On the Theory of Automorphic Forms and Applications

蒋迪华论文摘要（DMS-0400414）主要研究现代自同构形式和l -函数理论中的一些基本问题。在整体理论中，他研究了自同构形式的离散谱结构，并利用经典不变理论的思想，从自同构形式的周期出发，刻画了自同构l函数的中心值或残数的不灭性。在局部理论方面，他的研究攻击了p进群调和分析中的局部朗兰兹猜想和相关的基本问题。他的长期目标是了解自同构形式理论中的一般局部-全局自同构原理，这反映了算术和数论的基本原理之一。自同构形式理论的研究由来已久，其根源在于高斯和黎曼等人的工作。它很早就成为分析和数论的聚集地，并已发展成为分析数论、代数数论、丢芬图问题、算术和代数几何，以及最近的无穷维李代数和数学物理中不可或缺的工具。最近椭圆曲线上谷山-志村-韦尔猜想的证明和费马大定理就是典型的例子。现代自同构形式和l -函数理论是在对自同构形式的理解的基础上，根据过局部紧拓扑群的表示理论和调和分析发展起来的。朗兰兹对数论或代数几何中的l -函数与自同构形式理论中的l -函数之间的关系作了系统的推测描述。他的工作集中在朗兰兹的基本猜想和数论的应用上。

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.