Mathematical Sciences: Arithmetic of Automorphic Forms

本研究将集中于狄利克雷级数、l -函数、自同构形式及相关对象的积分表示的研究。积分表示的核心是约化代数群上的爱森斯坦级数或这种爱森斯坦级数的一些合适的“片段”。关于特殊值、解析延拓和泛函方程的性质的结果将从这种积分表示的存在中得到。本研究将著重于编码重要数论资讯的各种解析函数。这种技术允许人们应用分析的深层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容易理解的函数的积分。这正是私家侦探想要做的，而且在过去做得非常成功。

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.