A comprehensive treatment of elliptic functions is linked by these notes to a study of their application to elliptic cur
238 90 10MB
English Pages 255 Year 1973
Table of contents :
Cover
Title
Copyright
Contents
Preface
1. Introductory
2. The Weierstrass functions
3. Invariants and the modular function
4. The Jacobi functions
5. The Riemann surface and the integral of the first kind
6. Quasielliptic functions and integrals of the second and third kinds
7. The ternary elliptic functions
8. Transformations and modular relations
9. Trigonometric expansions
10. The theta functions
11. Elliptic curves
12. The Plane cubic
13. Elliptic quartic curves
Historical Note
Bibliography
Index
London Mathematical Society Lecture Note Series.
9
Elliptic Functions and Elliptic Curves PATRICK DUVAL Ordinarius Professor of Geometry, University of Istanbul
Cambridge· At the University Press ·1973
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521200363 © Cambridge University Press 1973 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1973 Reissued in this digitally printed version 2009 A catalogue recordfor this publication is available from the British Library Library of Congress Catalogue Card Number: 7283599
ISBN 9780521200363 paperback
Contents
Page Preface 1. 2.
Introductor __Weierslrass__iunctions
1 15
3
Invariants and the modular function
~
4.
The Jacobi functions
52
5.
The Riemann surface and the integral of the first kind
82
6.
Quasielli tic functions and integrals of the second and_ihird__kinds
9_3
7.
ill
8.
Transformations and modular relations
132
9.
Trigonometric expansions
ill
10.
The..Jheia_iunctions
l62
11.
Elliptic curves
184
12.
The lane cubic
1.9..9
13.
Elliptic quartic curves
213
Histori.caLNote
2J1l
Bibliography
2Al
IIm.x
2AA
Preface
The lectures on which the following notes are based were given in various forms in University College, London, from about 1964 to 1969. Generally they were an optional undergraduate course, containing the substance of Chapters .16, and part of Chapter 8. Once or twice they were given to graduate students in geometry, and then included also the bulk of Chapters 913. Chapter 7, with the part of Chapter 11 which depends on this, and the cubic transformations in Chapter 8, never figured in the course, but it seemed to me very desirable to add them to the published notes. There is of course much more that I would have liked to include (such as transformations at least of order 5, some study of the connexion between modular relations and the subgroups of finite index in the modular group, a general examination of rectification problems, and the parametrisation of confocal quadrics and of the tetrahedroid and wave surfaces); but a limit of length is laid down for this series of publications, which I fear I have already strained to the utmost. In my treatment of elliptic functions I have tried above all to present a unified view of the subject as a whole, developing naturally out of the Weierstrass function; and to give the essential rudiments of every aspect of the subject, while unable to enter in very great detail into any one of these. In particular I have been concerned to emphasize the dependence of the properties of the functions on the shape of the lattice; it is for this reason that the modular function is introduced at such an early stage, and that equal prominence is given throughout (except in the context of the Jacobi functions) to the rhombic and the rectangular lattices. The treatment of the theta functions will be seen to be rather slight. They are in themselves a large subject, of which our study is in a considerable measure independent, since our approach (based on Neville's) to the Jacobi functions obviates any need for the theta functions as a preliminary, except for the expression of invariants such as k, K, J in iv Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:57:51, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.001
terms of r or q, i. e. in terms of the lattice shape. I have kept the analytic apparatus required to a minimum, largely because I am no expert analyst myself; all that I assume ought, I think, to be familiar to any graduate or thirdyear honours student, and is to be found in any such general textbook as Whittaker and Watson [43] or Copson [5], For the study of elliptic curves I have of course had to assume some knowledge of algebraic geometry. The general theory sketched in Section 85 can be read up in detail in such works as van der Waerden [38] or Hodge and Pedoe [21]; and the properties of the genus used in Section 89 in any book on algebraic curves, such as Walker [40] or Semple and Kneebone [35]. For any assumed properties of the plane cubic and twisted quartic, probably the best sources are still the two classics of Salmon [32, 33], now available in modern reprints; and for the finite groups V, T, O etc. perhaps the easiest reference is my own monograph [10]. In conclusion, I would like to express my gratitude to the London Mathematical Society for making this publication possible; to the general editor of the series, Professor G. C. Shephard, for his patience; to Dr D. G. Larman for assistance with the bibliography; and particularly to my wife for her help in reading the proofs.
Istanbul, 1971
Patrick Du Val
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1 • Introductory
i.
For any complex number z = x + iy (x, y real, i = 1) we define Re(z) = x, Im(z) = y, z  = (x2 + y 2 ) 2 , z = x  iy. If y = 0 (i.e. if z = z), z is real, if y * 0, z is imaginary, if x = 0, z is pure imaginary (note that 0 is pure imaginary without being imaginary) and if z = 1, z is unimodular. The real and pure imaginary axes in the Argand plane are horizontal and vertical respectively. Lattices. A lattice Cl of complex numbers is an aggregate of complex numbers with the two properties: (i) fi is a group with respect to addition; (ii) the absolute magnitudes of the nonzero elements are bounded below, i. e. there is a real number k > 0 such that oo  > k for all a? * 0 in S2. Every lattice is either (i) trivial, consisting of 0 only; (ii) simple, consisting of all integer multiples of a single generating element, which is unique except for sign; or (iii) double, consisting of all linear combinations with integer coefficients of two generating elements co , co , whose ratio is imaginary. These are not unique; if co , co generate £2, so do
where p, q, r, s are any integers satisfying ps  qr = ±1. It is usual however to require o> , a> to be so ordered that Im(co /co ) shall be positive; and if coT, cof are to be similarly ordered, this requires ps  qr = +1.
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2.
Lattice shapes
If 12 is any lattice, and m any non zero complex number, m!2 denotes the aggregate of complex numbers mto for all co in 12. This is also a lattice, which is said to be similar to 12; similarity is an equivalence relation between lattices, an equivalence class being a lattice shape. All simple lattices are similar, i. e. constitute one lattice shape. The lattice points (i. e. elements of the lattice, represented as points in the Argand plane) are (for a simple lattice) at equal intervals along one line through the origin, but in general (for a double lattice) are the vertices of a pattern of parallelograms filling the whole plane, whose sides can be taken to be any pair of generators. The lattice point patterns for similar lattices are similar in the elementary sense. 12 denotes the aggregate of complex numbers co for all oo in 12; 12 is also a lattice. If 12 = 12, 12 is called real. This is the case if and only if either: (i) 12 is simple, its generator (and hence all its elements) being either real or pure imaginary; (ii) generators can be so chosen that co is real and co pure imaginary, in which case 12 is called rectangular, the lattice points being the vertices of a pattern of rectangles, whose sides are horizontal and vertical, i. e. parallel to the real and imaginary axes; or (iii) generators can be chosen which are conjugate complex, in which case 12 is called rhombic, the lattice points being the vertices of a pattern of rhombi, whose diagonals are horizontal and vertical. Any lattice similar to a rectangular or rhombic lattice is also rectangular or rhombic, but is only real if the sides of the rectangles (diagonals of the rhombi) are horizontal and vertical. The real rectangular or rhombic lattice will be called horizontal or vertical, according as the longer sides of the rectangles (longer diagonals of the rhombi) are horizontal or vertical. Besides the simple lattice, there are two special lattice shapes: (i) square (ordinary squared paper pattern); this is both rectangular and rhombic, and may be said to be in the rectangular or rhombic position if thesides or diagonals respectively of the squares are horizontal and vertical (it is real in both cases); (ii) triangular (pattern of equilateral triangles filling the plane); this is rhombic in three ways, a rhombus
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(with diagonals in the ratio V3:l) consisting of any two triangles with a common side. Every lattice satisfies 12 = 12; the only cases in which 12 = k!2, with k ± ±1, are the square lattice (12 = i!2) and the triangular lattice (12 = e 12, where e is a primitive cube root of unity; we shall throughout denote these cube roots by e , e 2 instead of the more usual co, co , to avoid confusion with the use of co for an element of a lattice). 3.
