n = q,

k = m + 1,

and



  G= 

d = n − k + 1 = n − m,

1 t1 .. .

1 t2 .. .

... ...

1 tq .. .

tm 1

tm 2

. . . tm q

    

1 x .. .

.

xm

This is a Reed–Solomon code and is MDS; see Section 13.2. E XAMPLE 13.6 As in Example 6.71, take q = 4 and F = v(X03 + X13 + X23 ). Let Q = (1, x, y) be a generic point of F. The Fq -rational points of F are given in Table 13.1. Let Si be the place associated with Si . Table 13.1 F4 -rational points of v(X03 + X13 + X23 )

0 1 1

0 1 ω

0 1 ω2

1 0 1

1 0 ω

1 0 ω2

1 1 0

1 ω 0

1 ω2 0

S0

S1

S2

S3

S4

S5

S6

S7

S8

Consider f1 = 1/(x + y) and f2 = x/(x + y). Then div f1 div f2

= =

(S0 + S1 + S2 ) − 3S0 = (S1 + S2 ) − 2S0 , (S3 + S4 + S5 ) − 3S0 .

(13.2)

With S1 , . . . , S8 as in Table 13.1, let E = 3S0 and D = S1 + . . . + S8 . The genus g = 1 and C(D, E) has parameters n = 8,

k = 3,

5 ≤ d ≤ 6.

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Also,



 1 1 1 1 1 1 1 1 1 0 1 ω 2 ω 1 ω 2 ω  1/ (x + y) G= 0 ω ω2 0 0 0 1 1 1 x/ (x + y) As the last row of G reveals a word of weight 5, so d = 5. Hence C is an [8, 3, 5]4 code.

E XAMPLE 13.7 As in the previous example, let q = 4 and F = v(X03 +X13 +X23 ). Here, let D = S0 + S1 + · · · + S8 . Also, with G = v(X0 X1 + X0 X2 + X1 X2 ), let E = F · G; that is, E is the intersection divisor of the curves F and G. So E is an F4 -rational divisor, even though no point in its support is F4 -rational. In fact, the six points of the divisor E are, with τ a primitive element of F64 , (τ 9i + 1, τ −9i + 1, 1), for i = 1, . . . , 6. As deg E = 6, so k = 6. Let v = x + y + xy. Then a basis for L(E) is the following: 1 x2 y 2 x y xy , , , , , . v v v v v v Projectively, with V = x0 x1 + x0 x2 + x1 x2 , this basis is as follows: x20 x21 x22 x0 x1 x0 x2 x1 x2 , , , , , . V V V V V V Evaluating the places Si at these functions of L(E) gives the generator matrix G below: S0 S1 S2 S3 S4 S5 S6 S7 S8 0 1 1 

0 1 ω

0 1 ω2

1 0 1

1 0 ω

1 0 ω2

1 1 0

1 ω 0

1 ω2 0

 0 0 0 1 ω2 ω 1 ω2 ω 1/v 2  2  1 ω2 ω 0 0 0 1 ω ω x /v   2  1 ω ω2 1 ω ω2 0 0  0 y /v  G=  0 0 0 0 0 0 1 1 1  x/v    0 0 0 1 1 1 0 0 0  y/v 1 1 1 0 0 0 0 0 0 xy/v Here, C is a [9, 6, 3]4 code. The dual C ⊥ is a [9, 3, 6]4 code. This follows from the geometry of the curve F. A generator matrix for C ⊥ is   0 0 0 1 ω ω2 1 ω ω2 0 1 ω2 ω  . H =  1 ω ω2 0 0 2 2 1 ω ω 1 ω ω 0 0 0 Here, the nine columns of H are the nine F4 -rational points of the cubic curve F. If the rows of H are h1 , h2 , h3 , then an element of C ⊥ is a1 h1 + a2 h2 + a3 h3 . So a coordinate of this codeword is zero if a1 x1 +a2 x2 +a3 x3 = 0 for the corresponding column t (x1 , x2 , x3 ). Any line of PG(2, 4) meets F in at most 3 points and some, namely 12, in precisely 3 points. Hence any line misses at least 6 points of F and 12 miss precisely 6; so d⊥ = 6.

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13.2 MAXIMUM DISTANCE SEPARABLE CODES Linear codes in V (n, q) and subsets of PG(k − 1, q) are connected as follows. D EFINITION 13.8 An [n, k] system is a multi-set of n points in PG(k − 1, q) with at most n − d in a hyperplane. R EMARK 13.9 Given a generator matrix G of an [n, k, d]q code, the columns form an [n, k] system of n points in PG(k − 1, q). The following four concepts are equivalent for n ≥ k: (1) (Coding theory) a maximum distance separable (MDS) linear code C of length n, dimension k and hence minimum distance d = n − k + 1, that is, an [n, k, n − k + 1] code over Fq ; (2) (Matrix theory) a k × (n − k) matrix A with entries in Fq such that every minor is non-zero; (3) (Vector space) a set K′ of n vectors in V (k, q), with any k linearly independent; (4) (Projective space) an n-arc in PG(k − 1, q), that is, a set K of n points with at most k − 1 in any hyperplane of the projective space of k − 1 dimensions over Fq . To show the equivalence, consider a generator matrix G for such a code C in canonical form:

k

    

1 0 0 1 .. .. . . 0 0

... 0 ... 0 .. .

n a11 a21 .. .

. . . 1 ak1

... ...

a1,n−k a2,n−k .. .

. . . ak,n−k



  = G.  

Since C has minimum distance n − k + 1, any linear combination of the rows of G has at most k − 1 zeros; that is, considering the columns of G as a set K′ of n vectors in V (k, q), any k are linearly independent. Regarding the columns of G as a set K of points of PG(n, q) means that no k lie in a hyperplane; equivalently, any k points of K are linearly independent. This, in turn, implies that every minor of A is non-zero. For given k and q, let M (k, q) be the maximum value of n for such a code. Then M (k, q) = k + 1 for q ≤ k. A suitable set of vectors in V (k, q) is (1, 0, . . . , 0), (0, 1, 0, . . . , 0), . . . , (0, . . . , 0, 1), (1, 1, . . . , 1); that is, for q ≤ k, every element of V (k, q) is a linear combination of at most k − 1 of these k + 1 vectors.

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R EMARK 13.10 Example 13.5 gives q points in PG(m, q) on a normal rational curve. The system can be extended to include the point (0, 0, . . . , 0, 1), the remaining rational point on the normal rational curve T = {(sm , sm−1 t, . . . , stm−1 , tm ) | s, t ∈ Fq }.

The code can be extended to the MDS code C ′ by adding the transpose of this vector as an extra column of G. For C ′ , n = q + 1,

k = m + 1,

d = n − m = q + 2 − k = n − k + 1.

The Main Conjecture MCk for MDS Codes, always taking q > k, is the following:  q + 2 for k = 3 and k = q − 1 both with q even, M (k, q) = q + 1 in all other cases. It is convenient to have the notation m(k − 1, q) = M (k, q). In the rest of this section, the main problems and related results on MDS codes are presented in terms of projective geometry. Three problems may be enunciated. (I) For given k and q, what is the maximum value of n such that an n-arc exists in PG(k − 1, q)? What are the n-arcs corresponding to this value of n? (II) For what values of k and q with q > k is every (q + 1)-arc of PG(k − 1, q) is a normal rational curve? (III) For given k and q with q > k, what are the values of n(≤ q) such that each n-arc is contained in a normal rational curve of PG(k − 1, q)? In how many such curves is the n-arc contained? An n-arc is complete if it is maximal with respect to inclusion; that is, it is not contained in an (n + 1)-arc. Implicit in Problem III is Problem IV, which may be enunciated as follows: (IV) What are the values of n for which a complete n-arc exists in PG(k − 1, q)? In particular, what is the size of the second largest complete arc in this space? From above, m(r, q) is the maximum size of an arc in PG(r, q); also, let m′ (r, q) denote the size of the second largest complete arc in PG(r, q). Then an n-arc in PG(r, q) with n > m′ (r, q) is contained in an m(r, q)-arc. This is an important inductive tool. E XAMPLE 13.11 Let Γ be the normal rational curve of PG(r, K) viewed as a curve over Fq . Then the set Γ(Fq ) of all Fq -rational points is a (q + 1)-arc of PG(r, K), the classical arc of PG(r, K). If q ≥ r + 2, then any (r + 3)-arc of PG(r, q) is contained in a unique classical arc. The next result gives simultaneous information on pairs of dimensions. T HEOREM 13.12

(i) The dual code of an MDS code is also MDS.

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(ii) An n-arc exists in PG(k − 1, q) if and only if an n-arc exists in the space PG(n − k − 1, q). Proof. If G = [Ik | A] is a generator matrix for the code C, then a generator matrix for the dual code C ⊥ is H = [−t A | In−k ], where t A is the transpose of A. 2 C OROLLARY 13.13 (i) A (q + 1)-arc exists in PG(k − 1, q) if and only if a (q + 1)-arc exists in PG(q − k, q). (ii) A (q + 2)-arc exists in PG(k − 1, q) if and only if a (q + 2)-arc exists in PG(q − k + 1, q). (iii) A (q + 3)-arc exists in PG(k − 1, q) if and only if a (q + 3)-arc exists in PG(q − k + 2, q). T HEOREM 13.14 Let an,r be the number of n-arcs in PG(r, q) and let νr be the number of normal rational curves in PG(r, q). Then an,n−2−r /an,r = νn−2−r /νr . An approach to the Main Conjecture is made through the fact that, in a certain dimension, a (q + 1)-arc is classical. In the case of q odd, this occurs in the plane, see Theorem 13.32, but for q even, it is necessary to go to a higher dimension. T HEOREM 13.15 In PG(4, q), q even, a (q + 1)-arc is a normal rational curve. The following result is discussed in Section 13.3.  q + 1 for q odd, m(2, q) = M (3, q) = q + 2 for q even. Theorems 13.32 and 13.15 imply a dependency of m(r, q) and m′ (2, q) for q odd and of m(r, q) and m′ (4, q) for q even. Define the integer functions F and G by the purely arithmetic conditions: q + 1 > m′ (2, q) + r − 2 ⇐⇒ q > F (r); q + 1 > m′ (4, q) + r − 4 ⇐⇒ q > G(r). There are two results that produce an inductive argument on dimension. T HEOREM 13.16 Let K be an n-arc in PG(r, q) with q + 1 ≥ n ≥ r + 3 ≥ 6 and suppose there exist P0 , P1 ∈ K and a hyperplane π containing neither P0 nor P1 such that, for i = 0, 1, the projection Ki of K onto π is Fq -rational in π. Then the arc K is contained in one and only one classical arc of PG(r, q). T HEOREM 13.17 Let K be a (q + 2)-arc in PG(r, q) with q + 1 ≥ r + 3 ≥ 6. If a hyperplane π of PG(r, q) contains neither of the points P0 , P1 of K, then it cannot happen that both projections Ki of K from Pi , i = 0, 1, onto π are Fq rational in π. In particular, if every (q + 1)-arc in PG(r − 1, q) is classical, then m(r, q) = q + 1.

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These two results have as corollaries the following way of approaching solutions to Problems I, II, and III. T HEOREM 13.18 In PG(r, q), q odd, r ≥ 3, (i) if K is an n-arc with n > m′ (2, q) + r − 2, then K lies on a unique classical arc; (ii) if q > F (r), then every (q + 1)-arc is classical; (iii) if q > F (r − 1), then m(r, q) = q + 1. T HEOREM 13.19 In PG(r, q), q even, q > 2, r ≥ 4, (i) if K is an n-arc with n > m′ (4, q) + r − 4, then K lies on a unique classical arc; (ii) if q > G(r), then every (q + 1)-arc is classical; (iii) if q > G(r − 1), then m(r, q) = q + 1. T HEOREM 13.20 If q is even and q > 2, then m′ (4, q) ≤ q −

1√ 2 q

+

13 4 .

Theorems 13.16 and 13.17 with the upper bounds for m′ (2, q) in Sections 13.6, 13.7, 13.8 can be applied to give the following results for q odd. T HEOREM 13.21 For q odd, r ≥ 3, (i) F (r) ≤ (4r −

23 2 4 ) ;

(ii) F (r) ≤ 5(9r − 19) for q prime; (iii) F (r) ≤ 4(r + 2)2 when p ≥ 5; (iv) F (r) ≤ 4r2 when p ≥ 5, q ≥ 232 , q 6= 36 , 55 ; (v) F (r) ≤ 4r2 when q = 32e , q > 36 ; (vi) F (r) ≤ 13 (4r +

55 2 4 )

when q = 32e+1 .

For q even, the results depend on Theorems 13.15 and 13.20. T HEOREM 13.22 For q even, q > 2, r ≥ 4,

G(r) ≤ (2r − 72 )2 .

However, it should be noted that the classification of (q + 1)-arcs for r ≥ 4 will not produce a simple result. T HEOREM 13.23 (Glynn) In PG(4, 9), there are precisely two projectively distinct 10-arcs, a normal rational curve and a non-classical arc. A result on the Main Conjecture is the following.

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T HEOREM 13.24 For q odd and q ≤ 27, the largest size of square matrix over Fq with every minor non-zero is 21 (q + 1). As an illustration,         A=       

1 1 10 13 5 4 16 11 12 17

1 10 13 5 4 16 11 12 17 2

1 13 5 4 16 11 12 17 2 7

1 5 4 16 11 12 17 2 7 8

1 4 16 11 12 17 2 7 8 3

1 16 11 12 17 2 7 8 3 15

1 11 12 17 2 7 8 3 15 14

 1 1 1 12 17 2   17 2 7   2 7 8   7 8 3   8 3 15   3 15 14   15 14 6   14 6 9  6 9 18

is a 10 × 10 matrix with the property that every minor is non-zero modulo 19; further, there is no square matrix of larger size over F19 with this property. Near-MDS codes are linear codes whose parameters differ only slightly from those of MDS codes. D EFINITION 13.25 A linear [n, k]q -code is a near-MDS code if d(C) = n− k and d(C ⊥ ) = k. Like MDS codes, near-MDS codes can be investigated within finite projective geometry, since a near-MDS code [n, k]q can be viewed as a point set C of size n in PG(k − 1, q) satisfying the following conditions: (i) every k − 1 points in C generate a hyperplane of PG(k − 1, q); (ii) there exist k points in C lying in a hyperplane of PG(k − 1, q); (iii) any k + 1 points in C generate PG(k − 1, q). In the special case of near-MDS codes of dimension 4, this geometric representation shows that a near-MDS [n, 4]q code is a point set C of size n in PG(3, q) with the following properties: (i) C is a cap, that is, no three points in C are collinear; (ii) C is not an arc, that is, C contains four coplanar points; (iii) no five points in C are coplanar. Let K = Fq . The set of Fq -rational points of an elliptic quartic Γ4 of PG(3, K) defined over Fq is a natural example of such a point set C with n equal to the number N1 of all Fq -rational points of Γ4 . Since such an elliptic quartic is birationally equivalent over Fq to a non-singular plane cubic defined over Fq , an [n, 4]q nearMDS code exists provided that there exists a non-singular plane cubic over Fq whose number of Fq -rational points is exactly n.

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13.3 ARCS AND OVALS As explained in Section 13.2, MDS codes and arcs in finite projective spaces are equivalent objects. In this section, arcs in planes are investigated by using previous results on curves with many Fq -rational points. Some standard notation from finite geometry used in this chapter is as follows. N OTATION 13.26 (i) In PG(2, q), the coordinates are x0 , x1 , x2 with corresponding indeterminates X0 , X1 , X2 . (ii) In PG(2, q), let U0 = (1, 0, 0), U1 = (0, 1, 0), U2 = (0, 0, 1), U = (1, 1, 1); u0 = v(X0 ), u1 = v(X1 ), u2 = v(X2 ), u = v(X0 + X1 + X2 ). (iii) Write Ns = {1, 2, . . . , s} for s > 0. (iv) For binomial coefficients, write c(n, r) =

  n . r

The concept of an oval in finite geometry arises from two combinatorial properties of a closed convex curve in the real plane. D EFINITION 13.27 A set Ω of points in PG(2, q) is an oval if (i) no three points in Ω are collinear; (ii) at every point P ∈ Ω there is exactly one unisecant, that is, a line of PG(2, q) that meets Ω only in P . In this setting, a k-arc in PG(2, q) is a set Ω of k-points that has property (i). In particular, there are three types of lines with respect to an arc: (a) bisecants or chords, (b) unisecants and (c) external lines or lines disjoint from the arc. A combinatorial argument shows that ( q + 1 when q odd, k≤ q + 2 when q even, for every k-arc in PG(2, q). A dual k-arc in PG(2, q) is a k-arc in the dual plane, that is, a set of k lines in PG(2, q) with no three concurrent. In the study of arcs, the set of points of PG(2, q) lying on exactly one of the lines of a dual k-arc K plays an important role. Such a set ∆(K) has size kt with t = q − k + 2, each line of K containing exactly t points from ∆(K). Ovals are exactly the (q + 1)-arcs in PG(2, q). An example of an oval is the set of all Fq -rational points of an irreducible conic defined over Fq . For q odd, every oval in PG(2, q) is a complete arc. This does not hold true for q even because an irreducible conic in even characteristic is a strange curve, and hence, adding the

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nucleus to the Fq -rational points of the conic, a (q + 2)-arc in PG(2, q), necessarily complete, is obtained. More generally, every oval Ω in PG(2, q) with q even has a nucleus; that is, its unisecants have a common point N ; the oval Ω, together with N , forms a complete (q + 2)-arc of PG(2, q), called a hyperoval. Therefore ( q + 1 when q is odd, m(2, q) = q + 2 when q is even, Note that, if K is a k-arc with k > m′ (2, q), then K is contained in an oval of PG(2, q). It is shown below that every oval in PG(2, q) with q odd consists of the Fq rational points of an irreducible conic defined over Fq . Although ovals for q even have been classified for q ≤ 32, the classification of ovals for q ≥ 64 appears difficult and is a major challenge in finite geometry. Menelaus’ classical theorem that gives a condition for three points one on each side of a triangle to be collinear is generalised to show the existence of a plane curve passing through each point in ∆(K), the point set arising from a dual arc K of PG(2, q). Such a curve F, which may be singular and even reducible, is defined over Fq ; it has degree t, as small as possible, for q even, and 2t, larger but still low, for q odd. If k is large, then F is irreducible and an interesting case occurs; namely, F has low degree, and hence low genus, compared to its large number of Fq -rational points. Referring to the results in Chapter 9 on the maximum number of Fq -rational points that a curve defined over Fq can have, it is conceivable that this is an unusual occurrence. Large complete arcs are rare, but they can be investigated by studying particular curves defined over Fq which have many Fq -rational points. This is done in Section 13.4. The starting point of the study of ovals in odd characteristic is a series of combinatorial results. L EMMA 13.28 In PG(2, q) for q odd, every point off an oval Ω lies on exactly two or no unisecants of Ω. Proof. Since Ω consists of q + 1 points, there is exactly one unisecant through each point of Ω. Let ℓ be the unisecant at P to Ω and let Q ∈ ℓ\{P }. If σi (Q), with i = 1, 2, denotes the number of i-secants to Ω through Q, then σ1 (Q) + 2σ2 (Q) = q + 1 and, since q is odd, σ1 (Q) is even. As ℓ itself is a unisecant through Q, it follows that σ1 (Q) ≥ 2. Since this is true for each of the q points Q in ℓ\{P } and since Ω has exactly q + 1 unisecants, σ1 (Q) = 2 for all such points Q; that is, through every point of ℓ\{P }, there is exactly one other unisecant. 2 A point of PG(2, q) is external or internal to the oval Ω according as it lies on two or no unisecants of Ω. Hence, with respect to Ω, the q 2 + q + 1 points of PG(2, q) are partitioned into three classes: (a) q + 1 points on Ω;

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(b)

1 2 q(q

+ 1) external points;

(c)

1 2 q(q

− 1) internal points.

Since a k-arc in the dual plane is a dual k-arc, the q 2 + q + 1 lines of PG(2, q) are partitioned similarly into three classes with respect to Ω: (a′ ) q + 1 unisecants; (b′ )

1 2 q(q

+ 1) bisecants;

(c′ )

1 2 q(q

− 1) external lines.

For any k-arc K with 3 ≤ k ≤ q + 1, choose three of its points as the triangle of reference U0 U1 U2 of the coordinate system. A unisecant to K through one of U0 , U1 , U2 has the respective form, v(X1 − dX2 ),

v(X2 − dX0 ),

v(X0 − dX1 ),

v(X1 − ai X2 ),

v(X2 − bi X0 ),

v(X0 − ci X1 ),

with d 6= 0. Here, d is the coordinate of such a line. Suppose the t = q + 2 − k unisecants to K at each of U0 , U1 , U2 are i ∈ Nt .

L EMMA 13.29 (Segre’s Lemma of Tangents) The coordinates ai , bi , ci of the unisecants at U0 , U1 , U2 to a k-arc K through these points satisfy the identity Qt i=1 ai bi ci = −1.

Proof. If P = (d0 , d1 , d2 ) is any point of K other than U0 , U1 , U2 , then the lines P U0 , P U1 , P U2 are, respectively, v(X1 − eX2 ),

v(X2 − f X0 ),

where e = d1 /d2 , f = d2 /d0 , g = d0 /d1 and so

v(X0 − gX1 ),

ef g = 1.

(13.3)

Through U0 there are q −1 lines other than U0 U1 , U0 U2 ; they consist of t unisecants v(X1 − ai X2 ), i ∈ Nt ,

and k − 3 bisecants

v(X1 − ej X2 ), j ∈ Nk−3 .

Since the product of the non-zero elements of Fq is −1, so Q Q ai ej = −1.

