The Arithmetic Of Function Fields: Proceedings Of The Workshop At The Ohio State University, June 17-26, 1991 [Hardcover ed.] 3110131714, 9783110131710

Citation preview

Ohio State University Mathematical Research Institute Publications 2 Editors: Gregory R. Baker, Walter D. Neumann, Karl Rubin

The Arithmetic of Function Fields Proceedings of the Workshop at The Ohio State University June 17-26, 1991

Editors

David Goss David R. Hayes Michael I. Rosen

w DE

G Walter de Gruyter · Berlin · New York 1992

Editors: DAVID GOSS

MICHAEL I. ROSEN

Department of Mathematics The Ohio State University Columbus, Ohio 43210-1174, USA

Department of Mathematics Brown University Providence, RI 02912, USA

D A V I D R . HAYES

Department of Mathematics and Statistics University of Massachusetts Amherst, MA 01003, USA Series Editors: Gregory R. Baker, Walter D. Neumann, Karl Rubin Department of Mathematics, The Ohio State University, Columbus, Ohio 43210-1174, USA 1991 Mathematics Subject Classification: Primary: 11G09. Secondary: 11R58. ©

Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress Cataloging in Publication Data The arithmetic of function fields : proceedings of the workshop at The Ohio State University, June 1 7 - 2 6 , 1991 / editors, David Goss, David R. Hayes, Michael I. Rosen. p. cm. — (Ohio State University Mathematical Research Institute publications, ISSN 0942-0363 ; 2) Includes bibliographical references. ISBN 3-11-013171-4 (alk. paper) 1. Drinfeld modules — Congresses. 2. Fields, Algebraic — Congresses. I. Goss, David, 1952— . II. Hayes, David (David R.) III. Rosen, Michael I. (Michael Ira), 1938 — IV. Series. QA247.3.A75 1992 512'.74 —dc20 92-29651 CIP

Die Deutsche Bibliothek — Cataloging in Publication Data The arithmetic of function fields : proceedings of the workshop at the Ohio State University, June 17-26, 1991 / ed. David Goss ... - Berlin ; New York : de Gruyter, 1992 (Ohio State University Mathematical Research Institute publications ; 2) ISBN 3-11-013171-4 NE: Goss, David [Hrsg.]; Ohio State University < Columbus, Ohio > ; International Mathematical Research Institute < Columbus, Ohio > : Ohio State University ...

© Copyright 1992 by Walter de Gruyter & Co., D-1000 Berlin 30. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form — by photoprint, microfilm, or any other means — nor transmitted nor translated into a machine language without written permission from the publisher. Printing: Gerike GmbH, Berlin. Binding: Dieter Mikolai, Berlin. Cover design: Thomas Bonnie, Hamburg. Printed in Germany.

Preface

This volume consists of contributions from participants in a workshop on the Arithmetic of Function Fields which took place under the auspices of the International Mathematical Research Institute at Ohio State University from June 17-26, 1991. It was attended by over 90 mathematicians from more than 15 countries. The workshop on the Arithmetic of Function Fields was generously funded by grants from the National Science Foundation, the National Security Agency, the International Research Institute at Ohio State, the Office öf the Dean of Arts and Sciences at The Ohio State University, and the Office of the Vice-President for Research at The Ohio State University. We wish to express our sincere appreciation to these organizations and individuals. A primary topic of the workshop was the arithmetic of Drinfeld modules which is still a new area of research. As such, many of the contributions are of an expository nature and serve as an introduction to non-experts. Therefore, this volume will be useful to researchers in the area — both new and old — and to those who are simply curious! Indeed, the last article in the volume is a "dictionary" to help the reader understand the remarkable similarities between Drinfeld modules and classical objects such as elliptic curves. This dictionary is only meant to serve as a guide; as the subject is evolving so rapidly it will, in fact, soon be out of date. The Research Institute. The International Mathematical Research Institute at Ohio State University was founded in 1989 to support a program of visiting research scholars in mathematics at Ohio State and to run Workshops and Special Emphasis Programs on topics of particular importance and timeliness. A Research Semester on Low Dimensional Topology was the first major program of the Institute. Since then the Institute has supported workshops on, among others, Nearly Integrable Wave Phenomena in Nonlinear Optics, Quantized Geometry, the Arithmetic of Function Fields, L-Functions Associated to Automorphic Forms, and Geometric Group Theory. The Institute is currently supporting about 20-30 other research visitors (mostly short term) per year. The Institute publishes a preprint series as well as this book series, which is devoted to research monographs, lecture notes, proceedings, and other mathematical works arising from activities of the Research Institute. Acknowledgements. First and foremost, the editors thank The Ohio State University for its support of this program through the Research Institute. We thank our participants for their enthusiasm and contributions to the workshop and this volume. We also thank the non-academic staff of the Mathematics Department for their help in the organization and running of the Research Semester, particularly Marilyn Howard (administration and visas), Marilyn Radcliff (expenses), and Terry England for the numerous papers she has typed into TgX. The TgX macros were written by Larry Siebenmann and edited by Walter Neumann, and we wish to heartily thank Walter Neumann for his wizardry with TgX.

vi

Preface

Without his help this volume would not be possible. We also wish to thank Jeremy Teitelbaum who contributed some graphics work on a paper not his own. Finally we wish to thank the referees of the proceedings papers for their invaluable comments and service. David Goss, for the editors, June 1992.

