ABELIAN VARIETIES
WITH COMPLEX
MULTIPLICATION
AND
MODULAR FUNCTIONS
Goro Shimura
PRINCETON UNIVERSITY PRESS
PRINCETON, NEW JERSEY
Contents
Preface
Preface to Complex Multiplication of Abelian
Varieties and Its Applications to Number Theory
(1961)
Notation and Terminology
CHAPTER I.
1.
2.
3.
4.
CHAPTER II.
CHAPTER IV.
14.
15.
16.
17.
18.
xiii
3
Homomorphisms and divisors
Differential forms
Analytic theory of abelian varieties
Fields of moduli and Kummer varieties
3
7
19
25
Abelian Varieties with Complex Multiplication
35
Structure of endomorphism algebras
Construction of abelian varieties with
complex multiplication
7. Transformations and multiplications
8. The reflex of a CM-type
9.
10.
11.
12.
13.
ix
Preliminaries on Abelian Varieties
5.
6.
CHAPTER III.
vii
35
40
47
58
Reduction of Constant Fields
68
Reduction of varieties and cycles
Reduction of rational mappings and differential forms
Reduction of abelian varieties
The theory "for almost all p"
The prime ideal decomposition of an /V(p)-th
power homomorphism
68
74
83
87
96
Construction of Class Fields
101
Polarized abelian varieties of type (K\ {&))
The unramified class field obtained from the field
of moduli
The class fields generated by ideal-section points
The field of moduli in a generalized setting
The main theorem of complex multiplication in the
adelic language
101
109
114
118
121
CONTENTS
VI
CHAPTER
V.
19.
20.
21.
22.
CHAPTER VI.
23.
24.
25.
26.
CHAPTER VII.
27.
28.
29.
30.
31.
32.
33.
The Zeta Function of an Abelian Variety with
Complex Multiplication
132
The zeta function relative to a field over which some
endomorphisms are defined
The zeta function over smaller
fields
Models over the field of moduli and models with
given Hecke characters
The case of elliptic curves
145
149
Families of Abelian Varieties and
Modular Functions
151
Symplectic and unitary groups
Families of polarized abelian varieties
Modular forms and functions
Canonical models
151
156
165
169
132
137
Theta Functions and Periods on Abelian Varieties
173
Theta functions
Proof of Theorem 27.7 and Proposition 27.9
Theta functions with complex multiplication
The periods of differential forms on abelian varieties
Periods in the Hilbert modular case
Periods on abelian varieties with complex
multiplication and their algebraic relations
Proof of Theorem 32.4
173
181
188
190
193
196
201
Bibliography
209
Supplementary References
213
Index
215