Automorphic Functions and Number Theory
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54 Goro Shimura Princeton University, Princeton, New Jersey
Automorphic Functions and Number Theory 1968
Springer-Verlag Berlin . Heidelberg · New York
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. © by Springer. Verlag Berlin' Heidelberg 1968 Library of Congress Catalog Card Number 68-25132. Printed in Germany. Title No. 7374
Preface
These notes are based on lectures which I gave at the Forschungsinstitut fur Mathematik, Eidgenossische Technische Hochschule, Zurich in July 1967.
I have attempted to make a
short comprehensible account of the latest results in the field, together with an exposition of the material of an elementary nature. No detailed proofs are given, but there is an indication of basic ideas involved.
Occasionally even the definition of fundamental concepts
may be somewhat vague. the reader.
I hope that this procedure will not bother
Some references are collected in the final section in
order to overcome these shortcomings.
The reader will be able to
find in them a more complete presentation of the results given here, with the exception of some results of §lO, which I intend to discuss in detail in a future publication. It is my pleasure to express my thanks to Professors K. Chandrasekharan and B. Eckmann for their interest in this work, and for inviting me to publish it in the Springer Lecture Notes in Mathematics.
I wish also acknowledge the support of the
Eidgenossische Technische Hochschule, Institute for Advanced Study, and the National Science Foundation (NSF-GP 7444, 5803) during the summer and fall of 1967.
Princeton, January 1968
G. Shimura
Contents
1.
Introduction
2.
Automorphic functions on the upper half plane, especially modular
1
functions 3.
2
Elliptic curves and the fundamental theorems of the classical theory of complex multiplication
4.
8
Relation between the points of finite order on an elliptic curve and the modular functions of higher level
13
5.
Abelian varieties and Siegel modular functions
16
6.
The endomorphism ring of an abelian variety; the field of moduli of an abelian variety with many complex multiplications
26
7.
The class -field-theoretical characterization of K' (
33
8.
A further method of constructing class fields
39
9.
The Hasse zeta function of an algebraic curve
48
10.
Infinite Galois extensions with 1 -adic representations
55
11.
Further generalization and concluding remarks
62
12.
Bibliography
65
r(z».
Notation
We denote by Z, Q, Rand C respectively the ring of rational integers, the rational nwnber field, the real nwnber field and the complex nwnber field. rnent,
Y
X
For an associative ring Y with identity ele-
denotes the group of invertible eleznents in Y, M (Y) the n
and GL (Y) the n The identity group of invertible eleznents in M (Y), i. e , , M (Y}X n n eleznent of M (Y) is denoted by 1 , and the transpose of an eleznent n t n A 9f M (Y) by A as usual. When Y is coznznutative, SL (Y) n n denotes the gr
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