On the Theory of Automorphic Forms and Applications

江迪华的建议摘要(DMS-0400414)主要研究人员江迪华致力于研究现代自同构形和L函数理论中的一些基本问题。在整体理论中，他利用经典不变量理论的思想，研究了自同构形式的离散谱的结构，并用自同构形式的周期刻画了自同构L函数的中心值或剩余不为零的性质。在局域理论中，他的研究攻击了局部朗兰兹猜想和p-进群调和分析中的相关基本问题。他的长期目标是理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中的基本原理之一。自同构形式理论已经研究了很长时间，并植根于高斯和黎曼等人的工作。它很早就形成了分析和数论的基础，并发展成为解析数论、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。最近证明的关于椭圆曲线的Taniyama-Shimura-Weil猜想和Fermat最后一个定理就是典型的例子。根据局部紧拓扑群的表示理论和调和分析，在理解自同构形式的基础上发展了现代的自同构形式和L-函数理论。朗兰兹对数论或代数几何中的L函数与自同构型理论中的L函数之间的关系进行了系统的猜想描述。江的工作集中在朗兰兹的基本猜想和数字理论的应用。

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.