Geometric approach to the local Jacquet-Langlands correspondence

TL;DR

This paper introduces a geometric method using l-adic etale cohomology of the Drinfeld tower to establish the local Jacquet-Langlands correspondence for GL(n) over p-adic fields, avoiding global automorphic techniques.

Contribution

It provides a purely local, geometric proof of the Jacquet-Langlands correspondence for GL(n), especially when n is prime, without relying on global methods or detailed supercuspidal classification.

Findings

Constructs the correspondence via l-adic etale cohomology of the Drinfeld tower.

Proves the correspondence preserves irreducible representations when n is prime.

Offers a local proof without global automorphic or supercuspidal classification techniques.

Abstract

In this paper, we give a purely geometric approach to the local Jacquet-Langlands correspondence for GL(n) over a p-adic field, under the assumption that the invariant of the division algebra is 1/n. We use the l-adic etale cohomology of the Drinfeld tower to construct the correspondence at the level of the Grothendieck groups with rational coefficients. Moreover, assuming that n is prime, we prove that this correspondence preserves irreducible representations. This gives a purely local proof of the local Jacquet-Langlands correspondence in this case. We need neither a global automorphic technique nor detailed classification of supercuspidal representations of GL(n).