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Automorphic Forms and Galois Representations

自守形式和伽罗瓦表示

基本信息

批准号：
1902307
负责人：
Matthew Emerton
金额：
$ 39万
依托单位：
University of Chicago
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902307&HistoricalAwards=false
关键词：
Automorphic Forms Galois Representations

项目摘要

Number theory is the branch of mathematics that studies phenomena related to properties of whole numbers. A typical number theoretic question is to determine the number of whole number solutions of some equation of interest. The answers to such questions can often be encoded in certain mathematical functions known as L-functions. The mathematician Robert Langlands has developed a series of conjectures (or mathematical predictions) regarding L-functions, which predict that any L-function should arise from another kind of mathematical function called an automorphic form. (Number theorists refer to the conjectured relationship between L-functions and automorphic forms as a "reciprocity law.") Langlands developed an array of powerful representation theoretic methods to study the conjectures. These are methods that exploit the many symmetries of automorphic forms and L-functions to analyze their mathematical properties; these methods have been incorporated into a body of mathematics known as "the Langlands program." A more recent approach to the study of automorphic forms and L-functions is the use of p-adic methods. These are methods that involve using divisibility properties with respect to some fixed prime number p to study the Taylor series coefficients of the automorphic forms and L-functions. Recently, the representation theoretic methods and p-adic methods have begun to be unified into a so-called "p-adic Langlands program." This project aims to develop new results and methods in the p-adic Langlands program, and to use them to establish new reciprocity laws.The goal of the research project is to investigate the p-adic aspects of the Langlands program. At the heart of the Langlands program is a conjectured reciprocity law relating automorphic representations to p-adic Galois representations. The precise description of the manner in which automorphic forms and Galois representations are supposed to correspond involves local reciprocity laws, that is, reciprocity laws that relate the behavior of the automorphic representation at a prime q to the behavior of the Galois representation at that same prime. These local laws are most subtle when q is taken to be the same prime p that governs the coefficients of the Galois representation; indeed, in this case such a local reciprocity law would constitute a p-adic local Langlands correspondence, and its existence remains conjectural other than in the abelian case, and the case of GL_2(Q_p). Together with various collaborators, the principal investigator aims to investigate this hoped-for but mysterious p-adic local reciprocity law in various ways. The PI's study of moduli stacks of p-adic Galois representations will provide new geometric insight into p-adic local Langlands and related problems such as the Breuil--Mezard conjecture. The PI's proposed trace formula for p-adically completed cohomology should yield new insight into local-global compatibility in the p-adic context. The PI's research on growth of cohomology will increase the range of applicability of stable trace formula, one of the most powerful tools available for studying reciprocity laws associated to automorphic forms.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，研究与整数性质有关的现象。一个典型的数论问题是确定一些感兴趣的方程的整数解的个数。这些问题的答案通常可以编码在某些称为L函数的数学函数中。数学家罗伯特·朗兰兹（Robert Langlands）提出了一系列关于L函数的猜想（或数学预测），这些猜想（或数学预测）预测任何L函数都应该源于另一种称为自守形式的数学函数。（数论家把L-函数和自守形式之间的关系称为“互反律”。朗兰兹发展了一系列强有力的表示论方法来研究这些结构。这些方法利用自守形式和L-函数的许多对称性来分析它们的数学性质;这些方法已被纳入称为“朗兰兹纲领”的数学体系。最近研究自守形式和L-函数的方法是使用p-adic方法。这些方法涉及使用关于某个固定素数p的整除性质来研究自守形式和L-函数的泰勒级数系数。最近，表示论方法和p-adic方法开始统一为所谓的“p-adic Langlands程序”。“本项目的目的是在p-adic朗兰兹纲领中发展新的结果和方法，并利用它们建立新的互惠律。研究项目的目标是研究朗兰兹纲领的p-adic方面。朗兰兹纲领的核心是一个将自守表示与p进伽罗瓦表示联系起来的互反律。自守形式和伽罗瓦表示的对应方式的精确描述涉及局部互反律，即把自守表示在素数q处的行为与伽罗瓦表示在同一素数处的行为联系起来的互反律。当q被取为支配伽罗瓦表示系数的素数p时，这些局部律是最微妙的;事实上，在这种情况下，这样的局部互反律将构成p-adic局部朗兰兹对应，并且除了阿贝尔情形和GL_2（Q_p）情形之外，它的存在仍然是自然的。与各种合作者一起，首席研究员的目标是以各种方式研究这种希望但神秘的p-adic局部互惠定律。PI对p-adic伽罗瓦表示的模栈的研究将为p-adic局部朗兰兹和相关问题（如Breuil-Mezard猜想）提供新的几何见解。 PI提出的p-adically完成上同调的迹公式应该在p-adic上下文中对局部全局相容性产生新的见解。 PI对上同调增长的研究将增加稳定迹公式的适用范围，稳定迹公式是研究自守形式相关互惠定律的最强大工具之一。该奖项反映了NSF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。