Residue classes
If z is any value of a complex variable, z + 12 denotes the aggregate of values z + co for all co in the lattice 12. This aggregate is called a residue class (mod 12). The residue classes (mod 12) form a continuous group under addition, defined in the obvious way, namely (z + 12) + (w + 12) = (z + w) + 12. 12 itself is a residue class (mod 12), the zero element of the group. By a fundamental region of 12 we mean a simply connected region of the Argand plane which contains exactly one member of each residue class (mod 12). If 12 is the trivial lattice, each residue class consists only of a single value of z, and the only fundamental region is the whole plane. If 12 is the simple lattice generated by co, a fundamental region is an infinite strip, bounded by two parallel lines, one of which is the locus of z + co for all z on the other; these bounding lines need not be perpendicular to co, nor straight, though it is usually convenient to take them so; but they must not intersect. One of the two lines is included in the fundamental region, and the other is not, i. e. the strip is closed on one side and open on the other. If 12 is a double lattice, a fundamental region can be chosen in many ways; the simplest, and usually the most convenient, is what is called a unit cell, i. e. a parallelogram with sides co , co (any pair of generators), including one of each pair of parallel sides, and one vertex, but excluding the rest of the boundary. We obtain a topological model of the residue class group by identifying the points congruent (mod 12) on the boundary of the fundamental region, i. e. joining up the open edges to the corresponding closed edges. For the simple lattice, identifying the points z, z + co throughout the bounding lines of the strip, we obtain an infinite cylinder, with generators
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perpendicular to oo. This is topologically equivalent to a sphere with two pinholes, corresponding to the open ends of the cylinder (compare the Mercator map of the sphere, rolled up thus into a cylinder, on which every point of the sphere is mapped uniquely, except the two poles). For the double lattice, identifying points z, z + w on the sides of the unit cell parallel to u> we obtain a finite cylinder of length  oo  with ends perpendicular to its generators; to identify corresponding points on these two ends, the cylinder must be bent round (and also twisted, unless the sides of the unit cell are perpendicular, i. e. unless £1 is rectangular) to form a ring surface or torus. In particular, the torus x = (a + b cos 0)cos 0, y = (a + b cos 0)sin 0, z = b sin 0, / 2
.
2
.
2 .
(x + y + z
2
, 2X2
+a  b )
, . 2 , 2 .
2
S
=4a(x + y )
(a > b > 0), obtained by rotating the circle (x a)2 + z2 = b 2 , y = 0 about the z axis, is not only a topological model of the residue class group, but a conformal model of the fundamental region, for a rectangular lattice whose generators satisfy w2/wi = ib/V(a2  b2) . This means that the angle between the transverse common tangents of the 2
2
2
two circles (x ± a) + z = b , which are the section of the torus by the meridian plane y = 0, is equal to that between the diagonals of the rectangular unit cell of £1. Proof.
The element of arc on the surface is given by
ds2 = (a + b cos 0)2d02 + b2d) the sum of f(w) over all elements co of ft, ft and by Zff(ct>) the sum over all nonzero elements, i. e. the same sum ft with the term for OJ — 0 omitted. Theorem 1.1. For any lattice ft and any integer n > 2, S (ft) = Z'ctT converges absolutely. Proof.
It is well known that for n > 1,
2 r~ converges r=l absolutely; denote this sum by s (it is in fact the Riemann zeta function ?(n); but the use of the letter £ here is unacceptable, since in the context of elliptic functions this letter has a quite different but equally well established meaning, to which we shall come later). If ft is simple with generator co, for even n, S (ft) = 2c*T s , and for odd n, S (ft) = 0, as the terms (rco) , (rw)"n cancel. If ft is a double lattice, the lattice points can be distributed into sets lying on the perimeters of a sequence of concentric parallelograms, similar to the unit cell, those on the r perimeter being of the form pco + qa> , where  p  , q both < r, and at least one of them = r. Denote by £ f(ct>) the sum of terms r
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perimeter; then 2 f f(^)= Z Z f(^). Now if h is the ft r=l r lesser diameter of the unit cell perpendicular to an edge, every co on the r perimeter satisfies  co  > rh, the inequality being strict for most of them; and they are 8r in number. Thus Z  CL>  < 8r(rh) , so that the series Z Z l w is majorised by the absolutely convergent series r=l r
with a) on the r
OO
T»
OO
n
8h" Z r " n , and is thus itself absolutely convergent. // r=l The quantities S (ft) thus defined clearly satisfy the homogeneity property S (kft) = k"nS (ft), for all complex numbers k * 0 and all integers n > 2, since every term in the series on the left is k" n times the corresponding term in that on the right. It follows that if n is odd, S (ft) = 0, for every lattice ft, since ft = ft, Sn(ft) = Sn(ft) = Sn(ft). Similarly, if ft is square, as ft = ift, S (ft) = 0 for all n not divisible by 4, and if ft is triangular, as ft = e ft, S (ft) = 0 for all n not divisible by 6. If ft is real, S (ft) is real for all n, conjugate complex elements of the lattice giving rise to conjugate complex terms in the sum, and real elements to real terms; and in general S (ft) = S (ft). The simple lattice generated by co can be regarded as the limit of a double lattice, of which one generator co = co remains constant, and the other co varies continuously in such a way that Im(co /w ) tends to infinity, as all the lattice points except the integer multiples of co recede to infinity, leaving the plane empty of lattice points except those of the simple lattice. The simple lattice will therefore be called a degenerate double lattice. Theorem 1. 2. When a double lattice ft, varying continuously, tends to the degenerate limit, with generator co, S (ft) tends, uniformly in Re(co /co ), to the limit 2oT s , its value for the simple lattice. Proof. Denote co /co by r; on account of the homogeneity, it is sufficient to prove the theorem for the lattice ft , generated by 1, T. Now for any even n, pairing off the equal terms for co, co, we can write OO
CO
S (ft ) = 2s + 2 £ I (p + q r ) " n . n T n q=l p=°°
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Now let k be any integer.
For each value of q, we can divide the values
of p into sets of kq consecutive integers, according as Re(p + qr) lies between consecutive multiples of kq. If Im(r) > k, whatever Re(r) may be, for the two such sets of values of p defined by rkq < Re(p + qT)kq, < (r + l)kq,
(r + l)kq < Re(p + qr) < rkq
we have p + qr > kqV(l + r 2 ), so that for each value of q, oo
oo
I
(P +
\n
J < 2(kq)xln 2 (1 + r ) "
P =  °°
replacing (1 + r ) ^ (r  I) 2 < 1 + r 2 . oo
2 2
by (r  1)
, in all but the first two terms, since
Hence
oo
2
q=l p=oo
(P +
\n
< 4k 1  n s
V '
irrespective of the value of Re(r). Thus by taking Im(r) greater than a sufficiently large integer k, we can make 1Is
(ft ) 2s  n T n
as small as we like, uniformly in Re(r); the theorem is thus proved for ft , and follows immediately for any ft = oo ft . / / 5.
Functions and periods We recall that a function f (u) of a complex variable u is analytic oo
at u = a if it has an expansion as a power series f (u) = 2 c (u  a) , r=0 r with constant coefficients c , c , . . . , converging absolutely and uniformly in some circle u  a < k, where k > 0. f(u) is meromorphic at u = a if for some integer n, (u  a) f(u) is analytic at u = a; if n > 0 is the least integer for which this holds, f(u) has an expansion f(u) = J b (u  a)" r + I c (u  a) r , r=l r=0
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with b ^ 0; in this case u = a is a pole of f(u), of order n; the terms n r r 7 b (u  a) are called the infinite part of the function f(u), b its n r=l r leading coefficient, and b its residue, at u = a. (This well established use of the word residue has of course nothing to do with residue classes, to which unfortunately we occasionally have to refer in the same contexts.) Similarly u = a is a zero of order n of f(u) if f(u) is analytic at u=a, f(a) = 0, and n is the greatest integer such that (u  a)"nf(u) is analytic at u = a, i. e. c is the first nonzero coefficient in the expansion of oo
f(u) at u = a, which is accordingly of the form f(u) = 2 c r (u  a) r . r=n A function is said to be analytic or meromorphic in a given region, or in the whole plane, if it is so at every point of the region or of the plane. If f(u) is analytic and nonzero at any point, in any region, or in the whole plane, so is JT—r ; if f (u) is meromorphic, so is JTX , the poles of each being the zeros of the other, and of the same order. The poles of a function meromorphic in any region are a discrete set, i. e. for each pole, the distances of other poles from it are bounded below; and if f(u) is meromorphic in any finite region, including its boundary, f(u) can only have a finite number of poles in the region. As jjyr is also meromorphic, f(u) can only have a finite number of zeros in the region; and as f(u)  c is meromorphic (for any constant c) f(u) can only assume a given value c in a finite set of points in the region. A period w of a function f(u) is a constant such that f(u + w) = f(u) for all u. The sum of two periods is also trivially a period, and if o> is a period, so is co. Thus the periods of any function form a group with respect to addition. On the other hand, unless the absolute magnitude of nonzero periods is bounded below, the function must be constant in any region in which it is differentiate, since ——r^—— = 0 for some arbitrarily small but nonzero values of h. Thus the periods of a nonconstant meromorphic function must be a lattice. Zero is of course a period of every function; if it is the only one, the lattice of periods is the trivial lattice, and the function is called nonperiodic. If a function has a simple or double lattice of periods, it is called simply or doubly periodic. Familiar examples of simply periodic functions are sin u, tan u, e11, with simple lattices of periods generated
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by 277, 77, 2i77 respectively. 6.
