There are two major topics treated in this volume :
I. Complex multiplication of elliptic or elliptic modular functions.
II. Applications of the theory of Hecke operators to the zeta-functions
algebraic curves and abelian varieties.
Although these will form the " raison d'etre" of the book, 1 have also attempted, in the first few chapters, to present an introductory account of the theory of automorphic functions of one complex variable, along with the fundamentals of Hecke operators. Our discussion is mainly concerned with
elliptic modular functions of arbitrary level and the geometric objects directly related to them, except that we consider automorphic functions of a more general type in the first two and the last two chapters, and abelian varieties of higher dimension with complex multiplication in a few places.