Automorphic Forms on Reductive Groups and Their Covers

本研究项目研究数论中出现的函数，这些函数在变换下表现出特殊的对称性，称为自同构形式。其中一些函数具有编码算术中重要信息的系数，最近一些函数与物理学中弦理论的各个方面联系在一起。基本的朗兰兹泛函猜想预测了自同构形式的不同空间之间的微妙关系，这种结构与数论和分析中的许多问题密切相关。本课题的研究重点是建立自同构形的性质以及自同构形的不同空间之间的新联系。该项目还将支持一名研究生，并允许PI和他的学生通过会议和研讨会传播他们的工作。本项目处理约化群上的自同构形式和表示及其任意程度的元复盖。约化群上的自同构形式是朗兰兹纲领的关键组成部分，而盖上的自同构形式与互易律密切相关，并与同余子群问题有关。研究的重点是对应和l -函数。在一系列项目中，首席研究员将开发和研究新的θ对应，它将推广经典的θ对应，但涉及两个小表示的张量积。第二个项目试图给出一种新的志村对应关系，这种对应关系是通过正交群上的theta函数的周期来检测的。这将通过一个新的相对示踪公式来建立。第三组项目涉及利用残差谱的l函数的积分表示的系统发展。这些项目将促进我们对自同构形式的认识，包括约化群和有限次覆盖，我们对小自同构表示的理解，以及我们对不同群上自同构形式之间关系的理解。它将推进我们对数论和表征理论的认识。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project studies functions arising in number theory that exhibit special symmetry properties under transformations, called automorphic forms. Some of these functions have coefficients that encode important information in arithmetic, and some have recently been connected to aspects of string theory in physics. The fundamental Langlands Functoriality Conjectures predict subtle relations between different spaces of automorphic forms, a structure that is closely related to many questions in number theory and analysis. This research project focuses on establishing properties of automorphic forms and new connections between different spaces of automorphic forms. The project will also support a graduate student and allow the PI and his students to disseminate the work through conferences and seminars.This project treats automorphic forms and representations on reductive groups and on their metaplectic covers of arbitrary degree. Automorphic forms on reductive groups are the key ingredients of the Langlands Program, and automorphic forms on covers are closely tied to reciprocity laws and related to the congruence subgroup problem. The research focuses on correspondences and on L-functions. In one series of projects, the principal investigator will develop and study new theta correspondences that generalize the classical theta correspondence but involve the tensor product of two small representations. A second project seeks to give a new Shimura correspondence that is detected by a period involving a theta function on an orthogonal group. This will be established by means of a new relative trace formula. A third set of projects concerns the systematic development of integral representations for L-functions which make use of the residual spectrum. These projects will advance our knowledge of automorphic forms, both on reductive groups and on finite degree covers, our understanding of small automorphic representations, and our understanding of the relations between automorphic forms on different groups. It will advance our knowledge of number theory and of representation theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.