Definition
An elliptic function is a function of a complex variable, which is meromorphic in the whole plane, and doubly periodic. Since it has the same value in all points of any residue class (mod Q), where £1 is its lattice of periods, it can be thought of as a function of the residue class, rather than of the individual value of u, i. e. a function of position on the torus model of the residue class group rather than of position in the plane. Before proving (by construction) the existence of some functions with these properties, it is convenient to prove some elementary consequences of the definition, assuming that such functions exist. 7.
Liouville's theorem
This states that any function which is analytic and bounded in the whole plane is a constant. Also, a function which is analytic in any finite region (including its boundary) is bounded in that region. Hence, an elliptic function which has no residue classes of poles is bounded in the fundamental region, and so in the whole plane, and is accordingly a constant. This principle is applied in two main ways to elliptic functions: Theorem 1. 3. If two elliptic functions have the same lattice of periods, the same residue classes of poles, and the same residue classes of zeros, of the same order in each case, the ratio of the two functions is a nonzero constant. Proof. If f(u), g(u) have either zeros or poles of the same order at u = a, v^ is analytic and nonzero at u = a. // Theorem 1. 4. If two elliptic functions have the same lattice of periods, and the same residue classes of poles, with the same infinite part in each pole, the functions differ by a constant.
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Proof.
If f(u), g(u) have the same infinite part at u = a,
f(u)  g(u) is analytic there, i. e. has no pole. / / 8.
Contour integration theorems For any function meromorphic in a simply connected region R
bounded by a closed contour C, we recall the three classical theorems on integration round the contour C: Let f(u) be meromorphic in R, with zeros of orders m , . . , , m, at u = a , . . . , a, and poles of order n , .. . , n 1
at u = b , . .. , b, , with residues r , . . ., r, K —
1
K
1
res
K
pectively, all these zeros and poles being in R but none on C. Then
From this we deduce Theorem 1. 5.
Let f(u) be an elliptic function with the lattice 12
of periods, zeros of order m , . . . , i r u i n the residue classes a +12, . .. , a,+12, and poles of order n , . . . , n, in the residue classes l
n
l
b +12, . . . , b,+12, with residues r , . . . , r, k I. I r = 0; j=l 3 h k II. J m = 2 n ; j=l 3 1=1 3 h k III. 2 m.a. = 2 n.b. (mod 12) . ]=1 J J ]=1 J 3 Proof.
K
respectively.
Then
Take the contour C to be the boundary of a unit cell,
starting from a chosen point u = c, and travelling along straight lines to u = c + ct>,c + a> + a ) , c + a ; , and back to u = c in turn,
c being
chosen so that the path does not pass through any zero or pole. If 0(u) is any function of u 10 Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:55:26, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.002
J c 0(u)du= J^+a>1(0(u)  0(u+co2))du + ^ + W 2(0( u + W i ) _ 0 ( u )) d u . (8.1) If f(u) is an elliptic function with period lattice Q, generated by a> , Co , so is JT^I , and in both the integrals I, II, the integrand in both terms on the right in (8.1) is identically zero, which gives the results I, II of the theorem. As for integral III, f/ \ is not of course an elliptic function; but as in this case 0(u)  0(u + u) ) = a> f T(u)/f(u), the first term on the right in (8.1) becomes
and as f(c + w ) = f(c), the difference between their logarithms as obtained from the integral must be an integer multiple of 277i, say 2q7ii; thus the first term in (8. 1) reduces to 277i. qa> ; and similarly the other term reduces to 27ri. pc^ . Thus the integral III is equal to 2?ri times an element po> + qa> of ft, which proves the result III of the theorem. // 9.
Order of an elliptic function
Just as an sple zero of a polynomial f(x) is commonly and conveniently regarded as being s coincident zeros of f(x), or roots of the equation f(x) = 0, and this convention enables us to say that an equation of degree n has exactly n roots, when we make due allowance for coincidences; so an sple zero or pole of any meromorphic function is conventionally to be regarded as s coincident zeros or poles; and in the case of elliptic functions, with period lattice ft, if u = a is an sple zero or pole, so is every member of the residue class a + ft, which is regarded as s coincident residue classes of zeros or poles. With this convention the results II, III of Theorem 1. 5 can be restated as Theorem 1. 6. An elliptic function f(u) assumes any value c in a number n of residue classes which is independent of c and charact e r i s t i c ^ f(u), making due allowance for coincidences among these n residue classes for some values of c; moreover, the sum of these n residue classes is independent of c. 11 Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:55:26, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.002
Proof,
Results II, III of Theorem 1. 5 mean that, making due
allowance for coincidences, the number of residue classes of zeros is equal to that of residue classes of poles, and that the sum of the former residue classes is equal to that of the latter.
Moreover, the residue
classes in which f(u) = c are those which are zeros of the function f(u)  c, which is an elliptic function with the same lattice of periods and the same poles as f(u); thus the number and sum of the residue classes in which f(u) = c are equal to the number and sum of the poles. / / The integer n of Theorem 1. 6 is called the order of f(u). We f(x) may compare the order of a rational function V^, where f(x), g(x) are polynomials; the order is the greater of the degrees of f(x), g(x), and f(x) is the number of solutions of the equation X~\ — c, for any c, when we make due allowance for coincident roots (and for infinite roots) of the polynomial equation f(x)  cg(x) = 0. Rather similarly, most important simply periodic functions have a definite order in this sense; sin x, for instance, assumes any given value c in precisely two residue classes (mod 27r), which coincide for c = 1 and for c =  1 . No elliptic function can have order 1. Proof.
Any function of order 1 would have just one residue class
of simple poles, whose residue cannot be zero, since the infinite part of the function at a simple pole u = a must consist of a single term b(u  a)" , b 4 0; and this contradicts result I of Theorem 1. 5. Even more simply however, if f(u) were an elliptic function of order 1, the relation z = f(u) would define a oneone continuous mapping of the torus, model of the residue class group, onto the z sphere, which is manifestly impossible topologically. / / An elliptic function of order 2 must have either one residue class of double poles, with residue 0 (i. e. at which the infinite part consists of a single term in (u  a)" 2 , and none in (u  a)" 1 ), or two residue classes of simple poles with equal and opposite residues. For any elliptic function f(u) of order n, there can only be a finite number of values c for which the n residue classes in which f(u) = c are not all distinct.
For apart from any multiple poles (c = °°),
the places in which f(u) = c has a multiple root for any finite c are
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the stationary points of f(u), i. e. the zeros of f'(u), which is also an elliptic function, and vanishes in only a finite number of residue classes. / / In fact Theorem 1. 7.
Let f(u) be an elliptic function of order n, and the
sum of the n residue classes in which f (u) assumes any given value be w + ft; and let f(u) = c. in s. coincident residue classes a. + ft ( i = 1, 2, . . . ) . Then S(s.  1) = 2n,
S(s.  l)a. = 2w (mod ft) ,
the summation being over all residue classes a. + ft for which s. > 1. Proof.
The result is clearly unaltered if we include in the
summation any further residue classes a. + ft for which s. = 1. We therefore specifically include all the poles of f(u), whether simple or multiple; let the poles (c. = °°) be given by i = 1, . . . , h, and other (finite) multiple values c. by i = h+1, . . . , k; then h Z s. = n , i=l x
h Z s.a. = w (mod ft) . i=l l x
As an s.ple pole of any meromorphic function is an (s.+l)ple pole of its derivative, f f(u) is of order h classes of poles is Z (s. + l)a. i=l a. + ft (i = h+1, .. . , k) are the
Z (s. + 1), and the sum of its residue i=l x + ft. The remaining residue classes zeros of f f(u), a. being an s.ple zero
of f(u)  c , and hence an (s.l)ple zero of ff(u).
i. e.
Thus
k h h Z (s.  1) = Z (s. + 1) = 2n  Z (s.  1) , i=h+l x i=l i=l k h h Z (s.  l)a. = Z (s. + 1) = 2w  Z (s.  l)a x l i=h+l 1 i=l 1 i=l 1 k k Z (s.  1) = 2n, Z (s.  l)a. = 2w (mod ft) , l t=l x i=l X
(mod ft) ,
which proves the theorem. / / In particular, if n = 2, there are four values of c (necessarily all distinct, and possibly including infinity) for which the two residue 13 Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:55:26, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.002
classes in which f(u) = c coincide. If a + ft, b + ft are two of these, 2a = 2b (mod ft); thus the four are a + ft, a + w + ft, a + co + ft, a + w + ^w
+ft,
for some a, and the union of all four is a residue class (mod ift).