Similarly, for the q − 1 lines through U1 and through U2 other than the sides of the triangle of reference, Q Q Q Q ci gj = −1. bi fj = −1, Hence

Q

Q

(ej fj gj ) = (−1)3 = −1. Q However, as ej fj gj = 1 for each j by (13.3), so (ai bi ci ) = −1. (ai bi ci )

2

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C OROLLARY 13.30 If the unisecants to a (q + 1)-arc K at the points U0 , U1 , U2 of K are v(X1 − aX2 ), v(X2 − bX0 ), v(X0 − cX1 ), then abc = −1.

L EMMA 13.31 For q odd, the triangles formed by three points of a (q + 1)-arc K and the unisecants at these points are in perspective. Proof. Choose the three points as U0 , U1 , U2 ; then the unisecants are the lines v(X1 −aX2 ), v(X2 −bX0 ), v(X0 −cX1 ), with abc = −1. The vertices V0 , V1 , V2 of the triangle with the unisecants as sides are (c, 1, bc), (ca, a, 1), (1, ab, b). Hence the lines U0 V0 , U1 V1 , U2 V2 are v(X2 − bcX1 ), v(X0 − caX2 ), v(X1 − abX0 ). These three lines are concurrent if and only if a2 b2 c2 = 1, which is satisfied since abc = −1. It may be noted that, since the three unisecants are not concurrent, by Lemma 13.28, so abc 6= 1. 2 T HEOREM 13.32 (Segre) In PG(2, q), with q odd, every oval consists of all Fq rational points of an irreducible conic defined over Fq . Proof. Let K be the oval and choose any three of its points as U0 , U1 , U2 . Also, let the unit point U be the point of perspectivity of the triangle U0 U1 U2 and its circumscribed triangle as in the previous lemma. As U is the intersection of v(X0 − caX2 ) and v(X1 − abX0 ), which is (ca, −a, 1), so a = b = c = −1. Thus the unisecants to K at U0 , U1 , U2 are v(X1 + X2 ), v(X2 + X0 ), v(X0 + X1 ). Let P = (y0 , y1 , y2 ) be any other point of K and let the unisecant to K at P be ℓ = v(d0 X0 + d1 X1 + d2 X2 ). Lemma 13.29 applied to the triangle P U1 U2 gives the condition (d0 − d1 − d2 )[d1 (y0 + y1 ) − d2 (y0 + y2 )] = 0. However, by Lemma 13.28, the line ℓ cannot pass through the meet of the unisecants at U1 and U2 ; this point is (1, −1, −1) and so d0 − d1 − d2 6= 0. Thus d1 (y0 + y1 ) = d2 (y0 + y2 ); similarly, from the triangles P U2 U0 and P U0 U1 , d2 (y1 + y2 ) = d0 (y1 + y0 ), d0 (y2 + y0 ) = d1 (y2 + y1 ). Hence d0 : d1 : d2 = y1 + y2 : y2 + y0 : y0 + y1 . As d0 y0 + d1 y1 + d2 y2 = 0 and 2 6= 0, so y1 y2 + y2 y0 + y0 y1 = 0. Hence all points of K lie on the irreducible conic C = v(X1 X2 + X2 X0 + X0 X1 ) defined over Fq . As the set C(Fq ) of all Fq -rational points of C has size q + 1, it coincides with K. 2 The preceding proof is typical in this subject. A known property of conics is proved for ovals and this property is used to show that an oval is a conic. In other words, a property which characterises conics among algebraic curves also characterises ovals among k-arcs.

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13.4 SEGRE’S GENERALISATION OF MENELAUS’ THEOREM An n-point in PG(2, K) is a set of n points, no three of which are collinear, together with the 21 n(n − 1) lines that are joins of pairs of the points. The points and lines are called vertices and sides of the n-point. The vertices form an n-arc. Dually, an n-line in PG(2, K) is a set of n lines, no three of which are concurrent, together with the 21 n(n − 1) points that are intersections of pairs of the lines. The points and lines are again called vertices and sides of the n-line. The sides form a dual n-arc. An n-gon is an ordered set (P1 , P2 , . . . , Pn ) of n points, no three of which are collinear: the Pi are the vertices and the n lines P1 P2 , P2 P3 , . . . , Pn−1 Pn , Pn P1 are the sides of the n-gon. For small n, the following terms are used: n=3 n=4 n=5 n=6

n-point triangle quadrangle pentastigm hexastigm

n-line triangle quadrilateral pentagram hexagram

n-gon triangle quadrangle, quadrilateral pentagon hexagon

A triangle with vertices A, B, C and sides a, b, c is denoted ABC or abc when there is no risk of confusing it with the entire plane. Let Cn be a curve of degree n defined over Fq . Certain problems on the intersection of two curves Cn and Ck are considered, particularly when Ck is completely reducible and its linear components are sides of a k-line. Menelaus’ Theorem gives a necessary and sufficient condition for three points, one on each side of a triangle, to be collinear. In its Euclidean form, this is expressed as a product of ratios of distances being −1. This is now generalised projectively. Let K = {l1 , l2 , . . . , lk } with li = v(Li ) and Li a linear polynomial over Fq . Let Ck = v(L1 · · · Lk ). Consider the k-line K′ that has K as its set of sides. Let Aij = li ∩ lj , let Ai = {Aij | j 6= i} be the set of vertices on li , and also let A = {Aij | i, j ∈ Nk } be the set of all the vertices of K′ . Consider on each li a multi-set Gin whose support is disjoint from A, and let G = ∪ki=1 Gin .

For any Ck such that Cn contains no point of A, the multi-set G consists of all common points P of Cn and Ck , each counted I(P, Cn ∩ Ck ) times. Conditions are required under which a given G is the complete intersection of Ck with some Cn . In fact, given G on K, necessary and sufficient conditions are found for the set MG = {Cn | Cn ∩ Ck = G} to be non-empty. Let L be the linear system of all curves Cn through the points of G; that is, I(P, Cn , Ck ) ≥ nP for every point P with weight nP in G. Geometrically speaking, Cn ∈ L if Cn meets K at least in G. Then, L contains as a linear subsystem the set MG consisting of all curves Cn that meets K exactly in G. The dimension of L is found, as well as the number of conditions for a plane curve Cn to belong to L.

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L EMMA 13.33 (i) Cn meets K in G and some further point P if and only if all the lines li of K are components of Cn . (ii) The set L\MG of reducible curves containing Ck as a component is a linear subsystem LK of L whose dimension is equal to c(n − k + 2, 2) − 1.

Proof. If P ∈ l1 , then Cn contains n + 1 points {P } ∪ G1n of l1 ; so Cn contains l1 . However, any other line li meets l1 in A1i . So each li meets Cn in at least n + 1 points. Hence Cn contains all li . Thus any curve of L not in MG contains K, and LK consists of all curves v(L1 L2 . . . Lk Fn−k ), where Fn−k is a form of degree n − k; such a curve has as components the k lines li and Cn−k = v(Fn−k ). Thus LK is a linear subsystem of the required dimension. 2 T HEOREM 13.34 Suppose L is not empty, and let Cn′ = v(F ′ ) be a curve in L. (i) If k > n, then LK = ∅ and L = MG = {Cn′ }. (ii) If LK 6= ∅, then MG is not a linear subsystem of L. However, if Cn ∈ L and Cn′ ∈ MG , there is a curve Cn−k = v(Fn−k ) such that Cn = v(F ′ + L1 L2 . . . Lk Fn−k ).

(iii) If d is the dimension of L, then d = c(n − k + 2, 2). Proof. (i) If k > n, no Cn can have Ck as a component. So LK 6= ∅. From Lemma 13.33, L = MG and L consists of the single curve Cn′ . (ii) If LK 6= ∅, then LK and MG form a partition of L. So MG cannot be a linear subsystem. Take a point P ∈ l1 not in G. If Cn = v(F ) ∈ L and Cn′ ∈ MG , then some member of the pencil v(sF + tF ′ ) contains and so is in LK . Hence λF = F ′ + L1 L2 · · · Lk Fn−k for some λ and some Fn−k . (iii) From (ii) and Lemma 13.33, d = dim LK + 1 = c(n − k + 2, 2). 2 C OROLLARY 13.35

(i) If some curve meets K exactly in G, then G imposes

c(n + 2, 2) − c(n − k + 2, 2) = 21 k(2n − k + 3) − 1 conditions on a Cn to meet K in G. (ii) If n ≥ k, then K imposes

c(n + 2, 2) − c(n − k + 2, 2) = 12 k(2n − k + 3)

conditions on a Cn to meet K in G. T HEOREM 13.36 If k = 1, then MG is never empty and the n conditions imposed by G are independent. Proof. If K = {l1 }, G = {P1 , . . . , Pn }, P 6∈ l1 , and P Pi = v(L′i ), all i, then v(L′1 L′2 · · · L′n ) is in MG . 2 T HEOREM 13.37 If k = 2, then MG is never empty and the 2n conditions imposed by G are independent.

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Proof. If K = {l1 , l2 }, Gin = {P1i , . . . , Pni }, and Pj1 Pj2 = v(L′j ), all j, then v(L′1 L′2 · · · L′n ) is in MG . 2 Let the coordinate system in PG(2, q) be fixed. If the point P lies on a side of U0 U1 U2 and is not a vertex, it is the meet of that side and one of v(X1 − cX2 ),

v(X2 − cX0 ),

v(X0 − cX1 );

then c is the coordinate of P . T HEOREM 13.38 Let K be the triangle U0 U1 U2 . (i) Then MG 6= ∅ if and only if

Q

c = (−1)n ,

where the product ranges over all not-necessarily-distinct 3n points P of G and c is the coordinate of P . (ii) If MG 6= ∅, then G imposes 3n − 1 conditions on the curves of order n to meet K in G; if n = 1 or 2, then MG consists of a single Cn .

Proof. If Cn ∈ MG and Cn = v(c0 X0n + c1 X1n + c2 X2n + · · · ), then c0 c1 c2 6= 0 since Cn contains no vertex of U0 U1 U2 . Let Cn0 = v(c1 X1n + · · · + c2 X2n ),

Cn1 = v(c0 X0n + · · · + c2 X2n ),

Cn2 = v(c0 X1n + · · · + c1 X1n ).

Then, for i = 0, 1, 2, Gin = {nP P | P ∈ ui ; nP = I(P, ui ∩ Cni )}. Hence Q

c = (−1)n c2 /c1 , (−1)n c0 /c2 , (−1)n c1 /c0 , Q according as the product is taken over G0n , G1n , G2n . So c = (−1)n , where the product is taken over all points of G. Now suppose that the condition is satisfied by the points of G; then 3n − 1 points of G determine the remaining one, P say. Let LP be the linear system of Cn through the points of G\{P }. Then L ⊂ LP and, if d is the dimension of LP , 0 ≤ c(n + 2, 2) − 1 − (3n − 1) = c(n − 1, 2) ≤ d ≤ c(n + 2, 2) − 1. Since the dimension of LK is c(n − 1, 2) − 1 < d, so LP \LK 6= ∅. Now, Cn′

MG = L\LK ⊂ LP \LK .

However, ∈ LP \LK , it meets K in G′ , where G′ = (G\{P }) ∪ {Q} and S if Q ∈ ( li )\A. But G′ also satisfies the given condition. So Q = P and G′ = G. Hence Cn′ ∈ MG and MG = LP \LK 6= ∅. 2 For n = 1, this is Menelaus’ Theorem and, for n ≥ 2, Carnot’s Theorem. It is now generalised to the case that k ≥ 4.

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T HEOREM 13.39 Let K = {l1 , l2 , . . . , lk } be a set of k lines of PG(2, K) and let Gj be a multi-set of size n whose support is on lj , such that no point of intersection of any two of the k lines lies in the support of any Gj . Suppose that, for each J ⊂ Nk of the following special types, there is a plane curve CnJ of degree n that meets lj in Gj exactly: (a) {lj | j ∈ J} is a triangle; (b) {lj | j ∈ J} is a pencil. Then (i) there is a plane curve Cn that meets lj in Gj exactly, for all j; (ii) if there is such a curve Cn and k > n, then Cn is unique. Proof. If all the lines are concurrent there is nothing to prove. Also the result is established for a triangle in Theorem 13.38. So assume that k ≥ 4 and that not all the lines are concurrent; the proof is by induction. Let Pi = li ∩ lk for i = 1, 2, . . . , k − 1 and suppose that P1 6= P2 . Let lk = u0 and Pi = (0, ci , di ), i ∈ Nk−1 . By induction, there is a curve Cni = v(Fi ) with the correct intersection with all lines lj , j 6= i. Now, F1 (0, X1 , X2 ) and F2 (0, X1 , X2 ) have degree n, and both meet lk in Gk ; hence F1 (0, X1 , X2 ) and F2 (0, X1 , X2 ) only differ by a constant factor, and may be assumed equal. Also, P2 does not lie on the curve Cn1 , and so F1 (0, c2 , d2 ) 6= 0; similarly, Fk (0, c2 , d2 ) 6= 0. Let β ′ = −Fk (0, c2 , d2 )/F1 (0, c2 , d2 ), F ′ (X0 , X1 , X2 ) = Fk (X0 , X1 , X2 ) + β ′ F1 (X0 , X1 , X2 ),

and let C ′ = v(F ′ ); if F ′ (X0 , X1 , X2 ) is the zero polynomial, the result is proved. Note that β ′ 6= 0. The curve C ′ meets l2 in G2 and contains P2 ; hence, by B´ezout’s Theorem, l2 is a linear component of C ′ . Then, for each j = 3, 4, . . . , k − 1, the curve C ′ meets the line lj in Gj and also contains the point lj ∩ l2 ; hence lj is a linear component of C ′ for j = 3, 4, . . . , k − 1. Thus, if Pi = v(X0 ) ∩ v(Ri ) with Ri = di X1 − ci X2 , then F ′ (0, X1 , X2 ) = Fk (0, X1 , X2 ) + β ′ F1 (0, X1 , X2 ) is divisible by R2 R3 · · · Rk−1 . Now, interchange the roles of l1 and l2 in the above argument. Put β ′′ = −Fk (0, c1 , d1 )/F2 (0, c1 , d1 ), F (X0 , X1 , X2 ) = Fk (X0 , X1 , X2 ) + β ′′ F2 (X0 , X1 , X2 ), ′′

and let C ′′ = v(F ′′ ). As before, if F ′′ = 0, the result is proved. Now,

F ′ (0, X1 , X2 ) − F ′′ (0, X1 , X2 ) = β ′ F1 (0, X1 , X2 ) − β ′′ F2 (0, X1 , X2 ) = (β ′ − β ′′ )F1 (0, X1 , X2 ),

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and so F ′ (0, X1 , X2 ) − F ′′ (0, X1 , X2 ) = 0 meets lk in Gk and the point P3 , since k ≥ 4. Hence (β ′ − β ′′ )F1 (0, X1 , X2 ) is the zero polynomial, and so β ′ = β ′′ . It follows that F ′′ (0, X1 , X2 ) = Fk (0, X1 , X2 ) + β ′′ F2 (0, X1 , X2 ) = Fk (0, X1 , X2 ) + β ′ F1 (0, X1 , X2 ) is divisible by R1 R3 R4 · · · Rk−1 . Hence F ′ (0, X1 , X2 ) = F ′′ (0, X1 , X2 ) is divisible by R1 R2 · · · Rk ; here, the fact that P1 6= P2 and so R1 6= cR2 is used. So Fk (0, X1 , X2 ) + β ′ F1 (0, X1 , X2 ) = R1 R2 · · · Rk−1 R. Now, let li = v(Li ) with Li = ei X0 + Ri for i ∈ Nk−1 and define Cn = v(F ), where F (X0 , X1 , X2 ) = Fk (X0 , X1 , X2 ) − L1 L2 · · · Lk−1 R. Then Cn meets lj in Gj for j ∈ Nk−1 . The curve Cn meets lk where Fk (0, X1 , X2 ) − R1 R2 · · · Rk−1 R = 0,

that is, where β ′ F1 (0, X1 , X2 ) = 0; hence Cn meets lk in Gk . Hence Cn is the required curve. Finally, suppose that k > n, and that Cn = v(F ) and Dn = v(G) are curves satisfying the hypotheses. To obtain a contradiction, suppose that H(X0 , X1 , X2 ) = F (X0 , X1 , X2 ) + λG(X0 , X1 , X2 ), with λ = −F (0, c1 , d1 )/G(0, c1 , d1 ), is not the zero polynomial. Then the curve En = v(H) has degree n. By B´ezout’s Theorem, En contains the line lk as a linear component and so contains the points P1 , P2 , . . . , Pk−1 . Again, by B´ezout’s Theorem, En contains the lines lj for j = 1, 2, . . . , k − 1. Hence En of degree n contains k > n distinct linear components, a contradiction. Hence H is the zero polynomial, whence Cn = Dn . 2 13.5 THE CONNECTION BETWEEN ARCS AND CURVES The aim of this section is to associate a plane curve defined over Fq to a dual k-arc in PG(2, q). This shows that the dual view is useful in the study of k-arcs. The essential result on dual k-arcs is stated and proved separately for q even and odd. As before, let K be a k-arc, let K′ be a dual k-arc, and let t = q − k + 2. Then ∆(K′ ) stands for the set consisting of the kt points in PG(2, q) which lie on one but not on two of the lines in K′ . T HEOREM 13.40 Let K′ be a dual k-arc in PG(2, q) with q even. Then the tk points of ∆(K′ ) belong to a plane curve Γt of degree t with the following properties: (i) Γt is unique if k > t, that is, k > 21 q + 1;

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(ii) the points of K′ distinct from those in ∆(K′ ) do not belong to Γt , and so no line of K′ is a component of Γt ; (iii) if ℓ is any line of K′ and P ∈ ℓ ∩ ∆(K′ ), then I(P, ℓ ∩ Γt ) = 1; in particular, P is a non-singular, and hence an Fq -rational point of Γt ; (iv) every linear component of Γt is defined over Fq ; if k ≥ 5, this also holds for every irreducible component of Γt of degree two. Q Proof. By Lemma 13.29, c = 1 for each triangle of K′ , where the product ranges over all points of ∆(K′ ) on the sides of the triangle and c is the coordinate of such a point. By Theorem 13.39, there exists a plane curve Γt define over Fq , which is unique if k > t. Also, by one of the hypotheses in Theorem 13.39, the points of the lines of K′ other than those of ∆(K′ ) do not belong to Γt , and so no line of K′ is component of Γt . Hence every line of K′ contains at most t points of Γt . As each of the t points in ∆(K′ ) lying on a line ℓ of K is in Γt , none is counted twice in the intersection of ℓ and Γt . If a line ℓ′ is a linear component of Γt , then it meets every line of K′ at a point in ∆(K′ ). Hence Γt contains a point in PG(2, q) from each line of K′ . Since K′ consists of more than one line, Γt contains two distinct points of PG(2, q), and hence is defined over Fq . The same argument works for a quadratic component C of Γt when k ≥ 5, the assumption being necessary as there exist irreducible conics not defined over Fq which contain only four points of PG(2, q). 2 C OROLLARY 13.41 In PG(2, q) with q even, (i) a (q + 1)-arc Ω is incomplete; (ii) the q + 1 unisecants to Ω are concurrent and Ω lies in a unique (q + 2)-arc. Proof. In the dual plane of PG(2, q), the set Ω is a dual (q + 1)-arc, and so t = 1. Therefore Γt is a line. In PG(2, q), the corresponding point added to Ω gives a (q + 2)-arc. 2 C OROLLARY 13.42 If K is an incomplete k-arc with 3 ≤ k ≤ q+1 in PG(2, q), q even, and if k > 21 q + 1, then (i) there is a point R on no bisecant of K; (ii) if K′ is the dual k-arc arising from K in the dual plane of PG(2, q) and Γt is a curve associated to K′ via Segre’s generalisation of Menelaus’ Theorem, then the line ℓ corresponding to R in the dual plane is a component of Γt ; (iii) Γt is unique. Proof. (i) Since K is incomplete, there is such an R. (ii) Let K′ be the dual k-arc arising from K in the dual plane of PG(2, q). In that plane, R defines a line ℓ such that the points of ℓ cut out by the lines of K′ are distinct. There are k such points, each lying in ∆(K′ ). By Theorem 13.40, there

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exists a curve Γt defined over Fq that contains these k points of ℓ. Since Γt has degree t and the hypothesis k > 21 q + 1 implies that k > t, the line ℓ must be a component of Γt . (iii) Again by Theorem 13.40, since k > t, so Γt is unique. 2 A result similar to Theorem 13.40 is established for q odd. Now, −1 = 1 for q even; (−1)2 = 1 for q odd. This is the simple reason that the results for q odd are weaker than for q even; Lemma 13.29 can no longer be directly applied to Theorems 13.38 and 13.39. T HEOREM 13.43 Let K′ be a dual k-arc in PG(2, q) with q odd. Then the tk points of ∆(K′ ), where t = q + k − 2, belong to a plane curve Γ2t defined over Fq of degree 2t with the following properties: (i) Γ2t is unique if k > 2t, that is, k > 13 (2q + 4); (ii) the points of the lines of K′ distinct from those in ∆(K′ ) do not belong to Γ2t and so no line of K is a component of Γ2t ; (iii) if ℓ is any line of K′ and P ∈ ℓ ∩ ∆(K′ ), then I(P, ℓ ∩ Γ2t ) = 2; (iv) Γ2t may contain components of multiplicity at most two but does not consist entirely of double components; (v) every linear component of Γ2t is defined over Fq ; if k ≥ 5, this also holds for every irreducible component of Γ2t of degree two. Q Proof. By Lemma 13.29, c = −1 for each triangle of K′ , where the product ranges over all points Q of ∆(K′ ) on the sides of the triangle and c is the coordinate of such a point. So c2 = 1. By Theorem 13.39, there exists a plane curve Γ2t define over Fq , which is unique if k > 2t. Also, by one of the hypotheses in Theorem 13.39, the points of the lines of K′ distinct those from ∆(K′ ) do not belong to Γ2t , and so no line of K′ is component of Γ2t . Hence every line of K′ contains at most 2t points of Γ2t . As each of the t points in ∆(K′ ) lying on a line ℓ of K′ is in Γ2t and counted twice in the intersection of ℓ and Γ2t , there are no components of multiplicity three. If Γ2t consisted entirely of double components, the points in ∆(K′ ) would belong Q to a curve of degree t and the condition d = 1 would be satisfied for any triangle of K′ . Finally, the proof of (iv) of Theorem 13.40 also shows (v). 2 C OROLLARY 13.44 Let K be an incomplete k-arc with 3 ≤ k < q + 1, q odd. If k > 23 (q + 2), that is, k > 2t, then (i) there is a point R on no bisecant of K; (ii) if K′ is the dual k-arc arising from K in the dual plane of PG(2, q) and Γ2t is the curve associated to K′ , then the line ℓ corresponding to R in the dual plane is a component of Γ2t ;

