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Functoriality in the Mod-p Langlands Program

Mod-p Langlands 程序中的功能性

基本信息

批准号：
2310225
负责人：
Karol Koziol
金额：
$ 12.41万
依托单位：
CUNY Baruch 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：
2022-12-01 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2310225&HistoricalAwards=false
关键词：
Functoriality Mod Langlands Program

项目摘要

Number theory is the branch of mathematics that deals with properties of whole numbers and whole number solutions to polynomial equations, and stands as one of the oldest mathematical disciplines. Representation theory, another equally influential branch of mathematics, quantifies symmetries of geometric objects (such as a square or a hydrogen atom), and has important uses in physics. Though seemingly unrelated, these two areas are intimately linked by the Langlands Program, a vast set of conjectures that allows for the transfer of results and theorems between number theory and representation theory. It is of paramount importance to understand these conjectures, since tools from one discipline can be imported to tackle previously intractable problems in another (the proof of Fermat's Last Theorem being a prime example). This has pushed the Langlands Program to the forefront of current research. The present project seeks to establish instances of a local version of the Langlands Program with mod p coefficients, so that information from representation theory can be transferred into arithmetic data.The setting of the current project lies within the representation theory of p-adic reductive groups (such as GL_2(Q_p)) on mod p vector spaces. Such representations are exceedingly intricate, and one of the main goals is to use derived categories in order to more precisely relate such representations to modules over differential graded Hecke algebras. This will allow for the use of new tools to understand the relationships between Langlands correspondences for varying groups. In addition to this, the PI and his collaborators plan to use known instances of automorphic base change and the global theory of automorphic forms to develop a mod p Langlands correspondence for p-adic unitary groups. This would enrich the known instances of mod p Langlands correspondences by showing that they are compatible with functorial constructions.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.

数论是数学的一个分支，研究整数的性质和多项式方程的整数解，是最古老的数学学科之一。表征理论，另一个同样有影响力的数学分支，量化几何物体（如正方形或氢原子）的对称性，在物理学中有重要的应用。虽然看起来毫无关联，但这两个领域被朗兰兹纲领紧密地联系在一起。朗兰兹纲领是一套庞大的猜想，它允许数论和表示论之间的结果和定理的转移。理解这些猜想是至关重要的，因为一个学科的工具可以被引入到另一个学科中来解决以前难以解决的问题（费马大定理的证明就是一个最好的例子）。这将朗兰兹项目推向了当前研究的前沿。本项目旨在建立具有mod p系数的朗兰兹程序的局部版本的实例，以便从表示理论中获得的信息可以转换为算术数据。本课题的背景是模p向量空间上的p进约化群（如GL_2(Q_p)）的表示理论。这种表示是非常复杂的，其中一个主要目标是使用派生范畴，以便更精确地将这种表示与微分梯度赫克代数上的模块联系起来。这将允许使用新的工具来理解不同群体的朗兰兹对应之间的关系。除此之外，PI和他的合作者计划利用已知的自同构碱基变化的实例和自同构形式的整体理论来发展p进酉群的模p朗兰兹对应。这将丰富已知的模p朗兰兹对应的实例，表明它们与泛函结构兼容。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。