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Let K be a finite extension of Qp and let X be Drinfel’d’s symmetric space of dimension d over K. Let Γ ⊂ SLd+1(K) be a cocompact discrete (torsionfree) subgroup and let XΓ = Γ\X , a smooth projective K-variety. In this paper we investigate the de Rham and log crystalline (log convergent) cohomology of local systems on XΓ arising from K[Γ]-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 (φ,N)-module) and the degeneration of the relevant Hodge spectral sequence.

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