A torsion Jacquet-Langlands Transfer via K-theory

Lead Research Organisation: University of Sheffield
Department Name: Mathematics and Statistics

Abstract

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.
 
Description The theory of theta correspondence is major theme in representation theory and the theory of automorphic forms. A crucial discovery of this project has been that there is an intimate connection between local theta correspondence and the theory of C*-algebras. Using this connection, we were able to prove that local theta correspondence can be promoted from a set theoretic bijection to a categorical equivalence.

There are three papers that came of out this project up to now. More are on their way. In our first paper, we proved the above mentioned connection in the setting of the so called "equal-rank dual pairs". We also provide representation theoretic applications of this new connection in our paper. In our second paper, we prove this connection in the setting of the so-called ''stable range dual pairs". In a third paper, we have proven that the results on local theta correspondence of the first two papers can be put together to establish a similar connection between global theta correspondence and the theory of C*-algebras.

This was the first part of the project. In the second part, we are seeking to adapt this machinery to the setting of the Jacquet-Langlands functoriality in order to obtain an integral version at the level of K-theory. This part of the project is still on going.
Exploitation Route The result that we have proven so far have been of much interest the mathematicians working in representation theory and in operator algebra theory. I have received numerous conference and seminar invitations to talk about these findings.
Sectors Other
URL https://arxiv.org/abs/2207.13484