On the Theory of Automorphic Forms and Applications

Dihua Jiang的提案摘要（DMS-0400414）主要研究员Dihua Jiang致力于自守形式和L-函数的现代理论中的一些基本问题。在整体理论中，他研究了自守形式的离散谱的结构，并利用经典不变理论的思想，用自守形式的周期来刻画自守L-函数的中心值或残差的非零性。在局部理论中，他的研究攻击了p-adic群调和分析中的局部Langlands图及相关的基本问题。他的长期目标是理解自守形式理论中的一般局部-整体-自守原理，自守形式理论反映了算术和数论中的基本原理之一。自守形式理论已经研究了很长时间，并且植根于高斯和黎曼等人的工作。它很早就成为分析和数论的交汇点，并已发展成为解析数论、代数数论、丢番图问题、算术和代数几何以及最近无限维李代数和数学物理中不可或缺的工具。椭圆曲线上的Taniyama-Shimura-Weil猜想的最新证明和Fermat最后定理就是典型的例子。现代自守形和L-函数理论是在局部紧拓扑群的表示理论和调和分析对自守形的理解基础上发展起来的。朗兰兹系统地描述了数论或代数几何中的L-函数与自守形式理论中的L-函数之间的关系。姜文的工作主要集中在朗兰兹的基本结构和对数理论的应用上。

Abstract of Dihua Jiang's proposal (DMS-0400414)The Principal Investigator, Dihua Jiang, works on some basic problemsin the modern theory of automorphic forms and L-functions. In the globaltheory, he investigates the structures of the discrete spectrum ofautomorphic forms and characterizes the non-vanishing of the central valuesor the residues of automorphic L-functions in terms of periods of automorphicforms, using ideas from the classical invariant theory. In thelocal theory, his research attacks the local Langlands conjectures andrelated basic problems in harmonic analysis of p-adic groups. His long termgoal is to understand the general local-global-automorphic principles in thetheory of automorphic forms, which reflects one of the basic principles inarithmetic and number theory.The theory of automorphic forms has been studied for a long time and wasrooted in the work of Gauss and Riemann among others. It became early on ameeting ground for analysis and number theory, and has developed as anindispensable tool in analytic number theory, algebraic number theory,Diophantine problems, arithmetic and algebraic geometry, and recently ininfinite dimensional Lie algebras and mathematical physics. The recent proofof the Taniyama-Shimura-Weil conjecture on elliptic curves and the Fermat'slast theorem are typical examples. The modern theory of automorphic forms andL-functions was developed based on the understanding of automorphic formsin terms of the representation theory of and the harmonic analysis overlocally compact topological groups. Langlands gave a systematic conjecturaldescription of relations between L-functions in number theory or algebraicgeometry and those arising in the theory of automorphic forms. Jiang's workis concentrated on the basic conjectures of Langlands and applications tonumber thoery.