Factorisation de la cohomologie de syst\`emes locaux $p$-adiques sur le demi-plan de Drinfeld

TL;DR

This paper calculates the first cohomology of the symmetric algebra of a universal p-adic local system on Drinfeld's p-adic half-plane, revealing a factorized structure linked to the p-adic Langlands correspondence.

Contribution

It extends previous trivial coefficient results by computing cohomology with nontrivial coefficients using p-adic Langlands correspondence and automorphic multiplicities.

Findings

Cohomology group has a factorized form.

Results generalize previous trivial coefficient cases.

Uses Kisin rings and automorphic multiplicities.

Abstract

We compute the first cohomology group of the symmetric algebra of the universal \'etale $p$-adic local system on the tower of coverings of Drinfeld's $p$-adic half-plane. The result takes a factorized form, using the $p$-adic Langlands correspondence in families over Kisin rings. This work extends the corresponding results of Colmez, Dospinescu, and Niziol for trivial coefficients. It relies on the computation of automorphic multiplicities in the \'etale cohomology group of the local system, done in a previous paper, as well as on the determination of the Kisin rings for the special type as functions on an analytic open subset of the projective line.