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2 • The Weierstrass functions
10.
The Weierstrass function $>u
Largely for historical reasons, it is usual to denote a pair of generators of the period lattice of an elliptic function by 2OJ , 2w ; we shall therefore denote the lattice itself by 2 ft, where ft is generated by co , co . When it is desirable to specify the period lattice with which a given elliptic function f(u) has been constructed we shall write it f(u2ft), but where only one lattice is under consideration we shall write simply f(u), or even in some cases fu. In the first place, we define for any integer n > 3 P (u2ft) = £ (u 2co)~n . n ft This series converges absolutely and uniformly in every finite region which finitely excludes all lattice points, such as u < K, u2co > k, where K, k are positive constants which can be taken as large and as small as we like. Proof. For all u in the region, and for all co such that  co > K (i. e. for all but a finite number of terms in the series) u2co > co, so that the series is majorized by that for S (ft). // By taking K large enough and k small enough, the region of convergence can be made to include any chosen point of the plane, except a lattice point u = 2co; thus P (u) is analytic except at the lattice points. These are poles of order n, with infinite part (u  2co)"n, since by omitting this one term from the series, the point u = 2co can be included in the region of convergence. If co is any element of ft, 2co is a period of P (u), since the substitution of u + 2co for u merely permutes the terms of the series amongst themselves, as the aggregates ft, ft  co are the same; and (the convergence being absolute) does not affect the sum. Thus P (u 12ft) is an elliptic 15 Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:28:06, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.003
function of order n, with period lattice 2ft. Further, differentiating the series term by term, we have Pf (u) = nP + , (u). For n = 2 however, the corresponding series does not converge, as that which might be expected to define S (2 ft) does not converge. To obtain an elliptic function of order 2 in this sequence we define (pu= k for all co in ft) as that in which the series defining P (u) (n > 3) converges. n
Proof.
•
As
and if u < K and a> > K, then l  i L  <  and l  iL  > f, we have for all u in the region, and all but a finite number of terms in the series (u 2co)~2  (2c*>)~2 < Ka)" 3 so that the series is majorised by that for KS (ft).// Thus (Pu is analytic in the whole plane, except for the lattice points 2ft, where it has poles of order 2, with infinite part (u  2o>)~ . It is an even function, i. e.
=
3e
< 14 ' 3>
i "&
by (12. 1). Thus ,2
. ,,2
2X
d e (pu+(d.e ) 1 ^ (U  co.) = 1— + e. = — —±0 v ; i (pue. I (pue. which is equivalent to the theorem, by (14. 3); obviously from the periodicity of
(pu,
.) = (p (u  co.). / /
Further, if u = u  co. (mod 2£2), i. e. if u = j>w. (mod Q), we have ( (pu  e.) 2 = d?, i. e.
( p u = e.± d..
There are four residue classes
(mod 20) satisfying this condition, namely ± co. + 20, ±(co. ± co.) + 2fi, where j ^ i; it does not matter which of the two suffixes other than i is used here, as co. ± co. = co. ± co, (mod 212). We 1
]
1
K
define the square root d. so that (p (iw.) = e. + d., 15.
(p (Ico. + co.) = e.  d.
(14. 4)
Addition and duplication formulae Theorem 2. 2.
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^H
 2 w
(p fu = f(u) + A
"u = (Pmu = 0, since they are zeros of P (u) except when n is divisible by 3, by the same argument as for (Pu. In fact the three residue classes in which (P'u assumes any given value are of the form v + 2 ft, ev + 2ft, e v + 2ft; if two of these coincide, all three do, and this occurs 2
in the three residue classes v = 0, ±a> . 22.
Invariants of the real lattice We have seen that g , g
generally g.(ft) = g.(ft),
are both real for a real lattice (more
i = 2, 3). Conversely, if g 2 , g 3 are both real,
so is the lattice, since (11. 4) defines a real function, and if w is a period of this, so is w. The discriminant of the equation 4x  g x  g 3
=0
is
2
A = g  27g . This means in the first place that two roots (at least) of 3
2
the equation coincide if and only if g = 27g ; but also, for real g , g , that the roots are all real or one real and two conjugate imaginary, according as g3 > or < 27g2. Thus for real g , g , g3 > 27g2 is the necessary and sufficient condition for a rectangular, and g3 < 27g2 that for a rhombic lattice; and g3 =£ 27g for any proper double lattice, since we have seen that in all cases e , e , e are all distinct. It is convenient to classify rhombic lattice shapes (other than the square and the triangular) into extreme and medium, according as the shorter diagonal of the rhombus is shorter or longer than its sides, i. e. according as the ratio of the lengths of the diagonals is greater than V3, or between V3 and 1; the limiting values V3, 1 of this ratio corresponding evidently to the triangular and square shapes. Theorem 2. 9. For the real lattice, whether rectangular or rhombic, g is positive or negative according as the lattice is vertical or horizontal, g is positive for all real rectangular lattices, and for the real rhombic lattice g is positive or negative according as the lattice is extreme or medium. Proof. Consider a variable lattice ft, generated by a> (constant and real) and variable a; — ra> . We obtain one sample of every rectangular lattice shape, in the vertical position, by keeping Re(r) = 0 33 Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:28:06, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.003
constant, and letting Im(r) increase steadily from 1 (square lattice) to + 00 (the degenerate limiting shape). In the same way we obtain one sample of every rhombic lattice shape, in the vertical position, by keeping Re(r) = \ constant, and letting Im(T) increase steadily from \ (square lattice), through V3 (triangular lattice), to +°° (again the degenerate limiting shape), so that \ < Im(r) < \^7> gives the medium and Im(r) > V3 the extreme rhombic shapes. By Theorem 1. 2, as Im(r) tends to infinity, g , g (for the period lattice 212) tend, independently of Re(r), to the limits g* = — s aT 4 , g* = — s aT 6 , both obvious2
2
4
1
3
8
6
1
ly positive. Thus in both the prescribed variation processes, with Re(T) = 0 and with Re(r) =  \ , g starts from the value 0 for the square lattice, and varies continuously remaining real and not passing again through the value 0, and tending ultimately to the positive limit g*; it thus remains positive throughout the process; i. e. g is positive for all vertical real lattices, whether rectangular or rhombic; and since if 12 is vertical i!2 is horizontal, and g (i£2) = g (17), g is negative for all horizontal lattices, rectangular or rhombic. As for g , it is the same for the vertical and horizontal lattices, as g (i£2) = g (12); it is positive for all rectangular lattices, since g > 27g > 0; and for the rhombic lattices, as Im(r) varies from \ through 1V3 to +°°, with Re(r) =   , g starts from a negative value (for the square lattice in the rhombic position), passes through the value 0 only at Im(r) = iV3 (for the triangular lattice), and tends ultimately to the positive limit g*; it is thus negative for \ < Im(r) < iV3, the medium rhombic shapes, and positive for Im(r) > iV3, the extreme rhombic shapes; and the theorem is proved. // A simple corollary from this is that the zeros of (Pu are in the longer symmetry axis of the fundamental unit cell, whether this is rectangular or rhombic. For the real rectangular lattice 12, as e > e > e and e + e + e = 0, e > 0 > e , and the sign of e is opposite to that of g = 4e e e ; thus according as 12 is vertical or horizontal, 0 is between e , e or between e , e . Similarly, for the real rhombic lattice 12, as e , e are conjugate imaginary, e has the same sign as g , i. e. 0 is < e or > e according as 12 is vertical or horizontal.
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Theorem 2.10. The limiting values g*, g* of the invariants, corresponding to the simple lattice with real generator u> as degenerate limit of a variable double lattice, satisfy g*3 = 27g*2. Proof. In the variation process with Re(r) = 0 described in the proof of the last theorem, g3 > 27g at all stages; hence in the limit g*3 > 27g*2. Similarly in the variation with Re(r) = £, g3 < 27g2 at all stages, so that g*3 < 27g*2. // This means that there is a positive real number e* such that 2 g* = 3e* , g* = e*3, and 4x3  g*x  g* = (2x+e*)2(xe*); thus the limits of e , e , e are e*, £e*, £e*. Similarly for the limiting form of the horizontal lattice (rectangular or rhombic), the simple lattice generated by iw , they are  e * , ie*, e*. We have also s = e* 2 w 4 , s = —e*3w ; and we may round off this result by anticipating what will 6
35
1
be proved when we come to study the Jacobi functions, that for w = TT, 7T 4
2
7T 6
e* = ; are thus theknown actualofvalues s , the s theory are s of= the ^, Riemann s = ^rczetaThese values also courseoffrom function. 23.
