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

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 $\Gamma \subset \operatorname {SL}_ {d+1}(K)$ 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.