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

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Abstract for Collaborative FRG proposals DMS- 0354534, DMS -0353964, DMS-0354662 and DMS-0354582 of Hoffstein, Bump Friedberg and GoldfeldThe object of this proposal is to continue todevelop the theory of multiple Dirichlet seriesalong a number of highly promising directions.These include the formulation of aclassification theory via Dynkin diagrams andmetaplectic forms, analysis of naturalconstructions as inner products of automorphic forms on GL(n),and investigating examples coming fromEisenstein series related to deformation theoryof universal elliptic curves. Manyapplications are expected to the analysis of various familiesof L-functions The theory of L-functions of one complex variable iscentral in modern number theory. Special values ofL-functions have provided links between such diverseareas of mathematics as algebraic geometry, topology,probability and statistics, the representation theory ofinfinite dimensional Lie groups, and mathematicalphysics. In contrast, the theory of L-functions ofseveral complex variables (multipleDirichlet series) is still in its infancy. A large part of thefoundational theory was developedby the PI's and Postdocs of this proposal, who havebeen collaborating in teams, over thelast twenty years. The accumulated scientific results,combined with a mass sustained joint effort of thePI's, now point to the possibility of major breakthroughs.There is also an additionaltraining component. Workshops will be held each yearas well as short courses aimed at attracting graduate students, postdocs, andmathematicians in related fields. The motivationwill be to categorize and advertisethe major accessible problems in the field, tomap out progress made, and to prepare theparticipants for research projects.