Functoriality for Relative Trace Formulas

朗兰兹函数性猜想，即“不同的算术鼓共享一些共同的本征频率”，在数论中有着巨大的应用，其中包括Ramanujan和其他人关于称为自同构型的特殊函数的系数大小的百年猜想。PI和他的合作者将这一猜想扩展到所谓的“相对”环境，包括研究L函数(也称为Zeta函数)的特定值的方法，例如在Gan，Gross和Prasad的突出的和更新的猜想中。到目前为止，证明函数性重要实例的主要工具是迹公式，但在目前的形式下，它几乎达到了极限。这个项目将利用量子化的思想来研究证明这些猜想的方法，量子化的思想起源于数学物理。这个想法将被用来构建比较(相对)轨迹公式的新方法，极大地扩大它们的潜在覆盖范围和适用性。该项目更广泛的影响包括对研究生的会议组织和指导。PI已经在先前的工作中表明，在一些低级别的情况下，人们可以通过一些新的相对迹公式之间的“转移算子”来建立相对功能。这种示踪公式的非标准比较是在朗兰兹的“超越内窥镜检查”建议中设想的；“相对”设置允许更多的灵活性和更多的潜在应用，以探索这种比较。以前的工作主要集中在与相对迹公式相关的L群具有一阶的情况。这个项目的主要目标将是研究如何将转移操作员的结构推广到更高的级别。其主要思想是将迹公式视为其余切堆叠的量子化，而余切堆叠又在很大程度上由L集团控制。利用这种余切堆栈之间的自然对应，该项目旨在构建它们的量化之间的转移运算符。在一个单独的轨道上，该项目将继续在PI最近与Ben-Zvi和Venkatesh的工作中猜想的哈密顿空间的对偶性方面的工作，目的是将这种对偶性扩展到超球面设置之外，并探索p-进群表示理论的应用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.