Relative Langlands Program: Periods, Heights and L-functions

该奖项支持首席研究员在数论方面的研究。数论起源于对自然数的研究。它是最古老的数学分支之一。在过去的半个世纪里，它已经成为一种不可或缺的工具，在数据传输和处理、通信系统和网络安全等领域有着广泛的应用。少年派是亚利桑那州冬季学校研讨会的组织者之一。他还在培训研究生与该奖项相关的主题。更详细地说，该项目以自同构形理论为中心，这是数论的一个分支，通过对称性研究自然数。L函数论及其特殊价值是这门学科特别感兴趣的内容。主要目的是应用表示论和调和分析的各种技巧来证明L函数的特殊值与其他对象之间的联系，例如周期积分或Shimura簇上的特殊循环的高度。更具体地说，PI将研究以下猜想：(1)Gan-Gross-Prasad猜想及其算术变量；(2)关于线性周期的Guo-Jacquet猜想。该奖项反映了NSF的法定使命，并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，被认为值得支持。

This award supports the principal investigator's research on number theory. Number theory has its historical roots in the study of natural numbers. It is among the oldest branches of mathematics. Within the last half century it has become an indispensable tool, with diverse applications in areas such as data transmission and processing, communication systems, and internet security. The PI is one of the organizers of a workshop, the Arizona Winter School. He is also training graduate students in topics related to this award.In more detail, the project centers around the theory of automorphic forms, a branch of number theory which studies natural numbers through symmetries. The theory of L-functions and its special values is of particular interest in this subject. The main goal is to apply various techniques from representation theory and harmonic analysis to prove identities linking special values of L-functions to other objects, e.g. period integrals, or heights of special cycles on Shimura varieties. More concretely the PI will study the following conjectures: (1) Gan--Gross--Prasad conjectures and its arithmetic variant; (2)Guo--Jacquet conjecture on linear periods.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.