Factorisation de la cohomologie de systèmes locaux $p$-adiques sur le demi-plan de Drinfeld
Arnaud Vanhaecke
TL;DR
The paper computes the first étale cohomology $H^1_{ ext{ét}}$ of the symmetric algebra of the universal $p$-adic local system on the Drinfeld $p$-adic half-plane tower, achieving a factorized description via the $p$-adic Langlands correspondence in families over Kisin rings. The approach extends prior results for trivial coefficients by incorporating all potentially crystalline deformations with positive Hodge–Tate weights, using Langlands in families and deformation rings $R_{ ext{B}}^{ ext{ps},oldsymbol{ heta}}$ together with Azumaya-structure of endomorphism algebras. The main achievement is a detailed, blockwise factorization of $H^1_{ ext{ét}}$ as a direct sum over Hodge–Tate weights $oldsymbol{ heta}$ and Weil–Deligne types $M$, involving universal Galois deformations, the Langlands correspondence in families, and the Jacquet–Langlands correspondences. This provides a concrete link between automorphic multiplicities, deformation theory (Kisin rings), and the arithmetic of Drinfeld's half-plane, yielding an explicit description with potential applications in $p$-adic analytic geometry and the spectral theory of $p$-adic representations. The results illuminate the structure of the cohomology as a rich arithmetic object encoding both Galois and automorphic information, with explicit factorization into principal components governed by Kisin-type deformation spaces.
Abstract
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.