A torsion Jacquet-Langlands Transfer via K-theory

通过 K 理论的扭转 Jacquet-Langlands 传递

• 批准号: EP/V049119/1
• 负责人: Mehmet Haluk Sengun
• 金额: $ 25.22万
• 依托单位: University of Sheffield
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FV049119%2F1
• 关键词: torsion; Jacquet; Langlands; Transfer; via

Principles of functoriality and reciprocity lie at the heart of the celebrated Langlands program. In a nutshell, functoriality predicts maps between spaces of automorphic forms on different algebraic groups. It is an extremely powerful tool in number theory with applications to numerous fundamental conjectures such those of Artin, Ramanujan, Selberg and Sato-Tate. A well-known result of Franke says that all of the complex cohomology of an arithmetic manifold can be accounted for by automorphic forms. Therefore studying the complex cohomology of arithmetic manifolds, together with the action of Hecke operators, falls directly within the Langlands program. Perhaps one of the most exciting recent developments in the Langlands program has been the emergence of an integral version of the Langlands program which is centered around the integral cohomology of arithmetic manifolds. While torsion classes in the integral cohomology of arithmetic manifolds are outside the scope of Franke's result, the landmark result of Scholze has shown that they should play an important role in the Langlands program. Functoriality in the integral context is a burgeoning and fundamental topic.In this project, we aim to establish an integral version of the Jacquet-Langlands transfer (perhaps the most fundamental instance of functoriality) using ideas and tools of operator K-theory and C*-algebras. The strategy is built on capturing the theta correspondence theory of Howe in the formalism of Kasparov's powerful KK-theory.

功能性和互惠性原则是著名的朗兰兹纲领的核心。简而言之，函性预测了不同代数群上自守形式空间之间的映射。它是数论中一个非常强大的工具，可以应用于许多基本理论，如Artin，Ramanujan，Selberg和Sato-Tate。Franke的一个著名的结果说，算术流形的所有复上同调都可以用自守形式来解释。因此，研究算术流形的复上同调，以及Hecke算子的作用，福尔斯直接属于朗兰兹纲领。也许最近最令人兴奋的发展之一，在朗兰兹计划已经出现了一个积分版本的朗兰兹计划是围绕积分上同调的算术流形。虽然扭类的积分上同调的算术流形的范围外的弗兰卡的结果，具有里程碑意义的结果Scholze表明，他们应该发挥重要作用的朗兰兹计划。积分背景下的函子性是一个新兴的基本课题。在这个项目中，我们的目标是利用算子K-理论和C*-代数的思想和工具建立Jacquet-Langlands转移的积分版本（也许是函子性的最基本的实例）。该策略是建立在捕获的theta对应理论豪在形式主义的卡斯帕罗夫的强大KK理论。

项目成果

Equal rank local theta correspondence as a strong Morita equivalence

等阶局部 theta 对应作为强 Morita 等价

• DOI: 10.48550/arxiv.2207.13484
• 发表时间: 2022
• 作者: Mesland B