Problems in Automorphic Forms Related to Nonvanishing of L Functions at Specific Values

9970342本提案涉及约化群的表示理论。这些问题涉及某些基本问题(出现在自同构形式的朗兰兹程序中)，它们与解析不变量(称为自同构L函数)的不消失与朗兰兹程序猜想的函数式提升的存在有关。这种现象的一个非常经典的例子是Waldspurger对应，它将半整权形式的存在与经典L函数在对称中心处的不消失联系在一起。我们考虑了构造函数式提升的各种方法。该计划提出了可供选择的研究方法。这些新方法的具体思想包括：(1)构造称为Gelfand Graev模型的自同构表示的新模型；(2)确定GL(N)与经典群的直积的张量L函数。研究了典型群的自同构L函数的某些特定值的公式以及经典群的调和L包的构造。数论起源于对自然数的研究。它是最古老的数学分支之一。在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。

9970342This proposal deals with the representation theory of reductive groups. The problems concern the relation of certain basic questions (arising in the Langlands program of automorphic forms) relating the nonvanishing of analytic invariants ( called automorphic L-functions) to the existence of functorial liftings conjectured in the Langlands program. One very classical example of such phenomenon is the Waldspurger correspondence relating the existence of half integral weight forms to nonvanishing of classical L-functions at centers of symmetry. We consider various methods of constructing functorial liftings. The program sets forth alternative methods to be studied. The specific ideas used in these new methods include (1)the construction of new models for automorphic representations called Gelfand Graev models and (2) determination of tensor L-functions for the case of the direct product of GL(N) with a classical group. We study formulae for certain specific values of automorphic L-functions and the construction of tempered L packets for classical groups.This research is in the area of number theory. Number theory has its historical roots in the study of natural numbers. It is among the oldest branches of mathematics. Within the last half century it has become an indispensible tool in diverse applications in areas such as data transmission and processing, and communication systems.