Hankel Transform, Langlands Functoriality and Functional Equation of Automorphic L-Functions

在1860年的一本著名的回忆录中，Riemann研究了一个被称为Zeta函数的函数，该函数的性质意味着小于给定数字的素数的数目的近似公式。最优公式仍未得到证明，它取决于关于Zeta函数零点位置的黎曼假设。在他著名的假设提出之前，黎曼以函数方程的形式规定了Zeta函数的对称性。多亏了黎曼回忆录之后数论的发展，我们知道有一大类函数，其中包括Zeta函数；这些函数应该编码关于多项式方程系统的数字信息，例如它们的解的数目随着质数的变化而模为质数。这些函数应满足黎曼假设的一般形式以及函数方程。20世纪70年代初，朗兰兹提出了一个富有远见的计划，预测所有这些功能都可以从一个完全不同的角度来看待。这个项目将开发新的工具来帮助朗兰兹程序的开发。朗兰兹的自同构函数猜想蕴含着L函数的亚纯延拓和函数方程。除了内窥镜框架外，功能性猜想在很大程度上仍未得到探索。在21世纪初，朗兰兹(一边)和布雷弗曼-卡兹丹(另一边)制定了不同的一般策略来攻击超越内窥镜的功能猜想和L函数的函数方程。该项目旨在将这些建议的不同要素结合起来，制定一项新的、可能更现实的战略。重点将集中在对Braverman-Kazhdan提出的某些猜想Schwartz空间、Langland等人研究的轨道积分空间以及这些空间上的傅里叶变换和汉克尔变换的理解上。预计无限维弧空间的几何学将在这些研究中发挥重要作用。

In a celebrated memoir of 1860, Riemann studied a function known as the zeta function, whose properties imply an approximate formula for the number of prime numbers less than a given number. The optimal formula, still unproved, depends on the Riemann hypothesis on the location of zeros of the zeta function. Prior to the formulation of his celebrated hypothesis, Riemann stipulated a symmetry of the zeta function in the form of a functional equation. Thanks to the development of number theory following Riemann's memoir, we know that there is a large family of functions which includes the zeta function; these functions should encode numerical information about systems of polynomial equations, such as the number of their solutions modulo a prime number as the prime number varies. These functions should satisfy a generalized form of the Riemann hypothesis as well as a functional equation. In the early 1970s, Langlands proposed a visionary program predicting that all these functions can be viewed from a radically different perspective. This project will develop new tools that will aid in the development of Langlands' program.Langlands' automorphic functoriality conjecture implies both meromorphic continuation and functional equation of L-functions. The functoriality conjecture remains largely unexplored beyond the endoscopic framework. In the early 2000s, Langlands (on one side) and Braverman-Kazhdan (on the other), formulated different general strategies to attack the functoriality conjecture beyond endoscopy and the functional equation of L-functions. This project aims to combine different elements of those proposals in a formulation of a new, possibly more realistic, strategy. The focus will be on the understanding of certain conjectural Schwartz spaces proposed by Braverman-Kazhdan, spaces of orbital integrals studied by Langlands and others, and the Fourier and Hankel transforms on those spaces. It is expected that geometry of arc spaces, which are infinite dimensional, will play a major role in these studies.