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

这个项目围绕自守形式，自守表示和算术代数几何时期之间的相互作用。 特别是PI建议工作的几个问题上的一般衍生猜想，分析和算术的傅立叶系数的例外群体，界限的高度，计算研究nontempered期间，平均/非零的衍生物的L系列。近年来，周期的研究在L-函数方面取得了新的结果和证明，在代数圈的证明方面取得了突破，在表示论的经典问题上也有了新的见解。与其他工具相结合，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处于这些发展的最前沿。拟议的框架提出了一项雄心勃勃的计划，旨在制定和制定纳入/连接这些领域最近开创性工作的计划。该研究课题是数学的几个领域（算术几何，自守表示理论，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。这个小组将包括15名博士生，目前由PI提供咨询。

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.