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(iii) Γ2t is unique. Proof. The arguments used to prove Corollary 13.42 remain valid for odd q when k > 2t, this assumption being necessary to ensure the validity of (ii). By Theorem 13.43, there exists a plane curve Γ2t defined over Fq such that I(P, Γ2t ∩ ℓ) ≥ 2 for every point P ∈ ℓ ∩ ∆(K′ ). Since Γ2t has degree 2t, the hypothesis k > 2t implies that the line ℓ must be a component of Γt . 2 R EMARK 13.45 Segre’s Theorem 13.32 can be deduced from Theorem 13.43. As k = q + 1, so t = 1. Hence, by Theorem 13.43, the points of the dual (q + 1)-arc K′ belong to a conic Γ2 defined over Fq which does not split into two, possibly coincident, lines, as otherwise the (q + 1)-arc K would be incomplete. Also, since I(P, Γ2 ∩ l) = 2 for P ∈ l ∩ ∆(K′ ), so the lines of K′ are tangents to Γ2 at the points of ∆(K′ ). Since the number of Fq -rational points of Γ2 is exactly q + 1, so K′ can be viewed as the envelope of the conic Γ2 at its Fq -rational points. After dualising, Theorem 13.32 follows. E XAMPLE 13.46 As in Example 11.58, in PG(2, 13) there exists exactly one complete 12-arc, up to a projectivity, as in Table 13.2. In this case, ∆(K′ ) has size 36 Table 13.2 The 12-arc K and its dual K′

K′

K P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12

(1, 1, 1) (1, −1, 1) (−1, 1, 1) (−1, −1, 1) (3, 4, 1) (3, −4, 1) (−3, 4, 1) (−3, −4, 1) (4, 3, 1) (4, −3, 1) (−4, 3, 1) (−4, −3, 1)

X +Y X −Y −X + Y −X − Y 3X + 4Y 3X − 4Y −3X + 4Y −3X − 4Y 4X + 3Y 4X − 3Y −4X + 3Y −4X − 3Y

+Z +Z +Z +Z +Z +Z +Z +Z +Z +Z +Z +Z

and its points are listed in Table 13.3. Table 13.3 The points of ∆(K′ )

(±1, ±2, 1) (±2, ±1, 1) (±2, ±5, 1) (±5, ±2, 1) (±3, ±5, 1) (±5, ±3, 1) (±4, ±6, 1) (±6, ±4, 1) (±6, ±6, 1)

t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12

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By Theorem 13.43, no plane cubic contains all points in ∆(K′ ), but there is a plane sextic that does, namely the non-singular plane curve Γ6 defined over F13 : Γ6 = v(X06 + X16 + X26 + 3X02 X12 X22 ). ′

(13.4)

′

The lines of K are each tangent to Γ6 at 3 points of ∆(K ). In addition, Γ6 contains 18 points on the coordinate axes; these are the points Bt = (0, t, 1),

Ct = (t, 0, 1),

Dt = (t, 1, 0),

6

with t = −1, that is, t = ±2, ±5, ±6. These 54 points in PG(2, 13) are all F13 -rational points of Γ6 . It should be noted that Γ6 is optimal in that it attains the upper bound in Theorem 8.65 applied to an F13 -rational linear series of dimension 2. Here, g = 10, q = 13, n = 6 and S13 = 54. Therefore 54 is the largest number of F13 -rational points of an irreducible plane curve defined over F13 that has genus 10. E XAMPLE 13.47 Let f (X) = a0 X 2 + a1 X + a2 be a primitive polynomial of Fq2 [X]; that is, f is irreducible with one root and so the other generating the group Fq4 \{0}. The projectivity T of PG(2, q 2 ) associated to the matrix   0 1 0  0 0 1  a2 a1 a0

is a Singer cycle; that is, T acts on the set of points of PG(2, q 2 ), as well as on the set of lines of PG(2, q 2 ), as a single cycle. In particular, T has order q 4 + q 2 + 1, 2 and hence S = Tq +q+1 is a projectivity of order q 2 − q + 1. Also, every point orbit of S is a complete (q 2 − q + 1)-arc K in PG(2, q 2 ), and every line orbit of S is a complete dual (q 2 − q + 1)-arc K′ in PG(2, q 2 ). For even q, the plane curve Γq+1 associated to K′ is the Hermitian curve; see Theorem 13.49. This does not hold true for odd q, the associated curve Γ2(q+1) being projectively equivalent over Fq6 to the irreducible plane curve in affine form, v(Y 2 + X 2 Y 2q + X 2q − 2(X q+1 Y q + X q Y + XY q+1 )).

However, Γ2(q+1) is birationally equivalent over Fq6 to the Hermitian curve; see Theorem 13.64.

13.6 ARCS IN OVALS IN PLANES OF EVEN ORDER In this section, the size m′ (2, q) of the second largest complete arc in PG(2, q), q even, is obtained. T HEOREM 13.48 In PG(2, q), q even, (i)

m′ (2, q) ≤ q −

√ q + 1;

(13.5)

√ q + 1.

(13.6)

(ii) when q is also square, then m′ (2, q) = q −

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√ Proof. Let K be a dual k-arc in PG(2, q) with k > q − q + 1. Without loss of generality, K is assumed to be complete. Let Γt be the associated curve of degree t √ as in Theorem 13.40. Note that t = q − k + 2 < q + 1. Suppose that Γt has an Fq -rational linear component; that is, there is a line ℓ of PG(2, q) contained in Γt . Then the common point of ℓ and any line in K is in ∆(K). Therefore ℓ added to K is a dual (k + 1)-arc, a contradiction to K being complete. Thus Γt does not contain an Fq -rational linear component. Let Sq be the number of Fq -rational points of Γt . From Theorem 13.40 (iii), |Sq | ≥ |∆(K)| = kt. On the other hand, Theorem 9.67 applied to Γt implies that Sq < kt. This con√ tradiction proves that m′ (2, q) ≤ q − q + 1. If q is square, then Example 13.47 implies (13.6). 2 √ It is still an open problem to determine all complete (q− q+1)-arcs in PG(2, q) with q even and square. The following result is a step in this direction. T HEOREM 13.49 Let K be a complete dual (q 2 − q + 1)-arc in PG(2, q 2 ) with q even. Then the associated curve Γt is the Hermitian curve Hq . Proof. Here, t = q + 1. So Γt has genus g ≤ 12 (q 2 − q). From the proof of Theorem 13.48, the number of Fq -rational points of Γt is at least kt = q 3 + 1. By the Hasse–Weil theorem, this is only possible when g = 12 q(q − 1) and the number of Fq -rational points of Γt is exactly q 3 + 1. In particular, Γt is a non-singular model of an Fq2 -maximal curve of genus g = 12 q(q − 1). Thus Γt is the Hermitian curve Hq . 2 13.7 ARCS IN OVALS IN PLANES OF ODD ORDER In this section, a similar treatment to the previous section is given for planes of odd order. An upper bound is obtained for m′ (2, q) in the case that q is odd. The next result is both weaker and stronger, in the sense that the arithmetic condition is stronger, as is the specification of the complete arc containing K. T HEOREM 13.50 Let K′ be a dual k-arc with 3 ≤ k ≤ q + 1 in PG(2, q), q odd, and let Γ2t be the associated curve. If (a) Γ2t has an irreducible conic C defined over Fq as a component, (b) k > 14 (3q + 5), then the lines of K′ are tangents to C. Proof. The lines of K′ are of two types with respect to C according as the line is a tangent to C or not. The lines of the first type form a dual k1 -arc K1′ , and those of the second type form a dual k2 -arc K2′ such that k1 + k2 = k = q − t + 2. Also, the Fq -rational tangents to C distinct from the lines in K1′ form a dual k3 -arc K3′ with

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k3 = q + 1 − k1 . Since C meets a line at most two points, so k1 + 2k2 ≤ q + 1. Hence k1 ≥ q + 3 − 2t,

k2 ≤ t − 1,

k3 ≤ 2t − 2.

Let ℓ be a line in PG(2, q) which is not a tangent to C. At least 21 (q − 1) points of ℓ in PG(2, q) are the intersections of two tangents to C at Fq -rational points. Also, k > 41 (3q + 5) implies that 21 (q − 1) > 2t − 2. As k3 ≤ 2t − 2, at least one point P ∈ ℓ must be the meet of two Fq -rational tangents to C not contained in K3′ . So both are lines in K1′ . As ℓ is not an Fq -rational tangent to C, so ℓ is not in K′ . Thus K′ consists of Fq -rational tangents to C. 2 The next result is a key one. T HEOREM 13.51 In PG(2, q), q odd, m′ (2, q) ≤ q −

1√ 4 q

+ 74 .

√ √ Equivalently, if K is a k-arc with k > q − 41 q + 74 , that is, q > 4t − 1, then K is contained in the set of Fq -rational points of a unique irreducible conic defined over Fq . √ Proof. If k > q − 41 q + 74 , then k ≥ 5. So, if K is contained in the set of Fq -rational points of a conic, then the conic is unique. Consider the dual plane of PG(2, q). Let K′ be the dual k-arc arising from K. Theorem 13.43 gives all the necessary information on the curve Γ2t defined over Fq associated to K′ . For t = 1, such a curve is an irreducible conic C defined over Fq , and K′ consists of lines tangents to C. So it may be assumed that t ≥ 2. Choose an irreducible component Γ′n of Γ2t , where n ≤ 2t, and distinguish the following three cases: (i) Γ′n is an Fq -rational linear component of Γ2t ; (ii) Γ′n is an Fq -rational component of degree two; (iii) Γ′n is an Fq -rational component of degree at least three or Γ′n is not defined over Fq . Case (i). Here, Γ′n is an Fq -rational line ℓ which is not in K′ . So K1′ = K′ ∪ {ℓ} is a dual (k + 1)-arc. If the envelope Γ′n associated with K1′ is again of type (i), the process is continued until the dual arc becomes the set of all tangents to an Fq -rational conic or the respective plane curve becomes one of the other two types. Case (ii). Now, Γ′n is an Fq -rational conic C. Also, √ k > q − 41 q + 74 ⇐⇒ q > 4t − 1 ⇒ q > 4t − 3 ⇐⇒ k > 41 (3q + 5).

So, by Theorem 13.50, K′ consists of Fq -rational tangents to C.

Case (iii). Here Γ′n is Fq -rational with 3 ≤ n ≤ 2t or Γ′n is not Fq -rational. Suppose that Γ′n has S non-singular and d double points in PG(2, q). From Theorem 9.69, S + d < 12 n(q − t + 2) = 21 nk.

(13.7)

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By Theorem 13.43, I(P, Γ2t ∩ ℓ) = 2 holds for each of the t points P ∈ ∆(K′ ) of a line ℓ in K′ . Therefore I(P, Γ′n ∩ ℓ) ≤ 2, and I(Q, Γ′n ∩ ℓ) = 0 for Q 6∈ ∆(K′ ). Since P ′ Q∈ℓ I(Q, Γn ∩ ℓ) = n,

this implies that Γ′n has at least 12 n, possibly double, points from ℓ in PG(2, q). Hence S + d ≥ kn/2,

contradicting the previous estimate (13.7). Thus K is ever contained in the set of Fq -rational points of an irreducible conic defined over Fq . 2 √ C OROLLARY 13.52 If K is a k-arc in PG(2, q), q odd, with k > q − 41 q + 74 , then K is contained in the set C(Fq ) of all Fq -rational points of an irreducible conic C defined over Fq , and the only points for which K ∪ {Q} is a (k + 1)-arc are the q + 1 − k points of C(Fq )\K. Proof. If K ∪ {Q} is a (k + 1)-arc, then it lies in some complete arc, which is necessarily C(Fq ) by the theorem. Hence Q ∈ C(Fq )\K. 2 C OROLLARY 13.53 In PG(2, q), q odd and q > 49, a q-arc is contained in the set of Fq -rational points of a conic. √ Proof. The inequality q > q − 14 q + 74 holds when q > 49. 2 R EMARK 13.54 This result in fact holds true for q ≤ 49, giving the similar result in the case that q is even. Next, a small improvement to Theorem 13.51 is given. There the result is equiv√ alent to t < 14 ( q + 1) for a k-arc that is contained in a conic, where t = q + 2 − k. T HEOREM 13.55 (Thas) In PG(2, q), q odd, √ √ (i) a k-arc K for which t < ( q + 1)2 /(4 q + 1) is contained in the set of Fq -rational points of an irreducible conic defined over Fq ; √ 25 (ii) m′ (2, q) < q − 14 q + 16 . Proof. (i) If t = 1, then K is a (q + 1)-arc, that is, an oval in PG(2, q), and the assertion follows from Segre’s Theorem 13.32. So assume that t > 1. In particular, q > 3. Let K′ be the dual k-arc of K. Assume that the plane curve Γ2t defined over Fq , associated to K′ via Segre’s generalisation of Menelaus’ Theorem, is irreducible. With R the number of points of Γ2t with coordinates in Fq , (iii) of Theorem 9.57 √ gives that R ≤ q + 1 + (2t − 1)(2t − 2) q. So, as K has kt unisecants, that is, |∆(K′ )| = kt, √ q + 1 + (2t − 1)(2t − 2) q ≥ kt, √ √ (t − 1){q + 1 + 2 q − t(1 + 4 q)} ≤ 0.

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615 √ √ Hence t ≥ ( q + 1)2 /(4 q + 1), a contradiction. Therefore Γ2t is reducible. At least one irreducible component Γ′n has degree n ≤ t. Consider the same three cases as in the proof of Theorem 13.51. If Γ′n is Fq -rational of degree n = 1, then Γ′n is a line ℓ not in K′ . So K′ ∪ {ℓ} is a dual (k + 1)-arc. If Γ′n is Fq -rational of degree n = 2, then Γ′n is an irreducible conic C. Since k < 41 (3q + 5), that is, t < 41 (q + 3), Theorem 13.50 implies that K′ is a set of tangents to C at Fq -rational points. Finally, suppose that Γ′n is not Fq -rational or is Fq -rational with n ≥ 3. Then √ √ √ {3( q + 1)2 /(4 q + 1)} − 1 ≤ q, √ since q > 3, and so 3t − 1 < q. By Theorem 9.69, APPLICATIONS: CODES AND ARCS

R < 21 n(q − t + 2) = 21 nk.

From the proof of Theorem 13.51, Γ′n has at least 21 n, possibly double, points from ℓ in PG(2, q), where ℓ is a line of K′ . Hence R ≥ 21 nk, a contradiction. Thus K′ is contained in a dual (k + 1)-arc. Continuing the process shows that ′ K is contained in a dual (q + 1)-arc. Then Theorem 13.50 implies that K′ is a set of tangents to C at Fq -rational points. √ √ (ii) The inequality t < ( q + 1)2 /(4 q + 1) is equivalent to √ √ 2 25 9 k > q − 14 q + 16 − { 16 /(4 q + 1)}. 13.8 THE SECOND LARGEST COMPLETE ARC: FURTHER RESULTS In this section similar techniques to the previous section are used, but the St¨ohr– Voloch theorem is applied instead of the Hasse–Weil theorem. This makes it possible to obtain improvements on upper bounds for m′ (2, q). T HEOREM 13.56 In PG(2, p), with p an odd prime and p ≥ 7, m′ (2, p) ≤

44 45 p

+ 89 .

44 p + 98 . Proof. Suppose that there exists a complete k-arc K with k > 45 As in the proof of Theorem 13.51, the same three possibilities for an irreducible component Γ′n of Γ2t are considered, where Γ2t is the Fq -rational curve associated to the dual k-arc K′ . (i) If Γ′n is Fq -rational and n = 1, then K′ is incomplete. (ii) If Γ′n is Fq -rational and n = 2, then Γ′n is an irreducible conic C defined over Fq . However,

k>

44 45 p

+

8 9

> 41 (3p + 5).

So, by Theorem 13.50, K is contained in the set C(Fq ) of all Fq -rational points of C. (iii) Suppose that Γ′n has S non-singular and d singular points in PG(2, q). Then, as in the proof of Theorem 13.51, S + d ≥ kn/2.

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If Γ′n is not Fq -rational, then S + d ≤ n2 ; so k ≤ 2n ≤ 4t = 4(p + 2 − k). Hence 8 k ≤ 54 (p + 2) < 44 45 p + 9 for p ≥ 5. ′ Now, let Γn be Fq -rational with n ≥ 3. The linear system of all conics cuts out on Γ′n the fixed-point-free Fq -rational linear series L2 that defines the Veronese map. The resulting curve is an irreducible Fq -rational curve Γ of PG(5, K) of degree 2n where K = Fq . Let ν0 = 0, ν1 , . . . , ν4 be the Frobenius orders of Γ. From the fact that νi ≤ deg Γ = 2n = 4t and k > 14 (3p + 8) for p ≥ 5, it follows that 2n < p. As p ≥ 5, Proposition 8.50 (ii) implies that Γ is a Frobenius classical curve, that is, νi = i for 0 ≤ i ≤ 4. Therefore the St¨ohr–Voloch Theorem 8.65 gives that 1 2 kn

So

− d ≤ 52 n{5(n − 2) + p} − 4d. k ≤ 54 {5(n − 2) + p} ≤ 45 {5(2t − 2) + p}

= 45 {10(p + 1 − k) + p} = 8(p + 1 − k) + 54 p.

44 Hence k ≤ 45 p + 89 . 1 If t > 4 p, then k = p + 2 − t < 34 p + 2 < ruled out by the hypothesis.

44 45 p

+ 89 , unless p = 3; but this is 2

Now this theorem is extended to arbitrary odd powers of the characteristic. T HEOREM 13.57 In PG(2, q), q = p2e+1 with p an odd prime and e ≥ 1, √ m′ (2, q) ≤ q − 41 pq + 29 16 p + 1. Proof. Let K be a complete k-arc with k greater than the bound in the theorem. Let K′ be the dual k-arc arising from K in the dual plane of PG(2, q). As in the proof of the previous theorem, it is only necessary to consider case (iii) and a component Γ′n of Γ2t of degree n ≥ 3 that is Frobenius non-classical with respect to the linear system of conics. If the points in ∆(K′ ) are are all double points of Γ′n , then 1 2 kn

so k ≤ n ≤ 2t, and hence

≤ 12 (n − 1)(n − 2) ≤ 21 n2 ;

k ≤ 32 (q + 2) < q −

1√ 4 pq

+

29 16 p

+ 1,

as required. Otherwise, some P ∈ ∆(K′ ) is a non-singular point of Γ′n . If ℓ is the line in K′ through P , then I(P, Γ′n ∩ ℓ) = 2. Therefore the (P, L1 )-orders are 0, 1, 2, where L1 is the linear series on Γ′n cut out by lines. By the argument in the third paragraph of Section 7.8, Γ′n has order sequence (0, 1, 2, 3, 4, ǫ5 ≤ 2n) with respect to the linear series L2 cut out by conics. If Γ′n is Frobenius non-classical for L2 , then the Frobenius order sequence is (0, 1, 2, 3, ǫ5) such that ǫ5 = pm unless p = 3 and ǫ5 = 6. From the St¨ohr–Voloch Theorem 8.65 applied to L2 , see also (8.65), if d ≥ 0 points in ∆(K′ ) are double points, then 1 2 kn

− d ≤ 51 [(6 + ǫ5 ){n(n − 3) − 2d} + 2n(q + 5)].

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Hence k ≤ 25 [(6 + ǫ5 )(n − 3) + 2(q + 5)] ≤ 52 [(6 + ǫ5 )(2t − 3) + 2(q + 5)].

As k = q + 2 − t, it follows that q 6ǫ5 + 26 q t≥ + ≥ + 1. 4ǫ5 + 29 4ǫ5 + 29 4ǫ5 + 29 If ǫ5 ≥ pe+1 then, as ǫ5 ≤ 2n ≤ 4t, it follows that √ k ≤ q + 2 − 41 pe+1 = q + 2 − 14 pq,

(13.8)

a contradiction. Suppose, therefore, that ǫ5 < pe+1 . If p 6= 3, then ǫ5 is a power of p; so ǫ5 ≤ pe and, from (13.8), t ≥ q/(4pe + 29) + 1, whence √ k > q − 14 pq + 29 16 p + 1,

the required contradiction. If p = 3, the same argument gives the result unless ǫ5 = 6, which also satisfies ǫ5 ≤ pe unless e = 1, that is, q = 27. However, the result is trivially true for q = 27. 2 T HEOREM 13.58 In PG(2, q), q = 22e+1 with e ≥ 1, p m′ (2, q) ≤ q − 2q + 2.

