Complex Multiplication

190 Pages·1983·2.95 MB·English

by  Serge Lang (auth.)

Grundlehren der
mathematischen Wissenschaften 255
A Series of Comprehensive Studies in Mathematics
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239. Elliott: Probabilistic Number Theory I
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255. Lang: Complex Multiplication
Serge Lang
Complex Multiplication
Springer-Verlag
New York Berlin Heidelberg Tokyo
Serge Lang
Department of Mathematics
Yale University
New Haven, CT 06520
U.S.A.
AMS Subject Classifications: 10025, 14K22
Library of Congress Cataloging in Publication Data
Lang, Serge, 1927-
Complex multiplication.
(Grundlehren der mathematischen Wissenschaften;
255)
Bibliography: p.
1. Multiplication, Complex.!. Title.
II. Series.
QA564.L28 1983 512'.33 82-19464
© 1983 by Springer-Verlag New York Inc.
All rights reserved. No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New
YOrk 10010, U.S.A.
Typeset by Composition House, Ltd., Salisbury, England.
Printed and bound by Halliday Lithograph, West Hanover, MA.
9 8 7 6 5 4 3 2 I
ISBN-13: 978-1-4612-5487-4 e-ISBN-13: 978-1-4612-5485-0
DOl: 10.1007/978-1-4612-5485-0
Preface
The small book by Shimura-Taniyama on the subject of complex multi
plication is a classic. It gives the results obtained by them (and some by Weil)
in the higher dimensional case, generalizing in a non-trivial way the method
of Deuring for elliptic curves, by reduction mod p.
Partly through the work of Shimura himself (cf. [Sh 1] [Sh 2], and
[Sh 5]), and some others (Serre, Tate, Kubota, Ribet, Deligne etc.) it is
possible today to make a more snappy and extensive presentation of the
fundamental results than was possible in 1961. Several persons have found
my lecture notes on this subject useful to them, and so I have decided to
publish this short book to make them more widely available.
Readers acquainted with the standard theory of abelian varieties, and
who wish to get rapidly an idea of the fundamental facts of complex multi
plication, are advised to look first at the two main theorems, Chapter 3, §6
and Chapter 4, §1, as well as the rest of Chapter 4. The applications of
Chapter 6 could also be profitably read early.
I am much indebted to N. Schappacher for a careful reading of the manu
script resulting in a number of useful suggestions.
S. LANG
Contents
CHAPTER 1
Analytic Complex Multiplication
I. Positive Definite Involutions . . . 4
2. CM Types and Subfields. . . . . 6
3. Application to Abelian Manifolds. 8
4. Construction of Abelian Manifolds with CM 14
5. Reflex of a CM Type . . . . . 21
6. Application to Cyclotomic Fields 24
7. An Example: The Fermat Curve 29
CHAPTER 2
Some Algebraic Properties of Abelian Varieties 35
1. Invariant Differential Forms . . . . . . . 35
2. Homomorphisms and Inseparability. . . . 40
3. Reduction mod p and l-adic Representations 43
4. Reduction of Functions . . . . 50
5. Reduction of Differential Forms . . . . . 51
CHAPTER 3
Algebraic Complex Multiplication 53
I. Fields of Definition. . . . . . . . 53
2. Transformations and Multiplications 55
3. The Congruence Relation . . . . . 61
4. Polarizations. . . . . . . . . . . 67
5. Change of Riemann Forms Under Various Maps 73
6. The Main Theorem of Complex Multiplication . 76
CHAPTER 4
The CM Character 84
1. The Second Main Theorem of Complex Multiplication
and the CM Character. 84
2. Finite Extensions. . . . . . . . . . . . . . . . . 94
Vlll Contents
3. Algebraic Properties of the Associated Characters. 102
4. The CM Character over a Quadratic Subfield. . 106
5. Shimura's I-adic Representations . . . . . . . 110
6. Application to the Zeta Function in the CM Case 115
CHAPTER 5
Fields of Moduli, Kummer Varieties, and Descents 122
1. Fields of Moduli . 122
2. General Descent . 130
3. Kummer Varieties 132
4. Class Fields as Moduli Fields. 136
5. Casselman's Theorem. . . . 139
6. Descent to a Quadratic Subfield. 142
7. Further Descent Theorems. . . 145
CHAPTER 6
The Type Norm. 148
1. The Rank of a Type. 148
2. The Type Norm as Lie Homomorphism 152
3. The Image N",(o~) . . . . . . . . . 155
4. The Type Norm as Algebraic Homomorphism 156
5. Application to Abelian Varieties . . . . . . 160
CHAPTER 7
Arbitrary Conjugations of CM Types 163
1. The Reflex Norm and the Type Transfer . 163
2. General Reciprocity and the Type Transfer. 167
3. Application to the Conjugation of Abelian Varieties. 170
4. Another Property Implying e", = 1 . . . . . . . . 175
Bibliography 179
Index .... 183
CHAPTER 1
Analytic Complex Multiplication
This chapter is essentially elementary, and lays the foundations for the
study of the endomorphisms of complex toruses known as complex multipli
cations. Let V be a vector space of dimension n over the complex numbers.
Let A be a lattice in V. The quotient complex analytic group VjA is called
a complex torus. We assume known the basic facts concerning Riemann
forms and the projective embedding of such toruses. A (non-degenerate)
Riemann form E on VIA is an alternating non-degenerate form on V such that
