On Square Integrable Automorphic Forms and Related Problems

本文作者蒋迪华计划继续研究自同构形、L函数及其在数论和算术中的应用。通常，PI的目的是利用调和分析和群表示来研究自同构形的基本结构。特别地，PI攻击了现代自同构形理论中R.朗兰兹的基本猜想。对PI的研究有助于更好地理解J.Arthur猜想的平方可积自同构形式空间的离散谱、自同构L函数中心值的相关问题以及局部理论中的相应问题。在Shimura变元的算术和伽罗华群的基本问题上的应用已经获得或有望在不久的将来获得。宇宙中的基本物体可以根据它们的内在模式进行分类。这些内在模式的基本类型被称为对称类型，它可以通过群表示在数学上进行分类。调和分析提供了通过函数研究对称结构的数学方法。自同构函数是具有丰富对称结构的特殊函数，与几何和数论有着天然的联系。作为数学学科，自同构函数几个世纪以来一直是一个非常活跃的研究领域，并将在未来几个世纪继续如此。主要研究员蒋迪华建议研究自同构函数的基本对称结构及其在数论和算术中的重要问题上的应用。

Abstract for the proposal DMS-0653742 of JiangThe Principal Investigator, Dihua Jiang, plans to continue his research on automorphic forms, L-functions and applications to number theory and arithmetic. In general, the PI aims to investigate basic structures of automorphic forms by using harmonic analysis and groups representations. In particular, the PI attacks basic conjectures of R. Langlands in the modern theory of automorphic forms. The proposed research of the PI leads to better understanding the discrete spectrum of the space of square integrable automorphic forms as conjectured by J. Arthur, the related problems on the central values of automorphic L-functions and the corresponding problems in the local theory. Applications to arithmetic of Shimura varieties and to basic problems on Galois groups have been obtained or are expected to be obtained in near future.Basic objects in the universe may be classified by their intrinsic patterns. The basic types of these intrinsic patterns are called types of symmetry, which may be classified mathematically by group representations. Harmonic analysis provides mathematically methods to investigate the symmetric structures via functions. Automorphic functions are special functions with abundant symmetric structures and have natural connections to geometry and number theory. As mathematical subject, automorphic functions have been a very active research area for centuries and will continue to be so for centuries to come. The Principal Investigator, Dihua Jiang, proposes to investigate basic symmetric structures of automorphic functions and their applications to important problems in number theory and arithmetic.