√ Proof. Assume that there exists a complete k-arc K with k > q − 2q + 2. As in Theorem 13.40, let Γt be the curve of degree t = q + 2 − k is associated to the dual k-arc. Such a curve Γt is defined over Fq and has a non-linear irreducible component Γ′n of degree n with n ≤ t. Let ν0 = 0, ν1 = ν be the Frobenius orders of Γ′n with respect to the linear series cut out by lines. If Γ′n is not defined√over Fq , then kn ≤ n2 . So k ≤ n ≤ t = q + 2 − n; that is, k ≤ 21 (q + 2) ≤ q − 2q + 2. If Γ′n is defined over Fq , then, using (8.20), Hence

kn ≤ 12 [ν(2g − 2) + n(q + 2)] ≤ 21 n[ν(n − 3) + q + 2]. 2k ≤ ν(n − 3) + q + 2 ≤ ν(t − 3) + q + 2

and, since k = q + 2 − n,

t≥

q + 2 + 3v1 . ν +2

p q/2, then, for q > 8,  √  √ p 2(q + 2) + 3 2q 2(q + 2) + 3 2q √ √ = 2q − 1. > t≥ 2q + 4 2q + 4 √ √ Therefore k = q + 2 − t < q + 3 − 2q. As 2q is an integer, the theorem is proved in this case. p √ Since ν ≤ √ n ≤ t and ν is a power of 2, if ν > q/2, then 2q ≤ ν1 ≤ t; so k ≤ q + 2 − 2q, as required. For q = 8, the bound is sharp as the only complete arcs other than hyperovals are 6-arcs. 2 If ν ≤

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Table 13.4 Size of the second largest complete arc in small planes

q m′

7 6

8 6

9 8

11 10

13 12

16 13

17 14

19 14

23 17

25 21

27 22

29 24

31 22

32 24

E XAMPLE 13.59 Table 13.4 gives the known values of m′ (2, q) apart from those with q ≥ 64 implicit in Theorem 13.48. Here, m′ is written for m′ (2, q). From Theorems 13.48 and 13.48 and Table 13.4, a q-arc in PG(2, q) is always incomplete. As an application of the theorems in this section, the status of (q − 1)-arcs can be clarified. To resolve the question of the completeness of (q − 1)-arcs, the plane sextic Γ6 associated to the dual (q − 1)-arc is considered. T HEOREM 13.60 If q > 9, the plane sextic Γ6 associated to a complete (q−1)-arc is irreducible. Proof. The possibilities for Γ6 are examined in three parts. (i) Γ6 contains no Fq -rational linear components. If Γ6 did contain such a component, K would be incomplete. (ii) Γ6 contains no multiple components. Suppose that Γ6 = v(F ) and F = G1 G2 , where C1 = v(G1 ) has no multiple components and C2 = v(G2 ) consists of double components. From Theorem 13.43, Γ6 does not consist entirely of double components. By (i), it may be assumed that Γ6 has just one double component, an Fq -rational irreducible conic counted twice. Hence C1 is an irreducible conic defined over Fq . By Theorem ′ 13.50, the lines of K′ are Fq -rational tangents to C2r , and K′ is incomplete. (iii) Γ6 is irreducible. If Γ6 were reducible, then one of the following three cases would apply: (a) Γ6 splits into three irreducible conics defined over Fq ; (b) Γ6 splits into an irreducible conic and an irreducible quartic, both defined over Fq (c) Γ6 splits into two irreducible, distinct cubics. Again, by Theorem 13.50, (a) and (b) are impossible, since the fact that an irreducible conic defined over Fq is a component of Γ6 implies that K is incomplete. If, in case (c), Γ6 has components Γ3 and Γ′3 , then Γ3 and Γ′3 have a set R of at most nine common points in PG(2, q). Also, each line in the dual (q − 1)-arc K′ passes through three points of Γ6 with intersection multiplicity two. If such a line ℓ passed through two points of Γ3 each with multiplicity two, then Γ3 would be reducible. Similarly, if ℓ met Γ3 at one point not in R with multiplicity two and passed through two points of R, then Γ3 would be reducible. So ℓ must pass through three points of R. Since no two lines in K′ intersect on Γ6 , it follows that q − 1 ≤ 3, a contradiction. 2 T HEOREM 13.61 In PG(2, q), a (q − 1)-arc is incomplete for q > 13 except

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perhaps for except possibly for the fourteen values of q consisting of 49, 81 and the twelve primes 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83. Proof. From Table 13.4, only q > 32 need be considered. For q even, Theorem 13.48 shows that m′ (2, q) < q − 1 for q ≥ 8. For q odd, Theorem 13.55 shows that m′ (2, q) < q − 1 for q > 103; Theorem 13.56 improves this to q > 83. 2 R EMARK 13.62 It is conjectured that the exceptions can be removed in this theorem. The corresponding irreducible sextic Γ6 has q − 1 triple tangents and so at least 3(q − 1) rational points. It needs to be shown that there must exist more rational points on Γ6 to contradict the upper bounds (8.20) or (8.25). C ONJECTURE 13.63 In PG(2, q), with q square and q > 9, √ m′ (2, q) = q − q + 1. Further, the corresponding arc is projectively unique. The lowest value of q for which this conjecture is unresolved is q = 49; that is, it must be shown that m′ (2, 49) = 43. Related to this conjecture are Example 13.47 and the following result analogous to Theorem 13.49. T HEOREM 13.64 Let K′ be the dual of the cyclic (q 2 −q+1)-arc in PG(2, q 2 ) with q odd, defined in Example 13.47. Then the associated curve Γ2(q+1) is birationally equivalent over Fq6 to the Hermitian curve Hq . The proof requires some preliminary results. For a primitive (q 2 + q + 1)-st root of unity t in Fq6 , the mapping α given by the equations, ρX0′ = tX0 ,

ρX1′ = tq

2

+1

X1 ,

ρX2′ = X2 ,

is a projectivity of order (q 4 + q 2 + 1) that fixes each vertex of the fundamental triangle U0 U1 U2 of a projective coordinate system in PG(2, K), where K = Fq . Let A denote the collineation group generated by α. The orbit Π of the point U = (1, 1, 1) under A has size q 4 + q 2 + 1 and consists 2 4 2 of all points Pc = (c, cq +1 , 1) with cq +q +1 = 1. Note that Π lies in PG(2, q 6 ) as the (q 4 + q 2 + 1)-st roots of unity are in Fq6 . The orbit Π may be viewed as a subgeometry of PG(2, K) induced by the lines meeting Π in at least two points. P ROPOSITION 13.65 The orbit Π ∼ = PG(2, q 2 ) with the following properties: (i) Π is a projective subplane of PG(2, q 6 ) lying in a non-classical position; that is, Π 6= PG(2, q 2 ) but there is a projectivity of PG(2, q 6 ) that maps Π to PG(2, q); 2

(ii) the lines of Π are v(tX0 + tq +1 X1 + X2 ), with t running over all the (q 4 + q 2 + 1)-st roots of unity; they form the line orbit of v(X0 + X1 + X2 ) under A.

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Proof. The line v(X0 + X1 + X2 ) meets Π in exactly q 2 + 1 distinct points, since 2 uq +1 + u + 1 = 0 implies that u is a (q 4 + q 2 + 1)−th root of unity. In fact, if 4 2 2 4 2 2 4 2 uq +q + uq + 1 = 0, then uq +q +1 + uq +1 + u = 0 whence uq +q +1 = 1. 2 Also, the polynomial F (X) = X q +1 + X + 1 has no multiple roots as F (X) and 2 its derivative F ′ (X) = X q + 1 have no common root. 2 For any (q 4 + q 2 + 1)−th root t ∈ Fq6 , the line lt = v(tX0 + tq +1 X1 + X2 ) meets Π in the points Pi = (ci , c(q

2

+1)i

, 1) with (ci t)q

2

+1

+ ci t + 1 = 0.

As before, there are exactly q 2 + 1 such points. Therefore Π equipped with all the lines lt is a projective plane of order q 2 . This is shown by a direct calculation. 2 Next, intersections of Π with conics and Hermitian curves are considered. P ROPOSITION 13.66 The conic C = v(X02 + X12 + X22 ) and the plane curve D = v(X0 X2q − X1q+1 ) have 2(q + 1) distinct points in common, and half of these points lie in Π. Proof. First, v(X2 ) ∩ C ∩ D = ∅. Now, choose any point P = (u, v, 1) off the line v(X2 ). Then P is a point of D if and only if u = v q+1 . In this case, P is also a point of C if and only if v 2(q+1) + v 2 + 1 = 0. The polynomial f (X), with f (X) = X 2(q+1) + X 2 + 1,

has no double roots since its derivative f ′ (X) = 2X 2q+1 + 2X = 2X(X 2 + 1)q has no root in common with f (X). Now, q + 1 of the roots v of f (X) give rise to points P = (v q+1 , v, 1) lying in Π. Since f (X)q X 2 = X 2(q

2

+q+1)

+ X 2(q+1) + X 2 , 2

each root v of f (X) is also a root of the polynomial g(X) = X 2(q +q+1) − 1. Since f (X) = f (−X) and g(X) = g(−X), half of the roots v of f (X) are also 2 roots of the polynomial h(X) = X q +q+1 − 1. This shows that the corresponding q + 1 points P = (v q+1 , v, 1) lie in Π. 2 P ROPOSITION 13.67 The conic C = v(X02 + X12 + X22 ) and the Hermitian curve Hq = v(X0 X1q + X1 X2q + X2 X0q ) have q + 1 distinct common points, and all lie in Π. Also, the curves C and Hq have the same tangent line at each of these points. Proof. From the proof of Proposition 13.66, the set Π contains q + 1 distinct points P = (v q+1 , v, 1) of C, where v is a common root of f (X) and h(X). Each of these points also lies on Hq , since v 2q+1 + v q

2

+q

+ v = v −1 (v q

2

+q+1

+ v 2(q+1) + v 2 ) = v−1(1 + v 2(q+1) + v 2 ) = 0.

q+1 A calculation shows , v, 1) has √ that the tangent line to Hq at the point P = (v slope equal to − v. The same value is obtained when Hq is replaced by C. Hence

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C and Hq are tangent at each of these q + 1 points. By B´ezout’s Theorem 3.14, there is no other common point of C and Hq , and this completes the proof. 2 Now, consider the family of conics Ct = v(Ft ), with

Ft = tX02 + tq+1 X12 + X22 ,

(13.9)

2

where t is a (q + q + 1)-st root of unity. This family consists of all conics which are images of the conic C = v(X02 + X12 + X22) under the action of the collineation group A. Since q 4 + q 2 + 1 = (q 2 + q + 1)(q 2 − q + 1), the group A has a subgroup B of order q 2 −q +1. The orbit of C under B consists of the conics Ct with t ranging over the (q 2 − q + 1)-st roots of unity. The next result connects the set Φ of points of Hq lying in Π with the sets ∆t of points of Ct lying in Π. P ROPOSITION 13.68 The sets ∆t partition Φ into q 2 − q + 1 point sets, each of size q + 1. Proof. It is first shown that B leaves Hq invariant. The image of H under αi is the curve v(X0 X1q + uq(q

2

−q+1)

X1 X2q + u(q+1)(q

2

−q+1)

X2 X0q ), 2

where u = ti . If αi is a generator of B, then i = q 2 + q + 1 and uq −q+1 = 1, whence the result follows. Proposition 13.65 remains valid when C is replaced by any conic Ct in the family. Therefore no point of Φ lies on more than one such conic. In fact, if both t and u are (q 2 −q+1)-st roots of unity, and P = (a, b, 1) is a point in Φ∩∆t ∩∆u , Proposition 13.65 implies that the conics Ct and Cu have the same tangent at P = (a, b, 1). But, this can only occur when −tq = −uq , and hence t = u. From Proposition 13.67, each conic Ct in the family meets Φ in q + 1 points. Since Φ has size q 2 + q + 1, the assertion follows. 2 A relationship between conics Ct and lines lt is given in the following result. L EMMA 13.69 The standard quadratic transformation τ given by the equations X0′ = X02 ,

X1′ = X12 ,

X2′ = X22

leaves Π invariant. The restriction of τ to Π is a bijection and transforms ∆t into the set Ψt of the common points of lt and Π. Proof. The invariance of Π follows from a direct calculation. The fact that the mapping x 7→ x2 is a permutation of the set of all (q 2 + q + 1)-st roots of unity shows that ∆t is transformed into Ψt . 2 The proof of Theorem 13.64 consists in showing that the transformation τ turns Hq into Γ2t . Let Γ be the image of Hq under τ . From Propositions 13.67, 13.68 and 13.69, each point of the q 3 + 1 points of Γ lying in Π is a non-singular point of Γ, and the line lt , where t is a (q 2 − q + 1)-st root of unity, is a (q + 1)-fold tangent to Γ; that is, lt is the tangent line to Γ at each of their common points. Such lines lt form a cyclic dual (q 2 − q + 1)-arc K′ in Π viewed as projective plane defined over Fq2 . By Theorem 13.43, Γ is the curve Γ2(q+1) associated to K′ .

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Proof of Theorem 13.64. It remains to show that τ maps H onto Γ birationally. Consider the linear system Φ of all conics, v(c0 X02 + c1 X12 + c2 X22 ), with c = (c0 , c1 , c2 ) ∈ PG(2, K).

2 It must be shown that the linear series g2(q+1) cut out on Hq by Φ is simple, that is, not composed of an involution. For a point P of Hq , the conics in Φ through P 2 form a linear subsystem of Φ. To show that g2(q+1) is simple, it suffices to prove the existence of a point P ∈ Hq such that the conics in ΦP have no further common point on Hq . It is shown that such a condition is satisfied by every point P = (a, b, c) ∈ Hq with abc 6= 0. The conics of ΦP have exactly four common points in PG(2, K), namely,

P = (a, b, c),

P1 = (a, −b, c),

P2 = (−a, b, c),

P3 = (a, b, −c).

None of the last three points belongs to Hq . Thus the point P = (a, b, c) is the only common point of the conics of ΦP which lies on Hq . This completes the proof.  A result that improves Theorem 13.51 is the following. T HEOREM 13.70 For q = ph with p ≥ 5,

m′ (2, q) ≤ q −

1√ 2 q

+ 5.

Theorem 8.104 can be applied to give better bounds for m′ (2, q). T HEOREM 13.71 Let q = ph with p ≥ 3, and let q = 32e when p = 3. (i) If q ≥ 232 and q 6= 36 or 55 , then

m′ (2, q) ≤ q −

 q − 22    q−9 ′ m (2, q) ≤ q−9    q−5

1√ 2 q

+ 3.

when q = 55 , when q = 36 , when q = 232 , when q = 192 . √ Proof. Given a k-arc K in PG(2, q) with k > q − 12 q + 3, the aim is to prove that K is contained in an irreducible conic defined over Fq . Let K′ be the complete dual k-arc arising from K in the dual plane of PG(2, q). From Theorems 13.43 and 13.50, the curve Γ2t associated to K′ does not have linear components. Further, if it has a quadratic component then K′ is a dual (q+1)arc consisting of all tangents to an Fq -irreducible conic C at its points in PG(2, q). So assume that each component of Γ2t has degree at least three. For such a component Γ = v(f (X, Y )) of degree d, two cases are distinguished according as either Γ is defined over Fq or over a non-trivial finite extension of Fq . In the latter case, f (X, Y ) is distinct from its first conjugate f (1) (X, Y ) and also from cf (1) f (X, Y ) for c ∈ K; see Section 8.3. Hence Γ is distinct from its first conjugate which is the irreducible plane curve Γ′ = v(f (1) (X, Y )) but an s-fold point P of Γ with coordinates in Fq is also an r-fold point of Γ′ . (ii)

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Let γ1 , . . . , γN be the branches of Γ centred at the points in ∆(K′ ), and let Pi be the corresponding places of K(D). If ri = j1 (Pi ) denotes the order of γi , then the centre of γi is a point of Γ whose multiplicity is at least ri . The same holds for the ′ branches γ1′ , . . . , γN of Γ′ centred at the points in ∆(K′ ). By B´ezout’s Theorem 3.14, PN 2 P √ √ ′ 2 1 q + 1). P ∈D(K′ ) I(P, Γ ∩ Γ ) ≤ d ≤ d( q − 2) < 2 d(q − i=1 ri ≤

If Mq and Mq′ , with respect to Γ, have the same meaning as in Definition 8.79, it follows that √ (13.10) 2Mq + Mq′ < d(q − q + 1). Now, let Γ be an Fq -rational irreducible component of Γ2t of degree d. If (13.10) does not hold, then Theorem 8.104 applies to Γ, showing that Γ is projectively √ equivalent over Fq to the Fermat curve of degree d = 21 ( q + 1); in particular, √ 2Mq + Mq′ = d(q − q + 1). Thus, for every irreducible component Γ of Γ2t , also √ 2Mq + Mq′ ≤ d(q − q + 1). Therefore P

(2Mq + Mq′ ) ≤ 2t(q −

√ q + 1),

(13.11)

where the summation is over all irreducible components of Γ2t . On the other hand, Theorem 13.43 provides a lower bound for the above sum. In fact, every point P ∈ ∆(K′ ) is either a non-singular point or a double point of Γ. In the former case, P counts in Mq as I(P, Γ2t ∩ ℓ) = 2 for the unique line ℓ in ∆(K′ ) which passes through P . In the latter case, P is the centre of either one branch of order 2 or of two branches of order 1. Thus P counts at least twice in 2Mq + Mq′ . Since ∆(K′ ) consists of kt points, so P 2kt ≤ (2Mq + Mq′ ). (13.12) √ Comparison of (13.11) and (13.12) gives k ≤ q − q + 1, contradicting the hy√ 2 pothesis that k > q − 12 q + 3. R EMARK 13.72 Since q is odd, in the case in which q is a square, the 3 can be lowered to 25 . The best result for the case q = 32e+1 is that √ √ m′ (2, q) ≤ q − 43 q + 103 16 .

13.9 THE THIRD LARGEST COMPLETE ARC By Theorem 13.48, the exact value of m′ (2, q) is known for every even square q. The next question is to find the maximum size m′′ (2, q) of the third largest complete arcs. The following result depending on Theorem 8.78 provides an upper bound.

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624 T HEOREM 13.73 Let q = 2h , q ≥ 16 and q square. Then  √ q−2 q+6 for q ≥ 64, ′′ √ m (2, q) ≤ q − 2 q + 4 = 12 for q = 16.

CHAPTER 13

(13.13)

Proof. Let K′ be a complete dual k-arc in PG(2, q) with q as given. It must be shown that, if k is larger than the right-hand side in (13.13), then one of the following holds: √ (i) k = q − q + 1; (ii) k = q + 2. Let Γt be the curve defined over Fq associated to K′ . It must be shown that Γt is irreducible. Let Γ be an irreducible component of Γ of degree n. From Theorem 13.40 (iii), Γ contains n distinct points on every line in K′ , and such points lie in ∆(K′ ). As in the proof of Theorem 13.71, let Γ′ be the first conjugate of Γ. These curves are distinct if and only if Γ is not defined over Fq . Since every point of Γ with coordinates in Fq is also a point of Γ′ , each of the kn points in ∆(K′ ) counts in the intersection of Γ and Γ′ . Since kn = k(q − k + 2) > n2 , B´ezout’s Theorem 3.14 shows that the two curves coincide, unless n2 ≥ nk. This inequality implies that √ n ≥ k. Since t ≥ n, also q + 2 − k ≥ k; but this contradicts that k > q − 2 q + 6. Therefore Γ is defined over Fq . Next, it is shown that Γt = Γ. For, if not, then √ √ 2n ≤ t = q + 2 − k < q + 2 − (q − 2 q + 6) = 2 q − 4 √ and n < q − 2. Let gn2 be the linear series cut out on Γ by lines. It may be that Γ is Frobenius non-classical, and if this occurs then its Frobenius order ν1 = q ′ is √ bounded above by 12 q as q ′ does not exceed the largest power of 2 smaller than √ n < q − 2. From Theorem 8.74 applied to gn2 , nk ≤ N ≤ 12 {n(n − 3)q ′ + (q + 2)n}; so 2k ≤ (n − 3)q ′ + (q + 2). √ √ √ Now, apply the conditions k > q − 2 q + 6, n < q − 2, q ′ ≤ 12 q: √ √ √ 2(q − 2 q + 6) < 2k ≤ ( q − 5) 12 q + q + 2, whence √ √ √ 4(q − 2 q + 6) < ( q − 5) q + 2(q + 2), and √ q − 3 q + 20 < 0, a contradiction. Hence n = t and Γt = Γ. √ Now, Theorem 8.78 is applied. Since k > q − 2 q + 6, so √ t = q + 2 − k < 2 q − 4. Hence, by Theorem 8.78, the number of Fq -rational simple points N0 of Γt satisfies N0 ≤ t(q + 2 − t). Since ∆(K′ ) has size t(q + 2 − t), so N0 ≥ t(q + 2 − t). Hence N0 = t(q + 2 − t) and again, by Theorem 8.78, Γt is projectively equivalent to the √ Hermitian curve of degree q + 1, which is case (i). 2

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625

13.10 EXERCISES 1. Show that, over F13 , the sextic curve Γ′6 = v(X06 + X16 + X26 + X03 X13 + X03 X23 + X13 X23 ) also has N1 = 54, but is not isomorphic to the curve Γ6 in (13.4). One method is to verify that N2 = 162 for Γ′6 but N2 = 198 for Γ6 .

