Local and Global Aspects of Automorphic L-functions

自同构 L 函数的局部和全局方面

• 批准号: 0968505
• 负责人: James Cogdell
• 金额: $ 18万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-15 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0968505&HistoricalAwards=false
• 关键词: Local; Global; Aspects; Automorphic; functions

The problem of Langlands' Functoriality is central in the theory of automorphic forms and representations. It is a ramification of Langlands' formulation of non-abelian class field theory, probably the most important problem in modern number theory, which can be tested in a self-contained manner within the context of the theory of automorphic representations. The successful approach to this problem taken by the proposer and his coworkers is via thetheory of L-functions of automorphic representations and the development of a Converse Theorem for these L-functions for the general linear group. These L-functions are analytic invariants that can be attached both arithmetic objects and analytic objects and are used to mediate between them; Langlands non-abelian class field theory is one such connection. Converse Theorems allow one to characterize the analytic side of this equation via the properties of these invariants. The problem of Functoriality comes from interpreting arithmetic phenomena on the analytic side in term of these L-function invariants. The main thrust of this proposal is to develop techniques that will allow for the extension of these efforts. The local projects are to extend the proposers previous work on Bessel functions and stability of local L-functions and related invariants and to develop techniques for computing local L-functions at ramified and infinite places. The global aspects of the project are to improve the Converse Theorem, which is the engine that drives the results on Functoriality.Besides applications to extending the proposers results on Functoriality, which is the main motivation, these results, local and global, should have applications to unveiling the arithmetic hidden in special values of L-functions.The projects in this proposal all fall under the broad rubric of analytic number theory. At its most basic level, number theory is interested in understanding the integers. Additively, the integers are quite simple, generated by 1, but from the point of view of multiplication and factoring they are quite complicated and mysterious. The multiplicative structure is generated by the prime numbers and a large swath of number theory is devoted to the study of prime numbers. This study is full of problems that are simple to state but with no apparent machinery with which to attack them. Over the ages a vast and subtle algebraic structure has been built around these problems -- this is algebraic number theory. But as with many problems, to bring in seemingly incongruous techniques from other areas can lead to new insights. One such ``incongruous'' area is analysis and the theory of group representations; this leads to the theory of automorphic forms, a type of analytic number theory. The connection between the two in its most basic guise is ``class field theory'' and is mediated by certain analytic invariants, called L-functions. Class field theory is a deep and hard problem and any light we can shed on this connection lets us bring the tools of analysis to bear on basic arithmetic problems. This proposal continues our investigations of these invariants, the L-functions, from both the algebraic and analytic points of view, in hopes of narrowing the gap between these two areas in the short term and impacting our understanding of class field theory in the long term.

朗兰兹函数问题是自同构形式与表示理论的核心问题。它是朗兰兹提出的非阿贝尔类场理论的一个分支，非阿贝尔类场理论可能是现代数论中最重要的问题，它可以在自同构表示理论的背景下以一种独立的方式进行检验。作者和他的同事成功地解决了这个问题，方法是利用自同构表示的L函数的理论，并发展了一般线性群上这些L函数的逆定理。这些L函数是解析不变量，既可以连接算术对象，也可以连接分析对象，并用于在它们之间进行调解；朗兰兹非阿贝尔类场理论就是这样一种联系。逆定理允许人们通过这些不变量的性质来刻画这个方程的解析面。函数性问题源于用这些L函数不变量从解析的角度解释算术现象。这项提议的主旨是开发能够扩大这些努力的技术。这些局部项目是为了推广前人在Bessel函数和局部L函数及其相关不变量的稳定性方面所做的工作，以及发展计算分支和无限区域上的局部L函数的技术。该项目的全局方面是改进逆定理，这是驱动函数论结果的引擎。除了应用于推广已有的关于函数性的结果，这是主要的动机，这些局部和全局的结果应该应用于揭示隐藏在L函数的特殊值中的算术。在其最基本的层面上，数论感兴趣的是理解整数。此外，整数非常简单，由1生成，但从乘法和因式分解的角度来看，它们相当复杂和神秘。乘法结构是由素数生成的，大量的数论都致力于素数的研究。这项研究充满了问题，这些问题很容易描述，但没有明显的机械来攻击它们。多年来，围绕这些问题建立了一个庞大而微妙的代数结构--这就是代数数论。但与许多问题一样，从其他领域引入看似不协调的技术可能会带来新的见解。一个这样的“不协调”领域是分析和群表示理论；这导致了自同构型理论，这是一种解析数论。这两者之间最基本的联系是“类场论”，它是由某些被称为L函数的解析不变量所调节的。类域理论是一个深刻而困难的问题，我们所能揭示的任何关于这一联系的光都能让我们把分析工具运用到基本的算术问题上。这一建议继续了我们从代数和解析的角度对这些不变量L函数的研究，希望在短期内缩小这两个领域之间的差距，并在长期内影响我们对类场理论的理解。