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
DOI: https://doi.org/10.1090/S1056-3911-10-00541-2
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}\sb {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
Email: gkloenne@math.hu-berlin.de

DOI: https://doi.org/10.1090/S1056-3911-10-00541-2
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

American Mathematical Society