Contents

Preface

ν

David R. Hayes A Brief Introduction to Drinfeld Modules

1

Y. Hellegouarch Galois Calculus and Carlitz Exponentials

33

G. W. Anderson A Two-Dimensional Analogue of Stickelberger's Theorem

51

D. S. Thakur On Gamma Functions for Function Fields

75

Hassan Oukhaba Groups of Elliptic Units in Global Function Fields

87

Keqin Feng Class Number "Parity" for Cyclic Function Fields

103

D. S. Dummit Genus Two Hyperelliptic Drinfeld Modules over F 2

117

David Goss A Short Introduction to Rigid Analytic Spaces

131

Marc Reversat Lecture on Rigid Geometry

143

E.-U. Gekeler Moduli for Drinfeld Modules

153

Yuichiro Taguchi Ramifications Arising from Drinfeld Modules

171

Jeremy Teitelbaum Rigid Analytic Modular Forms: An Integral Transform Approach

189

E. -U. Gekeler and M. Reversat Some Results on the Jacobians of Drinfeld Modular Curves

209

David Goss Some Integrals Attached to Modular Forms in the Theory of Function Fields . . . 227 Jing Yu Transcendence in Finite Characteristic

253

viii

Contents

Alain Thiery Independance Algebrique des P6riodes et Quasi-periodes d'un Module de Drinfeld

265

L. Denis Geometrie Diophantienne sur les Modules de Drinfeld

285

G. Damamme Transcendence Properties of Carlitz Zeta-values

303

David Goss L-series of ί-motives and Drinfeld Modules

313

R. J. Chapman Classgroups of Sheaves of Locally Free Modules over Global Function Fields

403

E. de Shalit Artin-Schreier-Witt Extensions as Limits of Kummer-Lubin-Tate Extensions, and the Explicit Reciprocity Law

413

Mireille Car The Circle Method and the Strict Waring Problem in Function Fields

421

L. N. Vaserstein Ramsey's Theorem and Waring's Problem for Algebras over Fields

435

G. Payne and L. N. Vaserstein Sums of Three Cubes

443

Daqing Wan Heights and Zeta Functions in Function Fields

455

C. Friesen Continued Fraction Characterization and Generic Ideals in Real Quadratic Function Fields

465

David Goss Dictionary

475

A Brief Introduction to Drinfeld Modules David

R.

Hayes

Throughout, we work over a global base-field k of characteristic ρ > 0 and field of constants F r , where r — pm. We distinguish a place oo of k called the place at infinity, and we let Ά ' denote the ring of elements of k which have only oo as a pole. The local ring of the completion Κ of k at oo is isomorphic to the ring of formal power series F r d [[π]], where d = doo = deg oo and π is a uniformizer at oo. Every non-zero element χ e Κ can be expanded in a Laurent series (0.1)

with each cu € F r d and η = v00(ar). We define deg χ = — d^ • v ^ z ) , and we put N(x) = r d e g x . We further set sgn(x) = c n , the leading coefficient in the expansion of x. It is convenient also to define sgn(O) = 0. We regard sgn as a multiplicative map from Κ onto the copy of Frd which is contained in Κ itself. The map sgn depends upon the choice of uniformizer π, but the reader can easily prove that there are only finitely many such functions. We say that the element χ G Κ is positive if sgn(x) = 1. The ring A is a Dedekind domain with finite class number h{ A) = #Pic(A) = d^h, where h is the class number of the function field k . Let 0 ( A ) be the group of fractional ideals of A . Since Qf( A) is freely generated by the prime ideals ρ in A , we can define a group morphism deg: 9 ( A ) —» Ζ by specifying that each deg ρ is the F r -dimension of the residue class field at p . For a € 9 ( A ) , we put Ν (a) = r d e g n . The algebraic closure Κ of Κ is not complete, but the completion C of Κ is algebraically closed. We view C as the function field analog of the complex numbers

c. A Tiny Bit of History The earliest papers in the arithmetic of function fields focused on the case k = F r (T) with A = F r [T] and with the place at infinity more or less taken for granted. Of these early efforts, the most important are Dedekind's [D] systematic development, Kornblum's [K] proof of the polynomial analog of Dirichlet's Theorem and Artin's [A] detailed exposition of the arithmetic of quadratic extensions of F r (T). Artin's paper (his doctoral dissertation) makes explicit use of the completion at infinity, and, following Dedekind, introduces a sign-function sgn. The class numbers which Artin computes in this paper are class numbers of rings, and his L-functions are all missing their Euler factors over oo.