Properties of 2, S q(p+qi)~
is the sum of 4nl terms, each of which is in  6
 6
absolute magnitude < n , and all but one in fact < n , so that S q(p+qi)~7 < 4n~5, and oo
Z n q(p+qi)" ?  < 4(2~ 5 +3~ 5 +4~ 5 +5~ 5 +...) ' ' ' ' ~~ 1 5
'
by the usual device of grouping together the first two terms, the next four, the next eight, and so on. Thus
00
oo
12
IZ
q(p+qi)~ differs from  i by q=l p=°° a quantity less than — in absolute magnitude, and hence cannot vanish. The lemma is thus proved. // We now define I(T) = g3(r)/27g (T), which is meromorphic in the whole of the open upper half plane, its poles being the zeros of g (T), double poles in the group set of r containing r = i; that they are double, and not of higher order, is of course what the lemma was required for. We are now in a position to prove: Theorem 3. 4. I(r) assumes every complex value C ± 1 in precisely one point of the fundamental region (26. 2), and hence in precisely one point of every fundamental region of F. Proof. This is easily seen for the real values of C; for I(r) = 0 at T = S , it is infinite at r = i, and is real and negative at all points of the unit circle between these (g being negative for the medium rhombic lattice), and hence assumes every real negative value at some point of this arc. Similarly I(T) tends to the limit 1, uniformly in Re(r), as Im(r) tends to infinity; and as I is between 0 and 1 for the extreme rhombic and > 1 for the rectangular lattice, I(r) assumes every real value between 0 and 1 at some point on the half line Re(r) = f, Im(r) > iV3, and every value > 1 at some point of the half axis Re(r) = 0, Im(T) > 1; and it can assume none of these values more than once, since no two of the values of r in question correspond to the same lattice shape. We therefore take any imaginary value of C, and consider the function I(r)  C, which certainly does not vanish on the boundary of the region. We construct a contour for integration as follows: choosing a real constant K (which may be as large
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as we like) we start from r= i + Ki along the horizontal line Im(r)= K to r =  ! + Ki; then down the vertical line Re(r) =  i to T = e ; then along the unit circle r = 1 to its first intersection with the circle  r  i  = k, where k is another arbitrary positive constant, which we can make as small as we like; then clockwise round this last circle to its second intersection with \r\ = 1; on along  T  = 1 to T =  e 2 ; and finally up the vertical line Re(r) = \ back to the starting point. Since as Im(r) tends to infinity I(r) tends to the value 1, V(r) tends to zero; thus even if C is very close to 1, we can choose K large enough to make  I ( T )  1  < i  C  l  , lf(r)  < i  C  l  , so that  l ( r )  c  > £  C  l  > l f (r) at all point of the line Im(r) = K. Also, however large C may be, we can choose k small enough to ensure that the leading term A(ri) in the expansion of I(T)  C at its pole r= i is as large as we like in comparison with the rest of the series. Round this circuit we integrate ,) J n . For the first segment, from r = i + Ki to T=   + Ki, as the total length of the path is 1, and the absolute magnitude of the integrand is everywhere < 1, the absolute value of the integral is < 1; but as this is the integral of d log(I(T)C) between two points at which I(r)C has the same value, the integral must be a whole multiple of 27ii, and can thus only be 0. The parts of the integral on the two vertical lines evidently cancel, as we are integrating d log(I(T)C) through the same values in opposite senses; and similarly the parts of the integral along the two arcs of the unit circle cancel. This only leaves the small semicircle  ri =k; and here, by taking k small enough, we can make the integral of d log(I(r)C) as close as we like to the contribution from the leading term A(ri)~ in the expansion of I(T)C, which is itself very close to 2ffi, as we are describing in the negative sense an angle very close to n. Here again however, as I(T)C has the same value at the two ends of this arc, the integral must be a whole multiple of 2ffi, and hence is precisely 27ri. The value of the integral round the whole contour is accordingly 2fii; and as I(T)C has no pole within the contour, it has just one zero. Also, by taking K large and k small enough, the contour can be made to enclose every internal point of the fundamental region; thus I(r) = C at just one point in the region, and the theorem is proved. // 50 Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:24:55, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.004
Corollary. The correspondence between complex values of the absolute invariant I(2£2) and lattice shapes is oneone without exception, provided infinity is included among the complex values, and the degenerate or simple lattice, corresponding to 1(2^2) = 1, among lattice shapes. Conjugate imaginary values of I correspond to lattice shapes which are mirror images of each other. As increasing real values of a complex variable have the upper and lower half planes to left and right respectively, the left and right hand halves of the fundamental region (26. 2) of r are maps (conformal except at the corners) of the upper and lower halves of the I plane; and similarly each white region in the modular pattern is a map of the lower, and each shaded region of the upper half of the I plane. In terms of the absolute invariant J = g 3 /A = 1/(11), this last rule is reversed, since real values of I, J everywhere increase in opposite senses along the same loci in the r plane. As a function of r, J has the advantage over I that it is analytic throughout the open upper half of the r plane, since 1 = 1 , °° correspond to J = °°, 1 respectively. The main reason why we have considered I rather than J in this chapter is that it is rather easier to see that the integral of d log(I(T)C) along the portion Im(T) = K of the contour is 0, than that the integral of the corresponding function for J is equal to 2xri along this line.
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4The Jacobi functions
28.
The three halflattices of O
A lattice O being a group with respect to addition, a subgroup of finite index is called a sublattice; one of index n may be called an n lattice of O. In particular, 20 is a quarter lattice of O; its three cosets, other than itself, are the residue classes (mod 20) of elements of O that are not in 20, i.e. to. + 20(i=l, 2, 3). The union of any one of these, co. + 20, with the zero residue class 20, is a half lattice of O, which we denote by O., and 20 is in its turn a half lattice of O.. S2 , O , O consist of those elements mco + nco of O in which respectively n = 0, m = 0, m = n (mod 2). We can take generators for these three half lattices as
respectively; and with this convention (O ) = (O ) = (O ) = 20; the generators of (O ) and (O ) being (2co , 2co ), and those of (O ) (2co , 2co ). If O is rectangular, with co /co pure imaginary, O and O are also rectangular, and O is rhombic; if O is rhombic, with co /u) unimodular, O and O are mirror images of each other, conjugate imaginary if O is real, and O is rectangular. For the square lattice, two half lattices are rectangular, the rectangle's length being twice its width, and the third is square; for the triangular lattice all three are rectangular, the length being V3 times the width. 29.
