Factorisation de la cohomologie de systèmes locaux $p$-adiques sur le demi-plan de Drinfeld

We compute the first cohomology group of the symmetric algebra of the universal étale $p$-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 étale 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.