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.