Book Review: Abelian varieties with complex multiplication and modular functions

Abstract

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