A torsion Jacquet-Langlands Transfer via K-theory

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.