# Abelian Varieties with Complex Multiplication and Modular Functions

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Dezember 1997

## Beschreibung

### Beschreibung

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.### Inhaltsverzeichnis

PrefacePreface to Complex Multiplication of Abelian Varieties and Its Applications to Number Theory (1961)

Notation and Terminology

Ch. I. Preliminaries on Abelian Varieties

Ch. II. Abelian Varieties with Complex Multiplication

Ch. III. Reduction of Constant Fields

Ch. IV. Construction of Class Fields

Ch. V. The Zeta Function of an Abelian Variety with Complex Multiplication

Ch. VI. Families of Abelian Varieties and Modular Functions

Ch. VII. Theta Functions and Periods on Abelian Varieties

Bibliography

Supplementary References

Index

### Pressestimmen

[This book] is a beautifully written, self-contained and complete treatment of a subject of which G. Shimura is a founding master, and is a fundamental reference for any researcher or student of the antimetric theory of abelian varieties and modular functions, and in particular of its applications to class field theory.EAN: 9780691016566

ISBN: 0691016569

Untertitel: 'Princeton Mathematical Series'.
Sprache: Englisch.

Verlag: Princeton University Press

Erscheinungsdatum: Dezember 1997

Seitenanzahl: 232 Seiten

Format: gebunden

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