2

D. Hayes

The great flowering of the theory of global function fields associated with the names of Artin, Chevalley, Hasse, F. Κ. Schmidt, Weil and many others that flourished in the 1920's and 1930's emphasized the symmetry of the set of places of k . Choosing a distinguished place at infinity was for them like puncturing a sphere, thereby losing both compactness and the aesthetics of the subject. The crowning achievement of this era was the Riemann Hypothesis which had been conjectured by Artin in [A], proved by Hasse for elliptic curves, and proved in general by Weil in 1940 [W], just as World War II was beginning in Europe. This symmetric point of view was studiously ignored, however, by Carlitz in all his papers on function fields during the 1930's. His willingness to work with a distinguished infinite place removed an obstacle that might have prevented him from introducing the first Drinfeld module, the Carlitz module, in 1938 [C4], and using it to give an explicit construction of the class field theory of Fr(T). Actually (see [HI]), Carlitz did not construct all of the maximal abelian extension of ¥r(T) by this method. Naturally enough, he missed the part that comes from the place at infinity! The moral of this bit of history is that you should always learn as many points of view as possible on any mathematical enterprise, and you should keep them all in mind when you think about a problem.

Part I: Basic Theory of Drinfeld Modules with Coefficients in a Field over A Drinfeld's papers [Drl] and [Dr2] provide the foundation for the theory. Other sources are the notes of Deligne and Husemoller [D-Η], the survey papers of Goss [Gol] and [Go2] and Gekeler's book [Gel]. The presentation below more or less follows that of

[H2], 1. Motivation: The Carlitz module. We find our first motivation in the rational function field k = F r ( T ) with A = F r [T] and oo equal to the unique pole of T. We ask how might one compute a factorial in F r [T] ? Carlitz answered this question in 1932 (see [C2]). For j > 0, put [j] =Tr' - Τ and define k Lk = Y[[j} j=ι

k-l and

Ffc =

(1-1) j=ο

The product of the monic polynomials in ¥r[T] of degree η is Fn; and the product of the non-zero polynomials of degree strictly less than η is (—l) n · Fn/Ln. Carlitz derived these results by a method that he used to prove much more. He showed that the product *»(*)=

Π A deg A < n

(Z~A)

(1-2)

Brief Introduction to Drinfeld Modules

3

expands into the F r -linear polynomial

i=0 in ζ where Fn .i -

Fr-K-i

Dividing (1.2) and (1.3) by ( - l ) n · Fn/Ln,

we obtain

- A^o Π ( i - ^ i=o B - D · ^n-i ^ deg

1

A 1, the series

and

are absolutely convergent and define holomorphic functions of s inthathalf-plane. As one can show easily from the Riemann-Roch Theorem, both these functions are holomorphic in the whole complex plane except for simple poles at the points s = 1 + 2 i r i k / d l o g r , k G Z. (Both are in fact rational functions of r~ds.) Further, Za(0, z) = 0 for every ζ G C - a, and K ( 0 ) = - 1 . We call Za(s, z) the partial zeta-function of a. From its definition, Za(s,z) is periodic in ζ with a as a group of periods. According to the conjectures of Stark, the derivatives of the functions Za(s,z) at s = 0 encode important information about those class fields of k in which oo splits completely. For Re(s) > 1, we can compute the derivative term by term: — s deg(a+z)

( l o g r ) " 1 · \^Za(s,

z)] = - ^ d e g ( a + z ) . r - s d e s ( a + 2 >

= -deg2-r-sdeg* -

H(s)=

Σ

deg(l +

Σ a€a—{0}

z/a)-r~sde^a+^-H(s)

deg(a)-r~sde^a+z\

Now deg(l + ζ/a) = 0 for all but finitely many elements a € α and so ds

α£α-{0}

s=0

=

dVoo(ea(z))-tf(0)

where α€α-{0}

Brief Introduction to Drinfeld Modules

5

is the exponential function of α. To evaluate / / ( 0 ) , we note that for Re(s) > 1 H{s)=

J2

dega-r-sdcs(-a+z)

+

ο,φ η dcgo /C"[F] that satisfies Do φ = δ and which is non-trivial in the sense that not all the twisted polynomials in the image of φ are constant polynomials. A Drinfeld module is always a monomorphism and therefore defines a faithful representation of A on G α / κ · Because δ is F r -linear, φ actually takes its values in the subring K[Fm] of K[F],

