Locally analytic representation theory and p-adic interpolation

该项目的目标是在自同构形式算术中的两个基本问题上取得进展：p-进位的Hecke特征形式解析族的构造和研究，以及p-进位的L函数的构造和研究；并利用群表示论的方法来做到这一点。在过去的几十年中，数论的发展导致了自同构形式的概念的扩展，包括可以在p-进位的解析族中内插的p-进位的自同构形式，以及由此产生的相关的p-进位的L函数。这些对象在数论中发挥着越来越重要的作用，但它们可能很难研究，因为在经典的自同构形理论中起着如此关键作用的强大的表示理论方法并不适用于它们。本课题将利用这些方法来构造任意约化群的自同构Hecke特征形式的p-进解析族，并构造和研究依附于这些自同构形的p-进L-函数。这个项目的一个组成部分将是这些结构基础上的表示理论工具的发展。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编码成某些数学函数。自同构型和L-函数是这样产生的两类函数，它们在数论中起着特别重要的作用。研究这两类函数的传统方法之一是使用表示论方法。这些方法利用了自同构型和L函数的许多对称性来分析它们的数学性质。在他们的研究中，一种更新的方法正在发挥着越来越重要的作用，那就是使用p-han方法。这些方法涉及到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。它们的很大一部分用处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函数分组到被称为“p-进数族”的族中，并同时研究这些族的成员。直到最近，表示理论方法和p-进方法仍然是截然不同的。拟议项目的目标是使用所谓的“局部解析表示论”(表示论的一个新兴分支)的方法将这两种方法结合起来。作者将在这一理论中发展新的工具，并应用它们来构造和研究自同构族和L函数族的新例子，以及加深我们对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.