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.
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.
People |
ORCID iD |
| Mehmet Haluk Sengun (Principal Investigator) |
| 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 theta correspondence and induction theory for representations of C*-algebras. Using this connection, we were able to prove that theta correspondence can be promoted from a set theoretic bijection to a categorical equivalence. There are now two papers on this: one is already on the arXiv and the other will be released within this week actually! These papers prove that our original speculation was correct. The second part of the project aims to find a global version of the above mentioned intimate connection that is now proven. This work is still under progress. |
| Exploitation Route | The result that we have just proven so far will be of interest the mathematicians working in representation theory and in operator algebra theory. |
| Sectors | Other |
| URL | https://arxiv.org/abs/2207.13484 |