6

D. Hayes

We write φχ for the twisted polynomial associated to χ G A by φ. If R is any Κ-algebra, then (taking points) φχ acts as an F r -linear map on the additive group of Κ. We write φχ(ί) for the action of ψχ on i G R. Of course, under the taking of points, multiplication in Ä"[F] corresponds to composition of linear maps. Example: The Carlitz module. Take A = F r [Τ], and let δ be the inclusion map from A into C . If r is any non-constant polynomial in C[F m ] with D(T) = T, then there is a unique Drinfeld module φ: A —ν C such that φτ = τ. If we take τ = Τ — F m , we obtain the Carlitz module mentioned in §1 above. The action of φτ on an element t in some Κ-algebra is just φτ(ί) — Tt — tr (cf. (1.6)). In [HI], the Carlitz module was defined by Τ ι—• Τ 4- F m . This choice is better for many purposes because it produces a normalized module (see §12). The two Carlitz modules are isomorphic in a sense that we will now explain. Definition. Let φ and φ' be Drinfeld A -modules over K. An isogeny from φ to φ' is a twisted polynomial r 6 Ji[F] such that τφχ = φ'χτ for all χ € A. Clearly a product of isogenies is again an isogeny. The Drinfeld A -modules over Κ constitute a category DrinA(/0 in which the morphisms are the isogenies. What is an isomorphism in DrinA(^0? Since only constant twisted polynomials are invertible in K"[F], φ and φ' are isomorphic if and only if there is an element ξ G K x such that ξφχ = φ'χξ for all χ Ε A. The two Carlitz modules are defined above as elements of DrinA(C). They are isomorphic via ξ — ζ, where ζ G C is an (r — l)-st root of —1. Had we defined the Carlitz modules as elements of DrinA(K), then for ρ φ 2 they would not be isomorphic because Κ does not contain an (r — l)-st root of - 1 .

The category DrinA^) can be defined for any scheme S over A . With two exceptions (see §§14 and 15), we consider only the case S = Spec(/C) in these notes. 4. The action of ideals in A on Οήη^Κ). Let Ö be an order in an imaginary quadratic number field. If 0 acts on an elliptic curve E, then each invertible ideal of 0 defines an isogeny from Ε onto another elliptic curve E'. If Ε is defined over C, this fact follows readily from the analytic theory. When Ε is defined over, say, a finite field, it is still true but not quite so obvious (cf. [S]). As we now show, the ideals of A act in a similar manner on D r i n A ^ ) · Proposition 4.1. The twisted polynomial ring Κ [F] is a left principal ideal ring. Sketch of proof. The proof goes just as in the familiar commutative theory except that one has to mind the rule (3.1). The degree of a twisted polynomial is well defined. One shows that there is a right division algorithm in Ä"[F] by monic polynomials, and this implies that left ideals are principal. • Given a Drinfeld A-module φ over Κ and a non-zero ideal α in A , let Ι0]ψ be the left ideal generated in K[F] by the twisted polynomials φα, a Ε a. As left ideals are principal, = K[F] · φα for a unique monic twisted polynomial φα. Clearly, /„ ^ is

Brief Introduction to Drinfeld Modules

7

carried into itself by multiplication on the right by the φχ, χ Ε A. Therefore, for every χ G A there is a uniquely defined φ'χ Ε K[F] such that Φα • Φχ = Φ'χ · Φ*.

(4.2)

One checks easily that the map φ': A —» AT[F] defined by χ t—> φ'χ is a Drinfeld A moduleover Κ except for one subtlety: Do φ' = 6pJ ° , where j(a) is the smallest power of F that occurs in φα with a non-zero coefficient. So φ' is indeed a Drinfeld module, j(a) but possibly for a different field over A . Actually, it is always true that δρ — δ, as we show in §5 below, but this is not immediately obvious. We introduce the notation φ' = α*φ. (4.3) Thus, we have an operation * of the non-zero ideals in A on DrinA(iO which we may characterize as follows: α * φ is the unique Drinfeld A-module which is isogenous to φ via the isogeny φα. If R is a Κ -algebra, the roots of φα(ί) in R are the points of α-torsion for A acting on R through φ. In fact, much more is true. The kernel of φα as an endomorphism of Qa/K is a finite subgroup scheme of Ga/K equal to the intersection of the kernels of all the endomorphisms φα for a Ε a. The basic properties of the action * are contained in the following lemmas. These all follow easily from the definitions and (5.8) below. Lemma 4.4. Let α = wA be a non-zero principal ideal, and let μ be the leading coefficient of φw. Then φα — μ~ι • and (α * φ)χ = μ - 1 · φχ • μ for all χ Ε Α. Lemma 4.5. Let α and b be non-zero ideals in A. Then 0ab = (b * φ)α · b

(4.6)

a * (b * φ) = (ab) * φ.

(4.7)

and

Let IsomA(^0 be the set of isomorphism classes of the Drinfeld A-modules over K . It is clear from the definitions and Lemma 4.4 that * also operates on IsomA(^) and that the principal ideals in A act trivially. It follows that the ideal class group Pic(A) also acts on IsomA(i^). 5. The rank and height of a Drinfeld A -module. As we prove in §8 below, Drinfeld A -modules over C exist in great profusion. Therefore, what we always knew was true for k = F r ( T ) with oo the unique pole of Τ is actually true in general—the ring A is a ring of (twisted) polynomials! For τ G Ä"[F], τ φ 0, let d e g r be the degree of r as a polynomial in F . If φ: A —> /C[F] is a Drinfeld A-module, then the map v Ζ defined by \φ{χ) = — deg φχ satisfies i) \φ(χν)

= \φ(χ)

+ νφ(ν)

and

ii) νφ(χ + y) > min{v^(a:), v 0 (t/)}.

