Introduction to the Arithmetic Theory of Automorphic Functions (Publications of the Mathematical Society of Japan 11)

287 Pages·1971·4.4 MB·English

by  Goro Shimura

PUBLICATIONS OF THE MATHEMATICAL SOCIETY OF JAPAN
11
INTRODUCTION
TO THE
ARITHMETIC THEORY OF
AUTOMORPHIC FUNCTIONS
BY
GORO SHIMURA
KANÔ MEMORIAL LECTURES 1
Princeton University Press
Princeton, New Jersey
Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, Chichester, West Sussex
Copyright © 1971 by the Mathematical Society of Japan
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Shimura, Gore), 1930—
Introduction to the arithmetic theory of automorphic functions /
by Gor6 Shimura.
p. (cid:9) cm. —(Publications of the Mathematical Society of Japan ; 11. Kant) memorial
lectures ; 1)
Originally published: Tokyo : Iwanami Shoten ; Princeton, N.J. : Princeton University
Press, 1971.
Includes bibliographical references and index.
ISBN 0-691-08092-5 (pbk. : acid-free)
1. Automorphic functions. I. Title. II. Series: Publications of the Mathematical
Society of Japan ; 11. III. Series: Publications of the Mathematical Society of Japan.
Kan6 memorial lectures ; 1.
QA353.A9S55 1994
515.9—dc20 (cid:9) 94-5898
Kanô Memorial Lectures
In 1969, the Mathematical Society of Japan received an anonymous donation to en-
courage the publication of lectures in mathematics of distinguished quality in commemora-
tion of the late K6kichi Kan6 (1865-1942).
K. Kan6 was a remarkable scholar who lived through an era when Western mathemat-
ics and philosophy were first introduced in Japan. He began his career as a scholar by
studying mathematics and remained a rationalist for his entire life, but enormously en-
larged the domain of his interest to include philosophy and history.
In appreciating the sincere intentions of the donor, our Society has decided to publish a
series of "Kan6 Memorial Lectures" as a part of our Publications. This is the first volume
in the series.
Originally copublished in 1971 by Iwami Shoten, Publishers, and Princeton University
Press; reprinted in paperback by arrangement with the Mathematical Society of Japan
First Princeton Paperback printing, 1994
Princeton University Press books are printed on acid-free paper and meet the guidelines for
permanence and durability of the Committee on Production Guidelines for Book Longevity
of the Council on Library Resources
10 9 8 7 6 5 4 3 2 1
Printed in the United States of America

