Locally analytic representation theory and p-adic interpolation

本课题的目标是在自同构形式算法的两个基本问题上取得进展：p进hecke特征型解析族的构造和研究，以及p进l函数的构造和研究；通过运用群体表征理论的方法来实现。在过去的几十年里，数论的发展扩大了自同构形式的概念，包括了p进自同构形式，它可以插入到p进解析族中，以及相关的p进l函数，它们由此产生。这些对象在数论中发挥着越来越重要的作用，但它们很难研究，因为在经典自同构形式理论中起着至关重要作用的强大的表示理论方法不适用于它们。作者的工作表明，它们可以用最近引入的p进群的局部解析表示理论的方法从理论上研究表征。本课题将利用这些方法构造任意约化群的自同构Hecke特征型的p进解析族，并构造和研究附于这些自同构型上的p进l函数。该项目的一个组成部分将是开发这些结构背后的表征理论工具。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数学函数进行编码。自同构形式和l函数是两种以这种方式产生的函数，它们在数论中起着特别重要的作用。研究这些函数的传统方法是用户表示理论方法。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。他们最近的一种研究方法是使用p进方法，这种方法发挥着越来越重要的作用。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。它们的很大一部分用途来自于这样一个事实，即它们允许人们将自同构形式和相关的l -函数归为称为“p进族”的族，并同时研究这个族的成员。直到最近，表示理论方法和p进方法仍然泾渭分明。该项目的目标是使用所谓的“局部分析表征理论”（表征理论的一个新兴分支）的方法来统一这两种方法。作者将在这一理论中开发新的工具，并应用它们来构建和研究自同构形式和l函数的p进族的新例子，以及提高我们对那些已知存在的族的理解。

The goal of the proposed project is to make progress on twofundamental problems in the arithmetic of automorphic forms:the construction and study of p-adic analytic families ofHecke eigenforms, and the construction and study of p-adic L-functions;and to do so by employing the methods of group representation theory.Developments in number theory over the last few decades have ledto a broadening of the concept of automorphic form, to include notionssuch as p-adic automorphic forms, which can be interpolated inp-adic analytic families, and the related p-adic L-functions to whichthey give rise. These objects play an increasingly importantrole in number theory, but they can be difficult to study, becausethe powerful representation theoretic methods that plays such a crucialrole in the classical theory of automorphic forms do not apply to them.Work of the proposer shows that they can be studied representationtheoretically, however, by using methods from the recently introducedlocally analytic representation theory of p-adic groups. The proposedproject will employ these methods to construct p-adic analytic familiesof automorphic Hecke eigenforms for arbitrary reductive groups, and toconstruct and study p-adic L-functions attached to these automorphicforms. An integral part of the project will be the development ofthe representation-theoretic tools that underlie these constructions.Number theory is the branch of mathematics that studies phenomenarelated to properties of whole numbers. A typical number theoreticquestion is to determine the number of whole number solutions of someequation of interest. The answers to such questions can often beencoded in certain mathematical functions. Automorphic forms andL-functions are two kinds of functions that arise in this way,and that play a particularly important role in number theory.One traditional approach to studying these functions is to userepresentation theoretic methods. These are methods that exploitthe many symmetries of automorphic forms and L-functions to analysetheir mathematical properties. A more recent approach to their study,that is playing an ever more important role, is to use p-adic methods. These are methods that involve using divisibility properties with respectto some fixed prime number p to study the Taylor series coefficientsof the automorphic forms and L-functions. A large part of theirusefulness comes from that fact that they allow one to groupautomorphic forms and the related L-functions into familiescalled ``p-adic families'', and study the members of the familiessimultaneously. Until recently, the representation theoretic approachand the p-adic approaches have remained quite distinct. The goal of theproposed project is to unite the two approaches using methods of so-called``locally analytic representation theory'' (an emerging branch ofrepresentation theory). The proposer will develop new tools in this theory,and apply them to construct and study new examples of p-adic families ofautomorphic forms and L-functions, as well as to improve our understandingof those families that are already known to exist.