It therefore extends to a valuation (also called ν φ) on k . Which place of k does it define? Since \φ is negative on A , it can only define the distinguished place oo. Hence, there

8

D. Hayes

is a rational number r ^ such that \φ(χ)

= Τφπιά,χ, • \ 1. Since y is transcendental over F p , the coefficient on the left above is never zero. Thus, given the initial value CQ = 1, (7.2) is uniquely solvable for the coefficients Ci, i > 1. • Corollary 7.3. If D{a) is transcendental over Fp> then λ σ · Κ • λ" 1 is the centralizer of σ in tf[[F]]. Proof. The centralizer of D(a) in /^[[F]] is clearly just K. Since σ is the image of D{a) under the inner automorphism through λ σ , the centralizer of σ is the image of Κ under the same automorphism. • Corollary 7.4. If D(a) is transcendental over F p , then for each w € K, TW = ΛΣ • W- X~L

is the unique power series in /i[[F]] with constant term w which commutes with σ.

12

D. Hayes

Proposition 7.5. Let φ be a formal k -module over K. Then there is a unique power series oo i=0 such that Co — 1 and φχ — λψ · δ {χ) · A^1 for all χ Ε k. Proof. Choose ζ Ε k so that twisted power series associated Φχζ — Φζ · Ψχ> which implies is a constant for all ζ Ε F r , Λψ

6(z) = y is transcendental over F p . Let Α ψ be the to σ — φζ by Lemma 7.1. If χ Ε k, then ψχ • φζ — that φχ — • δ(χ) · Λ^1 by Corollary 7.4. Since δ(ζ) belongs to the subring #[[F m ]] of K[\F]]. •

This proposition shows that a Drinfeld A -module φ over Κ of generic characteristic determines a unique twisted power series Χφ in ii[[F]] with Ο(λψ) — 1. Conversely, in order to construct a Drinfeld A -module over Κ, all we have to do (!) is to construct a non-trivial twisted power series Λ with D(Λ) = 1 such that λ · δ(χ) · A - 1 is in Ä"[F] for all χ Ε A. We show how to do this in the next section. Example: The Carlitz module. For the Carlitz module φ: Τ ^ Τ - F m of §§1 and 3, the reader is invited to check by direct calculation that

i=0 This fact also follows from (1.5) and (1.6). 8. The analytic construction of Drinfeld A -modules.

Throughout this section,

δ-.A^C is the inclusion map. The exponential functions associated with lattices in C provide a powerful means of constructing Drinfeld A -modules of any rank. We will use this analytic method to prove the existence theorem on which the whole theory is based. Except for the special case A = F r [T], writing down examples of Drinfeld A-modules by non-analytic means is not an easy task. If k is the function field of a hyperelliptic curve over F r with oo chosen so that the curve has its standard equation away from oo, then there is a fairly straightforward algebraic algorithm for computing Drinfeld A -modules of rank one. See [H7] for a description of this algorithm and some examples with r = 2 and r = 3. D. Dummit has recently used the algorithm to compute the Drinfeld modules of rank one associated to all elliptic curves over F r for 2 < r < 13. See Dummit's article in these proceedings for some examples of Drinfeld modules associated to curves of genus 2. When this hyperelliptic algorithm is successful, it proves the existence of a Drinfeld A-module. We know that it is, in fact, always successful because of the analytic constructions that we will now describe. Definition. A lattice is a discrete A -submodule Γ of C such that Κ Γ has finite Κ dimension r r . The integer r r is called the rank of Γ. Since C is an infinite dimensional Κ -vector space, lattices of every positive rank exist.

Brief Introduction to Drinfeld Modules

13

The following theorem is proved in the same way as its analog over E : Theorem 8.1. Let Τ be a lattice, and let χ be a non-constant element of A , Then Γ is an F r [x] -module of finite rank. Further, every F r [x] -basis of Γ is linearly independent over the closure of F r ( x ) in Κ . Definition. A power series f(z) with coefficients in C is F r -linear if only monomials with exponent a power of r appear in f(z) with non-zero coefficient. Let D(f ) denote the coefficient of ζ in f(z). We call D(f) the tangent vector of f(z). A basic theorem of non-archimedean analysis states that an entire function on C (i.e. a function defined by an everywhere convergent power series) is determined up to a multiplicative constant by its roots, multiplicities being counted. In particular, we have Lemma 8.2. If fi(z) and fi{z) are everywhere convergent Fr-linear power series with D(fi) = D(/2) φ 0 and if fi(z) and f2{z) have the same set of roots, then / i ( * ) = /a(*)· The exponential function associated to the lattice Γ is the function er(z) 2 G C by the infinite product er(z)

= z·

Π ί 1 " ^ ) · 7 / 7er-{o} ^

defined for

(8·3)

