Collaborative Research: FRG: Applications of Multiple Dirichlet Series to Analytic Number Theory

Brown University, Providence RI

Investigators

Abstract

Abstract for Collaborative FRG proposals DMS- 0354534, DMS -0353964, DMS-0354662 and DMS-0354582 of Hoffstein, Bump Friedberg and Goldfeld The object of this proposal is to continue to develop the theory of multiple Dirichlet series along a number of highly promising directions. These include the formulation of a classification theory via Dynkin diagrams and metaplectic forms, analysis of natural constructions as inner products of automorphic forms on GL(n), and investigating examples coming from Eisenstein series related to deformation theory of universal elliptic curves. Many applications are expected to the analysis of various families of L-functions The theory of L-functions of one complex variable is central in modern number theory. Special values of L-functions have provided links between such diverse areas of mathematics as algebraic geometry, topology, probability and statistics, the representation theory of infinite dimensional Lie groups, and mathematical physics. In contrast, the theory of L-functions of several complex variables (multiple Dirichlet series) is still in its infancy. A large part of the foundational theory was developed by the PI's and Postdocs of this proposal, who have been collaborating in teams, over the last twenty years. The accumulated scientific results, combined with a mass sustained joint effort of the PI's, now point to the possibility of major breakthroughs. There is also an additional training component. Workshops will be held each year as well as short courses aimed at attracting graduate students, postdocs, and mathematicians in related fields. The motivation will be to categorize and advertise the major accessible problems in the field, to map out progress made, and to prepare the participants for research projects.