13.11 NOTES Several books, such as [178], [336], [350], [364], [419], [428], [460], [486], offer a detailed treatment of algebraic-geometry codes. See also the exposition by Høholdt, van Lint, and Pellikaan [227]. Algebraic curves defined over Fq with many Fq -rational points have important applications to a wide variety of combinatorial problems, especially in finite geometry; see the survey paper of Sz˝onyi [444]. For a general account of finite projective spaces and their properties, see [216], [213], [224]. The three problems I, II, III were enunciated by Segre in his seminal paper [395]. For MDS algebraic-geometry codes, MCk has been shown for some particular cases, namely, for codes arising from elliptic curves and curves of genus two when q > 83, by Munuera [340]; see also [74]. For any fixed genus g there is a q0 such that for all q > q0 , the Main Conjecture holds for MDS algebraic-geometry codes arising from hyperelliptic curves defined over Fq . This was shown by de Boer [98] using Munuera’s approach; see also [48]. Theorem 13.23 is due to Glynn [175]. For surveys of further results on arcs and MDS codes, see [214], [215], [221], [222], and MacWilliams and Sloane [317, Chapter 11]. Bounds on the size of the second largest complete arcs in small planes of order q ≥ 37 are found in [126], [97]. The formal definition of near-MDS codes is due to Dodunekov and Landjev [106]. The exact values of the maximum length of a near-MDS code were calculated by Dodunekov and Landjev [107] for q = 2, 3, 4, 5. Marcugini, Milani, and Pambianco considered near-MDS codes for k = 3, 4 and obtained a complete classification of those of maximal length for q ≤ 11; see [323]. For more results on near-MDS codes, see Abatangelo and Larato [2], Ballico and Cossidente [36], Giulietti [169], and Landjev [290]. Sections 13.3, 13.4, 13.5 are all based on Segre’s work [394], [396]. Fisher, Hirschfeld, and Thas [129], and, independently, Boros and Sz˝onyi [57], Cossidente [87], Kestenband [267], and Ebert [112], showed that every point orbit of S is a complete (q 2 − q + 1)-arc K in PG(2, q 2 ), and every line orbit of S is a complete dual (q 2 − q + 1)-arc K′ in PG(2, q 2 ). In [129] it is also shown for even q that the plane curve Γq+1 associated with K′ , via Segre’s generalisation of Menelaus’ Theorem, is the Hermitian curve; see Theorem 13.49. For more results on cyclic arcs, see [443], [434], [324].

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Theorem 13.51 is due to Segre [398]. Theorem 13.55 is due to Thas [453]. Theorem 13.56 is due to Voloch [492]; Theorems 13.57 and 13.58 come from Voloch [495]. Theorem 13.60 is due to Bichara and Migliori [52]. For Theorem 13.70, see [217]; for Theorem 13.71, see [218]. For a proof that a q-arc is contained in the rational points of a conic, see [396]. Direct proofs that a q-arc in PG(2, q) is always incomplete are found in [395] for odd q and in [446] for even q. For the classical result that τ2 ≤ 21 d(d − 2)(d2 − 9), see, for example, [396]. Complete k-arcs can be constructed from elliptic curves over Fq . Let F be a non-singular plane cubic curve defined over Fq , equipped with the group GF (Fq ) isomorphic to Pic0 (Fq (X )), the zero-degree Fq -rational divisor class group of F. Assume that GF (Fq ) has even order, and let H be its subgroup of index 2. Then the the points of F associated to the elements of the coset GF (Fq ) \H form an arc in PG(2, q). Such an arc is either complete or may completed by adding a few points. For the proofs, see [493], [494], [167]. For Exercise 1, see [96].

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Appendix A Background on field theory and group theory

A.1 FIELD THEORY In this section, fundamental concepts and results from field theory are listed. Apart from two generalisations of the Theorem of the Primitive Element, all the material with proofs can be found in standard textbooks. (I) The characteristic of a field F is the smallest positive integer n, if it exists, such that 1 + 1 + . . . + 1 = 0. | {z } n times

In this case, n is prime and is denoted by p. If there is no such n, then F is of characteristic 0; write p = 0. (II) Every field has a prime field, that is, a subfield containing no subfield other than itself. If p > 0, the prime field is isomorphic to Zp = Z/pZ = Fp , the field of integers modulo p, while, if p = 0, the prime field is isomorphic to Q, the field of rational numbers. (III) Let L be a field that contains F as a subfield. Then L is an extension of F , and the symbol L/F is used to express this. If L is regarded as a vector space over F , its dimension is the degree of L/K and denoted by [L : K]. If [L : K] = n < ∞, then L/K is a finite extension. If both L/F and M/L are finite extensions then M/F is also finite, and its degree is [M : F ] = [M : L][L : F ]. (IV) An element α ∈ L is either algebraic or transcendental over a subfield F of L, according as there is or is not a non-constant polynomial f (X) ∈ F [X] satisfied by α; that is, f (α) = 0. Every algebraic element α satisfying the polynomial X n − 1 is an n-th root of unity. If, in addition, α does not satisfy any polynomial X k − 1 with 1 ≤ k < n, then α is a primitive n-th root of unity. (V) The field L/F is an algebraic extension, and L is algebraic over F , when all elements in L are algebraic over F ; otherwise, L is a transcendental extension. In the latter case, L has at least one transcendental element over F .

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APPENDIX A

(VI) If L1 /F and L2 /F are any two extensions, then a mapping σ : L1 → L2 is a field homomorphism if σ(x + y) = σ(x) + σ(y),

σ(xy) = σ(x)σ(y),

for all x, y ∈ L1 . (VII) A field homomorphism σ : L1 → L2 is an embedding of L1 into L2 over F provided that σ(x) = x for all x ∈ F .

Such a homomorphism must be injective, giving an isomorphism of L1 onto the subfield σ(L1 ) of L2 . A surjective, and hence bijective, embedding of L1 into L2 is an F -isomorphism.

(VIII) A field M is algebraically closed if every polynomial f (X) ∈ M [X] of positive degree has a root in M . For every field F , there exists an algebraic field extension F¯ /F such that F¯ is algebraically closed. Then, F¯ is uniquely determined up to an F -isomorphism, and it is the algebraic closure of F . If L/F is an algebraic extension, then there exists an embedding L → F¯ over F . There are several different embeddings but their number is at most [L : F ]. (IX) If α1 , . . . , αr ∈ L, then the smallest subfield M of L containing F and all elements α1 , . . . , αr is denoted by F (α1 , . . . , αr ), and α1 , . . . , αr are the adjoined elements of F (α1 , . . . , αr ). The field extension F (α1 , . . . , αr )/F is algebraic if and only if every αi is algebraic over F . (X) If α is an algebraic element of L/F , there is a unique, monic polynomial f (X) over F of minimal degree satisfied by α. This minimal degree is the degree of α over F . The polynomial f (X) is uniquely determined, and it is the minimal polynomial of α over F . Other polynomials g(X) ∈ F [X] can be satisfied by α, but this only occurs when f (X) | g(X). (XI) An element α ∈ L is algebraic over F if and only if [F (α) : F ] = n < ∞. Let f (X) be the minimal polynomial of α over F , and put r = deg f (X). Then [F (α) : F ] = r, and a basis of F (α)/F is {1, α, . . . , αr−1 }. (XII) Conversely, let f (X) ∈ F [X] be a non-constant polynomial; then, there exists an algebraic extension field L = K(α) such that f (α) = 0. If f (X) is irreducible over F , this extension field is uniquely determined up to an F isomorphism. In other words, if L′ = K(α′ ) is another algebraic extension field with f (α′ ) = 0, then there exists an F -isomorphism σ : L → L′ with σ(α) = α′ . (XIII) More generally, if f1 (X), . . . , fr (X) ∈ F [X] are monic polynomials of degree d1 , . . . , dr , then there is an algebraic extension M , unique up to an F -isomorphism, containing F such that all fi (X) split into linear factors over M ; that is, Q di fi (X) = j=1 (X − aij ), aij ∈ M.

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629

The field F (a11 , . . . , ardr ) is uniquely determined up to an F -isomorphism and is the splitting field of f1 , . . . , fr over F . Over the prime field of F , the decomposition Q X n − 1 = d|n Fd (X)

holds for every natural number n not divisible by p for p > 0, where Q Fn (X) = d|n (X d − 1)µ(n/d)

is a polynomial of degree ϕ(n) whose zeros in a splitting field of X n − 1 are exactly the primitive n-th roots of unity. Here µ(i) is the M¨obius function:  if i = 1,  1 0 if i is divisible by the square of a prime , (A.1) µ(i) =  (−1)j if i is the product of j distinct primes.

and ϕ(n) is the number of all primitive n-th roots of unity. The polynomial Fn (X) is the n-th cyclotomic polynomial.

(XIV) Let f (X) ∈ F [X] be a monic polynomial of degree d ≤ 1. There is an algebraic extension L of K in which f (X) splits into linear factors: Q f (X) = di=1 (X − αi ).

The polynomial f (X) is separable if the roots of f (X) are distinct. If, however, αi = αj for some i 6= j, then f (X) is inseparable.

If F has zero characteristic, then all irreducible polynomials over F are separable.PIn the case of positive characteristic, an irreducible polynomial, f (X) = ai X i ∈ F [X], is separable if and only if ai 6≡ 0 (mod p) for at least one index i. P (XV) The derivative of a polynomial f (X) = ai X i ∈ F [X] is the polynomial P f ′ (X) = iai X i−1 . A separability criterion is that f ′ (X) is not the zero polynomial.

(XVI) Given an algebraic extension L/K, an element α ∈ L is separable over F if its minimal polynomial f (X) ∈ F [X] is a separable polynomial. The field L/K is a separable extension if all α ∈ L are separable over F . In zero characteristic, every algebraic extension is separable. (XVII) If L/K is a finite extension of degree [L : K] = n, then L/K is separable if and only if there are n distinct embeddings σ1 , . . . , σn of L into F¯ over F . In this case, an element α ∈ L is in F if and only if σi (α) = α for i = 1, . . . , n.

For a tower M ⊃ L ⊃ F of algebraic extensions, the field M/F is separable if and only if both M/L and L/F are separable.

(XVIII) In positive characteristic, inseparable extensions exist. An element α ∈ L r is purely inseparable over F if some power αp of α with r ≥ 0 lies in F . This occurs if and only if the minimal polynomial of α over F is of type e X p − c with c ∈ F and e ≤ r. The extension L/F is purely inseparable if all elements α ∈ L are purely inseparable.

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APPENDIX A

(XIX) Given any algebraic extension L/K, there exists a unique intermediate field S, that is, F ⊂ S ⊂ M , such that S/K is separable but L/S is purely inseparable. This depends on the fact that the set of all separable elements form a subfield. (XX) If L is a field containing F as a subfield, the elements α1 , . . . , αr ∈ L are algebraically dependent over F if they satisfy a non-trivial polynomial relation over F . Otherwise, α1 , . . . , αr are algebraically independent over F . (XXI) An infinite subset A is algebraically independent over F if every finite subset in A is algebraically independent; otherwise, A is algebraically dependent. If α1 , α2 , . . . are algebraically independent over F and L/F (α1 , α2 , . . .) is an algebraic extension, then {α1 , α2 , . . .} is a transcendence basis of L over F. (XXII) If {α1 , . . . , αs } is a finite transcendence basis of L over F , then any s + 1 or more elements of L are algebraically dependent over F . Any two finite transcendence bases of L over F have the same number of elements, and this number is the transcendence degree of L over F . Note that the function field K(F) of an irreducible plane curve F is a finite extension of an algebraically closed field K of transcendence degree 1 over K. So, any two elements in K(F) are algebraically dependent over K. (XXIII) A field F is perfect if all algebraic extensions L/F are separable. In zero characteristic, every field is perfect. In positive characteristic p, a field F is perfect if and only if every element of F is a p-th power of some element in F. Important examples of perfect fields are the algebraically closed fields and finite fields. (XXIV) An algebraic field extension L/F is simple if L = F (α) for some α ∈ L. A primitive element of L over F is any element β ∈ L such that L = F (β).

It should be noted that the elements of a simple extension field F (α) may be written in the form g(α), where g(X) is a polynomial over F of degree at most n − 1, with n = [F (α) : F ].

Here, F (α) is an algebra of rank n with basis {1, α, . . . , αn−1 }. Using the minimal polynomial of α, say f (X) = X n +b1 X n−1 +. . .+bn with bi ∈ F , the product of two arbitrary elements of the algebra may be calculated.

(XXV) In positive characteristic, a finite algebraic extension L/F may not be simple. A sufficient condition for a finite extension L/F to be simple is that the adjoined elements α1 , . . . , αr be, with at most one exception, separable over F. (XXVI) A complex number is algebraic if it is algebraic over Q; that is, it satisfies a non-zero polynomial equation with coefficients in Q. The set A of algebraic

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numbers is a subfield of C. A subfield of A which is an algebraic extension of Q is an algebraic number field. An algebraic number is an algebraic integer if it satisfies a monic polynomial equation with coefficients in Z. The set B of algebraic integers is a subring of A. An algebraic integer is a rational number if and only if it is a rational integer; that is, B ∩ Q = Z. The following result is an essential ingredient of other chapters. T HEOREM A.1 (Theorem of the Primitive Element) Let F be a field containing infinitely many elements, and let F (α, β) in an algebraic extension of F . If α, β are separable over F, then there exists λ ∈ F such that α + λβ is a primitive element of F (α, β)/F . Proof. Let z be an indeterminate over F (α, β). Every irreducible polynomial G(X) ∈ F [X] is also irreducible when regarded as a polynomial in F (z)[X]. This can be shown indirectly. If G(X) = U (z, X)V (z, X)/d(z) with U, V ∈ F (z, X) and d(z) ∈ F (z), then G(X) = U (λ, X)(V λ, X)/d(λ), for any λ ∈ F such that d(λ) 6= 0, is a factorisation in F [X]. Such a λ exists as F contains infinitely many elements. Therefore α, β are separable over F (z) as well, because they satisfy the same irreducible equations over F (z) as over F . Then w = α + zβ is separable over F (z). Consider the simple extension F (z)(w)/F (z) and denote by g(W ) ∈ F (z) the minimal polynomial of w over F (z). Then g(W ) comes from a polynomial G(X, W ) ∈ F (X, W ) in the usual way; that is, g(W ) = G(z, W ). Since w is separable, ∂G/∂W 6= 0. From the fact that degW it follows that



∂G < degW G, ∂W

∂G ∂W



W =w

6= 0.

Now, G(z, α + zβ) = 0, whence     ∂G(z, W ) ∂G(z, W ) + · β = 0. ∂z ∂W W =α+zβ W =α+zβ From this, β=

H(z, α + zβ) , d1 (z)

with H(X, W ) ∈ F [X, W ] and d1 (z) ∈ F [z]. So, for λ ∈ F \{0}, β=

H(λ, α + λβ) . d1 (λ)

Thus β ∈ F (α + λβ). Since α ∈ F (α + λβ), the assertion follows. A major result on primitive elements is the following. T HEOREM A.2 Every finite separable extension L/K is simple.

2

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This theorem holds without the hypothesis on separability provided that L has transcendence degree 1. A proof is given, for which two lemmas are required. L EMMA A.3 For p > 0, let x be a transcendental element over K, let v be the p-th root of x in the algebraic closure K(x) of K(x), and put v = x1/p . Then [K(x1/p ) : K(x)] = p;

(i)

e

(ii) if e is a positive integer and x1/p is a root of x of order pe , e

[K(x1/p ) : K(x)] = pe . Proof. The first assertion is a consequence of the irreducibility of the polynomial Z p − x ∈ K(x)[Z] over K. Applying this to x1/p gives the following: 2

2

[K(x1/p ) : K(x)] = [K(x1/p ) : K(x1/p ] · [K(x1/p ) : K(x)] = p · p = p2 . e

Repeating the argument shows that K(x1/p : K(x)] = pe .

2

L EMMA A.4 Let K have the following properties: (a) p > 0; (b) the element u is in an extension field of K(x) which is algebraic over K(x), where x is transcendental; (c) the element x is not separable in K(x, u), where u is a root of an irreducible e polynomial F ∈ K(x)[U p ] for some e ≥ 1; e+1

(d) the polynomial F does not belong to K(x)[U p

].

Then e

(i) K(x, u) contains a root x1/p of order pe of x; e

(ii) u is a separable element over K(x1/p ). e

Proof. From the hypothesis, put G(x, U p ) = F (x, U ). If degV G(x, V ) = d, e then degU F (x, U ) = dpe and [K(x, u) : K(x)] loss of generality, P =i dppe. Without suppose that F ∈ K[x, U ] and write G = cij x (U )j . In K(x), the element u satisfies the polynomial P 1/pe e e G(1/p ) = cij x1/p U j e

of degree d which is the minimal polynomial of u over K(x1/p ). In fact, if u e e satisfied a polynomial H(x1/p , U ) of degree d′ < d, then u would satisfy H (1/p ) , which is a polynomial over K(x) of degree d′ ep < dep , a contradiction. Hence e

e

[K(x1/p , u) : K(x1/p )] = d and, from Lemma A.3, it follows that e

[K(x1/p , u) : K(x)] = dpe . e

e

Thus K(x, u) = K(x1/p , u), whence x1/p ∈ K(x, u). Finally, (ii) is a consequence of the definition of e. 2

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T HEOREM A.5 (Theorem of the Primitive Element for Function Fields of Transcendence Degree 1) Let Σ be a field of transcendence degree 1 over an algebraically closed subfield K. If x ∈ Σ is transcendental over K, then there is y ∈ Σ such that Σ = K(x, y). Proof. By Theorem A.2, take x to be not separating. Let u be a separating function of Σ. Then Theorem A.2 ensures the existence of v ∈ Σ such that Σ = K(u, v). e With the notation of Lemma A.3, from (ii) in that lemma, up is separable over n K(x) for some e ≥ 0. Similarly, there is an integer n ≥ 0 such that v p is separable over K(x). Let k = min{m, n}. By the sufficient condition for an extension to be e n k simple, K(x, up , v p , x1/p ) = K(x, y) for some y ∈ Σ. e n e To complete the proof, it suffices to show that u, v ∈ K(x, up , v p , x1/p ). In e proving this for u, the hypothesis e > 0 may be assumed. Then x1/p is a root of e e the polynomial Z p − x over K(x), and hence over K(x, up ). This polynomial is e irreducible over K(x, up ) by the following argument. Suppose e

0 < m1 , m2 < p m .

Z p − x = (Z m1 + · · · )(Z m2 + · · · ), pe

In the algebraic closure K(x, u ), e

e

e

Z p − x = (Z − x1/p )p ,

e

which shows that Z m1 + · · · and Z m2 + · · · must each be a power of Z − x1/p . Let ps be the largest power of p dividing m1 . Since m1 < pe , so s < e. Then ps = tm1 + rps with t, r integers. Therefore e−s

x1/p

e

e

= (xm1 /p )s xr ∈ K(x, up ). e−s

e−s

The minimal polynomial of x1/p over K(x) is Z p − x contradicting the pe separability of K(x, u ) over K(x). e e e It follows that [K(x, up , x1/p ) : K(x, up ] = pe . Putting e

d = [K(x, up ) : K(x)], e

e

e

e

this gives [K(x, up , x1/p ) : K(x)] = dpe . Since K(x, up , x1/p ) is a sube field of K(x1/p , u) and the latter coincides with K(x, u) by Lemma A.4, from e e [K(x, u) : K(x)] = dpe the assertion K(x, u) = K(up , x1/p ) follows. Hence e e e e u ∈ K(x, up , x1/p ). Similarly, v ∈ K(x, v p , x1/p ). 2 A.2 GALOIS THEORY (I) An extension L/F is normal if, whenever f (X) ∈ F [X] is an irreducible polynomial which has one root in L, then all roots of f (X) lie in L. A finite extension L/F is normal if and only if L is the splitting field of a polynomial in F [X]. In particular, a finite normal extension is separable if and only if L is the splitting field of a separable polynomial over F . (II) A finite, normal, separable extension L/F is a Galois extension. Galois theory relates field extensions to groups via the concept of an automorphism.

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(III) An isomorphism of a field L onto itself is an automorphism. The automorphisms of L form a group, the automorphism group of L, written Aut(L). If H ⊂ Aut(L), then the set of all fixed elements of H, written LH , is a subfield of L, the fixed field of H. (IV) If L/F is finite, then Aut(L/F ) is also finite as its order is less than or equal to [L : F ]. If L/F is a Galois extension then equality holds, and Aut(L/F ) is the Galois group of L/F . Usually, the Galois group is denoted by Gal(L/F ). (V) The most important result on Galois extensions is the Galois correspondence which is a connection between subgroups H of Gal(L/F ) and subfields M of L intermediate between F and L. More precisely, the maps M 7→ Gal(L/M ), H 7→ H L are mutually inverse bijections between the subfields of L containing F and the subgroups of Gal(L/F ). (VI)

(a) For every subgroup H of Gal(L/F ), [L : LH ] = |H|,

[LH : F ] = |Gal(L/F )/|H|,

H = Gal(L/LH ).

(b) For every two subgroups H, G of Gal(L/F ), H ⊂ G ⇐⇒ LH ⊃ LG . (c) For every intermediate field M of L/F , there is a subgroup H of Gal(L/F ) such that H = Gal(L/M ). (d) For every two intermediate fields M1 , M2 of L/F , Gal(L/hM1 , M2 i) = Gal(L/M1 ) ∩ Gal(L/M2 ), where hM1 , M2 i is the subfield generated by M1 and M2 , and Gal(L/(M1 ∩ M2 ) = hGal(L/M1 ), Gal(L/M2 )i, where hG1 , G2 i is the group generated by G1 and G2 .

