FRG: Collaborative Research: Periods of automorphic forms and applications to L-functions

该项目围绕自同构形式、自同构表示和算术代数几何之间的相互作用展开。特别地，pi建议研究一些问题，如一般导数猜想，特殊群上傅立叶系数的分析和算术，高度的界，非调质周期的计算研究，l级数导数的平均/不消失。近年来，周期的研究产生了关于l -函数的新结果和证明，在代数循环猜想方面取得了突破，并对表征理论的经典问题提出了新的观点。结合其他工具，周期也增强了我们对局部对称空间上的等分布问题和拓扑的理解。pi处于这些发展的前沿。拟议的框架提出了一个雄心勃勃的计划，将这些领域最近的突破性工作结合起来，进行研究和制定推测。研究课题是几个数学领域（算术几何、自同构表示理论、解析数论）的中心。该项目的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与拟议的研究务静会和年度研讨会。这个小组将包括目前由pi指导的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.