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Abelian Varieties with Complex Multiplication and Modular Functions : Princeton Mathematical - Goro Shimura

Abelian Varieties with Complex Multiplication and Modular Functions

Princeton Mathematical

Hardcover Published: 28th December 1997
ISBN: 9780691016566
Number Of Pages: 232

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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.

Industry Reviews

"[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."--Bulletin of the London Mathematical Society

Preface to Complex
Multiplication of Abelian Varieties and Its Applications to Number Theory (1961)
Notation and Terminology
Preliminaries on Abelian Varieties
Abelian Varieties with Complex Multiplication
Reduction of Constant Fields
Construction of Class Fields
The Zeta Function of an Abelian Variety with Complex Multiplication
Families of Abelian Varieties and Modular Functions
Theta Functions and Periods on Abelian Varieties
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691016566
ISBN-10: 0691016569
Series: Princeton Mathematical
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 232
Published: 28th December 1997
Country of Publication: US
Dimensions (cm): 24.0 x 16.0  x 1.91
Weight (kg): 0.51

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