Mathematical Sciences: Arithmetic of Automorphic Forms

本文主要研究Dirichlet级数、L函数、自同构型及相关对象的积分表示。积分表示的核心是约化代数群上的Eisenstein级数或这种Eisenstein级数的一些合适的“碎片”。关于特定值、解析延拓和函数方程的性质的结果将从这种积分表示的存在而得到。这项研究将集中在编码重要数论信息的各种解析函数上。这项技术允许人们应用深入的分析结果来获得数论信息。一个基本的分析工具是将这些函数表示为更好地理解的函数的积分。这正是公安局打算做的事情，过去也成功做到了这一点。

This research will concentrate on an investigation of integral representations of Dirichlet series, L-functions, automorphic forms and related objects. The kernels of interest for the integral representations are Eisenstein series on reductive algebraic groups or some suitable "fragments" of such Eisenstein series. Results on the nature of special values, analytic continuations and functional equations would follow from the existence of such integral representations. This research will focus on various analytic functions that encode important number theoretic information. This technique allows one to apply the deep results of analysis to obtain number theoretic information. A basic analytic tool is to represent these functions as integrals of functions which are better understood. That is precisely what the P.I. intends to do and has done so successfully in the past.