(e) A subgroup H of Gal(L/F ) is a normal subgroup if and only if LH /F is a Galois extension. (f) If the equivalent conditions of (e) hold, then Gal(M/F ) is isomorphic to the factor group Gal(L/F )/Gal(L/M ). ¯ be the algebraic closure of (VII) Let L/F be a finite separable extension, and let L L. Then there exists a unique field M with following properties: ¯ (a) L ⊂ M ⊂ L;

(b) M/F is a Galois extension; ¯ and N/F is Galois extension, then M ⊂ N . (c) if L ⊂ N ⊂ L The field M is the Galois closure of L/F .

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(VIII) A Galois extension L/F is cyclic if Gal(L/F ) is cyclic. Here, two relevant special cases are considered, namely n = [L : F ] with n prime to p, and [L : F ] = p. (IX) If gcd(n, p) = 1, then L/F is a Kummer extension if F contains all n-th roots of unity; that is, the polynomial X n − 1 splits into linear factors in F [X]. Every Kummer extension can be described as a simple algebraic extension in the following way. Take an element c ∈ F which is not an m-th power in F for any divisor m of n. Then the polynomial X n − c is irreducible in F [X], and L/F is the simple extension F (v) with v n − c = 0. The automorphisms σ in Gal(L/F ) are given by σ(u) = ξu as ξ ranges over the n-th roots of unity in F . Conversely, If F contains all n-th roots of unity, n being prime to p, and L = F (v) where v n − c = 0 with c satisfying the above condition, then L/F is a cyclic extension of degree n. (X) When [L : F ] = p, take an element c ∈ F that cannot be written as ap − a for any a ∈ F . Then X p − X − c is an irreducible polynomial in F [X], and the simple extension L = F (v) with v n − v − c = 0 is an Artin–Schreier extension of degree p. The automorphisms σ in Gal(L/F ) are given by σ(u) = u + η as η ranges over the p elements in the prime field of F . Conversely, if L = F (v) is a simple extension where v p − v − c = 0 and c satisfies the above condition, then L/K is cyclic of degree p.

A.3 NORMS AND TRACES (I) Let L/F be a finite extension of degree n = [L : F ]. Since L may be viewed as a vector space over F , every element u ∈ L induces a linear transformation νu of L given by νu (x) = ux for x ∈ L. (II) The norm of u with respect to the extension L/F is NL/F (u) = det(νu ) in F . (III) The trace of u with respect to L/F , written TL/F (νu ), is the trace of νu . In other words, if {u1 , . . . , un } is a basis of L/F and Pn uui = j=1 aij ui with aij ∈ F , then

(a) NL/F (u) = det(aij ), Pn (b) TL/F (u) = j=1 aii .

(IV) For x, y ∈ L and α ∈ F , with n = [L : F ],

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(a) NL/F (α) = αn ; (b) TL/F (α) = n · α;

(c) NL/F (x · y) = NL/F (x) · NL/F (y);

(d) NL/F (x) = 0 if and only if x = 0;

(e) TL/F (x + y) = TL/F (x) + TL/F (y); (f) TL/F (α · x) = α · TL/F (x);

(g) for any finite extension M/L and u ∈ M , (i) NM/F (u) = NL/F (NM/L (u)). (ii) TM/F (u) = TL/F (TM/L (u));

(V) A finite extension L/F is inseparable if and only if TL/F (x) = 0 for every x ∈ L. (VI) L/F is separable if and only if TL/F : L → F is surjective. (VII) If f (X) = X r + a1 X r−1 + . . . + ar ∈ F [X] is the minimal polynomial of a ∈ L over F , and [L : F ] = n = rs with s = [L : F (a)], then (a) NL/F (a) = (−1)n asr ; (b) TL/F (a) = −sa1 . (VIII) Given a separable field extension L/F of degree n, let σ1 , . . . , σn be the distinct embeddings of L into the algebraic closure F¯ of F . Then, for every a ∈ L, Qn (a) NL/F (x) = i=1 σi (x); Pn (b) TL/F (x) = i=1 σi (x). In the case of a Galois extension with Galois group G = Gal(L/F ), these become the following: Q (a) NL/F (x) = σ∈G σ(x); P (b) TL/F (x) = nσ∈G σ(x).

A.4 FINITE FIELDS Finite fields, also called Galois fields, are all of positive characteristic and have been classified. Their structure is described below. For q = pn , the splitting field Fq of the irreducible polynomial X q − X over the field Z/pZ has order q; that is, it consists of q elements, and is of characteristic p. (I) Up to isomorphism, the fields Fq are the only finite fields. (II) The multiplicative group of Fq is a cyclic group of order q − 1.

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(III) The additive group of Fq is an elementary abelian group of order q. (IV) Every subfield of Fq is isomorphic to some Fpi for a divisor i of n. Conversely, for every divisor i of n, the field Fq has a unique subfield of order pi . (V) The automorphism group of Fq is the cyclic group of order n, generated by σ : x 7→ xp . (VI) Fq ⊂ Fqm for m ≥ 1. (VII) The extension Fqm /Fq is a Galois extension of degree m. (VIII) The Galois group Gal(Fqm /Fq ) is cyclic and is generated by the Frobenius automorphism Φ of Fqm given by Φ(x) = xq . (IX) The norm, N(x) = NFqm /Fq (x) = x1+q+q

2

+···+qm−1

.

(X) The trace, 2

T(x) = TFqm /Fq (x) = x + xq + xq + · · · + xq

m−1

.

q

(XI) When x ∈ Fqm , then TFqm (x) = 0 if and only if x = y − y for some y ∈ Fqm . L EMMA A.6 (Lucas) (i) Let p be a prime and let the positive integers n, m have the p-adic expansions P P n = n i p i , m = mi p i , with 0 ≤ ni ≤ p − 1, 0 ≤ mi ≤ p − 1. Then   n 6≡ 0 (mod p) m if and only ni ≥ mi for all i.

(ii) With q = ph ,

  nq ≡n q

(mod p).

Proof. (i) For an indeterminate X, consider the following expansion over Fp : r

(1 + X)n = (1 + X)n0 +n1 p+···+nr p

r

= (1 + X)n0 (1 + X p )n1 · · · (1 + X p )nr . Now, pick out the coefficient of X m from both sides:        n n0 n1 nr = ··· . m0 m1 mr m This gives the result. P (ii) With n = ni pi and q = ph the p-adic expansions, compare the coefficient of X q on both sides of h r+h (1 + X)nq = (1 + X)n0 p · · · (1 + X)nr p . Also, note that n ≡ nq (mod p) to obtain the result. 2

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A.5 GROUP THEORY The purpose here is to state definitions and results from group theory which are applied in Chapter 11. (I) The order of a group G, written |G| or ord G, is the cardinality of the set G. If this cardinality if finite, then |G| is the number of elements in G. Let p be a prime. A finite group G is a p-group if its order is a power of p. (II) Two elements x, y in a group G are conjugate if y = g −1 xg for some g in G. Similarly, two subgroups G, H are conjugate if every element in H is conjugate to some element in G. (III) A cyclic group G is a group generated by a single element g; that is, G consists of all the powers of g, and is denoted G = hgi. The order of g is the order of hgi, written ord g. Hence G = {1, g, g 2 , . . . , g n−1 | g n = 1} if G is finite,

G = {. . . , g −2 , g −1 , 1, g, g 2, . . .}

if G is infinite.

A dihedral group G is a finite group Dn of order 2n generated by two elements g and h together with the following relations: ord g = 2,

ord h = n,

ghg = h−1 .

Other important groups generated by two elements g, h are the generalised quaternion groups and the semidihedral groups; both are 2-groups. The generalised quaternion group of order 2e+1 with e > 1 is defined by the relations ord g = 2e ,

e−1

h2 = g 2

h−1 gh = g −1 .

,

The semi-dihedral group of order 2e+1 with e > 1 is defined by the relations ord g = 2e ,

ord h = 2,

e−1

h−1 gh = g 2

−1

.

(IV) An group G is abelian or commutative if xy = yx for all elements x, y of G. (V) The centre Z = Z(G) of a group G is Z = {x ∈ G | xg = gx all g ∈ G}. So a group is abelian if it coincides with its centre. Cyclic groups are abelian. (VI) The direct product of the groups G and H is G × H = {(g, h) | g ∈ G, h ∈ H} such that (g1 , h1 )(g2 , h2 ) = (g1 g2 , h1 h2 ). The direct product of abelian groups is abelian. (VII) (Lagrange’s Theorem) If H is a subgroup of a finite group G, then |H| divides |G|, and |G|/|H| is the number of all cosets of H in G; in particular, if N is a normal subgroup of the finite group G, then |G|/|N | = |G/N |.

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(VIII) The converse of Lagrange’s Theorem holds for subgroups of prime power order by Sylow’s Theorem. Let d be a prime dividing |G|. If dn divides |G| but dn+1 does not, then G has a subgroup of order dn . Such a subgroup is a d-Sylow subgroup of G. Any two d-Sylow subgroups of G are conjugate in G and their number is congruent to 1 modulo d. (IX) Given a subgroup of H of G, the left coset of an element g ∈ G is the set gH = {gh | h ∈ H} and the right coset of g is the set Hg = {hg | h ∈ H}. If G is finite, then the number of right or left cosets is the index of H. A subgroup N of G is normal, written N ⊳ G, if gN = N g for every g ∈ G. For instance, subgroups of index 2 are normal. A subgroup is normal if and only it is conjugate only to itself. The trivial normal subgroups of G are G and {1} = I. If G has only trivial normal subgroups, then G is simple.

(X) If N is a normal subgroup of G, then the set of all left or right cosets of N in G form the quotient or factor group G/N under the operation (g1 N )(g2 N ) = (g1 g2 )N. (XI) Given two groups A and B, an extension of A by B is a group G which contains a normal subgroup N such that N is isomorphic to A while G/N is isomorphic to B. If G = N H and N ∩ H = {1}, then the extension is split, or, equivalently, G is the semidirect product of N by H, written G = N ⋊ H. In this context, H is a complement. A sufficient condition for an extension to be split is that |N | and [G : N ] have no common divisor. The latter condition also implies that complements are conjugate in G. If a normal subgroup S of G contains a Sylow d-subgroup Sd of G, then G=NG (Sd ) S. (XII) A minimal normal subgroup of G is a normal subgroup N with N 6= I which does not contain properly any other non-trivial normal subgroups of G. Every minimal normal subgroup N of a finite group G is a direct product of N = T1 × · · · × Tk , where the Ti are simple normal subgroups of N , pairwise conjugate under G. The socle of G is the subgroup generated by all minimal normal non-trivial subgroups of G, written soc(G). (XIII) If x, y ∈ G, the element x−1 y −1 xy ∈ G is the commutator of x and y. The subgroup G′ generated by all commutators of G is the derived group of G. The factor group G/G′ is abelian and, if N is a normal subgroup of G such that G/N is abelian, then N ⊃ G′ . (XIV) A group G is soluble if the sequence G > G′ > · · · > G(i) > · · · , in which each group G(i) is the derived group of the preceding, terminates with I. (XV) A group G is nilpotent if it has a finite sequence G = G0 ⊲ G1 ⊲ · · · ⊲ Gi ⊲ · · · such that Gi−1 /Gi < Z(G/Gi ) for every i ≥ 1. A finite group is nilpotent if and only if it is the direct product of its Sylow subgroups.

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(XVI) For g ∈ G, the centraliser CG (g) = {x ∈ G | xg = gx}. The intersection of all centralisers in G is the centre Z = Z(G) and is a normal subgroup G. If |G| = pn with p prime and n > 1, then Z(G) 6= I, and Z(G) ∩ N is non-trivial for every N ⊳ G. Also, a finite 2-group which contains an element of order 2 whose centraliser has order 4 is either dihedral or semidihedral or cyclic of order 4. (XVII) For a subgroup H of G, the normaliser NG (H) = {g ∈ G | gHg −1 = H}. Assume that H has odd order. If NG (H) has even order and g ∈ NG (H) has order 2, let H1 = {h ∈ H | ghg −1 = h},

H−1 = {h ∈ H | ghg −1 = h−1 }.

Then H = {h1 h−1 | h1 ∈ H1 , h−1 ∈ H−1 }. Also, if H1 is trivial, then H is abelian. (XVIII) The Frattini subgroup of G is the intersection of all maximal subgroups of G. (XIX) The group of all permutations of a finite set Ω = {ω1 , . . . , ωn } is the symmetric group Sn of Ω. Its order |Sn | is equal to n!.

The normal subgroup of Sn of index 2 consisting of all even permutations of Ω is the alternating group An of Ω.

A permutation group G is a group whose elements are permutations of a finite set, that is, a subgroup of a symmetric group. (XX) If H is a finite group and G is a permutation group with a homomorphism ρ : H → G, then H/ ker ρ ∼ = G and G is the permutation group induced by H on Ω, or G is a permutation representation of H. If ker ρ is trivial, then this representation is faithful. (XXI) Let G be a permutation group on a finite set Ω whose elements are called points. The degree of G is |Ω|. The orbit of ω ∈ Ω under G, written G(ω), consists of the images of ω under G; that is, G(ω) = {ω ′ | ω ′ = g(ω), g ∈ G}.

The orbit structure of G is the list of lengths of its orbits, each counted with multiplicity. (XXII) The stabiliser of a point ω ∈ Ω, written Gω , is the subgroup consisting of all elements in G that fix ω; that is, Gω = {g ∈ G | g(ω) = ω}.

More generally, if ∆ ⊂ Ω, then the stabiliser of ∆ under G, written G∆ , is the subgroup consisting of all elements in G which fix each point in ∆; that is, G∆ = {g ∈ G | g(ω) = ω, ω ∈ ∆}.

Similarly, the global stabiliser of ∆, written G{∆} , consists of all elements in G which preserve ∆; that is, G{∆} = {g ∈ G | g(ω) ∈ ∆, ω ∈ ∆}. If ∆ has size n, then G∆ is an n-point stabiliser of G.

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(XXIII) The length of an orbit and the stabiliser of a point in the orbit are connected: |G| = |G(ω)| |Gω |. In particular, if Gω = I, then |G(ω)| = |G| and the orbit is a long orbit; otherwise G(ω) is a short orbit. If every orbit is long, that is, no non-trivial element in G fixes a point in Ω, then G is a semiregular permutation group on Ω. The orbit structure of G is the list of the lengths of its orbits. (XXIV) The group G is transitive if it has only one orbit, and it is regular or sharply transitive if it is also semiregular. So, G is transitive on Ω if and only if, for any ordered pair (ω, δ) with ω, δ ∈ Ω, there exists g ∈ G such that δ = g(ω), and G is regular if there is exactly one such permutation g ∈ G. (XXV) More generally, G is n-fold transitive on Ω if, for every n ordered pairs (ωi , δi ), there exists g ∈ G such that δi = g(ωi ) for i = 1, . . . , n, and sharply n-fold transitive if there is exactly one such permutation in G. Note that, if G is (n − 1)-fold transitive on Ω and the n-point stabilisers are transitive on the remaining points, then G is n-fold transitive, and vice versa. Also, if G is sharply n-fold transitive, then its order is equal to |Ω|(|Ω| − 1) · · · (|Ω| − n + 1). (XXVI) Let G be a 2-transitive permutation group on a finite set Ω of size n and let N be a minimal normal subgroup of G. (i) If N is regular, then N is elementary abelian of order n = dk for some prime number d and G is a subgroup of GL(k, d). (ii) If N is not regular, then N is a non-abelian simple group such that N ≤ G ≤ Aut(N ). When (i) occurs and n = 2k , the group G is a subgroup of AΓL(1, 2k ). This does not hold true for every odd n. However, for n = 9, the group G is isomorphic to one of AΓL(1, 9),

GL(1, 9),

AγL(1, 9).

Here, AγL(1, 9) is the subgroup of AΓL(1, 9) consisting of all permutations x → axσ + b where b ∈ F9 , where a is a non-zero square in F9 and where σ ∈ Aut(F9 ). Both GL(1, 9 and AγL(1, 9) are sharply 2-transitive, but the 2-point stabiliser is cyclic only if G ∼ = GL(1, 9). (XXVII) A Frobenius group is a transitive group which is not regular, but in which only the identity has more than one fixed point. Those elements in a Frobenius group F which have no fixed point together with the identity constitute a regular normal nilpotent subgroup N , the kernel of F . Also, G = N ⋊ Gω for any ω ∈ Ω.

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The subgroup Gω , and every subgroup isomorphic to Gω , is a Frobenius complement. If Gω is even then it has only one element of order 2, and N is commutative. If Gω is odd, then it is metacyclic. In any case, Z(Gω ) is non-trivial. (XXVIII) Let G be a permutation group of Ω and d a prime number. Assume that every point of Ω is fixed by a permutation in G of order d which has no other fixed point; then G is transitive on Ω. If, in addition, d = 2, every point in Ω is fixed by exactly one involutory element in G, and no involutory element in G fixes more than one point of Ω then the subgroup of G consisting of all products of even number of involutory elements in G has odd order and acts transitively on Ω. (XXIX) For n ≥ 2, all n-transitive permutation groups are known, their classification being a corollary to that of all finite simple groups. In Chapter 11, doublytransitive permutation groups whose 2-point stabilisers are cyclic play an important role. Below, the relevant classification theorems are stated after describing the permutation groups involved given in their usual doublytransitive permutation representations. (XXX) A finite group G admits a partition if it contains a set {G1 , . . . , Gk } of subgroups satisfying the following properties: Sk G = i=1 Gi , Gi ∩ Gj = {1}.

The subgroups Gi are the components of the partition. A counting argument Pk shows that n = i=1 |Gi | − (k − 1).

E XAMPLE A.7 Projective linear groups

1. Let Ω = Fq ∪ {∞}. The linear fractional mappings

ax + b , ad − bc 6= 0, cx + d with a, b, c, d ∈ Fq , form a sharply triply-transitive group, the projective general linear group PGL(2, q). ϕ(a,b,c,d) :

x 7→

2. The 2-point stabilisers are isomorphic to the multiplicative group F∗q of Fq . 3. If q is odd, then PGL(2, q) contains a normal 2-transitive subgroup consisting of all ϕ(a,b,c,d) with ad − bc square in Fq , the projective special linear group PSL(2, q); it has order 21 (q 3 − q). 4. The 2-point stabiliser of PSL(2, q) is isomorphic to the subgroup of all square elements in F∗q . 5. Apart from the cases q = 2, 3, the group PSL(2, q) is simple. T HEOREM A.8 For q = pm , the following is a complete list of subgroups of the group PGL(2, q) up to conjugacy:

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(i) the cyclic group of order n with n | (pm ± 1); (ii) the elementary abelian p-group of order pf with f ≤ m; (iii) the dihedral group Dn of order 2n with n | (q ± 1); (iv) the alternating group A4 for p > 2, or p = 2 and m even; (v) the symmetric group S4 for p > 2; (vi) the alternating group A5 for 5 | (q 2 − 1); (vii) the semidirect product of an elementary abelian p-group of order ph by a cyclic group of order n with h ≤ m and n | (q − 1); (viii) PSL(2, pf ) for f | m; (ix) PGL(2, pf ) for f | m. E XAMPLE A.9 Projective unitary groups 1. Let U be the classical unital in PG(2, q 2 ), that is, the set of all self-conjugate points of a unitary polarity Π of PG(2, q 2 ). The projective unitary group PGU(3, q) comprises the projective transformations in PG(2, q 2 ) commuting with Π. Equivalently, PGU(3, q) preserves U and can be viewed as a permutation group on U, since the only collineation in PGU(3, q) fixing every point in U is the identity. 2. With Ω = U, the group PGU(3, q) is a doubly-transitive permutation group on Ω and its 2-point stabiliser is isomorphic to F∗q . 3. |PGU(3, q)| = (q 3 + 1)q 3 (q 2 − 1). 4. With d = gcd(3, q + 1), the group PGU(3, q) contains a normal subgroup PSU(3, q), the special unitary group, of index d which is still 2-transitive. If q > 2, the group PSU(3, q) is non-abelian and simple. 5. For another representation of PGU(3, q), let M = {m ∈ Fq2 | mq + m = 0}. Take an element c ∈ Fq2 such that cq +c+1 = 0. A homogeneous coordinate system in PG(2, q 2 ) can be chosen so that U = {X∞ } ∪ {U = (1, u, cuq+1 + m) | u ∈ Fq2 , m ∈ M }. Then U consists of all Fq2 -rational points of the Hermitian curve Hq = v(cX0q X2 + cq X0 X2q + X1q+1 ).

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APPENDIX A

6. A Sylow p-subgroup Sp of PGU(3, q) consists of the projective transformations associated to the matrices   1 aq (1 + cq−1 ) aq+1 + c−1 m  1 a Ta,m =  0 0 0 1 for a ∈ Fq2 and m ∈ M .

7. The set {T0,m | m ∈ M } is an elementary abelian p-group, which is Z(Sp ). Also, Sp is a normal subgroup of PGU(3, q)Y∞ and acts transitively on the other q 3 points of U. 8. The point O is in U, and the stabiliser T of O, Y∞ under PGU(3, q) consists of the projective transformations associated to the matrices  q+1  a 0 0  0 a 0  0 0 1 for a ∈ Fq2 \{0}.

9. The group PGU(3, q) is generated by Sp and T , together with the projective transformation W associated to the matrix   0 0 c  0 1 0  c−1 0 0 that interchanges the points Y∞ and O.

