Functoriality for Relative Trace Formulas

相对迹公式的函数性

• 批准号: 2401554
• 负责人: Ioannis Sakellaridis
• 金额: $ 31.2万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401554&HistoricalAwards=false
• 关键词: Functoriality; Relative; Trace; Formulas; Functoriality; Relative; Trace; Formulas

The Langlands functoriality conjecture, that "different arithmetic drums share some common eigenfrequencies," has immense applications in number theory, among others to the century-old conjectures due to Ramanujan and others about the size of coefficients of special functions called automorphic forms. The PI and his collaborators have broadened this conjecture to the so-called "relative" setting, which includes methods of studying special values of L-functions (also called zeta functions), such as in the prominent, and more recent, conjectures of Gan, Gross, and Prasad. The main tool for proving important instances of functoriality so far has been the trace formula, but in its current form it has nearly reached its limits. This project will examine ways to prove these conjectures by use of the idea of quantization, whose origins lie in mathematical physics. This idea will be used to construct novel ways of comparing (relative) trace formulas, drastically expanding their potential reach and applicability. The broader impacts of the project include conference organization and mentoring of graduate students.The PI has already shown, in prior work, that in some low-rank cases one can establish relative functoriality via some novel "transfer operators" between relative trace formulas. Such non-standard comparisons of trace formulas were envisioned in Langlands's "Beyond Endoscopy" proposal; the "relative" setting allows for more flexibility, and more potential applications, for the exploration of such comparisons. Prior work was focused mostly on the case when the L-groups associated to the relative trace formulas are of rank one. The main goal of this project will be to examine ways to generalize the construction of transfer operators to higher rank. The main idea is to view a trace formula as the quantization of its cotangent stack, which in turn is largely controlled by the L-group. Using natural correspondences between such cotangent stacks, the project aims to construct transfer operators between their quantizations. On a separate track, the project will continue work on the duality of Hamiltonian spaces conjectured in the PI's recent work with Ben-Zvi and Venkatesh, with the aim of extending this duality beyond the hyperspherical setting, and exploring applications for the representation theory of p-adic groups.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.

Langlands的功能性猜想，“不同的算术鼓共享一些常见的本本征”，在数字理论中具有巨大的应用，以及其他关于Ramanujan和其他人所引起的世纪猜想，以及其他关于特殊功能系数的特殊功能系数大小，称为自动形式。 PI和他的合作者将这种猜想扩大到了所谓的“相对”设置，其中包括研究L功能的特殊值（也称为Zeta功能）的方法，例如在突出的，最新的GAN，GROSS和PRASAD的猜想中。迄今为止，证明功能性重要性重要的实例的主要工具是痕量公式，但是以当前形式几乎达到了限制。该项目将通过使用量化概念来研究证明这些猜想的方法，其起源在于数学物理学。该想法将用于构建比较（相对）痕量公式的新型方法，从而大大扩展了其潜在的影响力和适用性。该项目的更广泛的影响包括会议组织和研究生的指导。PI在先前的工作中已经表明，在某些较低的案例中，可以通过一些相对痕量公式之间的一些新颖的“转移操作员”来建立相对功能。在兰兰兹的“超越内窥镜”提案中设想了这种非标准的痕量公式的比较； “相对”设置允许更大的灵活性和更多的潜在应用，以探索这种比较。先前的工作主要集中在与相对痕量公式相关的L组相关的情况下。该项目的主要目标是研究将转移操作员构建升级为更高排名的方法。主要思想是将痕量公式视为其cotangent堆栈的量化，而该堆栈反过来又由L组在很大程度上控制。该项目使用此类Cotangent堆栈之间的自然对应关系，旨在在其量化之间构建转移操作员。 On a separate track, the project will continue work on the duality of Hamiltonian spaces conjectured in the PI's recent work with Ben-Zvi and Venkatesh, with the aim of extending this duality beyond the hyperspherical setting, and exploring applications for the representation theory of p-adic groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader影响审查标准。