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

9970342这个建议涉及约化群的表示理论。这些问题涉及的关系的某些基本问题（产生于朗兰兹计划的自守形式）有关的非零的分析不变量（称为自守L-函数）的存在函提升证明朗兰兹计划。这种现象的一个非常经典的例子是Waldspurger对应，它将半积分权形式的存在与经典L-函数在对称中心的非零性联系起来。我们考虑构造函子提升的各种方法。该方案提出了可供研究的替代方法。这些新方法中使用的具体思想包括：（1）构造称为Gelfand Graev模型的自守表示的新模型;（2）确定GL（N）与经典群直积的张量L-函数。本文研究了自守L-函数的某些特定值的计算公式和典型群的调和L-包的构造，这是数论领域的研究。数论的历史根源在于对自然数的研究。它是最古老的数学分支之一。在过去的半个世纪，它已成为一个不可或缺的工具，在不同的应用领域，如数据传输和处理，通信系统。

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.