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，称为自守形式。其中一些函数的系数在算术中编码了重要信息，还有一些最近与物理学中的弦理论有关。 基本的朗兰兹功能性猜想预言了自守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。 这个研究项目的重点是建立自守形式的性质和自守形式的不同空间之间的新连接。 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工作。该项目将自守形式和表示约化群及其任意度的元复盖。约化群上的自守形式是朗兰兹纲领的关键组成部分，覆盖上的自守形式与互反律密切相关，并与同余子群问题有关。研究的重点是对应关系和L-函数。在一系列项目中，主要研究者将开发和研究新的theta对应，这些对应概括了经典的theta对应，但涉及两个小表示的张量积。 第二个项目旨在提供一个新的志村对应关系，这是检测到的一个时期，涉及一个正交组上的θ函数。这将通过一个新的相对迹公式来建立。第三套项目涉及系统的发展，利用剩余频谱的L-函数的积分表示。这些项目将推进我们的知识自守形式，无论是在约化群和有限度覆盖，我们的理解小自守表示，我们的理解自守形式之间的关系，对不同的群体。 该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.