E(x, Y) E Z for x, YEA, and such that the form E(ix, y) is symmetric positive
definite. Equivalently, one may say that E is the imaginary part of a positive
definite hermitian form on V, and takes integral values on A. The torus
admits a projective embedding if and only if it admits a Riemann form,
and such a projective embedding is obtained by projective coordinates
given by theta functions. We shall not need to know anything about such
theta functions aside from their existence. An abelian manifold is a complex
torus which admits a Riemann form.
In this chapter we assume the basic analytic theory of such abelian
manifolds, cf. for instance [L 2J, and prove additional facts, in an analytic
setting which makes the theory elementary. In the next chapter, we assume
that the reader is acquainted with the basic algebraic theory of abelian
varieties as in [L 7J, for instance.
An abelian variety A, defined over a field k, is a projective non-singular
variety, with a group law whose graph is an algebraic subvariety of the
product, also defined over k. It is a basic fact that then the group law is
commutative.
Suppose that A is defined over the complex numbers. Then the group of
complex points A(e) is a compact complex Lie group, and as such is complex
analytically isomorphic to a torus, so there is a complex analytic isomorphism
8: VjA --+ A(C)
where dim V = dim A, and A is a lattice in V. A fundamental fact is that
complex analytic homomorphisms of abelian varieties must be algebraic,
so if
8': V'jA' --+ A'(C)
2 1. Analytic Complex Multiplication
is an analytic parametrization of another abelian variety A', and ()(: A -+ A'
is an algebraic homomorphism, then there is an analytic homomorphism &
making the following diagram commutative:
A(C)
l~
-
V'/A' A'(C)
8'
Furthermore, the association
gives an isomorphism between Hom(A, A') and Hom(V/A, V'/A'). The
homomorphism ()( is in fact represented by a C-linear map
&:V-+V'
sending A into A', that is &A c A'.
We shall assume Poincare's complete reducibility theorem. A homo
morphism A: A -+ A' is called an isogeny if Ah as finite kernel and is surjective.
Given any abelian manifold, or abelian variety, there exists an isogeny
with a product
A: A -+ A 1 X ... x Am
with each Ai simple, and the factors are determined up to isogeny. To say
that A is simple means that A has no abelian subvariety of dimension> 0
unequal to A. In light of the complete reducibility theorem, this amounts to
saying that End(A) has no divisors of 0; or again that End(A) 0 Q is a
division algebra.
We abbreviate End(A) ® Q = End(A)Q' If A: A -+ B is a homomorphism
of abelian varieties, then the image of A is an abelian subvariety of B. It is a
fact that Hom(A, B) has no torsion, which is easily seen over the complex
numbers from the structure of a complex torus. Thus the natural map of
End(A) into End(A)Q is injective. If A is an isogeny, then A has an inverse in
Hom(B, A)Q = Hom(B, A) 0 Q, and we have an isomorphism
End(A)Q ~ End(B)Q given by
In particular, if A is isogenous to a product B x ... x B taken In times, then
we have an isomorphism
of End(A)Q with the In x In matrices over End(B)Q'
3
1. Analytic Complex Multiplication
Let A: VIA -> V'IA' be a homomorphism of complex toruses. Then A
lifts to a C-linear homomorphism making the following diagram com
mutative.
x
V -----> V'
I I
VIA ~ V'IA'
The association A ~ ~ is called the complex representation of
Hom(VIA, V'/A'),
and we may denote ~ by Rc(A).
On the other hand, ~ maps A into A', and A, A' are free modules over Z,
so we obtain an integral representation. Using Q as a subscript to indicate
tensoring with Q, we then get a rational representation
In particular, we get a rational representation RQ of End(VIA)Q' which is of
dimension 2n, where n = dim VIA. This applies to any complex torus A
represented as a quotient of V by some lattice A. We assume known the
elementary fact that the rational representation of End(A)Q is equivalent to
the direct sum of the complex representation and its complex conjugate. For
these basic analytic facts, cf. for instance [L 2].
For an arbitrary abelian variety in characteristic p, there is no representa
tion in a vector space over the rationals, but there is something almost as
good: for each prime I #- p there is a representation R = R/ in a vector space
V over Q/ having the following properties:
R 1. dim V = 2· dim A.
R 2. R is faithful (i.e. injective) on End A.
R 3. The characteristic polynomial of R(a) has coefficients in Z for all
a E End(A), independent of I.
R 4. The constant term of this characteristic polynomial, that is
det R(a), is equal to the degree v(a) ~ 0.
If a: A -> A is an isogeny, then the degree is the usual degree of a finite
covering. If the kernel of a has dimension> 0, then the degree v(a) is defined
to be 0. In light of R 3, we say that R is a rational representation. Such
representations will be discussed in more detail when we use them later. For
this chapter, we concentrate on the analytic theory.