1997年12月8日 志村五郎 (著) Abelian Varieties with Complex Multiplication and Modular Functions (Hardcover ed.). Princeton University Press. (1997-12-08).
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.
Goro Shimura is Professor of Mathematics at Princeton University. He was awarded the Leroy P. Steele Prize in 1996 for lifetime achievement in mathematics by the American Mathematical Society. He is the author of Introduction to Arithmetic Theory of Automorphic Functions (Princeton).
CHAPTER I Preliminaries on Abelian Varieties (pp. 3-34)
CHAPTER II Abelian Varieties with Complex Multiplication (pp. 35-67)
CHAPTER III Reduction of Constant Fields (pp. 68-100)
CHAPTER IV Construction of Class Fields (pp. 101-131)
CHAPTER V The Zeta Function of an Abelian Variety with Complex Multiplication (pp. 132-150)
CHAPTER VI Families of Abelian Varieties and Modular Functions (pp. 151-172)
CHAPTER VII Theta Functions and Periods on Abelian Varieties (pp. 173-208)
//
Preliminaries on Abelian Varieties Homomorphisms and divisors
Differential forms
Analytic theory of abelian varieties Fields of moduli and Kummer varieties
ix xiii
3 3 7
19 25
Contents
Preface vii Preface to Complex Multiplication of Abelian
Varieties and Its Applications
(1961)
Notation and Terminology
to Number
Theory
Abelian Varieties with Complex Multiplication 35
Structure of endomorphism algebras 35 Construction of abelian varieties with
complex multiplication 40 Transformations and multiplications 47 The reflex of a CM-type 58
Reduction of Constant Fields 68
Reduction of varieties and cycles 68 Reduction of rational mappings and differential forms 74 Reduction of abelian varieties 83 The theory "for almost all p " 87 The prime ideal decomposition of an /V(p)-th
power homomorphism 96 Construction of Class Fields 101
Polarized abelian varieties of type (K\ {&)) 101 The unramified class field obtained from the field
of moduli 109 The class fields generated by ideal-section points 114 The field of moduli in a generalized setting 118 The main theorem of complex multiplication in the
adelic language 121
he Zeta Function of an Abelian Variety with
Complex Multiplication 132
The zeta function relative to a field over which some endomorphisms are defined 132 The zeta function over smaller fields 137 Models over the field of moduli and models with
given Hecke characters 145 The case of elliptic curves 149
Families of Abelian Varieties and
Modular Functions 151
Symplectic and unitary groups 151 Families of polarized abelian varieties 156 Modular forms and functions 165 Canonical models 169
Theta Functions and Periods on Abelian Varieties 173
Theta functions 173 Proof of Theorem 27.7 and Proposition 27.9 181 Theta functions with complex multiplication 188 The periods of differential forms on abelian varieties 190 Periods in the Hilbert modular case 193 Periods on abelian varieties with complex
multiplication and their algebraic relations 196 Proof of Theorem 32.4 201
//
https://pdfs.semanticscholar.org/80a2/666bb38cd37b0479d3975414673db800b232.pdf
//////
志村五郎 (著) Shimura, Goro 書籍 英語版
志村五郎 (著) Automorphic Functions and Number Theory. Lecture Notes in Mathematics. 54 (Paperback ed.). Springer. (1968).
志村五郎 (著) Introduction to the Arithmetic Theory of Automorphic Functions (Paperback ed.). Princeton University Press. (1971-08-01).
志村五郎 (著) Euler Products and Eisenstein Series. CBMS Regional Conference Series in Mathematics (Paperback ed.). American Mathematical Society. (1997-07-01).
志村五郎 (著) Abelian Varieties with Complex Multiplication and Modular Functions (Hardcover ed.). Princeton University Press. (1997-12-08).
志村五郎 (著) Arithmeticity in the Theory of Automorphic Forms. Mathematical Surveys and Monographs (Paperback ed.). American Mathematical Society. (2000-08-22).
志村五郎 (著) Arithmetic and Analytic Theories of Quadratic Forms and Clifford Groups. Mathematical Surveys and Monographs (Hardcover ed.). American Mathematical Society. (2004-03-01).
志村五郎 (著) Elementary Dirichlet Series and Modular Forms. Springer Monographs in Mathematics (Paperback ed.). Springer New York. (2009-12-28).
志村五郎 (著) Arithmetic of Quadratic Forms. Springer Monographs in Mathematics (Hardcover ed.). Springer. (2010-07-15).
//////
論文集
Collected Papers. I: 1954-1965 (Hardcover ed.). Springer. (2002).
Collected Papers. II: 1967-1977 (Hardcover ed.). Springer. (2002).
Collected Papers. III: 1978-1988 (Hardcover ed.). Springer. (2003).
Collected Papers. IV: 1989-2001 (Hardcover ed.). Springer. (2003).
//////
備忘録 メモ
