Automorphic Galois Representations and Automorphic L-functions

自同构伽罗瓦表示和自同构 L 函数

• 批准号: 1404769
• 负责人: Michael Harris
• 金额: $ 22.08万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1404769&HistoricalAwards=false
• 关键词: Automorphic; Galois; Representations; functions

Number theory originates from the study of solutions to equations in whole numbers. It is one of the oldest branches of mathematics, and its methods have for millenia been based on the interaction between the divisibility properties of whole numbers and their size. Twentieth-century number theory formalized these two properties in two different ways. Divisibility and the measure of size can be seen both as geometric and as dynamical properties, the latter rooted in the equations of mathematical physics. The branch of mathematics concerned with their geometric relations is called arithmetic geometry; the branch concerned with their dynamical relations is called automorphic forms. Symmetry plays a central role in both arithmetic geometry and automorphic forms; the hypothetical Langlands correspondence unifies these two branches by showing how each kind of symmetry encodes the other. The PI proposes to use this coding to understand objects on one side of the correspondence in terms of properties of the corresponding object on the other side. A particular focus is the transfer of divisibility properties of automorphic forms to arithmetic geometry, which often leads to surprisingly precise information about solutions of equations.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 arising in the cohomology of Shimura varieties, directly or by application of congruence methods. The long-term goals are the identification of all such motives and all such Galois representations (the modularity problem) and the proof of outstanding conjectures on the arithmetic of motives, notably Deligne's conjecture on special values of L-functions, and the conjectures of Greenberg, Coates, Perrin-Riou, and others on the existence of p-adic L-functions, for the motives obtained in this way. Special attention is given to special values of tensor product L-functions. This project fits into an international program to use the full range of available techniques to extend to all such motives results established for L-functions of elliptic modular forms; the PI is actively collaborating with colleagues in France, Austria, Israel, Japan, and Hong Kong, as well as in the United States and Canada. The methods involved in the present project combine standard techniques from arithmetic geometry and automorphic forms, the differential-geometric approach to cohomological automorphic forms on which the PI has worked for many years, with p-adic analysis, and representation theory.

数字理论源自整体方程的解决方案的研究。 它是数学最古老的分支之一，其对千禧一代的方法是基于整数的划分属性及其大小之间的相互作用。 20世纪的数字理论以两种不同的方式正式化了这两种属性。 分裂性和大小的度量可以看作是几何和动力学特性，后者植根于数学物理方程。 与它们的几何关系有关的数学分支称为算术几何。与他们的动态关系有关的分支称为自动形式。 对称性在算术几何形式和自动形式中都起着核心作用。假设的兰兰兹对应通过显示每种对称性如何编码另一个分支来统一这两个分支。 PI建议使用此编码来理解信函一侧的对象，从另一侧的相应对象的属性来理解对象。一个特定的重点是自身形式形式的划分特性转移到算术几何形状，通常会导致有关方程解决方案解决方案解决方案的精确信息。该项目是对自动态形式的算术理论的贡献，在兰格兰计划的环境中，特别关注了与他们相关的动力的算法，或者直接涉及其相关性的代表，或者是在其关联的范围内，或者是在其相关的范围内，或一致方法的应用。 长期的目标是确定所有此类动机和所有此类GALOIS表示（模块化问题）以及关于动机算术的出色猜想证明，特别是Deligne对L功能的特殊价值的猜想，以及Greenberg，Greenberg，Greenberg，Greenberg，Coates，Coates，Coate，Perrin-riou等，以及其他对p-ad lunctions的存在，以获取该工具，以获取典型的态度，以获取迫切的态度。特别注意张量产品L功能的特殊值。该项目符合国际计划，以使用各种可用技术来扩展到为椭圆形模块形式的L功能所建立的所有此类动机结果； PI正在与法国，奥地利，以色列，日本和香港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形式和自动形式的标准技术，这是PI在PI多年工作多年的共生体型形式的差分几何方法，并通过P-ADIC分析和表示理论。

项目成果

Derived Hecke Algebra for Weight One Forms

重量一形式的导出赫克代数

• DOI: 10.1080/10586458.2017.1409144
• 发表时间: 2018
• 期刊: Experimental Mathematics
• 影响因子: 0.5
• 作者: Harris, Michael;Venkatesh, Akshay
• 通讯作者: Venkatesh, Akshay

Period Relations and Special Values of Rankin-Selberg L-Functions

• DOI: 10.1007/978-3-319-59728-7_9
• 发表时间: 2016-08
• 期刊: arXiv: Number Theory
• 作者: M. Harris;Jiezhu Lin

Chern classes of automorphic vector bundles

自守向量丛的陈氏类

• DOI: 10.4310/pamq.2017.v13.n2.a1
• 发表时间: 2017
• 期刊: Pure and Applied Mathematics Quarterly
• 影响因子: 0.7
• 作者: Esnault, Hélène;Harris, Michael
• 通讯作者: Harris, Michael