Cohomologie de de Rham du rev\^etement mod\'er\'e de l'espace de Drinfeld

TL;DR

This paper investigates the De Rham cohomology of the first Drinfeld cover, providing a local proof that connects supercuspidal representations with the local Jacquet-Langlands correspondence via comparison with Deligne-Lusztig varieties.

Contribution

It offers a new local proof linking supercuspidal parts of cohomology to the Jacquet-Langlands correspondence, using explicit descriptions of the cover and generalized excision techniques.

Findings

Supercuspidal parts realize the local Jacquet-Langlands correspondence.

Comparison with rigid cohomology of Deligne-Lusztig varieties.

Explicit cyclic cover description aids the proof.

Abstract

In this article, we study the De Rham cohomology of the first cover in the Drinfel'd tower. In particular, we get a purely local proof that the supercuspidal part realizes the local Jacquet-Langlands correspondence for ${\rm GL}_n$ by comparing it to the rigid cohomology of some Deligne-Lusztig varieties. The representations obtained are analogous to the ones appearing in the $\ell$-adic cohomology if we forget the action of the Weil group. The proof relies on the generalization of an excision result of Grosse-Kl\"onne and on the explicit description of the first cover as a cyclic cover obtained by the author on a previous work.