Abelian Varieties With Complex Multiplication and Modular Functions

by Shimura, Goro

• ISBN13: 9780691016566
• ISBN10: 0691016569
• Format: Hardcover
• Copyright: 1997-12-08
• Publisher: Princeton Univ Pr

Summary

Reciprocity laws of various kinds play a central role in number theory. In the easiest case, one obtains a transparent formulation by means of roots of unity, which are special values of exponential functions. A similar theory can be developed for special values of elliptic or elliptic modular functions, and is called complex multiplication of such functions. In 1900 Hilbert proposed the generalization of these as the twelfth of his famous problems. In this book, Goro Shimura provides the most comprehensive generalizations of this type by stating several reciprocity laws in terms of abelian varieties, theta functions, and modular functions of several variables, including Siegel modular functions. This subject is closely connected with the zeta function of an abelian variety, which is also covered as a main theme in the book. The third topic explored by Shimura is the various algebraic relations among the periods of abelian integrals. The investigation of such algebraicity is relatively new, but has attracted the interest of increasingly many researchers. Many of the topics discussed in this book have not been covered before. In particular, this is the first book in which the topics of various algebraic relations among the periods of abelian integrals, as well as the special values of theta and Siegel modular functions, are treated extensively.

Table of Contents

Preface	vii	(2)
Preface to Complex Multiplication of Abelian Varieties and Its Applications to Number Theory (1961)	ix	(4)
Notation and Terminology	xiii
CHAPTER I. Preliminaries on Abelian Varieties	3	(32)
1. Homomorphisms and divisors	3	(4)
2. Differential forms	7	(12)
3. Analytic theory of abelian varieties	19	(6)
4. Fields of moduli and Kummer varieties	25	(10)
CHAPTER II. Abelian Varieties with Complex Multiplication	35	(33)
5. Structure of endomorphism algebras	35	(5)
6. Construction of abelian varieties with complex multiplication	40	(7)
7. Transformations and multiplications	47	(11)
8. The reflex of a CM-type	58	(10)
CHAPTER III. Reduction of Constant Fields	68	(33)
9. Reduction of varieties and cycles	68	(6)
10. Reduction of rational mappings and differential forms	74	(9)
11. Reduction of abelian varieties	83	(4)
12. The theory "for almost all p"	87	(9)
13. The prime ideal decomposition of an N (p)-th power homomorphism	96	(5)
CHAPTER IV. Construction of Class Fields	101	(31)
14. Polarized abelian varieties of type (K: XXX)	101	(8)
15. The unramified class field obtained from the field of moduli	109	(5)
16. The class fields generated by ideal-section points	114	(4)
17. The field of moduli in a generalized setting	118	(3)
18. The main theorem of complex multiplication in the adelic language	121	(11)
CHAPTER V. The Zeta Function of an Abelian Variety with Complex Multiplication	132	(19)
19. The zeta function relative to a field over which some endomorphisms are defined	132	(5)
20. The zeta function over smaller fields	137	(8)
21. Models over the field of moduli and models with given Hecke characters	145	(4)
22. The case of elliptic curves	149	(2)
CHAPTER VI. Families of Abelian Varieties and Modular Functions	151	(22)
23. Symplectic and unitary groups	151	(5)
24. Families of polarized abelian varieties	156	(9)
25. Modular forms and functions	165	(4)
26. Canonical models	169	(4)
CHAPTER VII. Theta Functions and Periods on Abelian Varieties	173	(36)
27. Theta functions	173	(8)
28. Proof of Theorem 27.7 and Proposition 27.9	181	(7)
29. Theta functions with complex multiplication	188	(2)
30. The periods of differential forms on abelian varieties	190	(3)
31. Periods in the Hilbert modular case	193	(3)
32. Periods on abelian varieties with complex multiplication and their algebraic relations	196	(5)
33. Proof of Theorem 32.4	201	(8)
Bibliography	209	(4)
Supplementary References	213	(2)
Index	215