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

Hoffstein、Bump Friedberg和Goldstein的联邦德国合作提案DMS-0354534、DMS-0353964、DMS-0354662和DMS-0354582摘要该提案的目的是沿着一些非常有前途的方向继续发展多重狄利克雷级数的理论。这些方向包括通过Dynkin图和元形式来表述分类理论，分析了GL（n）上自守形式内积的自然结构，并研究了与泛椭圆曲线形变理论有关的Eisenstein级数的例子。L-函数在分析各种L-函数族中有着广泛的应用。单复变L-函数理论是现代数论的核心。L-函数的特殊值为代数几何、拓扑学、概率和统计、有限维李群的表示论和物理学等不同的数学领域提供了联系。相比之下，多复变量L-函数（多重狄利克雷级数）的理论仍处于起步阶段。基础理论的很大一部分是由这个提议的PI和博士后开发的，他们在过去的二十年里一直在团队中合作。积累的科学成果，再加上PI的大规模持续的共同努力，现在指向了重大突破的可能性。 每年将举办讲习班以及旨在吸引毕业生的短期课程， students学生，postdocs博士后，andmathematicians数学家in related相关fields领域.动机是对该领域的主要问题进行分类和分析，绘制出所取得的进展，并为参与者的研究项目做好准备。

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.