Modular Forms and Functional Equations

Abstract

Bernhard Riemann’s epoch-making 8-page paper “Über die Anzahl der Primezahlen unter einer gegebenen Große” of 1859, which is the starting point of the systematic study of integral representations, functional equations and analytic continuations of Dirichlet series, attracted enormous attention because of the beautiful symmetry about the critical line the functional equation possessed. The problem of categorising those Dirichlet series qualifying functional equations of Riemann type was taken up by Hecke and his fundamental contribution to the theory of modular forms was to study the functional equation and analytic properties of the L-function L(f, s) attached to a modular form f of weight k. Hecke’s work however provides a complete and satisfactory answer in one direction only. Later on, Weil in his famous paper of 1967 took up the so-called converse theorem for modular forms thereby accomplishing a perfect picture about Dirichlet series possessing functional equations. Since then Hecke-Weil theory was the centre of interest and several generalisations to modular forms of other types including some of higher degree have been done so far, with a lot more waiting for the future. In this expository article, we present without proofs a bird’s eye view of several such results including some global ones.