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

合作研究：FRG：多重狄利克雷级数在解析数论中的应用

• 批准号: 0353964
• 负责人: Solomon Friedberg
• 金额: $ 12万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-15 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0353964&HistoricalAwards=false
• 关键词: Collaborative; Research; FRG; Applications; Multiple

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.

Hoffstein、Bump Friedberg 和 Goldfeld 的合作 FRG 提案 DMS-0354534、DMS -0353964、DMS-0354662 和 DMS-0354582 的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理论。这些包括通过 Dynkin 图和元波形式制定分类理论，分析自然结构作为 GL(n) 上自守形式的内积，并研究了与通用椭圆曲线变形理论相关的爱森斯坦级数的例子。许多应用有望用于分析各种 L 函数族。一个复变量的 L 函数理论是现代数论的核心。 L 函数的特殊值提供了数学不同领域之间的联系，例如代数几何、拓扑、概率和统计、无限维李群的表示论以及数学物理学。相比之下，多个复变量的L函数（多重狄利克雷级数）理论仍处于起步阶段。基础理论的很大一部分是由该提案的 PI 和博士后开发的，他们在过去二十年里一直在团队中合作。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。 每年都会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。其动机是对该领域的主要可解决问题进行分类和宣传，规划所取得的进展，并为参与者做好研究项目的准备。