FRG: Collaborative Proposal: Periods of Automorphic Forms and Applications to L-Functions

FRG：协作提案：自同构形式的周期及其在 L 函数中的应用

• 批准号: 1065807
• 负责人: Akshay Venkatesh
• 金额: $ 34.89万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1065807&HistoricalAwards=false
• 关键词: FRG; Collaborative; Proposal; Periods; Automorphic

This project centers around the interactions between periods of automorphic forms, automorphic representations, and arithmetic algebraic geometry. In particular the PIs propose to work on several problems on the general derivative conjecture, analysis and arithmetic of Fourier coefficients on exceptional groups, bounds on heights, computational study of nontempered periods, averages/nonvanishing of derivatives of L-series. Recently the study of periods has yielded new results and proofs about L-functions, breakthroughs towards conjectures about algebraic cycles, and new perspectives on classical questions of representation theory. Combined with other tools, periods have also enhanced our understanding of equidistribution problems and topology on locally symmetric spaces. The PIs are at the forefront of these developments. The proposed framework presents an ambitious plan to work on and formulate conjectures incorporating/connecting the recent groundbreaking works in these areas. The research topic is central to several areas of mathematics (arithmetic geometry, automorphic representation theory, analytic number theory). A long range goal of the project is to establish a network of scientists working in automorphic representations, number theory, and arithmetic geometry. The PIs envision a group of PhD students and post-docs participating actively in the proposed Research Retreats and Annual Workshops. This group would include the 15 PhD students presently advised by the PIs.

该项目以自同构形式、自同构表示和算术代数几何周期之间的相互作用为中心。 特别是，PI建议研究一般导数猜想、例外群上傅里叶系数的分析和算术、高度界限、非调和周期的计算研究、L系列导数的平均/不消失等几个问题。近年来，周期研究取得了L函数的新结果和证明，代数圈猜想的突破，表示论经典问题的新视角。与其他工具相结合，周期还增强了我们对等分布问题和局部对称空间拓扑的理解。 PI 处于这些发展的最前沿。拟议的框架提出了一项雄心勃勃的计划，旨在结合/连接这些领域最近的突破性工作并制定猜想。该研究课题是多个数学领域（算术几何、自同构表示论、解析数论）的核心。该项目的长期目标是建立一个研究自守表示、数论和算术几何的科学家网络。 PI 设想有一组博士生和博士后积极参与拟议的研究务虚会和年度研讨会。该小组将包括目前由 PI 指导的 15 名博士生。