The series 7

Σ i

er-{o} '

is absolutely convergent in the oo -adic topology on C , as one sees without difficulty. The infinite product (8.3) therefore converges for all 2 e C and so defines an entire function on C . In fact, we can multiply out this product into an everywhere convergent F r -linear power series oc er(z) = z + ^2cizr' (8.4) i=l

by considering the partial products of (8.3) over the 7 e r with — v o c (7) less than a given bound b. Since such 7 ' s constitute a finite dimensional F r -vector space, each partial product multiplies out into an F r -linear polynomial, and so the F r -linear series (8.4) appears in the limit as b —> 00. All this is fairly easy because we are dealing with a non-archimedean valuation. Theorem 8.5. The function er(z) is an entire function on C with the properties: 1) ev(z) is a surjective F r -linear endomorphism of C , and 2) e r ( z ) is periodic with Γ as a group of periods. Proof. The theorem is immediate from (8.3) and (8.4) if we recall that entire functions on the non-archimedean field C are necessarily surjective. •

14

D. Hayes

This theorem shows that C/Γ is isomorphic to C itself by an F r -linear map. We now show that er(z) admits complex multiplications by the elements of A . Let Γ' and Γ be a pair of lattices with Γ of finite index in Γ ' . As er(z) induces an isomorphism of the F r -vector spaces Γ'/Γ and ε Γ ( Γ ' ) , β Γ ( Γ ' ) is a finite set. Put Ρ(Γ'/Γ; 0 = ί · Π ί ν σεβ Γ (Γ')-{0}

1

"^)· 7

(8·6)

Then Ρ ( Γ ' / Γ ; ί ) is an F r -linear polynomial of degree # ( Γ ' / Γ ) associated to the pair Γ'/Γ in a canonical way. Theorem 8.7. Let Γ " D Γ ' D Γ be lattices with # ( Γ " / Γ ) finite. Then er,(z)

= P(T'/T-er(z))

(8.8)

for all ζ e C, and Ρ ( Γ " / Γ ; t) = Ρ (Γ"/Γ'; P(Γ'/Γ; t)).

(8.9)

Proof. From the definitions, both sides of (8.8) have the same set of roots in C . Since each is defined by an everywhere convergent power series with tangent vector one, equality holds by Lemma 8.2. The identity (8.9) is an immediate consequence of (8.8) • Given a lattice Γ and any χ e A, the index # ( χ _ 1 Γ / Γ ) is finite by Theorem 8.1. If we set φτχ{ί) = χ·Ρ{χ-λΤ/Τ·,ί),

(8.10)

then φζ(ί) is an F r -linear polynomial of degree #(x _ 1 Γ / Γ ) and with tangent vector x. Now from the definitions, if Γ ' = ζ _ 1 Γ , then er,(z) = x_1 • er(xz). Therefore, (8.8) implies that er(z) admits the "complex multiplication" eT(xz)

= φΓχ{ετ{ζ)).

(8.11)

The F r -linear polynomial is unique and canonically defined. From (8.9), (8.11) and the uniqueness,

and =

M(t))

for all elements χ and y in A . These two identities show that the polynomials (t) provide a faithful representation of A as a ring of F r -linear endomorphisms of the F r vector space C . It is clear that this representation factors through the twisted polynomial ring C [ F m ] . Thus φΓ:χ ι—> φζ is a Drinfeld A-module over C ! What is the rank of φτΊ Let us compute d e g i n terms of v ^ ( x ) : d e g ^ ( i ) = # ( χ - 1 Γ / Γ ) = # ( Γ / χ Γ ) = rk by Theorem 8.1 where k = r a n k e r = r r · rankFr[!B] A = rT • [k: ¥r(x)]. Now since oo is the only pole of x, -d^

• \oo(x) — [k: F r (x)]. Thus, we have proved

Brief Introduction to Drinfeld Modules

Theorem 8.12. Each lattice Γ defines a Drinfeld A-module φΓ over C via the multiplications (8.11). The rank of φΓ equals the rank r r of Γ. We may now state our fundamental result, the analytic Drinfeld A -modules over C .

uniformization

complex

theorem

Theorem 8.13. Let φ be a Drinfeld A -module over C . Then there is a uniquely mined lattice Γ in C such that φ = φτ. Sketch of proof

15

for deter-

Let oc

i=0 be the formal twisted power series associated to φ by Proposition 7.5. One proves first that the F r -linear power series oo λ φ { ζ ) = τ : Cjz ri 1=0

converges for all ζ e C. This follows from any one of the recurrences (7.2) with σ = φν, y a non-constant element of A . The identity λψ · χ = φχ\φ is then equivalent to the complex multiplications (8.11) with \ψ{ζ) replacing e r ( z ) . It remains then to show that the set of roots of Χψ is a lattice Γ. This follows from the fact that Χψ{ζ) admits complex multiplications by the elements of A . For the details, see §§4 and 5 of [H2], • We leave it to the reader to check that ξ G C is an isomorphism from φΓ to φΓ (i.e. φς = ξ φ ^ ξ - 1 ) if and only if Γ' = ξΓ. In fact, there is a natural isomorphism of categories between DrinA(C) and the category whose objects are the lattices in C and whose set of morphisms from Γ to Γ' is given by Hom(r, Γ") = 0 if r r φ r r , and otherwise by Hom(r, Γ') = {z € C : zT C Γ'}. Finally, we ask how can one understand the action * of §4 in this analytic setting? The answer is given by the following theorem, which we state without proof. Theorem 8.14. Let Γ be a lattice, and let a be a non-zero ideal in A. Put φ' = α*φΓ, and let Γ' be the lattice associated to φ' by the analytic uniformization theorem. Then V = Ώ{φΙ)·α~ιΤ.