10. The group PGU(3, q) also fixes Hq , and hence is a K-automorphism group of Σ = K(Hq ). T HEOREM A.10 The following is the list of maximal subgroups of PSU(3, q) up to conjugacy: (i) the one-point stabiliser of order q 3 (q 2 − 1)/d; (ii) the non-tangent line stabiliser of order q(q 2 − 1)(q + 1)/d; (iii) the self-conjugate triangle stabiliser of order 6(q + 1)2 /d; (iv) the normaliser of a cyclic Singer group of order 3(q 2 − q + 1)/d; further, for q = pk with p > 2, (v) PGL(2, q) preserving a conic; (vi) PSU(3, pm ), with m | k and k/m odd; (vii) the subgroup containing PSU(3, pm ) as a normal subgroup of index 3 when m | k, k/m is odd, and 3 divides both k/m and q + 1;

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(viii) the Hessian groups of order 216 when 9 | (q + 1), and of order 72 and 36 when 3 | (q + 1); √ (ix) PSL(2, 7) when either p = 7 or −7 6∈ Fq ; √ (x) the alternating group A6 when either p = 3 and k is even, or 5 ∈ Fq but Fq contains no cube root of unity; (xi) the symmetric group S6 for p = 5 and k odd; (xii) the alternating group A7 for p = 5 and k odd; for q = 2k , (xiii) PSU(3, 2m ) with k/m an odd prime; (xiv) the subgroups containing PSU(3, 2m ) as a normal subgroup of index 3 when k = 3m with m odd; (xv) a group of order 36 when k = 1. E XAMPLE A.11 Suzuki groups 1. An ovoid O in PG(3, q) is a point set with the same combinatorial properties as an elliptic quadric in PG(3, q); namely, O consists of q 2 + 1 points, no three collinear, such that the lines through any point P ∈ O meeting O only in P are coplanar. 2. Now, assume that q = 2q02 with q0 = 2s and s ≥ 1. Then xϕ = x2q0 is an 2 automorphism of Fq , and xϕ = x2 . Let T be the Suzuki–Tits ovoid in PG(3, q), which is the only known ovoid in PG(3, q) other than an elliptic quadric. The Suzuki group Sz(q), also written 2 B2 (q), is the projective group of PG(3, q) fixing T . The group Sz(q) can be viewed as a permutation group on T as the identity is the only projective transformation in Sz(q) fixing every point in T . 3. With Ω = T , the group Sz(q) is a 2-transitive permutation group on Ω, and its 2-point stabiliser is isomorphic to F∗q . 4. |Sz(q)| = (q 2 + 1)q 2 (q − 1). 5. The group Sz(q) is simple. If S2 is a 2-Sylow subgroup of Sz(q), the elements of S2 of order 2 together with the identity form an elementary abelian normal subgroup N of order q, and the quotient group S2 /N is isomorphic to N . 6. In a suitable coordinate system of PG(3, q) with Z∞ = (0, 0, 0, 1), T = {Z∞ } ∪ {(1, u, v, uv + u2ϕ+2 v ϕ ) | u, v ∈ Fq }.

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APPENDIX A

A 2-Sylow subgroup S2 consists of the projective transformations associated to the matrices   1 0 0 0  a 1 0 0    ϕ  c a 1 0  ac + aϕ+2 + cϕ aϕ+1 + c a 1

for a, c ∈ Fq .

7. The subgroup S2 is normal in Sz(q)Z∞ and regular on the remaining q 2 points of T . The stabiliser Sz(q)Z∞ ,O with O = (1, 0, 0, 0) consists of the projective transformations associated to diagonal matrices diag(d−q0 −1 , d−q0 , dq0 , dq0 +1 ) for d ∈ Fq2 \{0}. 8. The group Sz(q) is generated by S2 and SzZ∞ ,O together with the projective transformation W associated to the matrix   0 0 0 1  0 0 1 0     0 1 0 0  1 0 0 0 that interchanges Z∞ and O.

In the next result, Sz(2) = GL(1, 5). T HEOREM A.12 Up to conjugacy, Sz(q) has the following maximal subgroups: (i) the one-point stabiliser of order q 2 (q − 1); (ii) the normaliser of a cyclic Singer group of order 4(q + 2q0 + 1); (iii) the normaliser of a cyclic Singer group of order 4(q − 2q0 + 1); (iv) Sz(q ′ ) for every q ′ such that q = q m with m prime. Further, the subgroups listed below form a partition of Sz(q) : (v) all subgroups of order q 2 ; (vi) all cyclic subgroups of order q − 1; (vii) all cyclic Singer subgroups of order q + 2q0 + 1; (viii) all cyclic Singer subgroups of order q − 2q0 + 1. E XAMPLE A.13 Ree groups The Ree group can be introduced in a similar way using the combinatorial concept of an ovoid, this time in the context of polar geometries.

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1. An ovoid in the polar space associated to the non-degenerate quadric Q in the space PG(6, q) is a point set of size q 3 + 1, with no two of the points conjugate with respect to the orthogonal polarity arising from Q. 2. Now, let q = 3q02 and q0 = 3s with s ≥ 0. Then xϕ = x3q0 is an automor2 phism of Fq , and xϕ = x3 . Let K be the Ree–Tits ovoid of Q. The Ree group Ree(q), also written G2 (q), is the projective group of PG(6, q) fixing K.

2

3. The group Ree(q) can be viewed as a permutation group on K as the identity is the only projective transformation in Ree(q) fixing every point in K. With Ω = K, the group Ree(q) is a 2-transitive permutation group on Ω, and its 2-point stabiliser is isomorphic to F∗q . 4. |Ree(q)| = (q 3 + 1)q 3 (q − 1). 5. Ree(3) ∼ = PΓL(2, 8) has a normal subgroup of index 3. 6. For q0 > 1, the group Ree(q) is simple. Let S3 be a 3-Sylow subgroup of Ree(q). The elements of order 3 in Z(S3 ) together with the identity constitute an elementary abelian normal subgroup N of order q 2 , and the factor group S3 /N is an elementary abelian group of order q. 7. In a suitable coordinate system of PG(6, q) with Z∞ = (0, 0, 0, 0, 0, 0, 1), the quadric Q = v(X32 + X0 X6 + X1 X5 + X2 X4 ) and K = {Z∞ } ∪ {(1, u1 , u2 , u3 , v1 , v2 , v3 )}, with ϕ+3 v1 (u1 , u2 , u3 ) = u21 u2 − u1 u3 + uϕ , 2 − u1

ϕ ϕ 2ϕ+3 2 v2 (u1 , u2 , u3 ) = uϕ , 1 u2 − u3 + u1 u2 + u2 u3 − u1 v3 (u1 , u2 , u3 ) = ϕ+1 ϕ u1 uϕ u2 + uϕ+3 u2 + u21 u22 − uϕ+1 − u23 + u2ϕ+4 , 3 − u1 1 2 1

for u1 , u2 , u3 ∈ Fq . 8. Put w1 (u1 , u2 , u3 ) = −uϕ+2 + u1 u2 − u3 , 1 ϕ 2 u w2 (u1 , u2 , u3 ) = uϕ+1 2 + u1 u3 − u2 , 1

w3 (u1 , u2 , u3 ) =

ϕ+2 ϕ uϕ u2 − u1 u22 + u2 u3 − uϕ+1 u3 − u2ϕ+3 , 3 + (u1 u2 ) − u1 1 1 ϕ+3 ϕ 2 w4 (u1 , u2 , u3 ) = u1 − u1 u2 − u2 − u1 u3 .

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Then a Sylow 3-subgroup S3 of Ree(q) consists of the projective transformations associated to the matrices   1 0 0 0 0 0 0  a 1 0 0 0 0 0    ϕ  b a 1 0 0 0 0     c b − aϕ+1 −a 1 0 0 0     v1 (a, b, c) w1 (a, b, c) −a2 −a 1 0 0     v2 (a, b, c) w2 (a, b, c) ab + c b −aϕ 1 0  v3 (a, b, c) w3 (a, b, c) w4 (a, b, c) c −b + aϕ+1 −a 1 for a, b, c ∈ Fq . Here, S3 is a normal subgroup in Ree(q)Z∞ and regular on the remaining q 3 points of O.

9. The stabiliser Ree(q)Z∞ ,O with O = (1, 0, 0, 0, 0, 0, 0) is the cyclic group consisting of the projective transformations associated to the diagonal matrices, diag(1, d, dϕ+1 , dϕ+2 , dϕ+3 , d2ϕ+3 , d2ϕ+4 ) for d ∈ Fq2 . 10. The group Ree(q) is generated by S3 and Ree(q)Z∞ ,O , together with the projective transformation W of order 2 associated to the matrix   0 0 0 0 0 0 1  0 0 0 0 0 1 0     0 0 0 0 1 0 0     0 0 0 1 0 0 0     0 0 1 0 0 0 0     0 1 0 0 0 0 0  1 0 0 0 0 0 0 that interchanges Z∞ and O.

T HEOREM A.14 Up to conjugacy, Ree(q) has the following maximal subgroups: (i) the one-point stabiliser of order q 3 (q − 1); (ii) the centraliser of an involution z ∈ Ree(q) isomorphic to hzi × PSL(2, q) of order q(q − 1)(q + 1); (iii) a subgroup of order 6(q + 3q0 + 1), the normaliser of a cyclic Singer group of order q + 3q0 + 1; (iv) a subgroup of order 6(q − 3q0 + 1), the normaliser of a cyclic Singer group of order q − 3q0 + 1; (v) a subgroup of order 6(q + 1), the normaliser of a cyclic subgroup of order q + 1; (vi) Ree(q ′ ) with q = q ′m and m prime.

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T HEOREM A.15 Ree(q) has a unique conjugacy class of involutions and each involution has at least one fixed point on K. T HEOREM A.16 (Zassenhaus) Let G be a 2-transitive permutation group whose 3-point stabiliser is trivial, and suppose that G has no regular normal subgroup. If G has degree n + 1 and order (n + 1)nd with d even and d ≤ 21 (n − 1) then G is PSL(2, n) in the usual 2-transitive permutation representation. T HEOREM A.17 (Kantor–O’Nan–Seitz) Let G be a permutation group whose 2point stabiliser is cyclic. Then G has either a regular normal subgroup, or G is one of the following groups in their natural doubly-transitive permutation representations: PSL(2, q), PGL(2, q), PSU(3, q), PGU(3, q), Sz(q), Ree(q). T HEOREM A.18 (Huppert) assume that (i) occurs in (XXVI). If n is a power of 2, then G is a subgroup of AΓL(1, 2k ). This holds true for every odd n, except for n = 32 , 52 , 72 , 112 , 232 , 34 . R EMARK A.19 In Theorem A.18, if n = 9 and G is sharply 2-transitive, then either G ∼ = GL(1, 9) and the 1-point stabiliser is cyclic, or G ∼ = G(1, E) with E the unique proper nearfield of order 9 and the 1-point stabiliser the quaternion group.

A.6 NOTES Proofs of Theorems A.1 and A.2 come from Seidenberg [400, Chapter 23]. For Lemma A.6, see Lucas [314] and Dickson [103, Vol. I, p.271]. A systematic development of the theory of finite groups is found in [239, 240, 241]; see also the survey paper [268]. Frobenius groups as in Section A.5 (XXVII) are treated in Huppert [239, Chapter I.8]. A recent textbook on permutation groups is Cameron [63], in which the classification of finite simple groups is formulated, and, as a corollary, the classification of finite 2-transitive groups is given. For more detail on projective unitary groups, see [238, II.8], [239, II.10], [226] and [268]. For more details on Suzuki groups, see [437] [439], [454], [455], [316]. For more details on Ree groups, see [454], [367], [241], [121], [304]. For more on ovoids, see [224, Appendix VI] and [362]. Theorem A.16 is found in [513]. Theorem A.17 is the main result in [257]. For a proof of Theorem A.18, see [241, Chapter XII.7].

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Appendix B Notation A⊂B t M N Nm Z R C Q K p q Fq Fq AG(2, K) AG(r, K) AG(2, q) AG(r, q) PG(2, K) PG(r, K) PG(2, q) PG(r, q) PGL(2, K) PGL(r, K) PGL(2, q) PGL(r, q) K[X] K[X1 , X2 , . . . , Xn ] K[X][Y ] Fq [X] Fq [X1 , . . . , Xn ] Fi F ∗ (X0 , X1 , X2 ) C, F, G, X . . . ∆, Γ, . . .

A is a subset of B tranpose of the matrix M {1, 2, . . .}, set of natural numbers {1, 2, . . . , m} ring of integers field of real numbers field of complex numbers field of rational numbers any field in Section 1.1, algebraically closed field, otherwise characteristic of K, either p = 0 or p ≥ 2 prime a power of p (for p 6= 0) finite field with q elements algebraic closure of Fq affine plane over K r-dimensional affine space over K affine plane over Fq affine space over Fq projective plane over K r-dimensional projective space over K projective plane over Fq projective space over Fq two-dimensional projective group over K r-dimensional projective group over K two-dimensional projective group over Fq r-dimensional projective group over Fq polynomial ring over K in the indeterminate X polynomial ring over K in X1 , X2 , . . . , Xn polynomial ring over K[X] in the indeterminate Y polynomial ring over Fq in the indeterminate X polynomial ring over Fq in X1 , . . . , Xn partial derivative ∂F/∂Xi for F ∈ K[X1 , . . . , Xn ] homogeneous polynomial associated to F (X, Y ) ∈ K[X, Y ] algebraic curves

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NOTATION

v(F (X, Y )) F = v(F (X0 , X1 , X2 )) F = v(F (X, Y )) R(f, g) RX (f, g) (F, G) I(P, F ∩ G) K[[t]] Aut(K[[t]]) K[[X1 , . . . , Xn ]] K((t)) K((X1 , . . . , Xn )) ordt x X∞ , U1 Y∞ , U2 O, U0 E, U I(P, F ∩ γ) OP K(F) K(ξ, η) Σ (F; (ξ, η)) Σ/Σ′ [Σ : Σ′ ] D(Σ/Σ′ ) P, Q, R, . . . P(Σ) ordP x Σq Dt f Dζ f dx g (i) Dt f (i) Dζ f Rest y ResP (ydx)

651 {(x0 , x1 , x2 ) ∈ PG(2, K) | F ∗ (x0 , x1 , x2 ) = 0} = v(F ∗ (X0 , X1 , X2 )) plane projective curve with homogeneous equation F = 0 plane projective curve with affine equation F = 0 resultant of f, g ∈ K[X] resultant of f, g ∈ K[X, Y ] considered as polynomials in X ideal generated by F and G intersection number of F and G at the point P ring of all formal power series in t over K K-automorphism group of K[[t]] ring of all formal power series over K in X1 , . . . , Xn quotient field of K[[t]] quotient field of K[[X1 , . . . , Xn ]] order of x ∈ K((t)) (0, 1, 0) (0, 0, 1) (1, 0, 0) (1, 1, 1) intersection number of F and a branch γ centred at the point P local ring at the point P function field of F subfield of K(F) generated by ξ, η ∈ K(F) field of transcendency degree 1 over K model of the curve F arising from a generic point P = (ξ, η) algebraic extension of Σ′ to Σ degree of Σ over Σ′ different divisor of Σ/Σ′ places of Σ set of all places of Σ order of x ∈ Σ subfield of all q-powers in Σ derivative of f ∈ K((t)) derivative of f in f ∈ Σ with respect to ζ differential of x genus of an irreducible curve i-th Hasse derivative of f ∈ K((t)) i-th Hasse derivative of f ∈ Σ with respect to ζ residue of y ∈ K((t)) residue of ydx in Σ

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APPENDIX B

P

D = nP P Supp (D) A≡B A≻B deg A div x div(x)0 div(x)∞ Div(Σ) Prin(Σ) Div0 (Σ) Pic(Σ) Pic0 (Σ) L dim L deg L gnr |C| F ·G L(C) ℓ(C) i |dx| K (j0 (P ), . . . , jr (P )) (ǫ0 , ǫ1 , . . . , ǫr ) vP (R) resP c0 (d, r) c1 (d, r) Φ Φn Fq (x, y) Fq (F) Fq [[t]] Aut(Fq [[t]]) Fq ((t)) f (n) Σ(Fq ) X (Fq ) Sq = N 1 Ni Rq

a divisor of Σ support of the divisor D equivalence of divisors A, B divisor A − B is effective degree of a divisor principal divisor of x zero divisor of x pole divisor of x divisor group of Σ group of principal divisors of Σ group of zero divisors of Σ divisor class group (Picard group) of Σ zero divisor class group (Picard zero group) of Σ virtual linear series of Σ dimension of L degree or order of L linear series of dimension r and order or degree n linear series of all effective divisors equivalent to C intersection divisor cut out on F by G Riemann–Roch space of C dimension of L(C) index of speciality of a gnr canonical series canonical curve order sequence of Γ at P order sequence of Γ weight of P in the ramification divisor R residue map Castelnuovo’s number Halphen’s number Frobenius map n-th Frobenius map Fq -rational subfield of Σ generated by x, y Fq -rational subfield of an Fq -rational curve F ring of formal power series over Fq K-automorphism group of Fq [[t]] quotient field of Fq [[t]] n-th conjugate of f set of all Fq -rational places of Σ set of all Fq -rational points of X number of Fq -rational points of X number of Fqi -rational points of X number of points P ∈ F that lie in PG(2, q)

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NOTATION

Rq∗ eq R

Bq Eq bP cP

mP F(Fq )′ Ng (q) A+ (q) A− (q) Pic(Fq (F)) Pic0 (Fq (F)) vP (S) (ν0 , ν1 , . . . , νr−1 ) ζ(X , s) = Z(X , t) L(t) = Lq (t) DLS curve DLR curve Zl JX Tl (A) PSL(2, q) PGU(3, q) PSU(3, q) Sz(q) Ree(q) H 0, Comm. Algebra 15 (1987), 1469–1501. [230] M. Homma, Nonreflexive plane curves of a certain type, Ryukyu Math. J. 1 (1988), 22–37. [231] M. Homma, Nonreflexive plane curves of a certain type. II, Ryukyu Math. J. 2 (1989), 21–28. [232] M. Homma, A souped-up version of Pardini’s theorem and its application to funny curves, Compositio Math. 71 (1989), 259–302. [233] M. Homma, Linear systems on curves with no Weierstrass points, Bol. Soc. Bras. Mat (N.S.) 23 (1992), 91–108. [234] M. Homma, On Esteves’ inequality of order sequences of curves, Comm. Algebra 21 (1993), 3685–3689. [235] M. Homma, On duals of smooth plane curves, Proc. Amer. Math. Soc. 118 (1993), 785–790. [236] M. Homma, Plane curves whose tangent lines at collinear points are concurrent, Geom. Dedicata 53 (1994), 287–296. [237] E.W. Howe and K.E. Lauter, Improved upper bounds for the number of points on curves over finite fields, Ann. Inst. Fourier (Grenoble) 53 (2003), 1677–1737; Corrigendum, to appear. [238] D.R. Hughes and F.C. Piper, Projective Planes, Graduate Texts in Mathematics 6, Springer, New York, 1973, x+291 pp. [239] B. Huppert, Endliche Gruppen. I, Grundlehren der Mathematischen Wissenschaften 134, Springer, Berlin, 1967, xii+793 pp. [240] B. Huppert and B.N. Blackburn, Finite Groups. II, Grundlehren der Mathematischen Wissenschaften 242, Springer, Berlin, 1982, xiii+531 pp. [241] B. Huppert and B.N. Blackburn, Finite Groups. III, Grundlehren der Mathematischen Wissenschaften 243, Springer, Berlin, 1982, ix+454 pp. [242] N.E. Hurt, Many Rational Points. Coding Theory and Algebraic Geometry, Mathematics and its Applications 564, Kluwer, Dordrecht, 2003, xxii+346 pp. [243] D. Husem¨oller, Elliptic Curves, Springer, Berlin, 2004, xxi+487 pp. ¨ [244] A. Hurwitz, Uber algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1893), 403–492.

