Survey of Modular Functions of One Variable

Abstract

In this paper a survey is given of the theory of modular functions of one variable, including Dirichlet series, functional equations, compactifications, Hecke operators, Eisenstein series and the Petersson product. They are notes of an introductory course on the subject at the International Summer School 1972, held at Antwerp University. It is to be noted that much of the material presented is to be found in [7], where complete proofs of most of the theorems occur. However, the material goes beyond [7], and is complementary to it, in the sense that more attention is paid to the relation between Eisenstein series and elliptic curves. Also included is an exposition of some basic features of the work of Artin and Lehner on old and new forms. Little, if no attention at all, is paid to the relation between modular functions and quadratic forms.

Notes by F. Van Oystaeyen