p-adic aspects of the Langlands program

拟议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，它将某种类型的自同构表示（即那些朗兰兹参数在阿基米德位置为“积分”或“代数”的自同构表示）与由数域上的代数变异产生的伽罗瓦表示（即“动机”的伽罗瓦表示）联系起来。伽罗瓦表示理论中的一个基本事实是，这种表示被某些p进解析空间（伽罗瓦变形空间）自然地参数化，其中动机伽罗瓦表示漂浮在更大的非动机表示的海洋中。研究整个伽罗瓦表示的海洋是一个有趣而重要的问题，即使一个人最终只对这个空间的动机点感兴趣，它也被证明是至关重要的，因为大多数成功的建立互惠的方法都涉及到在所有伽罗瓦表示的海洋中p- ady插值信息。对于Q的绝对伽罗瓦群的二维表示，通过建立模曲线的p进完全上同调与Breuil， Colmez和Paskunas的p进局部Langlands对应的局部-全局相容结果，证明了在大多数情况下期望的互易性。提议的项目涉及将这些结果推广到其他环境，例如完全实域的/绝对伽罗瓦群的表示。由于在这种情况下没有p进局部朗兰兹对应的构造，因此项目的主要部分将涉及构造这样的对应。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函数与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”。本论文旨在发展p进朗兰兹规划的新结果和新方法，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律。

The goal of the proposed 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 of a certain type (namely those whose Langlands parameter at the archimedean place is ``integral'', or ``algebraic'') to Galois representations arising from the etale cohomology of algebraic varieties over number fields (i.e. Galois representations that are ``motivic''). A basic fact in the theory of Galois representations is that such representations are naturally parametrized by certain p-adic analytic spaces (Galois deformation spaces), in which the motivic Galois representations float in a much larger sea of non-motivic representations. Studying this entire sea of Galois representations is an interesting and important problem, which turns out to be of fundamental importance even if one is ultimately interested only in the motivic points of this space, since most successful approaches to establishing reciprocity have involved p-adically interpolating information across the sea of all Galois representations. In the case of two-dimensional representations of the absolute Galois group of Q, the proposer has proved the desired reciprocity in most cases, by establishing a local-global compatibility result relating the p-adically completed cohomology of modular curves to the p-adic local Langlands correspondence of Breuil, Colmez, and Paskunas. The proposed project involves generalizing these results to other contexts, such as to representations of the/absolute Galois group of a totally real field. Since there isno construction of a p-adic local Langlands correspondence in this case, a major part of the project will involve constructing such a correspondence.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, in particular, to establish new reciprocity laws.