The three root functions
If a function f (u) is analytic and nonzero at u = a, its two square roots are analytic and wholely distinct in some circle contained in the circle of convergence of f(u); hence if either of these square roots is 52 Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:32:04, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.005
carried by analytic continuation round any closed path contained in this circle, it returns to its original value. If f(u) has on the other hand a zero of order n, or a pole of order n, at u = a, f(u) = (ua)ng(u), where g(u) is analytic and nonzero at u=a; and if the square roots are carried by analytic continuation round a small circuit surrounding u= a, since Im(log(ua)) is increased by 2TJ\, Im(log(ua)2n) is increased by n?ri, so that the two values of (ua) 2n , and hence also the two square roots of f(u), return to their original values or are interchanged on returning to the starting point, according as n is even or odd. In the latter case we say that the square root is branched at u= a. It follows that if f(u) is analytic and nonzero on a contour C bounding a simply connected region R, and is meromorphic throughout R, the two square roots of f(u) return to their original values on being carried round C by analytic continuation, or are interchanged on returning to the starting point, according as the number of points in R where they are branched is even or odd, i. e. according as the sum of the orders of all the zeros and poles of f(u) in R is even or odd. Now for i = 1, 2, 3, the function $> u e. has in the whole plane exclusively poles and zeros of order 2, in the residue classes 2£2, (x). + 20, respectively. Its square roots are accordingly not branched anywhere, and the choice of one square root at any point, not a zero or a pole, uniquely determines by analytic continuation a single valued function in the whole plane, everywhere meromorphic and a square root of (pue., with simple poles and zeros at the double poles and zeros of #> u e.. Since the leading term of $> u e. at the origin is u" 2 , those of its two square roots are ±u~ ; we define the root function f.u to be that square root of #>u e. (i = 1, 2, 3) whose leading term at the origin is +u , i. e. whose residue there is 1. f.u is evidently not an elliptic function with the period lattice 212, or it would be of order 1. It is in fact an elliptic function of order 2, with period lattice 212., with simple poles in the two residue classes (mod 212.) that make up the zero residue class (mod 212). Proof. It is easily seen that if an even function has a zero or pole at the origin of order 2n, its square roots are even or odd functions accor
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ding as n is even or odd. Thus as
$>u e. is an even function, not
only of u, but of u  a>., and also of u  a>. (j ± i), f .u is an odd function of u and of u  o>., but an even function of u  co. (j =£ i). From f .(2co.u) = f.u = f.(u), f.(2o).u) = f.u = f.(u), we have on replacing u by u f.(u+2co.) = f.u, f.(u+2co.) = f.u (j * i) ,
(29. 1)
whence also of course f.(u+4a>.) = f.u. Thus f.u has the lattice 2ft. of periods, generated by 2co., 4co..// Obviously, as
, e.(2mft) = m" 2 e.(2ft),
#>(mu2mft) = m " 2
we have f.(mu2mft) = m~4.^12ft) (i = 1, 2, 3) l
(29. 2)
l
where the lattice indicated after the lying function
is the period lattice of the under
$>u, not of f.u itself; we shall call 2ft the underlying
lattice of the three root functions f.u (i = 1, 2, 3). 1
2
Putting
2 of 212* are always denoted by 2K, 2iK' (K, KT real and positive). 2i!2*, also normal, is generated by 2KT, 2iK; and the assigned generators for the other normal lattices 2kl2*, 2ikl2*, 2kf12*, 2ik'12* are 2k(K+iKf), 2ikKf; 2kK\ 2k(K'+iK); 2k'K, 2k'(K+iKt); 2kt(KT+iK), 2ikTK. For the rhombic lattice on the other hand, none of the normal lattices is real; since one of e , e , e is real and the other two conjugate imaginary for the real rhombic lattice, one pair of equal and opposite values of h are pure imaginary, and the other two pairs are conjugate imaginary; and hence the diagonals of the rhombi are inclined to the horizontal at angles ± TT for one pair, a, a + \n for another 4
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pair, and G?, \TICL for the third pair, where we can take 0< a< 77; it is easily seen that a = ^ for the triangular lattice, so that 0 < a < ~ for the extreme, and ^ < a < j for the medium rhombic lattice. For this reason, the theory of the Jacobi functions, in which the lattice used is normal throughout, though it is formally applicable to lattices of any shape, is only really useful and simple when the lattice is rectangular. We shall therefore throughout this chapter, when anything involving reality or complex conjugacy is concerned, assume ft* to be rectangular, with real and pure imaginary generators K, iKT assigned, so that k, kT are between 0 and 1. 31.
The JacobiGlaisher functions We now define csu = fl(u2O*), nsu = f2(u2ft*), dsu = f3(u2ft*),
(31.1)
whose period lattices are 2ft*, 2ft*, 2ft*, generated respectively by (2K, 4iKf), (4K, 2iKT), (2K2iKT, 2K+2iKf). We define further spu = —u, ps where p is any one of the letters c, n, d; and pqu = psu/qsu, where p, q are any two of c, n, d. It follows that if p, q are any two of the four letters s, c, n, d, pqu=l/qpu, and if p, q, r are any three of the four letters, pqu = pru/qru. The poles of csu, nsu, dsu are all in the residue class 2ft*, and their zeros are in the residue classes K+2ft*, iK'+2ft*, K+iKf+2ft* respectively, spu has accordingly zeros in the residue class 2ft*, and poles in the zeros of psu; and if p, q are any two of c, n, d, pqu has neither zeros nor poles in 2ft*, its value at the origin being 1, since psu, qsu both have the leading term u" ; but pqu has zeros in those of psu, and poles in the zeros of qsu. It follows that the four letters s, c, n, d are associated with the residue classes 2ft*, K+2ft*, iK'+2ft*, K+iK'+2ft* respectively, in such a way that each of the twelve functions pqu, where p, q are any two of s, c, n, d, has zeros in the residue class associated with p, and poles in that associated with q. The four functions obtained by any partition of the four letters s, c, n, d into two pairs have the same period lattice 2ft* (i=l, 2 or 3). 56 Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:32:04, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.005
Proof. Let pqr be any permutation of end; if co is in 212*, 2co is a period of all the twelve functions; if it is in ft* but not in 2 ft*, it is in precisely one of ft* ft*, ft*; thus if ps(u+2w) = psu, either qs(u+2a>) = qsu and rs(u+2co) = rsu, or qs(u+2w) = qsu and rs(u+2co) = rsu; and in either case qr(u+2co) = qru. In exactly the same way, if ps(u+2o>) = psu, qr(u+2a>) = qru. // The functions snu, cnu, dnu, with their poles in the residue class iK' + 2ft*, are the Jacobi functions properly so called. They were defined by Jacobi under the names sin amu, cos amu, Aamu; the notation snu, cnu, dnu was introduced later by Gudermann, who also used tnu for JacobiTs tan amu, the modern scu. Later still Glaisher defined the other nine functions, as reciprocals and quotients of the three already known, just as has been done here in terms of csu, nsu, dsu, and he devised the notation now in use. Although it is in many ways convenient to treat all twelve functions symmetrically, it can still be said that the three original functions of Jacobi are the most important. From the quadratic identities (29. 4) we have at once 0
1
1
1
0
1
1
1
0
k2
k< 2
1
k
\
k 2
'
1
1
0
/
cs u ns u ds u
= 0 ,
1
four equations of which any three are linearly dependent. Multiplying all these in turn by sc2u, sn2u, sd2u, they become equations between each set of three functions with common poles. In particular, for the three Jacobi functions properly so called we have cn2u + sn2u = dn2u + k 2 sn 2 u = 1 , J2
,22
,.2,2
2
(31. 2) ,.22
dn u  k cn u = k' , dn u  en u = kT sn u , of which the most important to remember are the first two, (31. 2); all the others follow from these. The relations (31. 2) make intelligible Jacobi's notation referred to above, though the motivation will only appear when we consider the integrals of the first kind; there is in fact an angle, denoted by amu (the amplitude of u) whose sine and cosine are snu, cnu respectively; 57 Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:32:04, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.005
2
2
and for any angle 0, V(lk sin 0) is denoted by A0. 32.
Stationary values
The poles of each of these twelve functions, and likewise its zeros, being a residue class (mod 20*), are two residue classes (mod 212*), the actual period lattice of the function, which is thus of order 2. The poles and zeros together are a residue class (mod O*); and the other residue class (mod O*) which with this makes up the lattice O*, and which like it is the union of two residue classes (mod 20*), and of four (mod 20*), consists of stationary points of the function; for each of the twelve functions is either an odd or an even function of u  a>, for all o> in O*; it is odd if u = co is a pole or a zero; otherwise it is even, and its value at u = co is stationary. Since the stationary values of f .u are the square roots of e.e., e, e., those of csu, nsu, dsu are in each case the square roots of two 2
2
of the six quantities ±1, ±k , ±kT . The signs to be given to the square roots are seen from the fact that as each of these functions has the leading term u" at the origin, it is real and positive for small real positive values, and negatively pure imaginary for positively pure imaginary small values of u; thus as u travels round the rectangle from u = 0 to K, K+iKT, iKT, and back to 0, the function diminishes through real positive values to 0 on one angle, and from there on is negatively pure imaginary. Hence the values of csu at u = K, K+iKf, iK' are 0, ikT, i; those of nsu are 1, k, 0; and those of dsu are kT, 0, ik. Their values at the remaining lattice points O* are found immediately from these by the periodicity and parity properties of the functions; and the values of the remaining nine functions from these by division. Figure 8 shows the values of all twelve functions in the lattice points O* (poles, zeros, and stationary values) for 2K < Re(u) < 2K and 2Kf < Im(u) < 2Kf, and in other respects (loci of real and pure imaginary values, direction of increase, and the quadrant of the function plane mapped on each rectangle of the pattern) is constructed like Figures 1, . .. , 6. These stationary values determine the moduli of the half lattices O* (i = 1, 2, 3) in terms of those of O*. Since in fact 58 Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:32:04, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.005
one modulus for ft* is
R(a, a, b, b) =
one
Ik 1k 2 for ft* is TTt = ("L^") , and one for ft* (which we recall is rhombic, k+ik'
2
not rectangular) is , ., t = (k+ik') . The common periods of all twelve functions form the lattice 4ft*, and this is accordingly the period lattice of a general linear combination of any three of them that have the same poles, which is a function of order 4 with four residue classes of simple poles. 33.
Half period translations
If f(u) is any one of the twelve functions,
f(u+K), f(u+iK'), or
f
f(u+K+iK ) is another of them, apart from a constant multiplier; for the three that have the same period lattice as f(u) have their zeros and poles displaced from those of f(u) by the vectors K, iKf, K+iKf r e s pectively.
