P-adic aspects of the Langlands program

拟议项目的目标是调查朗兰兹计划的一些p-adic方面。Langlands程序研究了几个主题之间的相互关系，包括约化群的表示理论（局部域和全局域的阶上）、自同构形式、全局伽罗瓦表示和l函数。每一个主题都有一个p进的方面：人们可以考虑作用于拓扑p进向量空间的p进群或阿德尔群，p进自同构形式（或它们的近亲，对称空间的算术商上具有p进系数的上同调类），伽罗瓦群的p进表示及其变形理论，以及p进l函数。拟议的调查将包括所有这些。本研究预计将得到一些特殊的结果：关于对称空间的算术商上的p基完全上同调的结构的新结果，以及关于模形式上的l函数的特殊值与相关伽罗瓦表示的算术性质的新结果。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。（数论学家将朗兰兹猜想的l函数和自同构形式之间的关系称为“互易律”。）朗兰兹发展了一系列强大的表征理论方法来研究他的猜想。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。近年来，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”。在p进朗兰兹规划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，并开发建立新的互易律的工具。

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 regarding the structure of p-adically completed cohomology on arithmetic quotients of symmetric spaces, and new results relating special values of L-functions attached to modular forms to the arithmetic properties of the associated Galois representations.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-adic methods 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, and to use them to establish new results about L-functions, and to develop tools for establishing new reciprocity laws.