权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Endoscopy in the Relative Langlands Program

相关朗兰兹计划中的内窥镜检查

基本信息

批准号：
2200852
负责人：
Winston Leslie
金额：
$ 16.17万
依托单位：
Boston College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-09-15 至 2025-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200852&HistoricalAwards=false
关键词：
Endoscopy Relative Langlands Program

项目摘要

The Langlands program of mathematics, at its core, is a vast collection of conjectures which posits that two seemingly distinct areas of mathematics, number theory and representation theory, are connected in deep ways via the study of arithmetically rich functions known as L-functions. These functions arise both in the study of solutions to polynomial equations and in the study of highly-symmetric functions known as automorphic forms. A central insight is that the symmetry properties of automorphic forms may be used to prove properties of L-functions. The so-called relative Langlands program now gives us a conjectural framework describing the relationship between L-functions and certain invariants of automorphic forms known as periods. Analyzing these objects has combined tools from harmonic analysis, algebraic geometry, representation theory, and mathematical physics. In this project, the PI will develop and generalize a novel theory, known as endoscopy, to further study these period integrals of automorphic forms. The project will also support the PI as he mentors graduate and undergraduate students, organizes conferences and workshops, and contributes to the Automorphic Project. The relative trace formula is a central tool in the relative Langlands program for studying automorphic periods. Despite this, its theory has not yet been developed to meet the needs of the broader theory, and it has largely been used in contexts with strong multiplicity-one properties. This project will develop a novel theory of endoscopy and stabilization for relative trace formulae. This can be seen as a generalization of the Langlands-Shelstad-Kottwitz program of stabilization of the Arthur-Selberg trace formula, and will open the technique of comparison of trace formulas to periods where local multiplicity one may fail. The main goals of the project are: (1) to pre-stabilize the elliptic part of the relative trace formula associated to a large family of periods, (2) to develop a robust local and global theory of endoscopy in this setting and consider the stabilization of the relative analogues of the Hitchin fibration following the work of Ngo, (3) establish the central harmonic-analytic conjectures to affect new global comparisons of relative trace formulae in several cases of interest to arithmetic applications.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.

朗兰兹数学纲领的核心是一个巨大的猜想集合，它假设数学中的两个看似截然不同的领域--数论和表示论--通过研究被称为L函数的算术丰富的函数，以深度的方式联系在一起。这些函数既出现在多项式方程的解的研究中，也出现在被称为自同构形式的高度对称函数的研究中。一个重要的见解是，自同构形的对称性可以用来证明L函数的性质。所谓的相对朗兰兹程序现在给了我们一个猜想框架，描述了L函数和某些自同构形式的不变量(称为周期)之间的关系。分析这些对象结合了调和分析、代数几何、表示理论和数学物理的工具。在这个项目中，PI将发展和推广一种新的理论，称为内窥镜，以进一步研究这些自同构形的周期积分。该项目还将支持PI指导研究生和本科生，组织会议和研讨会，并为自构项目做出贡献。相对迹公式是相对朗兰兹程序中研究自同构周期的核心工具。尽管如此，它的理论还没有发展到满足更广泛的理论的需要，而且它在很大程度上被用于具有强烈的多重性-一的性质的语境中。该项目将开发一种新的内窥镜检查和相对痕量配方稳定的理论。这可以看作是亚瑟-塞尔伯格迹公式稳定化的朗兰兹-谢尔斯塔德-柯特维茨程序的推广，并将开启迹公式与局部重数一可能失效的周期的比较技术。该项目的主要目标是：(1)预先稳定与一大类周期相关的相对轨迹公式的椭圆部分，(2)在这种情况下发展一个健壮的局部和全球内窥镜理论，并考虑在NGO的工作之后希钦纤维的相对类似物的稳定性，(3)建立中心谐和解析猜想，以影响在几个与算术应用有关的情况下相对轨迹公式的新的全球比较。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，认为值得支持。