Langlands Correspondences and Motivic L-Functions

朗兰兹对应和动机 L 函数

• 批准号: 1701651
• 负责人: Michael Harris
• 金额: $ 20.96万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1701651&HistoricalAwards=false
• 关键词: Langlands; Correspondences; Motivic; Functions

The study of the abstract properties of numbers and their relations has appeared at an early stage in the history of every civilization, and reflection on the problems of number theory is consistently found at the root of most of the ideas that characterize contemporary life, from timekeeping, to the symmetry concepts of modern physics, to the logic of computers. Numbers can be studied in two different ways that are nearly independent: they can be used for measurement and they can be used to do arithmetic. The interaction between these two properties has always been the basis of number theory. The second half of the twentieth century saw the formulation of several ambitious research programs that aimed at obtaining a systematic understanding of these interactions by studying numbers and their relations with the help of symmetry. The branch of mathematics concerned with their geometric symmetries is called arithmetic geometry; the branch concerned with their dynamical symmetries is called automorphic forms. The Langlands program aims to unify these two branches by showing how each kind of symmetry encodes the other. Contemporary theoretical physics has introduced the concept of higher order symmetries, based on new notions of space that themselves owe a great deal to earlier developments in number theory; more recently, higher order symmetries have been of increasing importance in the Langlands program. This project explores the role of higher order symmetries in connection with several specific questions in the Langlands program, with the ultimate aim of contributing to the understanding of solutions of equations in whole numbers.The project is a contribution to the arithmetic theory of automorphic forms, in the setting of the Langlands program, with special attention to the arithmetic of motives and their associated Galois representations, directly or by application of congruence methods. The specific goals of the project are the study of the local Langlands parametrizations for general groups, using trace formula methods; the proof of Deligne's conjecture on special values of L-functions, especially tensor product L-functions; the verification of Venkatesh's conjecture on derived Hecke algebras for modular forms of weight 1, using an unexpected relation with p-adic L-functions; and the development of a character theory for mod p representations of p-adic groups. The methods involved in the present project combine standard techniques from arithmetic geometry and automorphic forms, an approach to cohomological automorphic forms based on differential geometry and representation theory, categorical representation theory, as well as new methods.

对数字及其关系的抽象属性的研究已经出现在每个文明历史的早期阶段，并且对数字理论问题的反思始终是在大多数表征当代生活的思想的根源上，从定时式到现代物理学的对称性概念，再到计算机的逻辑。 可以以几乎独立的两种不同方式研究数字：它们可用于测量，并且可以用于进行算术。 这两个属性之间的相互作用一直是数字理论的基础。 二十世纪下半叶看到了一些雄心勃勃的研究计划的制定，旨在通过对对称性的帮助研究数字及其关系来系统地了解这些相互作用。 与它们的几何对称性有关的数学分支称为算术几何。与它们的动态对称性有关的分支称为自动形式。 Langlands计划旨在通过展示每种对称性如何编码对方来统一这两个分支。 当代理论物理学介绍了高级对称性的概念，基于空间的新概念，这些概念本身归功于数量理论的早期发展。最近，在兰兰兹计划中，高级对称性的重要性越来越重要。 This project explores the role of higher order symmetries in connection with several specific questions in the Langlands program, with the ultimate aim of contributing to the understanding of solutions of equations in whole numbers.The project is a contribution to the arithmetic theory of automorphic forms, in the setting of the Langlands program, with special attention to the arithmetic of motives and their associated Galois representations, directly or by application of congruence methods. 该项目的具体目标是使用痕量公式方法研究一般组的局部兰兰群岛参数。 Deligne对L功能的特殊值的猜想证明，尤其是张量产品L功能；使用与P-Adic L功能的意外关系，对Venkatesh在衍生的Hecke代数中的猜想验证；以及针对P-ADIC群体的Mod P表示的性格理论的发展。 本项目所涉及的方法结合了算术几何形式和自动形式的标准技术，这是一种基于差异几何学和表示理论，分类代表理论以及新方法的共同体自体形式的方法。

项目成果

$\hat{G}$-local systems on smooth projective curves are potentially automorphic

• DOI: 10.4310/acta.2019.v223.n1.a1
• 发表时间: 2016-09
• 期刊: Acta Mathematica
• 影响因子: 3.7
• 作者: Gebhard Bockle;M. Harris;Chandrashekhar B. Khare;J. Thorne

Minimal modularity lifting for nonregular symplectic representations

非正则辛表示的最小模块化提升

• DOI: 10.1215/00127094-2019-0044
• 发表时间: 2020
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Calegari, Frank;Geraghty, David
• 通讯作者: Geraghty, David

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

Chern classes of automorphic vector bundles, II

自守向量丛的陈氏类，II

• 发表时间: 2019
• 期刊: Épijournal de géométrie algébrique
• 作者: Esnault, H.;Harris, M.
• 通讯作者: Harris, M.

p-ADIC L-FUNCTIONS FOR UNITARY GROUPS

酉群的 p-ADIC L 函数

• 发表时间: 2020
• 期刊: Forum of mathematics
• 作者: Eischen, Ellen;Harris, Michael;Li, Jian-Shu;Skinner, Christopher
• 通讯作者: Skinner, Christopher