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

这个项目围绕着自同构形式、自同构表示和算术代数几何的周期之间的相互作用展开。特别是，PI提出了关于广义导数猜想、例外群上傅里叶系数的分析和算法、高度的界、非调和周期的计算研究、L级数的导数的平均/非零化等几个问题。近年来，周期研究对L函数的研究有了新的结果和证明，对代数圈猜想的研究有了突破，对表示论经典问题的研究有了新的视角。结合其他工具，周期还加深了我们对局部对称空间上的等分布问题和拓扑的理解。私人投资倡议处在这些发展的前列。拟议的框架提出了一项雄心勃勃的计划，以工作和制定猜想，纳入/联系这些领域最近的开创性工作。该研究课题是数学的几个领域(算术几何、自同构表示理论、解析数论)的中心。该项目的一个长期目标是建立一个研究自同构表示、数论和算术几何的科学家网络。PIS设想一群博士生和博士后积极参加拟议的研究务虚会和年度研讨会。这一组将包括目前由私人投资促进机构建议的15名博士生。

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.