p-adic Aspects of the Langlands Program

拟议项目的目标是调查朗兰兹方案的一些P-ADY方面。朗兰兹计划研究几个学科之间的相互关系，包括约化群的表示理论(局部域上和全局域上的Adeles表示)、自同构形式、整体伽罗瓦表示和L函数。这些主题中的每一个都有一个p-进的方面：人们可以考虑作用在拓扑p-进向量空间上的p-进或adelic群、p-进自同构形式(或者它们的近亲，上同调类，在对称空间的算术商上具有p-进系数的上同调类)，Galois群的p-进表示及其形变理论，以及p-进L-函数。拟议的调查将涵盖所有这些内容。这项研究有望得到的一些特殊结果是：关于有理数的绝对伽罗瓦群的二维p-进表示的模性的新结果，以及几个依附于p-进模族的变量p-进L函数的构造。数论是数学的一个分支，研究与整数的性质有关的现象。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被编码成称为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式的数学函数产生。(数学家将朗兰兹猜想的L函数与自同构形之间的关系称为“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他的猜想。这些方法利用了自同构形和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-adic方法开始被统一为所谓的“p-进朗兰兹程序”。作者的目的是在p-adic朗兰兹程序中发展新的结果和方法，并利用它们建立关于L函数的新结果，特别是建立新的互易定律。

The goal of the proposed project is to investigate some of the p-adic aspects of the Langlands program. The Langlands program studies the interrelations between several subjects, including the representation theory of reductive groups (both over local fields and over the adeles of a global field), automorphic forms, global Galois representations, and L-functions. Each of these subjects has a p-adic aspect:one may consider p-adic or adelic groups acting on topological p-adic vector spaces, p-adic automorphic forms (or their close cousins, cohomology classes with p-adic coefficients on arithmetic quotients of symmetric spaces), p-adic representations of Galois groups and their deformation theory, and p-adic L-functions. The proposed investigation will encompass all these. Some of the particular results expected to follow from this investigation are: new results on the modularity of two-dimensional p-adic representations of the absolute Galois group of the rational numbers, and the construction of several variable p-adic L-functions attached to p-adic families of modular forms.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 Langlands conjectured relationship between L-functions and automorphic forms as a ``reciprocity law''.) Langlands developed an array of powerful representation theoretic methods to study his 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-adicmethods have begun to be unified into a so-called ``p-adic Langlands program''. The proposer aims to develop new results and methods in the p-adic Langlands program, to use them to establish new results about L-functions, and, in particular, to establish new reciprocity laws.