Converse Theorems and Functoriality

朗兰兹的功能性问题是自守形式和表示理论的核心。它是朗兰兹非交换类场论公式的衍生物，可能是现代数论中最重要的问题，这可以在自守表示理论的范围内以独立的方式进行测试。这里考虑的解决这个问题的方法是通过自守表示的L-函数理论和这些L-函数的匡威定理的发展。一般线性群的函数。这些L-函数是解析的不变量，可以附加在算术对象和解析对象上，并用来在它们之间进行调解;朗兰兹非交换类场论就是这样一种联系。匡威定理允许人们通过这些不变量的性质来表征这个方程的解析侧。功能性问题来自于根据这些L-函数不变量在分析方面解释算术现象。在功能性问题上，提出者与Kim，Piatetski-Shapiro和Shahidi合作取得了很大进展，并为用这些技术证明这些结果设定了范式。这项建议的主旨是继续这些努力。它包括改进匡威定理的项目，扩展L-函数的某些技术成果和开发适用范围更广的新技术的项目，以及旨在应用于算术的项目。这个建议中的项目都属于解析数论的大标题。在最基本的层面上，数论对理解整数感兴趣。加法，整数很简单，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。乘法结构是由素数产生的，大量的数论研究都致力于素数的研究。这项研究充满了问题，这些问题很容易陈述，但没有明显的机制来解决它们。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以带来新的见解。一个这样的“不协调”领域是分析和群表示理论;这导致了自守形式理论，一种解析数论。两者之间的联系在其最基本的伪装是“类域理论”，并通过某些分析不变量，称为L-函数。类域理论是一个深奥而又困难的问题，我们在这一联系上所能得到的任何启示，都使我们能够把分析的工具运用到基本的算术问题上。这个建议调查这些不变量，L-函数，从代数和分析的角度来看，希望在短期内缩小这两个领域之间的差距，并影响我们的理解类场理论的长期。

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 approach to this problem considered here is via the theory 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. Much progress on the problem of Functoriality has been made by the proposer in collaboration with Kim, Piatetski-Shapiro, and Shahidi and has set the paradigm for proving such results with these techniques. The main thrust of this proposal is to continue these efforts. It includes projects toimprove the Converse Theorem, projects to extend certain technical results on L-functions and develop new techniques that are more widely applicable, and finally projects aimed towards applications to arithmetic. 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 investigates 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.