The local Langlands correspondence via endoscopy, geometry, type theory, and their interplay

通过内窥镜检查、几何学、类型理论及其相互作用得出的当地朗兰兹对应

• 批准号: 1161489
• 负责人: Tasho Kaletha
• 金额: $ 17.66万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161489&HistoricalAwards=false
• 关键词: local; Langlands; correspondence; via; endoscopy

The aim of this project is to investigate the local Langlands correspondence for p-adic groups using the tools of type-theory, algebraic and rigid geometry, and spectral theory. The local Langands correspondence is a mysterious conjectural relationship between two kinds of symmetries -- a geometric one, offered by groups of matrices with coefficients in real, complex, or p-adic fields, and an arithmetic one, offered by the Galois group of the real or a p-adic field. The last decade has seen a tremendous progress in this area, but the theory is still largely conjectural. The PI and his collaborators will attempt to establish new cases of the conjectural correspondence through the use of type-theory, as well as to elucidate the connection between types and the cohomology of Rapoport-Zink spaces, in particular in the case of low-rank unitary groups. The already existing constructions will be investigated more deeply using the theory of endoscopy. Furthermore, the PI and his collaborators will study both the local and the global Langlands correspondences for unitary groups of arbitrary rank, using the trace-formula techniques developed by Arthur.The Langlands program brings together several areas of mathematics, including arithmetic, geometry, analysis, and symmetry. By establishing deep and surprising ties between them, it provides the means for answering very hard questions about the arithmetic of numbers, as exemplified by the recent proof of Fermat's Last Theorem. A sophisticated understanding of the arithmetic of numbers is in turn fundamental to many recent technological developments, including error-correcting codes, compression, encryption and secure communication. This project aims at deepening our understanding of the Langlands program by exploring connections between several approaches to it, as well as by exploiting techniques which have only recently become available.

这个项目的目的是利用类型论、代数和刚性几何以及谱理论的工具来研究p进群的局部朗兰兹对应。局部朗朗兹对应是两种对称之间的一种神秘的推测关系——一种是几何对称，由实数、复数或p进域中的系数矩阵群提供，另一种是算术对称，由实数或p进域的伽罗瓦群提供。在过去的十年里，这一领域取得了巨大的进步，但理论在很大程度上仍然是推测性的。PI和他的合作者将尝试通过使用类型论建立推测对应的新案例，并阐明类型与Rapoport-Zink空间上同调之间的联系，特别是在低秩酉群的情况下。已经存在的结构将使用内窥镜理论进行更深入的研究。此外，PI和他的合作者将使用Arthur开发的追踪公式技术，研究任意秩酉群的局部和全局朗兰兹对应。朗兰兹计划汇集了数学的几个领域，包括算术、几何、分析和对称。通过在它们之间建立深刻而令人惊讶的联系，它为回答关于数字算术的非常困难的问题提供了方法，最近对费马大定理的证明就是一个例子。对数字运算的复杂理解反过来又对许多最近的技术发展至关重要，包括纠错码、压缩、加密和安全通信。该项目旨在通过探索几种方法之间的联系，以及利用最近才可用的技术，加深我们对朗兰兹纲领的理解。