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Index

abelian group, 638 adjoint curve, 173 canonical, 176 special, 176 adjoint of a curve, 176 affine collineation, 5 affine curve, 6–9, 37 component, 6, 7 degree, 6 irreducible, 6 rational, 9 affine group, 5 affine plane, 4 affine point, 110 affine space, 4 affine subspace, 4 affine transformation, 5 algebraic closure, 628 algebraic element, 627 algebraic extension, 627 algebraic transform, 51, 84 algebraic variety, 18, 268–270 algebraic-geometry code, 590–593 algebraically dependent, 630 algebraically independent, 630 analytic branch, 99–107 analytic curve, 99–107 analytic cycle, 99–107 centre, 100 arc, 594–625 complete, 599–625 dual, 607–611 Artin–Schreier curve, 396, 501, 502 Artin–Schreier extension, 502 asymptotic bounds, 353–356 automorphism Frobenius, 5 of a curve, 458–542 order, 470–473 automorphism group, 473–475, 480–482, 509–542 abelian, 492–497 bound, 480–482, 513–532, 535–538 elliptic curve, 503 hyperelliptic curve, 505–508 non-tame, 481, 486–501

orbit, 475 plane, 468–469 stabiliser, 475 tame, 481, 483–486 B´ezout’s Theorem, 12, 21, 45–49, 55, 56, 83, 92, 159, 202, 292, 361, 362, 367, 424, 606, 607, 624 Bertini’s Theorem, 196 bidual curve, 156–159 birational isomorphism, 118, 204 birational transformation, 49–55, 116–118, 213, 459–464 birationally equivalent, 118 Bound Castelnuovo’s, 257–260, 407, 412, 428, 431, 436, 448 Drinfeld–Vl˘adut¸, 353–354 Hasse, 340 Hasse–Weil, 360, 395, 396 Ihara, 353 Serre, 348–349, 360, 386, 585–587 branch analytic, 99–107 linear, 201 of a curve, 81–92, 278–281 rational, 278–281, 287–289 branch representation, 75–81, 115, 199 centre, 75 imprimitive, 77–81 order, 75 order sequence, 76 primitive, 81, 121 special, 75 special affine coordinates, 76–81 tangent, 76 Brill–Noether Theorem, 183 Burnside’s Theorem, 533 canonical adjoint, 176 canonical class, 170 canonical curve, 216–217, 466–467 canonical series, 170, 172–177, 182, 187, 215, 217, 218, 511 Carnot’s Theorem, 605 Cartesian product, 3 Cartier operator, 191

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690 Cartier–Manin matrix, 192 Castelnuovo’s Bound, 257–260, 407, 412, 428, 431, 436, 448 Castelnuovo’s number, 430–439 Cayley–Bacharach Theorem, 109 centraliser, 640 characteristic, 627 class number, 290, 385 class number of quadratic form, 383 classical curve, 228–229, 363–366 for lines, 228–229 classical gap sequence, 186 Clifford’s Theorem, 183, 260–262 closed place, 285 closed point, 342, 346 code algebraic-geometry, 590–593 error-correcting, 590–598 MDS, 592, 594–598 near-MDS, 598 Reed–Solomon, 592 collineation, 5 affine, 5 Frobenius, 5, 292 linear, 5 companion automorphism, 464 companion group, 464 complement, 639 Frobenius, 641 complete arc, 599–625 second largest, 611–623 third largest, 624–625 complete linear series, 169–170, 173–179, 215, 241 component of affine curve, 6 conic osculating, 231, 233 conjugate elements, 638 conjugate map, 281 conjugate subgroups, 638 conormal variety, 272 constant point, 110–111 coordinate function, 200 coset, 639 covariant, 10–14, 16 covering, 207, 278, 356 Galois, 209 Cremona transformation, 49 cubic curve, 191, 193–196, 241–243, 358, 378–385, 388, 503 singular, 358 curve adjoint, 173–176 affine, 6–9, 37 analytic, 99–107 Artin–Schreier, 396, 501, 502 asymptotic bounds, 353–356

INDEX bidual, 156–159 canonical, 216–217, 466–467 classical, 228–229, 363–366 classical for lines, 228–229 cubic, 191, 193–196, 241–243, 358, 378–385, 388, 503 DLR, 352, 395, 398, 456, 575–584 DLS, 118, 303, 352, 366, 395, 398, 447–452, 456, 532, 564–572 dual, 155–159, 238–241, 326, 327 elliptic, 191, 193–196, 216, 224, 241–243, 378–385, 388, 467–468, 501, 503 elliptic quartic, 598 Fermat, 16, 237, 319, 323, 325–326, 386, 436–469 Frobenius classical, 296, 363–366 Frobenius non-classical, 296–327 Hermitian, 16, 82, 187, 197, 210, 230, 254, 284, 301–302, 305, 308, 366, 469, 489, 532, 572–574, 611, 619–622 Hessian, 13–18, 234 Hurwitz, 442–445 hyperelliptic, 216, 243–254, 406–407, 465, 501, 505–509, 525, 532 index, 56–58 irreducible, 200 Klein quartic, 204, 341–342, 404 Kummer, 396 maximal, 395–452 non-classical, 228–237, 239, 363–366 non-classical for conics, 230–237 non-classical for lines, 228–230 non-reflexive, 157–159, 327 normal, 214 normal rational, 214, 410, 595–597 nucleus, 12 number of points, 358–369 optimal, 395, 447–452 ordinary, 501 plane, 6–17, 277–280, 358–366, 369, 439–442, 468–469, 546–587 polar, 10–12 projective, 9–17 quartic, 184, 204 quotient, 473, 481, 485, 489 rational, 9, 18, 159, 224, 229, 230, 239, 271, 501 reflexive, 156–158 sextic, 182, 618–619 singular, 366 space, 238, 287–291 strange, 12–13, 18, 59–61, 159, 229, 230, 271 supersingular, 191, 378 curve as algebraic variety, 268–270 curves linear system, 171

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INDEX net, 171 pencil, 171 cusp, 7–10 cycle analytic, 99–107 cyclotomic polynomial, 629 degree of curve, 6, 10 degree of divisor, 162 degree of hypersurface, 201 degree of monomial, 6 derivation, 125–130 derivative Hasse, 144–155 Hasse partial, 148–155 derived group, 639 Descartes, xii Deuring–Shafarevich Theorem, 485 Dickson’s Theorem, 642 different, 209, 461, 471, 488 differential, 126–129, 170 exact, 128–129 differential form, 138–144 residue, 138–144 dihedral group, 638 direct product, 638 discriminant, 30–31 divisor, 96, 161–191 degree, 162 double-point, 172 effective, 96, 161 equivalence, 162 fixed, 163 Frobenius, 374–378 positive, 161 principal, 162 ramification, 222 rational, 284 St¨ohr–Voloch, 294–305 virtual, 96, 161 divisor class, 162, 289 divisor class group, 162, 190–191, 289, 369–373 divisor group, 161, 285, 369–373, 460 divisor of poles, 162 divisor of zeros, 162 DLR curve, 352, 395, 398, 456, 575–584 DLS curve, 118, 303, 352, 366, 395, 398, 447–452, 456, 532, 564–572 double point, 7 isolated, 7 double-point divisor, 172 Drinfeld–Vl˘adut¸ Bound, 353–354 dual arc, 607–611 dual curve, 155–159, 238–241, 326, 327 dual hypersurface, 272 effective divisor, 96, 161

691 element special, 73 elliptic curve, 191, 193–196, 216, 224, 241–243, 378–385, 388, 467–468, 501, 503 automorphism group, 503 supersingular, 504 endomorphism Frobenius, 374–378 error-correcting code, 590–598 Euler function, 35 Euler product, 332 Euler’s formula, 155 exact differential, 128–129 extension Galois, 633 normal, 633 purely inseparable, 122–123, 629 separable, 122–123, 629 simple, 630 Fermat curve, 16, 232, 237, 319, 323, 325–326, 386, 436–439, 469 field finite, 636–637 of rational functions, 112 perfect, 630 prime, 627 projective, 264 field homomorphism, 628 field isomorphism, 628 field theory, 627–637 finite field, 636–637 formal power series, 63–74 field of rational functions, 65 quotient field, 64 Fq -rational branch, 278–281 Fq -rational covering, 278 Fq -rational divisor class, 289 Fq -rational function field, 278 Fq -rational place, 281–287 Fq -rational transformation, 278 Fqn -rational divisor, 284 Fqn -rational divisor group, 284 Frattini subgroup, 640 Frenet frame, 219 Frobenius automorphism, 5 Frobenius classical curve, 296, 363–366 Frobenius collineation, 5, 292 Frobenius complement, 641 Frobenius divisor, 374–378 Frobenius endomorphism, 374–378 Frobenius group, 641 Frobenius linear series, 374–378, 399–417, 421–423 Frobenius non-classical curve, 296–327 Frobenius order, 294–327

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692

INDEX

Frobenius order sequence, 294–327 Frobenius transformation, 123–125 function field, 112, 278 rational, 282, 463 functional equation, 336 Fundamental Equation, 373–378

Suzuki, 513, 525, 528–531, 541, 564–572, 645–646, 649 transitive, 641 group extension, 639 group partition, 642 group theory, 638–649

Galois closure, 634 Galois covering, 209 Galois theory, 633–635 gap number, 186 Gauss map, 155–159, 229, 235, 238, 239 Gaussian dual, 239 general linear group, 4 general semilinear group, 4 generalised quaternion group, 638 generator matrix, 590 generic point, 110–138, 155, 156, 200, 239, 245, 269, 278 isomorphism, 111–112 genus, 130–138, 159 virtual, 55–58, 132–138 geometric transform, 52–61, 85–92 global stabiliser, 640 gonality, 357–358 group abelian, 638 affine, 5 derived, 639 dihedral, 638 divisor class, 162, 190–191 divisors of degree zero, 163 Frobenius, 641 general linear, 4 general semilinear, 4 generalised quaternion, 638 nilpotent, 639 order, 638 ordered, 262 permutation, 640 projective special linear, 649 projective special unitary, 643, 649 projective unitary, 643–649 projective general linear, 5, 513, 529, 642 projective linear, 525, 541, 642–643, 649 projective semilinear, 5 projective special linear, 513, 525, 527, 529, 541, 642 projective special unitary, 541, 644–645 projective unitary, 469, 503, 525, 529–531, 541, 572–574, 643, 645 ramification, 488–501 Ree, 525, 528, 530, 541, 575–584, 646–649 semidihedral, 638 sharply transitive, 641 Singer, 511 soluble, 639

Halphen’s number, 262, 439, 453 Halphen’s Theorem, 261–262, 407, 428, 430 Hasse Bound, 340 Hasse derivative, 16–17, 144–155 Hasse partial derivative, 148–155 Hasse–Weil Bound, 343, 346, 348, 353, 360, 395, 396 Hasse–Weil Theorem, 343–352 Hasse–Witt invariant, 191–192 Hasse–Witt matrix, 191–192 Hensel’s Lemma, 69–71, 80, 102, 141 Hering’s Theorem, 533 Hermitian P-invariant, 218 Hermitian curve, 16, 82, 187, 197, 210, 230, 254, 284, 301–302, 305, 308, 366, 469, 489, 532, 572–574, 611, 619–622 Hermitian variety, 407–421 Hessian curve, 13–18, 234 Hilbert Different Formula, 486–501 Hilbert’s Nullstellensatz, 36, 93 Hoffer’s Theorem, 644 homaloidal net, 50 Huppert’s Theorem, 540, 649 Hurwitz curve, 442–445 Hurwitz’s Theorem, 208–211, 471, 478, 481, 483, 489 hyperelliptic curve, 216, 243–254, 406–407, 465, 501, 505–509, 525, 532 automorphism group, 505–508 hyperelliptic involution, 506 hyperoval, 600 hyperplane, 18, 201 osculating, 217–227 hypersurface, 200–202 degree, 201 dual, 272 projective, 18 Ihara Bound, 353 imprimitive branch representation, 77–81 index of a curve, 56–58 index of speciality, 177 inflexion, 8, 15 inflexional tangent, 8 inseparable degree, 123 inseparable polynomial, 629 inseparable variable, 123 intersection divisor, 202 intersection multiplicity, 201 intersection number, 7, 37–49, 83 involution, 211–216

ac2 October 19, 2007

693

INDEX irreducible curve, 200 isolated double point, 7 isomorphism, 204 birational, 118 Jacobian variety, 373–378 K-automorphism of power series, 65–67 K-homomorphism of power series, 65 K-monomorphism of power series, 65–67 K-morphism, 265–268 Kantor–O’Nan–Seitz Theorem, 526, 540 Klein quartic, 204, 341–342, 404 Kummer curve, 396 (L, P)-order, 218–227 l-adic integers, 377 L-order, 222–227 L-ordinary, 224 L-polynomial, 339, 343 L-Weierstrass point, 224–226 Lagrange’s Theorem, 638 Lam´e’s Theorem, 62 Lemma Hensel’s, 69–71, 80, 102, 141 Lucas’s, 637 Lemma of Tangents, 601 line, 4 linear branch, 201 linear collineation, 5 Linear General Position Principle, 257, 407 linear series, 163–191, 202, 211–227, 239, 241, 464–466 r , 168 gn classical, 225, 226 complete, 169–170, 173–179, 215, 241 composed of an involution, 211–216 difference, 169 dimension, 164 Frobenius, 374–378, 399–417, 421–423 non-special, 177 normalised, 165 order, 163 rational, 286 simple, 213 special, 177 subseries, 165 sum, 169, 183 virtual, 163–168 linear system of curves, 171 linear transformation, 4 local parameter, 120, 139 local ring, 92 Lucas’s Lemma, 637 L¨uroth semigroup, 356–357 L¨uroth’s Theorem, 270 Maclaurin’s Theorem, 61

maximal curve, 395–452 non-isomorphic, 446–447 on Hermitian variety, 407–421 on quadric surface, 421–428 plane, 439–447 MDS code, 592, 594–598 Menelaus’ Theorem, 603–611 minimal normal subgroup, 639 minimal polynomial, 628 monomial degree, 6 morphism, 204, 264–268 trivial, 265 multiple point, 7 ordinary, 8 multiplicity of a point in a divisor, 161 M¨obius Inversion Formula, 346 n-th conjugate of a power series, 278 Natural Embedding Theorem, 412, 430 near-MDS code, 598 net of curves, 171 Newton, xii nilpotent group, 639 node, 7 Noether’s conditions, 93 Noether’s Reduction Theorem, 179 Noether’s Theorem, 92–98, 108 non-classical curve, 228–237, 239, 363–366 for conics, 230 for conics, 237 for lines, 230 non-classical curve for lines, 228 non-gap, 137 non-reflexive curve, 157–159, 327 norm, 635–636 normal curve, 214 normal rational curve, 214, 410, 595–597 normal subgroup, 639 normaliser, 640 nucleus of a curve, 12 optimal curve, 395, 447–452 orbit, 475 long, 641 short, 641 orbit structure, 640 order sequence, 222–227, 481 ordered group, 262 orders, 222 ordinary curve, 501 ordinary multiple point, 8 ordinary point, 186 ordinary singularity, 55, 177 osculating i-space, 219 osculating conic, 231, 233 osculating hyperplane, 217–227 osculation point, 220

ac2 October 19, 2007

694 oval, 599–602 ovoid, 575, 645, 647 p-group, 638 p-rank, 191–192, 373, 485, 499 partition of a group, 642 Pascal’s Theorem, 61, 109 pencil of curves, 171 perfect field, 630 permissible r-ple, 163 permutable position, 424 permutation group, 640 permutation representation, 640 faithful, 640 Picard group, see divisor class group place, 119–122 closed, 285 degree, 285 fixed, 163 rational, 283, 288 stabiliser, 476–480 place representation, 119 primitive, 119 plane affine, 4 projective, 4 plane curve, 6–17, 277–280, 358–366, 369, 439–442, 468–469, 546–587 point, 4, 199–201 affine, 110 closed, 342 constant, 110–111 double, 7 generic, 110–138, 155, 156, 200, 239, 245, 269, 278 infinite, 9–10 inflexion, 8 multiple, 7 non-singular, 7 ordinary multiple, 8 rational, 288 simple, 7 singular, 7 terrible, 59–61 triple, 7 variable, 110–111 Weierstrass, 224–226, 461, 466, 467, 481, 506, 511 point at infinity, 9–10 point of inflexion, 8, 15 polar curve, 10–12 pole number, 186 pole of a differential, 130–138 pole of a function, 120–122 pole of a place, 133 polynomial

INDEX cyclotomic, 629 degree, 6 homogeneous, 6 inseparable, 629 irreducible, 5 minimal, 628 reducible, 5 regular, 73 separable, 629 special, 73 positive divisor, 161 power series, 63–74 order, 64 quotient field, 64 subdegree, 64 prime field, 627 primitive branch representation, 121 primitive place representation, 119 primitive representation, 279 primitive root of unity, 627 principal divisor, 162 Principle Linear General Position, 257, 407 Uniform Position, 261–262 projective curve, 9–17 projective field, 264 projective general linear group, 5, 513, 529, 642 projective hypersurface, 18 projective linear group, 525, 527, 541, 642–643, 649 projective plane, 4 projective semilinear group, 5 projective space, 4 projective special linear group, 513, 525, 527, 529, 541, 642, 649 projective special unitary group, 541, 643–645, 649 projective subspace, 4 projective transformation, 5 projective unitary group, 469, 503, 525, 529–531, 541, 572–574, 643–645, 649 projective variety, 18 projectivity, 5 purely inseparable extension, 122–123, 629 quadratic transformation, 50–55, 57 local, 84–92 standard, 51–55 quadric surface, 421–428 quartic Klein, 204, 341–342, 404 quotient curve, 473, 481, 485, 489 quotient field of formal power series, 64 quotient group, 639 ramification divisor, 222, 228, 376, 423, 483 ramification group, 488–501

ac2 October 19, 2007

INDEX ramification index, 77, 206, 474–475, 491 ramification number, 490–499, 521 rational branch, 278–281, 287–289 rational curve, 9, 18, 159, 224, 229, 230, 239, 271, 501 rational divisor, 284 rational function field, 282 rational linear series, 286 rational place, 283, 288 rational point, 288 rational transformation, 49–55, 112–118, 125, 203–207, 213, 278 inseparable, 125 separable, 125 redundancy, 107, 206 Ree group, 351, 525, 528, 530, 541, 575–584, 646–649 Ree’s Theorem, 648 Ree–Tits ovoid, 647 Reed–Solomon code, 592 reflexive curve, 156–158 regular function at a place, 121 regular polynomial, 73 residue map, 265–268, 461, 473 residue of differential form, 138–144 resolution of singularities, 55–61 resultant, 21–35 Riemann zeta function, 332 Riemann’s Theorem, 177 Riemann–Roch space, 180, 465, 472, 591 Riemann–Roch Theorem, 177–183, 189, 206, 212, 215–218, 243, 257, 260, 333, 337, 357, 472 Salmon’s Theorem, 242–243, 381 semidihedral group, 638 semigroup L¨uroth, 356–357 semilinear transformation, 4 separable degree, 123 separable extension, 122–123, 629 separable polynomial, 629 separable variable, 123 Serre Bound, 348–349, 360, 386, 585–587 Serre’s Explicit Formula, 350–351 sextic curve, 182, 618–619 sharply transitive group, 641 simple extension, 630 simple point, 7, 203 Singer cycle, 611 Singer group, 511 singular cubic curve, 358 singular curve, 366 singular point, 7, 203 singularities resolution, 55–61 singularity

695 ordinary, 55 socle, 639 soluble group, 639 space curve, 238, 287–291 special adjoint, 176 special affine coordinates, 76–81 special branch representation, 75 special element, 73 special polynomial, 73 special representation, 199 specialisation, 268 split extension, 639 St¨ohr–Voloch divisor, 294–305, 405, 406 St¨ohr–Voloch Theorem, 292–305, 344, 345, 440 stabiliser, 475, 640 place, 476–480 standard quadratic transformation, 51–55 Stokes’ Theorem , 144 strange curve, 12–13, 18, 59–61, 159, 229, 230, 271 Study’s Theorem, 27, 36, 47 subseries, 165 subspace projective, 4 supersingular curve, 191, 378 support of a divisor, 161 surface quadric, 421–428 Suzuki group, 351, 513, 525, 528–531, 541, 564–572, 645–646, 649 Suzuki’s Theorem, 646 Suzuki–Tits ovoid, 645 symmetric polynomial, 22–27 tangent line, 7–8 Tate group, 377–378 terrible point, 59–61 Theorem B´ezout’s, 12, 21, 45–49, 55, 56, 83, 92, 159, 202, 292, 361, 362, 367, 424, 606, 607, 624 Bertini’s, 196 Brill–Noether, 183 Burnside’s, 533 Carnot’s, 605 Cayley–Bacharach, 109 Clifford’s, 183, 260–262 Deuring–Shafarevich, 485 Dickson’s, 642 Halphen’s, 261–262, 407, 428, 430 Hasse–Weil, 343–352 Hering’s, 533 Hoffer’s, 644 Huppert’s, 540, 649 Hurwitz’s, 208–211, 471, 478, 481, 483, 489

ac2 October 19, 2007

696 Kantor–O’Nan–Seitz, 526, 540 Lagrange’s, 638 Lam´e’s, 62 L¨uroth’s, 270 Maclaurin’s, 61 Menelaus’, 603–611 Natural Embedding, 412 Noether’s, 92–98, 108 Noether’s Reduction, 179 of the Nine Associated Points, 62 of the Primitive Element, 631–633 Pascal’s, 61, 109 Ree’s, 648 Riemann’s, 177 Riemann–Roch, 177–183, 189, 206, 212, 215–218, 243, 257, 260, 333, 337, 357, 472 Salmon’s, 242–243, 381 St¨ohr–Voloch, 292–305, 344, 345, 440 Stokes’, 144 Study’s, 27, 36, 47 Suzuki’s, 646 Weierstrass Division, 73 Weierstrass Gap, 184–190, 399, 405, 406, 472, 512, 550 Weierstrass normal form, 189 Weierstrass Preparation, 71–75, 99–104 Zassenhaus’, 649 tower of curves, 354–355 asymptotically good, 354 recursive, 354 trace, 635–636 trace map, 285, 291 transcendence basis, 630 transcendence degree, 630 transcendental extension, 627 transcendental element, 627 transform algebraic, 51 geometric, 52–61, 85–92 transformation affine, 5 birational, 49–55, 116–118, 213, 459–464 Cremona, 49 Frobenius, 123–125 linear, 4 local quadratic, 84–92 projective, 5 quadratic, 50–55, 57 rational, 49–55, 112–118, 125, 203–207, 213, 278 semilinear, 4 standard quadratic, 51–55 transitive group, 641 triple point, 7 trivial intersection set, 533–534 Uniform Position Principle, 261–262

INDEX uniformising element, 120, 475 valuation, 262–268 d-adic, 263 normalised, 263 trivial, 263 valuation ring, 262–268 Vandermonde determinant, 24 variable point, 110–111 variety algebraic, 18, 268–270 conormal, 272 Hermitian, 407–421 projective, 18 vector space, 4, 590 Veronese map, 230 virtual divisor, 96, 161 virtual genus, 55–58, 132–138 virtual linear series, 163–168 Weierstrass Division Theorem, 73 Weierstrass Gap Theorem, 184–190, 399, 405, 406, 472, 512, 550 Weierstrass normal form, 189 Weierstrass point, 186, 224–226, 461, 466, 467, 481, 506, 511 Weierstrass Preparation Theorem, 71–75, 99–104 Weierstrass semigroup, 186 weight of a point in a divisor, 161 Weil’s Explicit Formula, 351, 385 Wronskian determinant, 219–227, 293 Zassenhaus’ Theorem, 649 zero of a differential, 130–138 zero of a function, 120–122 zero of a place, 133 zero-degree Fq -rational divisor class group, 289, 290 zero-degree divisor class group, 369–373 zero-degree divisor group, 369–373 zeta function, 332–356, 369–373 Riemann, 332