Applications of Double Dirichlet Series to Automorphic Forms and Number Theory

Brown University, Providence RI

Investigators

Abstract

The investigator continues his investigation of applications of double Dirichlet series to the study of automorphic L-series and number theory. He and various collaborators have been developing the theory of double Dirichlet series for several years. It offers a simple replacement for the Rankin-Selberg method in many instances. For example, it provides a very short proof of the entirety of the symmetric square L-series of a generic automorphic form on GL(2). The investigator hopes to extend this technique as far as possible to understand higher symmetric power L-functions and the collective behavior of higher order twists of standard L-functions. As an additional source of input for trying to guess potential applications of the double Dirichlet series method, the investigator studies the Mellin transforms of certain generalizations of metaplectic forms. These are constructed from other non congruence subgroups. The theory of these forms is correspondingly rich and should contribute further insights into applications of double Dirichlet series. The study of L-series has been an essential part of the development of mathematics over the last 100 years. For example, certain L-series provided vital links in the chain that led to the recent proof of Fermat's last theorem. The study of double Dirichlet series is leading to an improved understanding of L-series which in turn leads to further knowledge of some very deep areas of mathematics. It is impossible to predict the ultimate impact of this investigation, but potential areas of application include cryptography.