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

Hoffstein、Bump Friedberg和Goldfield的协同FRG建议DMS-0354534、DMS-0353964、DMS-0354662和DMS-0354582本建议的目的是沿着几个很有希望的方向继续发展多重狄利克雷级数理论，包括通过动态图和亚正则形式建立类化理论，分析GL(N)上作为自同构形式内积的自然结构，以及研究来自Eisenstein级数的与万能椭圆曲线的变形理论有关的例子。L函数族的分析可望有更多的应用。一复变数的L函数理论是现代数论的核心。L-函数的特殊值在代数几何、拓扑学、概率和统计学、无限维李群的表示理论和数学物理等不同的数学领域之间提供了联系。相比之下，多元复变数(多重狄利克雷级数)的L函数理论还处于起步阶段。基本理论的很大一部分是由该提案的PI和博士后开发的，他们在过去的20年里一直在团队中合作。积累的科学成果，再加上国际和平组织的大规模持续共同努力，现在表明有可能取得重大突破。此外，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中的主要可接触问题进行分类和宣传，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.