Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



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

Author: Elmar Grosse-Klönne
Journal: J. Algebraic Geom. 20 (2011), 151-198
Published electronically: January 25, 2010
MathSciNet review: 2729278
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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 $\Gamma \subset \operatorname {SL}_ {d+1}(K)$ 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.

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Elmar Grosse-Klönne
Affiliation: Humboldt-Universität zu Berlin, Institut für Mathematik, Rudower Chaussee 25, 12489 Berlin, Germany

Received by editor(s): August 31, 2008
Received by editor(s) in revised form: April 21, 2009
Published electronically: January 25, 2010
Communicated by: John Coates