PREFACE
There are two major topics treated in this volume :
I. Complex multiplication of elliptic or elliptic modular functions.
II. Applications of the theory of Hecke operators to the zeta-functions
of algebraic curves and abelian varieties.
Although these will form the "raison d'être" of the book, I have also
attempted, in the first few chapters, to present an introductory account of
the theory of automorphic functions of one complex variable, along with the
fundamentals of Hecke operators. Our discussion is mainly concerned with
elliptic modular functions of arbitrary level and the geometric objects directly
related to them, except that we consider automorphic functions of a more
general type in the first two and the last two chapters, and abelian varieties
of higher dimension with complex multiplication in a few places.
As to the first topic, we shall give two formulations, both in terms of
adeles. One is concerned with the behavior of an elliptic curve and its
points of finite order under automorphisms of the number field in question.
The other is closely connected with the structure of the field of all modular
functions of all levels whose Fourier coefficients belong to cyclotomic fields.
It will be shown that the group of all automorphisms of is isomorphic to
the adelization of GL2(Q) modulo rational scalar matrices and the archimedean
part. Then the reciprocity-law in the maximal abelian extension of an imagi-
nary quadratic field is given as a certain commutativity of the action of the
adeles with the specialization of the functions of .
The second topic is a development of the result of Eichler in his paper
appeared in the Archly der Mathematik vol. 5, 1954. The conjecture of Hasse
and Weil will be verified for the algebraic curves uniformized by modular
functions. Further we shall show that if a cusp form of weight 2 is a
common eigen-function of the Hecke operators, then the product of several
Dirichlet series associated with it coincides, up to finitely many Euler factors,
with the zeta-function of a certain abelian variety which is specifically given.
As an application of this result, it will be shown that the arithmetic of
a real quadratic field—its units, abelian extensions, etc.—is closely con-
nected with the modular forms of " Neben "-type in Hecke's sense. My excuse
for including this rather immature subject is that I think it gives a positive,
if not complete, answer to the question "Can one construct abelian extensions
of a real quadratic field by an analytic means? ", which arises naturally after
the detailed discussion of the corresponding problem for an imaginary quad-
vi (cid:9) PREFACE
ratic field in Chapters 5 and 6.
The present book has grown out of my lectures at Princeton University
and the University of Tokyo on various occasions during 1963-69. The notes
taken by Larry Goldstein (Fall Term 1965) and by Alain Robert (Spring Term
1969) were most helpful in preparing the first draft. Here I gratefully ac-
knowledge my indebtedness to them. I wish to express my hearty thanks to
K. Doi, H. Naganuma, and H. Trotter who made the table of eigen-values of
Heck e operators in § 7.7; and to W. Casselman, S. Lang, T. Miyake, A. Robert,
and A. Weil, who read the manuscript as a whole or in part. Many of their
suggestions have been incorporated in the present volume. My thanks are
also due to S. Iyanaga and Y. Kawada, who took an interest in this work,
and invited me to publish it in Publications of the Mathematical Society of
Japan. Finally I would like to extend thanks to the audience of my lectures,
whose enthusiasm was very encouraging.
Princeton, May 1970 (cid:9) Goro Shimura
(cid:9)
CONTENTS
Preface I/
Notation and terminology (cid:9) xi
List of symbols (cid:9) xii
Suggestions to the reader (cid:9) xiv
Chapter 1. Fuchsian groups of the first kind (cid:9) 1
1.1. Transformation groups and quotient spaces (cid:9) 1
1.2. Classification of linear fractional transformations (cid:9) 5
r\tyr (cid:9)
1.3. The topological space 10
1.4. The modular group SL,(Z) (cid:9) 14
r\o*
1.5. The quotient as a Riemann surface (cid:9) 17
1.6. Congruence subgroups of SL,(Z) (cid:9) 20
Chapter 2. Automorphic forms and functions (cid:9) 28
2.1. Definition of automorphic forms and functions (cid:9) 28
22. Examples of modular forms and functions (cid:9) 32
2.3. The Riemann-Roch theorem (cid:9) 34
2.4. The divisor of an automorphic form (cid:9) 37
r\ei (cid:9)
2.5. The measure of 40
2.6. The dimension of the space of cusp forms (cid:9) 45
Chapter 3. Hecke operators and the zeta-functions associated with
modular forms (cid:9) 51
3.1. Definition of the Hecke ring (cid:9) 51
3.2. A formal Dirichlet series with an Euler product (cid:9) 55
3.3. The Hecke ring for a congruence subgroup (cid:9) 65
3.4. Action of double cosets on automorphic forms (cid:9) 73
3.5. Hecke operators and their connection with Fourier coefficients (cid:9) 77
3.6. The functional equations of the zeta-functions associated with
modular forms (cid:9) 89
Chapter 4. Elliptic curves (cid:9) 96
4.1. Elliptic curves over an arbitrary field (cid:9) 96
4.2. Elliptic curves over C (cid:9) 98
4.3. Points of finite order on an elliptic curve and the roots of unity (cid:9) 100
4.4. Isogenies and endomorph isms of elliptic curves over C (cid:9) 102
(cid:9)(cid:9)
viii(cid:9) CONTENTS
4.5. Automorphisms of an elliptic curve (cid:9) 106
4.6. Integrality properties of the invariant j (cid:9) 107
Chapter 5. Abelian extensions of imaginary quadratic fields and complex
multiplication of elliptic curves (cid:9) 111
5.1. Preliminary considerations (cid:9) 111
5.2. Class field theory in the adelic language (cid:9) 115
5.3. Main theorem of complex multiplication of elliptic curves (cid:9) 117
5.4. Construction of class fields over an imaginary quadratic field (cid:9) 121
5.5. Complex multiplication of abelian varieties of higher dimension (cid:9) 124
Chapter 6. Modular functions of higher level (cid:9) 133
6.1. Modular functions of level N obtained by division of elliptic
curves (cid:9) 133
6.2. The field of modular functions of level N rational over Q(e2x") (cid:9) 136
6.3. A generalization of Galois theory (cid:9) 141
6.4. The adelization of GL, (cid:9) 143
6.5. The action of U on 146
6.6. The structure of Aut() (cid:9) 149
r\o*
6.7. The canonical system of models of for all congruence sub-
r
groups of G I. 2(Q) (cid:9) 152
6.8. An explicit reciprocity-law at the fixed points of GQ.,. on 0 (cid:9) 157
6.9. The action of an element of GQ with negative determinant (cid:9) 163
Chapter 7. Zeta-functions of algebraic curves and abelian varieties (cid:9) 167
7.1. Definition of the zeta-functions of algebraic curves and abelian
varieties ; the aim of this chapter (cid:9) 167
7.2. Algebraic correspondences on algebraic curves (cid:9) 168
7.3. Modular correspondences on the curves Vs (cid:9) 172
7.4. Congruence relations for modular correspondences (cid:9) 176
7.5. Zeta-functions of Vs and the factors of the jacobian variety of Vs 179
7.6. 1-adic representations (cid:9) . (cid:9) 189
7.7. Construction of class fields over real quadratic fields (cid:9) 197
7.8. The zeta-function of an abelian variety of CM-type (cid:9) 211
7.9. Supplementary remarks (cid:9) 220
Chapter 8. The cohomology group associated with cusp forms (cid:9) 223
8.1. Cohomology groups of Fuchsian groups (cid:9) 223
8.2. The correspondence between cusp forms and cohomology classes 230
8.3. Action of double cosets on the cohomology group (cid:9) 236
8.4. The complex torus associated with the space of cusp forms (cid:9) 239
CONTENTS ix
Chapter 9. (cid:9) Arithmetic Fuchsian groups (cid:9) 241
9.1. (cid:9) Unit groups of simple algebras (cid:9) 241
9.2. (cid:9) Fuchsian groups obtained from quaternion algebras (cid:9) 243
Appendix (cid:9) 253
References (cid:9) 260
Index (cid:9) - 265
Errata (cid:9) 269