(8.15)

9. Uniformizing isomorphism classes in DrinA(C). We call two lattices Γ and Γ' homothetic if Γ' — ξΓ for some element ξ e C x . Let Lat f c be the set of lattices of rank k in C , and let -C^ be the set of homothety classes in Latfc . The map Γ ι—• α _ 1 Γ

(9.1)

defines an action of fractional ideals α on Lat^ , and this action clearly descends to an action of Pic(A) on Lk . Since every rank one lattice is homothetic to a fractional ideal of A , we have

16

D.Hayes

Proposition 9.2. The set of homothety classes -Ci is a principal homogeneous Pic(A). In particular, L\ is finite and #(£i) = h(A).

space for

We remind the reader that a principal homogeneous space for a given group G is a space on which G acts transitively and faithfully. Let Dk be the set of isomorphism classes of rank k Drinfeld A-modules over C . The analytic uniformizing theorem implies that the map Γ H-> φΓ descends to a natural bijective correspondence from to T>k . Theorem 8.14 says that this map is an isomorphism from as a representation space for Pic(A) via (9.1) to T>k as a representation space for Pic( A ) under the * operation. We may then restate Proposition 9.2 as follows. Theorem 9.3. There are exactly h(A) isomorphism classes of rank one Drinfeld A modules over C, and Di is a principal homogeneous space for Pic(A) under the * action.

Part II: The Rank One theory with Applications to the Class Field Theory of k In problem twelve of his famous list, Hilbert asks for an explicit construction of the class field theory over any given number field k. Hilbert was aware of the cyclotomic theory over k = Q, in which class fields are constructed by adjoining the special values of β2πιζ at all ζ e Q. He was aware also of the construction of the class fields of an imaginary quadratic number field k by adjoining special values of elliptic functions. Nowadays we understand that, in both examples, the class fields are constructed by adjoining the division points of the action of the ring of integers Ok of k on some algebraic group G. In the cyclotomic theory, G is the multiplicative group Gm. Over an imaginary quadratic number field k, G is an elliptic curve which admits complex multiplications by 0 k . For the function field k , the analog of Hilbert's problem twelve is solved by adjoining division points of rank one Drinfeld modules acting on G = Ga. More precisely, the class fields of k in which our fixed place oo splits completely are constructed by adjoining division points of a suitably normalized Drinfeld A -module over C of rank one (see §16). One obtains all the class fields of k by varying oo. Throughout Part II, except where explicitly stated otherwise, • C is the inclusion map. 10. Fields of definition in C . Let φ be a Drinfeld A -module over C of any rank. The Galois group Gal(C/k) acts as a group of automorphisms of the twisted polynomial ring C[F], Following the map χ — ι > φχ by the action of an element σ Ε Gal(C/k) therefore defines a new Drinfeld module σφ of the same rank as that of φ. Thus, Gal(C/k) acts on DrinA(C), and this action induces an action of Gal(C/k) on the set of isomorphism classes of Drinfeld A -modules over C . As the reader can check from the definitions, a * σφ = σ(α * φ)

for all σ e Gal(C/k) and every non-zero ideal ο in A .

(10-1)

Brief Introduction to Drinfeld Modules

17

We say that φ is defined over a subextension Kf k of C / k if φχ € K\F] for every χ € A . A subfield if of C is a field of definition for φ if φ is isomorphic over C to a Drinfeld module φ' which is defined over K. As is clear from the definitions and (5.8), if φ is defined over K, then α * φ is also defined over Κ for any non-zero ideal α in A. Proposition 10.2. If φ has rank one, then the completion Κ of k at oc is a field of definition for φ. Proof By Theorem 8.13, φ — φΓ for a lattice Γ of rank one over A . Because r r = 1, Γ is isomorphic to a fractional ideal of A , and so there is an element ξ € C such that Γ' = ( Γ C k C K . Therefore, the analytic constructions in §8 all take place within K , which shows that φΓ = ξφξ-1 is indeed defined over Κ . • Theorem 10.3. There exists a field of definition Κ φ, finitely generated over k, which is contained in every field of definition for φ. Proof

For χ e A, let τ·φ deg χ φχ=χ+

Σ u=1

Fm".

(10.4)

Then for any ξ 6 C , we have = ξ1-Γ'' ·α„(φ,χ).

ον(ξφξ-\χ)

(10.5)

Fix a non-constant element y 6 A , and let ν = v\,... ,vs be the indices of the coefficients οι/(φ,ι/) of φυ which are not zero. Let g be the greatest common divisor of the set of integers { r — 1 : 1 < i < s} and write s 0 =

· (r"' - 1 )

(ej-eZ).

