This book is a concise and easily readable introduction to the theory of abelian functions. The first chapter presents some preliminaries on compact Riemann surfaces, especially the Riemann-Roch theorem and Abel's theorem, as well as a short survey of elliptic functions and some background on functions of several complex variables. In Chapter 2 an elementary approach to Weil's proof of the existence of theta-functions for an arbitrary positive divisor on a complex torus is given. Then the author proves the classical theorem that a non-trivial theta-function exists if and only if there is a non-trivial positive semi-definite Riemann form on the given torus. Frobenius' formula for the dimension of the space of theta-functions of a given type is derived. Consequences are concerned with the structure of the field of abelian functions and the projective embedding of an abelian variety. The last chapter deals with morphisms, polarizations and the duality theory of abelian varieties; especially Poincaré's complete reducibility theorem for abelian varieties is proved and the ring of endomorphisms of an abelian variety is studied.

This book can be recommended to beginning graduate students and presupposes not much more than a basic complex variable course.

Reviewed by H. Klingen