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

Hoffstein， Bump Friedberg和goldfeld的协作FRG提案DMS- 0354534， DMS- 0353964， DMS-0354662和DMS-0354582。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Dirichlet级数理论。这些包括通过Dynkin图和元塑形式的分类理论的表述，作为GL(n)上自同构形式的内积的自然结构的分析，以及与通用椭圆曲线变形理论相关的梅森斯坦级数的例子的研究。一元复变的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、概率论和统计学、有限维李群的表示理论和数学物理等不同数学领域提供了联系。相比之下，复数变量的l函数理论（多重狄利克雷级数）仍处于起步阶段。基础理论的很大一部分是由该提案的PI和博士后在过去20年里以团队合作的方式发展起来的。积累的科学成果，加上pi的大规模持续共同努力，现在表明有可能取得重大突破。还有一个额外的培训部分。每年将举办研讨会和短期课程，旨在吸引相关领域的研究生、博士后和数学家。其动机将是对该领域的主要问题进行分类和宣传，绘制出取得的进展，并为研究项目的参与者做好准备。

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.