Converse Theorems and Functoriality

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，它可能是现代数论中最重要的问题，可以在自同构表示理论的背景下以自包含的方式进行检验。这里考虑的方法是通过自同构表示的L函数理论和发展关于一般线性群的这些L函数的逆定理来解决这个问题。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。该提出者与Kim、Piatetski-Shapiro和Shaidi合作，在函数性问题上取得了很大的进展，并为用这些技术证明这些结果设定了范例。这项提议的主旨是继续这些努力。它包括改进逆定理的项目，推广L函数的某些技术成果和开发更广泛应用的新技术的项目，以及针对算术应用的项目。这项提案中的项目都属于解析数论的广义范畴。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。本文从代数和解析两个角度对L函数这一不变量进行研究，以期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。

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.