Automorphic Forms and Galois Representations

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某个感兴趣的方程的整数解的个数。这类问题的答案通常可以编码成称为L函数的某些数学函数。数学家罗伯特·朗兰兹发展了一系列关于L函数的猜想(或数学预测)，这些猜想(或数学预测)预言，任何L函数都应该由另一种称为自同构形的数学函数产生。(数学家将L函数和自同构形之间的猜想关系称为“互易定律”。)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；这些方法已经被合并到一个被称为“朗兰兹计划”的数学体系中。研究自同构型和L函数的一种较新的方法是使用p-进方法。这些方法涉及利用关于某个固定素数p的可除性来研究自同构型和L函数的泰勒级数系数。最近，表示理论方法和p-进方法开始被统一为所谓的“p-进朗兰兹方案”。该项目旨在开发p-adic朗兰兹计划的新结果和新方法，并利用它们来建立新的互惠定律。朗兰兹计划的核心是一个猜想的互易律，它将自同构表示与p-进伽罗瓦表示联系起来。自同构形式和伽罗瓦表示应该对应的方式的精确描述涉及局部互易律，即将素数q处的自同构表示的行为与同一素数处的伽罗瓦表示的行为联系起来的互易律。当Q被认为是支配伽罗瓦表示的系数的同一素数p时，这些局部定律是最微妙的；事实上，在这种情况下，这样的局部互易定律将构成p-进局部朗兰兹对应，并且它的存在仍然是猜想的，而不是在阿贝尔情况下，也不是GL_2(Q_P)的情况。首席研究员与不同的合作者一起，致力于以不同的方式研究这一希望中但神秘的局部互惠定律。PI对p进位Galois表示的模堆叠的研究将提供对p进位局部朗兰兹及其相关问题，如Breuil-Mezard猜想的新的几何洞察。PI提出的p-ad完全上同调的迹公式应该会对p-进背景下的局部-全局相容产生新的见解。PI对上同调增长的研究将增加稳定迹公式的适用范围，稳定迹公式是研究与自同构形相关的互易律的最强大的工具之一。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.