L'espace de Drinfeld et correspondance de Langlands locale II

TL;DR

This paper investigates the geometry and cohomology of tamely ramified covers of Drinfeld's p-adic symmetric space, proving key conjectures about their $\, ext{l}$-adic cohomology structure and torsion properties.

Contribution

It provides a local proof of Harris's conjecture on the form of $\, ext{l}$-adic cohomologies and establishes torsion freeness for integral cohomology in this setting.

Findings

Proved Harris's conjecture on $\, ext{l}$-adic cohomology structure.

Established torsion freeness of integral cohomology.

Derived new results on coefficient systems over Bruhat-Tits buildings.

Abstract

We study the geometry and the cohomology of the tamely ramified cover of Drinfeld's $p$-adic symmetric space. For this tame level, we prove, in a purely local way, most of a conjecture of Harris on the form of the $\ell$-adic cohomologies, as well as the torsion freeness of the integral cohomology. During the study, we obtain some results of independent interest on the coefficient systems over the Bruhat-Tits building associated to $GL_d(K).$