The constant multiplier can be found by comparing the values
at any of the lattice points ft* that is neither a pole nor a zero. The results can be read off directly from Figure 8. Thus as sn(u+K) has poles in K+iKT+2ft*, and zeros in K+2ft*, which are the poles and zeros of cdu, sn(u+K) = Acdu, for some constant A, by Liouville's theorem; and as snK = 1 and cdO = 1, A = 1. The results, all of which can be found similarly, are tabulated below: f(u)
f(u+K)
f(u+iKf)
f(u+K+iKT)
cs u
k'sc u
i dn u
ik'nd u
sc u
—
i nd u
F
i cs u
ik'sc u
p c 11
\
dnu
dn u
k'Cs u k'ndu
nd u
idnu
i sc u
"F csu
ns u
dc u
k sn u
k cd u
dc u
ns u
k sn u
sn u
cd u
k cd u 1
cd u
sn u
Idcu
idcu
ds u
k'nc u
ik en u
 I ns u k ikk' sd u
nc u
j T ds u
ik sd u
genu
en u
k'sd u
idsu
ik'
sd u
rrCnU
(33. 1)
nc u
is d s u 59
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2
I
T
»ik «
I
!
4
3
_
!
3
i
2
kf i i
i
•ooo —
 OO ^
sc u —1—
CS U — oo < 
I—'—1 !
1

4
I'M
• 1
l
(35. 1)
J
Residues of the functions
36.
These follow at once from the known residues of the root functions f.u at the origin, which are 1 in each case by definition, using the trans1 i lation formulae (33.1). Thus as sn(u+iKf) = ^ nsu, the residue of snu 1 at the simple pole u = iKf is j times that of nsu at u = 0. In the table below typical members of each of the two residue classes of poles are given for each function, with the residue at each; in the first, second and third columns, the residue classes are (mod 212*, 212*, 212*) respectively. csu
2iK
f
{ KK+2iK
scu
T
T± i
dnu
1iK' j K+iK ^KiK'
dcu
i
"F i
1
2K
1
iK lK
1
snui
i f
ndu
1 1
0
nsu
ncu
(36. 1)
1
l2K+iKf K+iK
0
1
iKt
cdu I
dsu
'
lKHiK T
cnu
"E I "k 1 k
sdu
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37.
The addition theorem
This has always been regarded as one of the most central and essential features of the theory, and many different proofs have been given. From our point of view, much the most natural proof is directly from the addition theorem for the Weierstrass function. We begin by proving for the three root functions for a general lattice: Theorem 4.1.
f .(u+v) = 1
f.u. f .v. f,v  f.v. f.u.f, u J 2  — 2 2 f v  f u I
(37.1)
I
where ijk is any permutation of 123. Proof.
Note first that
f 2 u.f^u  f*v.f£v = ( (pue.)( . + 2ft, which are simple zeros of the denominator; and it has simple zeros in those of the numerator, the two residue classes w. + 2ft (j * i). It is thus of order 2. It is an odd function, not only of u (all three factors being odd), but of u  co. and of u  a>. (j * i), since in each of these cases just one factor is odd. By the homogeneity of the root functions, g.u also has the homogeneity property g.(mu12mft) = m^g.fa12ft). We note also that
Ag.
u =
(
f
.
u
.
k
k
j
k
i
i
]
^
in particular the derivatives of g.u at its zeros u= LO. + 2ft (j ^ i) are g.'o). = g.Tco = ek~e> where ijk is any permutation of 123. Since g.u, g.(u±co.) have the same poles, with the residues ±1 interchanged, their sum is constant by Liouville's theorem; and as they have also the same zeros, this constant sum is 0. Thus g.(u±a>.) = g.u .
(39.1)
Again, for j * i, the product g.u. g.(u±a>.) is constant, the poles of each
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factor coinciding with the zeros of the other; and as at u = co. the two —1
factors have expansions with the leading terms (e,e.)(uct>.), (uco.) K J J J respectively, the constant product is e,e.; thus K J g.(u*«.) = (e,e )/g u (j * i) . (39. 2) 1
J
K
J
1
It is clear that the twelve functions ±g.u, ±(e, e.)/g.u are all those with X
K
J
1
the period lattice 2Q that have simple poles with residues ±1 in two of the residue classes 20,, a>. + 2£2 (i = 1, 2, 3) and simple zeros in the other two. These functions satisfy four linear identities, each linking three that have a zero in common; taking the signs of these so that the residues, in each case ±1 in two of the other residue classes, are in cyclic order, their sum has total residue 0 in each of these, and hence has no poles, and by Liouville's theorem is a constant, which must be 0, as the three terms have a common zero. These four relations are g u  g u + (e e )/g u = 0 (ijk  123) (e 2 e 3 )/ g i u + (e 3  e i )/g 2 u + ( e ^ e ^ u = 0 . If we substitute for the three functions here their defining expressions in terms of the root functions, clear of fractions and remove common factors, they reduce to the quadratic identities (29. 4). For the normal lattice 2£2* these functions are gTu = g^uiK) = nsu. dcu = dsu. ncu =
sn
^(^u
gi(u±iKT)^g1(u±K±iKt) = k 2 snu. cdu = k2cnu. s d u = k g u = g (u±iKT) = dsu. cnu = csu. dnu = 2
2
cnu
dnu
SnU 2
g2(u±K±iK')=g2(u±K) = k' sdu. ncu=k' scu.
,.,2 ^
> (39.4) ^
g3u = g3(u±K±iK') = csu. ndu = nsu. cdu = g3(u±iKf) = g3(u±K) = scu. dnu = dcu. snu = As g (Ku) = g (u) = g u, g u is an even function of u  ^K; its stationary points are accordingly u = ^K + £2*, four residue classes (mod 2Q*). Similarly those of g2u, g3u are u = £ i respectively. Taking from (34. 1) the squares of the factors in (39. 4), we see that 73 Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:32:04, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.005
g 2 (K)=(l+k') 2 2 g
(iK+iK')=(lk') 2
g 2 (iK')=(l+k) 2 2 g
2 g
(K+liK')=(lk) 2
(K+^iK')=(kik') 2 (39. 5)
g 2 (KiK')=(k+ik') 2
The differential equations satisfied by g u, g u, g u are accordingly H"5C ?
&
(—j —x ?H+k T
7
QU
?
)x +k
4
c\"X ?
4
P
o
d
.
^—) — v + ? H + k jy +kT Q.U
Hv 9
4.
C_r—Y QU
?
?Ck
9
9
kT IV +1
respectively, the roots of the right hand member being in each case the stationary values of the function; no constant multiplier is required, as x having the leading term u" at the origin, (v)2 and x both have 4
the leading term u . As g l u.g 1 (iK f u) =  g ^ . g ^ u  i K ' ) ^ k 2 , g 2 (iK') = k 2 ; and similarly g 2 (!K+iiK') = g 2 (iK') = k2,
2 g
(iK)  g 2 (K+iK0 = k'2
The signs to be given to the square roots in (39. 5), (39. 6) can be determined from considerations of continuity, starting from the fact that as, for the real normal lattice, f u, f u, f u are all real on the real u axis, and positive between u = 0, K, and are pure imaginary on the imaginary axis, negatively between u = 0, iKf, g u, g u, g u have these properties also. From this it is easy to obtain the distributions of values shown in Figure 9, which is constructed like Figure 8, but on twice the scale, so that each diagram shows only the unit cell K < Re(u) < K, KT 0
i i
i i i
I 1 I
1k
1
l T
i
lkf
_
1k
1+k
i 0
J
1 t 4
1k
0
"F"
i i i *
•0
J 1 2
I
1k "k f
• oo 
kik f kik T
i
i t
i
kik
kik f k+ikf
k+ikT
kikT
k+ikT
k+ik'
1
i
k+ik
OO
>  1
3
kik'
Figure 9
i
k+ik
f
i
kik
Figure 10
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the pairs (k+4O, and is thus equal to 2g.(u4O) = g.(u2O), by the homogeneity of g.u; and similarly f .uf, u = g.(iu±w.). Putting 2u in 1 J K I j place of u, g.u = f .(2u)+fk(2u), g.(u±w.) = f .(2u)fk(2u)
(39.10)
and for the normal lattice 77 Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:32:04, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.005
gxu = ns(2u)+ds(2u) g2u = cs(2u)+ds(2u) g u = ns(2u)+cs(2u)
g^u+iK1) = ns(2u)ds(2u) g2(u+K) = cs(2u)ds(2u) g (u+K) = ns(2u)cs(2u)
(39.11)
Substituting 2u+K, 2u+iKT, 2u+K+iKf for 2u and making use of the half period translation formulae (33.1), we obtain expressions for all the 96 functions, each as a linear combination of two of the twelve JacobiGlaisher functions of 2u. In particular kh i u = dc(2u)+ktnc(2u), k'h2u = kcn(2u)+dn(2u), h3u=kcd(2u)+ik'nd(2u) (39.12) 40.
