Relative Langlands Program: Periods, Heights and L-functions

相关朗兰兹纲领：周期、高度和 L 函数

• 批准号: 1901862
• 负责人: Hang Xue
• 金额: $ 15万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-01 至 2023-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901862&HistoricalAwards=false
• 关键词: Relative; Langlands; Program; Periods; Heights

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.

这个奖支持首席研究员对数论的研究。数论的历史根源在于对自然数的研究。它是数学最古老的分支之一。在过去的半个世纪里，它已经成为一个不可或缺的工具，在数据传输和处理、通信系统和互联网安全等领域得到了广泛的应用。PI是一个研讨会的组织者之一，亚利桑那冬季学校。他也在培训与该奖项相关的研究生。更详细地说，该项目以自同构形式理论为中心，自同构形式是数论的一个分支，通过对称性研究自然数。l函数及其特殊值的理论在这门学科中具有特殊的意义。主要目标是应用表示理论和调和分析的各种技术来证明l函数的特殊值与其他对象的联系，例如周期积分，或Shimura变体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猜想及其算术变体；(2)关于线性周期的Guo—Jacquet猜想。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。

项目成果

Linear intertwining periods and epsilon dichotomy for linear models

线性模型的线性交织周期和 epsilon 二分法

• DOI: 10.1007/s00208-023-02615-9
• 发表时间: 2023
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Suzuki, Miyu;Xue, Hang
• 通讯作者: Xue, Hang

Epsilon dichotomy for linear models

线性模型的 Epsilon 二分法

• DOI: 10.2140/ant.2021.15.173
• 发表时间: 2021
• 期刊: Algebra & Number Theory
• 影响因子: 1.3
• 作者: Xue, Hang

Bessel models for real unitary groups: The tempered case

• DOI: 10.1215/00127094-2022-0018
• 发表时间: 2023-01
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Hang Xue

Orbital integrals on

轨道积分

• DOI: 10.4153/s0008414x21000122
• 发表时间: 2022
• 期刊: Canadian Journal of Mathematics
• 作者: Xue, Hang