The Langlands Conjectures for Connected and Disconnected Groups

本项目研究朗兰兹程序的各个方面--数学、桥数理论和几何学的一个领域，系统地研究代数方程的解的内在对称性。通过将这些对称性与几何联系起来，该程序创建了一种获取算术信息的强大方法。该项目特别关注明确获取此类信息。深入而明确的数字算术知识反过来对于我们今天认为理所当然的许多过去几十年的技术进步是必不可少的，包括加密和数字安全。更详细地，该项目将为分裂在驯服扩张和所有超尖参数上的一般连通约化p-ady群建立精化局部朗兰兹对应的统一和显式构造，其剩余特征为Weyl群的阶素数。该项目还旨在将这种构造的结果与其他更一般但不太明确的构造进行比较，例如针对局部函数域的Genstier-Lforgue型构造。这一奖项反映了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。

This project investigates facets of the Langlands program -- an area of Mathematics bridging number theory and geometry that examines in a systematic way the symmetries inherent in solutions of algebraic equations. By relating these symmetries to geometry, the program creates a powerful way of obtaining arithmetic information. The project is particularly concerned with obtaining such information explicitly. Deep and explicit knowledge of the arithmetic of numbers is in turn essential for many technological advances over the last decades that we take for granted today, including encryption and digital security.In more detail, the project will establish a uniform and explicit construction of the refined local Langlands correspondence, for general connected reductive p-adic groups splitting over a tame extension and all supercuspidal parameters, in residual characteristic prime to the order of the Weyl group. The project further aims to compare the result of this construction with other more general but less explicit constructions, such as the one of Genestier-Lafforgue for local function fields. The role of disconnected groups will also be considered.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.