Abelian Varieties with Complex Multiplication and Modular Functions / Goro Shimura.

Format:

Book

Author/Creator:

Shimura, Goro, author.

Series:

Princeton mathematical series ; 46.

Princeton Mathematical Series

Language:

English

Subjects (All):

Abelian varieties.

Modular functions.

Physical Description:

1 online resource (237 p.)

Place of Publication:

Princeton, NJ : Princeton University Press, [2016]

Language Note:

English

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.

Contents:

Frontmatter

Contents

Preface

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

Notation and Terminology

I. Preliminaries on Abelian Varieties

II. Abelian Varieties with Complex Multiplication

III. Reduction of Constant Fields

IV. Construction of Class Fields

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

VI. Families of Abelian Varieties and Modular Functions

VII. Theta Functions and Periods on Abelian Varieties

Bibliography

Supplementary References

Index

About the Author

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on online resource; title from PDF title page (publisher's Web site, viewed 08. Jul 2019)

ISBN:

9781400883943

1400883946

OCLC:

950463572