L-Functions and Geometric Methods in Langlands Duality

朗兰兹对偶中的 L 函数和几何方法

• 批准号: 2001369
• 负责人: Michael Harris
• 金额: $ 21.36万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001369&HistoricalAwards=false
• 关键词: Functions; Geometric; Methods; Langlands; Duality

Although the ancient Greeks saw arithmetic -- what we now call number theory -- and geometry as two distinct sciences, with very different methods, contemporary mathematicians have understood for many years that they are really two aspects of a single science. This science, whose ramifications extend throughout mathematics, and whose applications are manifest in every corner of modern physics, computer science, and even the most recent developments in statistics, is as old as civilization, but it also takes on a completely new character at least once per generation. The current project is a study of Langlands duality, a synthesis of number theory and the geometry of symmetry that has been one of the most familiar names for this science for nearly two generations. For most of this period, the emblem of this synthesis has been the theory of L-functions, a computable system for encoding properties in number theory and geometry that matches the two aspects of this common science. More recently, new and sophisticated geometric techniques promise to reframe the synthesis from above, as part of a larger synthesis that ultimately extends to the most speculative ideas in mathematical physics. The PI plans to revisit Langlands duality from this new geometric standpoint, and to understand the place of L-functions in this broader picture.The project is a contribution to the arithmetic theory of automorphic forms, in the setting of the Langlands program, motivated in part by the geometric Langlands program. The specific goals of the present project are the study of the local Langlands correspondence, using trace formula methods, L-functions, and Galois theory to reduce the local Langlands correspondence to a conjecture on "incorrigible" representations; the study of derived aspects of the Langlands program, inspired by both in specific cases arising from coherent cohomology and in the development of a unified geometric framework for both Langlands reciprocity for cohomology of locally symmetric spaces and Venkatesh's conjectures on the actions of derived deformation rings on cohomology; the construction of p-adic square root L-functions in families, using the higher Hida theory introduced by Pilloni and Boxer; the completion of the proof of an automorphic version of Deligne's conjecture on special values of tensor product L-functions; and the development of a character theory for mod p representations of p-adic groups. In connection with this project, the PI is actively collaborating with colleagues in France, Austria, Germany, England, and Spain, as well as in the United States and Canada. The methods involved in the present project combine techniques from arithmetic geometry and automorphic forms, especially the trace formula, categorical representation theory, and the new methods developed by Venkatesh, Darmon, and Gaitsgory.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.

虽然古希腊人把算术（我们现在称之为数论）和几何学看作是两门截然不同的科学，有着非常不同的方法，但当代数学家多年来已经明白，它们实际上是一门科学的两个方面。 这门科学，其分支延伸到整个数学，其应用在现代物理学的每一个角落，计算机科学，甚至是统计学的最新发展，是一样古老的文明，但它也采取了一个全新的性格至少每一代人一次。 目前的项目是朗兰兹对偶的研究，朗兰兹对偶是数论和对称几何的综合，这是近两代人以来这门科学最熟悉的名字之一。 在这一时期的大部分时间里，这一综合的标志是L函数理论，这是一个可计算的系统，用于编码数论和几何中的属性，与这一普通科学的两个方面相匹配。 最近，新的和复杂的几何技术承诺从上面重新构建综合，作为最终扩展到数学物理中最投机的想法的更大综合的一部分。 PI计划从这个新的几何角度重新审视朗兰兹对偶，并理解L函数在这个更广阔的图景中的位置。该项目是对自守形式的算术理论的贡献，在朗兰兹纲领的背景下，部分受到几何朗兰兹纲领的激励。本项目的具体目标是研究局部朗兰兹对应，使用迹公式方法、L-函数和伽罗瓦理论将局部朗兰兹对应归结为关于“无可救药”表示的猜想;对朗兰兹纲领的派生方面的研究，在特定情况下，从相干上同调和发展一个统一的几何框架两个朗兰兹启发局部对称空间上同调的互反性和Venkatesh关于导出变形环对上同调的作用的说明，利用Pilloni和Boxer引入的高阶希达理论构造了族中的p进平方根L-函数，完成了Deligne关于张量积L-函数特殊值的猜想的自守版本的证明;以及发展了p进群的模p表示的特征标理论。 在该项目中，PI与法国、奥地利、德国、英国和西班牙以及美国和加拿大的同事积极合作。 本项目所涉及的方法结合了算术几何和自守形式的联合收割机技术，特别是迹公式、分类表示理论以及Venkatesh、Darmon和Gaitsgory开发的新方法。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。

项目成果

Shimura varieties for unitary groups and the doubling method

酉群的志村品种和倍增法

• DOI: 10.1007/978-3-030-68506-5_6
• 发表时间: 2021
• 期刊: Relative Trace Formulas
• 作者: Harris, Michael

The Derived Hecke Algebra for Dihedral Weight One Forms

二面体权重一式的导出赫克代数

• DOI: 10.1307/mmj/20217221
• 发表时间: 2022
• 期刊: Michigan Mathematical Journal
• 影响因子: 0.9
• 作者: Darmon, Henri;Harris, Michael;Rotger, Victor;Venkatesh, Akshay
• 通讯作者: Venkatesh, Akshay

A local Langlands parameterization for generic supercuspidal representations of $p$-adic $G_2$

$p$-adic $G_2$ 的通用超尖角表示的局部 Langlands 参数化

• DOI: 10.24033/asens.2533
• 发表时间: 2023
• 期刊: Annales scientifiques de l'École Normale Supérieure
• 作者: HARRIS, Michael;KHARE, Chandrashekhar B.;THORNE, Jack A.
• 通讯作者: THORNE, Jack A.

Square root p-adic L-functions, I : Constructionof a one-variable measure

平方根 p 进 L 函数，I ：单变量测度的构造

• DOI: 10.2140/tunis.2021.3.657
• 发表时间: 2021
• 期刊: Tunisian Journal of Mathematics
• 影响因子: 0.9
• 作者: Harris, Michael