P-adic Aspects of the Langlands Program

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）发展了一系列关于L-函数的理论（或数学预测），预测任何L-函数都应该产生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”。“这个项目的目的是在p-adic Langlands程序中开发新的结果和方法，并使用它们来建立新的互惠定律。该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约束互易律，该表示产生于数域上代数簇的etale上同调。这种互易性的描述是根据局部律，也就是将自守表示在素数k处的行为与伽罗瓦表示在同一素数处的行为联系起来的互易律。当k取为支配伽罗瓦表示的系数的素数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）的情形之外，它的存在性仍然是不确定的。与合作者，主要研究者的目的是调查这种结构的p-adic本地朗兰兹对应在各种方式。工作的一部分旨在构建p-adic伽罗瓦表示的模堆栈。这将允许引入新的几何方法到研究伽罗瓦理论方面的对应关系。该项目的另一部分将尝试在各种新的背景下构建p-adic局部朗兰兹对应。更确切地说，主要研究者和合作者以前的工作使用全局方法来构建一个候选的p-adic局部朗兰兹对应在某种程度上的一般性;这个项目的目的是建立这样构建的对应是真正的本地。

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.