Geometric and Microlocal Study of Automorphic Periods

自守周期的几何和微局域研究

• 批准号: 2101700
• 负责人: Ioannis Sakellaridis
• 金额: $ 29万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101700&HistoricalAwards=false
• 关键词: Geometric; Microlocal; Study; Automorphic; Periods

In harmonic analysis, one represents functions on a space as a superposition of waves with varying frequencies. In number theory and the Langlands program, one is interested in functions on certain homogeneous spaces, where the "waves" are special eigenfunctions of Laplacians and Hecke operators, called automorphic forms. Some of the most mysterious and important invariants of the automorphic forms are the L-functions, a vast class of generalizations of the Riemann zeta function. A significant, but not well-understood, principle is that their superposition often represents a function that can be described independently, in terms of what are known as spherical varieties that give rise to a distribution called the period distribution. The amplitudes of the spectral decomposition of this distribution turn out to be special values of L-functions. The project will investigate conjectural connections between period distributions and L-functions using ideas of quantization (whose roots lie in mathematical physics). The PI also plans yearly meetings to train students and postdocs on the topics related to this proposal. According to the visionary program developed since the '60s by Abel Prize recipient Robert P. Langlands, L-functions should be understood as invariants of automorphic representations; those are the "eigenfrequencies" of "arithmetic manifolds", or else the representations of a (reductive) Lie group G, and of its algebra of Hecke operators, which appear as functions on a quotient L\G, where L is an arithmetic lattice. The precise incarnation of L-functions in this setting is by means of certain distributions called "periods", which the PI and others have studied and organized into a coherent theory in recent years. The present award aims to utilize ideas of symplectic geometry in the study of these periods. Among other goals, this project will study: (1) the duality between periods and L-functions as a duality between Hamiltonian spaces for the group and its dual group (building up on recent work with Ben-Zvi and Venkatesh); (2) the local spectrum of spherical varieties by combining the relative trace formula of Waldspurger with the geometry of the moment map studied by Knop; (3) the "transfer operators" of functoriality, in the spirit of Langlands' "beyond endoscopy", between the relative trace formulas of different groups and spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在谐波分析中，一个人表示在空间上的函数是具有不同频率的波的叠加。在数字理论和兰兰兹计划中，人们对某些均匀空间的功能感兴趣，其中“波”是拉普拉斯人和Hecke操作员的特殊特征，称为自动形态。汽车形式的一些最神秘，最重要的不变式是L功能，这是Riemann Zeta功能的大量概括。一个重要但不被理解的原则是，它们的叠加通常代表一个可以独立描述的功能，就所谓的球形品种而言，产生称为时期分布的分布。 该分布的光谱分解的幅度证明是L功能的特殊值。该项目将使用量化思想（其根源在数学物理学）中研究周期分布与L功能之间的猜想联系。 PI还计划年度会议，培训与该提案相关的主题的学生和博士后。根据Abel奖获得者Robert P. Langlands自60年代以来开发的有远见的计划，应将L功能理解为自动形式表示的不变式。这些是“算术流形”的“本本频”，或者是（还原性的）Lie G组的表示及其Hecke操作员代数的表示，它们在商L \ g上的函数表现为L，其中L是Arithmetic晶格。在这种情况下，L功能的确切化身是通过某些称为“时期”的分布，PI和其他人近年来研究并将其研究为连贯的理论。本奖项旨在在研究这些时期的研究中利用符号几何形状的思想。除其他目标外，该项目还将研究：（1）该小组的汉密尔顿空间与其双重组之间的双重性之间的二元性（在与Ben-Zvi和Venkatesh的最新工作基础上）； （2）通过将waldspurger的相对痕量公式与诺普研究的时刻图的几何形状相结合，将球形品种的局部频谱结合在一起； （3）在不同群体和空间的相对痕量公式之间，在兰兰兹的“超越内窥镜”的精神上，功能的“转移操作员”。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的审查标准通过评估来通过评估来支持的。