Automorphic Forms and Galois Representations

数论是研究与整数性质有关的现象的数学分支。一个典型的数论问题是确定某个感兴趣的方程的整数解的个数。这些问题的答案通常可以用某些称为l函数的数学函数进行编码。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（或数学预言），预言任何l函数都应该由另一种称为自同构形式的数学函数产生。（数论学家将l函数和自同构形式之间的推测关系称为“互易律”。）朗兰兹发展了一系列强有力的表征理论方法来研究这些猜想。这些方法利用自同构形式和l函数的许多对称性来分析它们的数学性质；这些方法已被纳入一个被称为“朗兰兹纲领”的数学体系。最近研究自同构形式和l函数的一种方法是使用p进方法。这些方法涉及到对固定素数p的可整除性来研究自同构形式和l函数的泰勒级数系数。最近，表示理论方法和p进方法开始统一为所谓的“p进朗兰兹程序”。本项目旨在发展p进朗兰兹程序的新结果和新方法，并利用它们建立新的互易律。研究项目的目标是调查朗兰兹计划的p-adic方面。朗兰兹纲领的核心是一个关于自同构表示和p进伽罗瓦表示的推测性互易律。对自同构形式和伽罗瓦表征对应方式的精确描述涉及到局部互易律，也就是说，将自同构表征在素数q处的行为与伽罗瓦表征在同一素数处的行为联系起来的互易律。当q与支配伽罗瓦表示法系数的素数p相同时，这些局部律就显得最微妙了；事实上，在这种情况下，这样的局部互易律将构成p进局部朗兰兹对应，并且除了在阿贝尔情况下和GL_2（Q_p）情况下，它的存在仍然是推测性的。与各种合作者一起，首席研究员旨在以各种方式研究这一令人期待但神秘的p进局部互惠定律。PI对p进伽罗瓦表示的模堆栈的研究将为p进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进完全上同调的迹公式将对p进环境中的局部-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同构形式相关互易律的最有力工具之一。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。

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.