The Arithmetic Of Function Fields: Proceedings Of The Workshop At The Ohio State University, June 17-26, 1991

492 Pages·1992·17.013 MB·English

by  David Goss, David R. Hayes, Michael I. Rosen

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
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Walter de Gruyter · Berlin · New York 1992
Editors:
DAVID GOSS MICHAEL I. ROSEN
Department of Mathematics Department of Mathematics
The Ohio State University Brown University
Columbus, Ohio 43210-1174, USA Providence, RI 02912, USA
DAVID 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 17-26, 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 Dimen-
sional 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 sup-
porting 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 mono-
graphs, 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 117
2
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, where r — pm. We distinguish a place oo of k called the place at infinity,
r
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 Fd [[π]], where d = doo = deg oo and π is a uniformizer at oo. Every non-zero
r
element χ e Κ can be expanded in a Laurent series
(0.1)
with each c € Fd and η = v (ar). We define deg χ = — d^ • v^z), and we put
u r 00
N(x) = rdeg x. We further set sgn(x) = c , the leading coefficient in the expansion of
n
x. It is convenient also to define sgn(O) = 0.
We regard sgn as a multiplicative map from Κ onto the copy of Fd which is
r
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 -dimension
r
of the residue class field at p . For a € 9(A), we put Ν (a) = rdegn.
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 (T) with
r
A = F[T] and with the place at infinity more or less taken for granted. Of these early
r
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 (T). Artin's paper (his doctoral
r
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 F(T). Actually (see [HI]), Carlitz did not
r
construct all of the maximal abelian extension of ¥(T) by this method. Naturally
r
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(T) with A = F[T] and oo equal to the unique pole of T. We
r r
ask how might one compute a factorial in F[T] ? Carlitz answered this question in 1932
r
(see [C2]). For j > 0, put [j] =Tr' - Τ and define
k k-l
L = Y[[j} and F = (1-1)
k fc
j=ι j=ο
The product of the monic polynomials in ¥[T] of degree η is F ; and the product
r n
of the non-zero polynomials of degree strictly less than η is (—l)n · F /L . Carlitz
n n
derived these results by a method that he used to prove much more. He showed that the
product
*»(*)= Π (Z~A) (1-2)
A
deg A <n