The Weierstrass function for the half lattices
The functions dnu, cdu, cnu have the period lattices 2£2* (i = 1, 2, 3) respectively, and all have the stationary value 1 at the origin; and from the derivatives (35. 1) dn"(0) = k 2 , cd"(0) = k f2 , cn"(0) = I ; thus 1dnu, 1cdu, 1cnu are functions with the stationary value 0 at the origin, and expansions there beginning with the terms  k u , k f 2 u 2 , iu 2 . It follows that l, 2 1+dnu l, T2 1+cdu l 1+cnu 4 1dnu ' "4 1cdu ' 4 1cnu are functions of order 2 with the period lattice 2 £2* (i = 1, 2, 3) respectively, each with a double pole at the origin with leading term u~2, and hence by Liouville's theorem differing only by a constant from the corresponding Weierstrass function $>(u2£2*). Moreover, as the three functions vanish at u=2iKf, 2K, 2K respectively, they are the functions (pue , (pue , #>ue for the three lattices, taking the generators of these in the standard form (2K, 4iKT), (4K, 2iKT), (2K2iKf, 2K+2iKT). Taking the values of dnu at u=K, K+2iK!, of cdu at u=iKT, 2K+iKT, and of cnu at u=K=FiK' from Figure 8, we see that the remaining stationary values of these functions are
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e e
r 24(1+k')2
e e
r 3=T (k+lk ' )2
e e=4(lk) 2
e e=i(lkf 2 4
2 e e3 =i(kik') 4
and a s for any lattice e + e + e = 0 , this gives
k'2k2
1+k'2 e
i
=
for 212*, 212*, 212* respectively. v i, 2 1+dnu ) = Tk T^TT;
/ l
Thus
1+k' 2 * i 003k , i, f 21+cdu , 1+k2 r— > 9 (u212*)^k T x _ c d u +g— , (40. 1)
4 1cnu and the stationary values e , e , e
of the Weierstrass function for all
three half lattices are 212*: 212*: 212*:
(l+6k f +k T *)/12, (l+k f )/6, (l6kT+kT (l+k 2 )/6, (l+6k+k 2 )/12, (l6k+k 2 )/12 (k 2 +6ikk'k' 2 )/12, (k 2 6ikk f k t 2 )/12, (k' 2 k 2 )/6.
(40. 2)
The results for 212* enable us to extend to the rhombic lattice the expression of the Weierstrass function in terms of the Jacobi functions, which is often useful, especially in view of the fact that excellent tables of the Jacobi functions exist for real values of u, k (0 < k < 1), whereas practically no useful tables of the Weierstrass function have been made. For the real (rectangular) normal lattice 212*, from e  e = 1 , e e =k 2 , e  e =kT , e +e +e =0, we have at once 2k'
k f2 2
l+k f
and of the expressions
2k 1
l2k T '
(40. 3)
#>u = cs u+e = ns u+e = ds u+e , the second
is generally more convenient in use; we have thus for its loci of real values, making use of
fu, and f(u) = snu, ff(u) = cnu. dnu. In the former case we have F(x) = 4x3g xg , and in the latter F(x) = (lx 2 )(lk 2 x 2 ), since en u = 1sn u, dn u = 1k sn u. In the latter case the elliptic function field C(f(u), ff(u)) is E(2to*), not E(2to*); and similarly it is the residue class group (mod 2to*), not (mod 2to*), that is in oneone correspondence with the aggregate of pairs of complex numbers (x, y) satisfying y2 = (lx 2 )(lk 2 x 2 ). 42.
The two sheeted Riemann surface
At this stage it is convenient to consider the aggregate of values of the complex variable x as mapped, not only on the Argand plane, but on a sphere, say X 2 +Y 2 +Z 2 = 1, by writing
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(42.1) 1+xx
1+xx
1+xx
This has the advantage that x = °°, like every finite value, has a single representative point on the sphere, namely (0, 0, 1); the ordinary mapping on the Argand plane is obtained by stereographic projection from (0, 0, 1) onto the plane Z = 0; and the point of the sphere representing the value x = tan \
. + 2£2, or by w +o>. a>. (mod + 2£2,2S2), w + co +2ft. Now the cross ratio of four values of x is real, if and only if their representative points on the sphere are coplanar; thus if Q is rectangular, as R(x , x , x , x ) is real, the points x , x , x , x are in a plane section of the sphere, which divides it into two regions which we may conventionally call hemispheres, even if they are unequal. Each sheet over each of these hemispheres is the image of a rectangle with sides a> , w , a quarter of a unit cell; x , x , x , x being in this order on the circle, the arcs x x , x x are the images of the sides co , and x x , x x those of the sides co ; and the mapping of the hemisphere and rectangle on each other is conformal everywhere except at the branch points, where the angle n between consecutive arcs of the circle corresponds to the angle  between consecutive sides of the rectangle. The crossing (cutting and rejoining) lines of the Riemann surface can conveniently be taken to be the arcs x x , x x of the circle, or equally well the arcs x x , x x ; the change from one arrangement to 86 Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:32:27, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.006
the other merely interchanges the names inner, outer, given to the two sheets over one hemisphere. 43.
The integral of the first kind
If R(x, y) is any rational function of (x, y) over C, where still y = F(x), we define the integral JR(X, y)dx along any path on the Riemann surface to be the same as this integral along the same path on the x sphere or plane, with the condition that the value of y = V[F(x)] that is to figure in the integrand at each point of the path is that denoted by the corresponding point on the Riemann surface. If the path does not pass through a branch point, this is equivalent to assigning a particular square root of F(x) at the beginning of the path, and requiring y to vary continuously with x throughout the integration. In particular, as x = f(u) satisfies (^) 2 = F(x), 2
to within a constant of integration. More precisely, if the lower and upper bounds of integration, as points on the Riemann surface, not merely values of x, correspond to the residue classes u + 212, u + 2£2, the value of the definite integral is in the residue class u u +2Q; it can be any member of this residue class, according to how the path is chosen on the Riemann surface; for taking u , u to be any members we choose of their residue classes, any path in the u plane from u to u has as image on the Riemann surface a path joining the assigned end points, and the definite integral J — along the path on the Riemann surface is the same thing as the definite integral Jdu along the path in the u plane, namely u  u . The indefinite integral (43. 1) is known as the elliptic integral of the first kind. The important cases are of course
( ^P
J ^^ 3
,
V(4x g2xg3)
1
fJ
f 2
22
(43.2)
2 2
V[(lx )(lk x )]
these we may call the Weierstrass and classical integrals of the first kind. A distinctive feature of these integrals, which is denoted by the expression 87 Downloaded from https:/www.cambridge.org/core. Cornell University Library, on 15 May 2017 at 17:32:27, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781107359901.006
T
of the first kind', is that they do not diverge anywhere, i. e. there is no 7 dx value x = a (finite or infinite) such that J — tends to infinity as z c y
tends to a. This is obvious, since every path from c to a on the Riemann surface is the image of some finite path in the u plane between two points u , u satisfying f(u ) = c, f(u ) = a, and the limit of the integral as z tends to a is simply u  u . Somewhat more generally, if y is any algebraic function of x, defined by a polynomial equation G(x, y) = 0 (in the present case y  F(x) = 0), an integral JR(X, y)dx (where R(x, y) is any rational function of (x, y) over C ) is said to be of the first kind if it does not diverge for any value of x (finite or infinite) as upper or lower bound. It may be remarked that there is no integral of the first kind of the form jR(x)dx, where R(x) is a rational function of x only (such an integral being always either a rational function or a constant multiple of the logarithm of a rational function of x). Further, if y is (as in the case before us) the square root of a cubic or quartic polynomial in x, the only integrals of the first kind are the constant multiples of J — ; this last result however cannot be proved satisfactorily till we have learned something about the integrals of the second and third kinds, which do diverge (algebraically and logarithmically respectively) for certain values of x. The value of the integral J — along any path on the Riemann surface being the difference between the end values of u for the corresponding path in the u plane, its values along any two paths that can be continuously deformed into each other, keeping the end points fixed, are the same. In particular, if the path on the Riemann surface is closed, the value of the integral is an element of the period lattice 2£2, and is the same for all closed paths that are continuously deformable into each other on the surface. The value of the integral is zero, if and only if the path is not only closed, but is continuously deformable into a point, i. e. if and only if it cuts the Riemann surface into two disjoint parts, as for instance any closed path does on a one sheeted sphere, but a meridian or parallel circle on the torus does not. The integral whose study originated the whole subject of elliptic functions was J2 V(lk sin2