Automorphic forms and automorphy preserving differential operators on products of halfplanes

Abstract

The theory of automorphic functions of one complex variable is now nearly a century old, and has antecedents -the theories of elliptic and elliptic modular functions -which can be t raced through ABEL'S far-reaching paper of 1827 [1] to the addition theorems for certain elliptic integrals discovered by FAO~A_~O in 1714 [18]. I t can be said to have at ta ined a state of matur i ty and to be, today, well understood in general al though a number of impor tan t specific problems still remain unsolved. Despite its origin as a branch of Analysis, and its reliance on essentially analyt ical techniques for the establishment of its major results, the theory maintains to this day a curiously algebraic outlook: its main results concern the algebraic structure of certain graded rings of functions, and the algebraic correspondences that persist amongst