On the $p$-adic cohomology of some $p$-adically uniformized varieties

TL;DR

This paper studies the p-adic cohomology of certain p-adically uniformized varieties, proving the monodromy weight conjecture and establishing admissibility and spectral sequence degeneration for their cohomology.

Contribution

It proves the monodromy weight conjecture for these varieties and describes the monodromy operator's iterates, also showing admissibility and spectral sequence degeneration in key cases.

Findings

Proved the monodromy weight conjecture for p-adically uniformized varieties.

Provided a rigid analytic description of the monodromy operator.

Established admissibility and degeneration of the Hodge spectral sequence.

Abstract

Let $K$ be a finite extension of ${\mathbb Q}_p$ and let $X$ be Drinfel'd's symmetric space of dimension $d$ over $K$. Let $\Gamma\subset {\rm SL}_{d+1}(K)$ be a cocompact discrete (torsionfree) subgroup and let ${{X}}_{\Gamma}=\Gamma\backslash {X}$, a smooth projective ${{K}}$-variety. In this paper we investigate the de Rham and log crystalline (log convergent) cohomology of local systems on $X_{\Gamma}$ arising from $K[\Gamma]$-modules. (I) We prove the monodromy weight conjecture in this context. To do so we work out, for a general strictly semistable proper scheme of pure relative dimension $d$ over a cdvr of mixed characteristic, a rigid analytic description of the $d$-fold iterate of the monodromy operator acting on de Rham cohomology. (II) In cases of arithmetical interest we prove the (weak) admissibility of this cohomology (as a filtered $(\phi,N)$-module) and the degeneration of the relevant Hodge spectral sequence.