A torsion Jacquet-Langlands Transfer via K-theory

功能性和互惠原则是著名的朗兰兹项目的核心。简而言之，功能预测了不同代数群上自同构形式空间之间的映射。它是数论中一个非常强大的工具，应用于许多基本猜想，如Artin， Ramanujan， Selberg和Sato-Tate。Franke有一个著名的结论：算术流形的所有复上同调都可以用自同构形式来表示。因此，研究算术流形的复上同调，以及Hecke算子的作用，直接属于Langlands规划。朗兰兹纲领最近最令人兴奋的发展之一就是朗兰兹纲领的积分版本的出现它以算术流形的积分上同调为中心。虽然算术流形的积分上同调中的扭转类不在Franke的结果范围内，但Scholze的里程碑式的结果表明，它们应该在Langlands规划中发挥重要作用。整体语境中的功能是一个新兴的基础性课题。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迁移（可能是泛函性的最基本实例）的积分版本。该策略是建立在捕捉Howe的θ对应理论在卡斯帕罗夫的强大的kk理论的形式主义。

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.