Automorphic L-Functions, Endoscopy, and Representation Theory

自同构 L 函数、内窥镜检查和表示理论

• 批准号: 9970156
• 负责人: Freydoon Shahidi
• 金额: $ 8.3万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2002-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970156&HistoricalAwards=false
• 关键词: Automorphic; Functions; Endoscopy; Representation; Theory

Professor Shahidi will continue his study of the holomorphy of symmetric cube L-functions for non-monomial cusp forms on the general linear group on dimension two vector spaces. This is the first step in establishing the existence of the symmetric cube lift to the general linear group on dimension four spaces. The investigator will also consider the questions of poles of all L-functions that are obtained from a method previously established in conjunction with R. Langlands. Also under consideration is the boundedness or finiteness of order on finite strips of theses functions. In a related project Professor Shahidi will attempt to increase the applications for Arthur's trace formula to the most general cases by establishing new identities satisfied by normalized intertwining operators. This project involves questions in number theory and representation theory. Historically there have been many examples in mathematics where important information about the fundamental properties of numbers is encoded in the functions of calculus. An important advance in number theory over the past twenty years has been the increased understanding that these examples were part of a very general theory. The full general theory is still quite a mathematical mystery. It is so powerful, however, that each time a step toward the general theory is made, the repercussions are felt in all parts of number theory and representation theory. L-functions are the functions that often encode the information that number theorists need, and the information appears at points called the poles of the function. In this project, Professor Shahidi will investigate several methods of extracting this information from the function.

Shahidi 教授将继续研究二维向量空间上一般线性群上非单项尖点形式的对称立方 L 函数的全纯性。这是在四维空间上建立一般线性群的对称立方体升力存在性的第一步。 研究人员还将考虑从先前与 R. Langlands 联合建立的方法获得的所有 L 函数的极点问题。 还要考虑这些函数的有限条上的阶数的有界性或有限性。 在一个相关项目中，Shahidi 教授将尝试通过建立规范化交织算子满足的新恒等式，将 Arthur 迹公式的应用范围扩大到最一般的情况。该项目涉及数论和表示论的问题。 历史上有许多数学例子，其中有关数字基本属性的重要信息被编码在微积分函数中。 过去二十年来数论的一个重要进步是人们越来越了解这些例子是非常普遍的理论的一部分。 完整的一般理论仍然是一个数学谜团。然而，它是如此强大，以至于每次向一般理论迈出一步，数论和表示论的所有部分都会受到影响。 L 函数通常对数论学家所需的信息进行编码，这些信息出现在称为函数极点的点处。在这个项目中，Shahidi 教授将研究从函数中提取此信息的几种方法。