Abelian varieties with complex multiplication and modular functions
By: Goro Shimura.
Series: Princeton mathematical series ;; 46
Publisher: Princeton University Press
Published: 1998
Place of publication: Princeton, N.J
Language: eng
Number of pages: 217
Subjects: Abelian varieties.; Modular functions.
Description: 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.
Notes: Includes bibliographical references (p. 209-214) and index.