3=1

From (10.5), it is clear that the elements \ (1 Π ^ , ϊ , Η

ίνΑΦ,ν) = ζνΛΦ,ν) •

GC

(10.6)

for 1 < i < s depend only on the isomorphism class of φ and hence belong to every field of definition for φ. Let Κ ψ be the field generated over k by the elements /„, (φ, y). We will show that Κφ is a field of definition for φ. Choose ξ e C so that s

= IK (*¥)"· 3= 1

For 1 < i < s, we may rewrite (10.6) as ι»ΛΦ,ν) =

•^ ( ^ y ) ,

(10·7)

18

D.Hayes

so that ξφυξ 1 has coefficients in Κψ . It follows now from (7.2) and Proposition 7.5 that ξφχξ~1 has coefficients in Κψ for all χ e A. Thus, φ is defined over Κψ . • We call Κψ the smallest field of definition for φ. For φ of rank one, we prove in §15 below that Κ φ is the maximal unramified abelian extension Ηχ of k in which oo splits completely. For rank one A -modules, therefore, Κφ is independent of the choice of φ. We call Ηχ the Hilbert Class Field of A . We go further in §15 and prove the following analog of Hasse's Theorem (cf. [S]). Theorem 10.8. Let p be a non-zero prime ideal in A , and let Frobp be the Frobenius automorphism of Ηχ at p. For any rank one Drinfeld module φ defined over Ηχ, we have Frobp (φ)

^ρ*φ.

(10.9)

The proof of (10.9) depends upon the theory of reduction of Drinfeld modules, which we introduce in the next section. 11. Reduction of Drinfeld modules over C . Let Kjk be a subextension of C / k . A discrete valuation ring in C is finite if it contains A . Let Οφ be a finite discrete valuation ring in C with field of fractions K , let φ be the maximal ideal of Οφ , and let νφ be the normalized discrete valuation on Κ with valuation ring 0

then μφ(π'ι)

= 1

(12.4)

by (6.6). Since π is positive for sgn, for any χ G K x we have sgn(x) = c n G Foo where η = v o c (x) < 0 and cn is defined by (0.1). Therefore, for χ G A we have μφ{χ) = μφ(οηπη)

= ϊφ(οη) = i 0 (sgn(x))

by (6.4) and (12.4). Thus, φ is sgn-normalized with the isomorphism invariant map ίφ determining the twist of sgn. • If φ is sgn-normalized, then the leading coefficient function μφ: A —> Foo extends naturally to k. In subsequent sections, we will write μφ(χ) for χ G k without further explanations. 13. The narrow class group Pic + (A). How many sgn-normalized A-modules are there? Let us first compute the number of sgn-normalized A -modules in each isomorphism class in Vi (see §9). Proposition 13.1. Let φ be sgn-normalized. If φ' = ξφξ-1 ξ G F£, and μφ = μφ>.

is also sgn-normalized, then

Proof. Since μφ(π~ι) = μφ>(π~ι) — 1, we have ξ G F ^ by (6.6). That μφ = μφ> is clear since deg φχ = deg χ is divisible by d. • Corollary 13.2. Let r

d

— 1

W

Then each isomorphism class in Ί)χ contains exactly κ sgn-normalized

A-modules.

Proof. We know by Theorem 12.3 that each class in D\ contains a sgn-normalized A module φ. For each ξ G F ^ , ξφξ_1 is clearly sgn-normalized. Since Aut(^) = F*, such ξ produce exactly κ distinct A-modules isomorphic to φ. By Proposition 13.1, these are the only possibilities. • Corollary 13.4. There are exactly κ • h( A) sgn-normalized A -modules. Let Γ be any rank one lattice in C . By Theorem 12.3 and the analytic theory, there is an element ξ(Γ) in C such that Γ' = ξ(Γ) · Γ is the lattice of a sgn-normalized A module φΓ . By Proposition 13.1, the element £(Γ) is determined up to multiplication

22

D. Hayes

by elements of . In fact, the κ lattices ζΤ' for ζ G F ^ are precisely the lattices associated to the sgn-normalized A -modules which are isomorphic to φΓ . In spite of this ambiguity, we call ξ(Υ) the invariant of Γ. The element Δ(Γ) = ξ(Γ)* 2, and ^ € F (any given field). By the same procedure I also obtained an abstract logarithm in the same ring satisfying exp · log = log · exp = σ°. Then the question arose to give a realization of our exponential as an additive map between certain groups and it seemed that a good candidate to consider was the NewtonPuiseux field Ρ of F (see below §11, 1): Ρ is almost algebraically closed and σ acts canonically on it. Moreover we have σ(Ρ) = Ρ in the typical case where a(t) = td with d φ 0 mod char(F). So it turns out that exp defines, through a new type of "entire" series, a local linear homeomorphism Ρ —> Ρ and that its kernel Η* (P) is a discrete A-submodule of Ρ (see [9]). However, many questions remain open and our original motivation was to know whether we have r a n M f f . ( P ) ) > deg