P-adic Aspects of the Langlands Program

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。朗兰兹程序的核心是一个推测的互易律，它将自同构表示与p进伽罗瓦表示联系起来，这种互易律是由数域上代数变异的上同调产生的。这种互易性的描述是用局部定律来描述的，也就是说，互易性定律将自同构表示在一个素数k处的行为与伽罗瓦表示在同一素数处的行为联系起来。当k与控制伽罗瓦表示法系数的素数p相同时，这些局部定律就显得尤为微妙；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。与合作者一起，首席研究员的目标是以各种方式调查这种推测的p进局部朗兰兹对应。工作的一部分旨在构建p进伽罗瓦表示的模堆栈。这将允许引入新的几何方法来研究对应的伽罗瓦理论方面。项目的另一部分将尝试在各种新的语境中构建p进的局部朗兰兹对应关系。更准确地说，以前的主要研究人员和合作者使用全局方法来构建p进局部朗兰兹对应的候选者；这个项目旨在建立这样构建的通信是真正的本地。

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 arising from the etale cohomology of algebraic varieties over number fields. The description of this reciprocity is in terms of local laws, that is, reciprocity laws that relate the behavior of the automorphic representation at a prime k to the behavior of the Galois representation at that same prime. These local laws are most subtle when k 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). With collaborators, the principal investigator aims to investigate this conjectural p-adic local Langlands correspondence in various ways. One part of the work aims to construct moduli stacks of p-adic Galois representations. This will allow for the introduction of new geometric methods into the study of the Galois-theoretic side of the correspondence. Another part of the project will attempt to construct the p-adic local Langlands correspondence in various new contexts. More precisely, previous work of the principal investigator and collaborators used global methods to construct a candidate for the p-adic local Langlands correspondence in some generality; this project aims to establish that the correspondence so constructed is truly local.