The cohomology of local systems on \(p\)-adically uniformized varieties
type: Item
instance of: scholarly article
title: The cohomology of local systems on \(p\)-adically uniformized varieties
zbMATH Open document ID: 0774.14022
DOI: 10.1007/BF01444738
author: Peter Schneider
published in: Mathematische Annalen
publication date: 1993-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/164979
review text: Let \(K\) be a finite extension of \(\mathbb{Q}_ p\) and let \(\Omega^{(d+1)}=\mathbb{P}^ d_ K-\) (union of all \(K\)-rational hyperplanes). Let \(\Gamma\subset PGL_{d+1}(K)\) be a cocompact discrete subgroup which has no fixed points in \(\Omega^{(d+1)}\). Then \(X_ \Gamma=\Omega^{(d+1)}/\Gamma\) is a proper smooth analytic variety over \(K\). Let \(M\) be a finite \(K[\Gamma]\)-module. It gives rise to a locally constant sheaf \({\mathcal M}\) on \(X_ \Gamma\). -- The author computes the De Rham cohomology groups \(H^*_{DR}(X_ \Gamma,{\mathcal M})\). The dimension of the groups as \(K\)-vector spaces is calculated. The author makes a conjecture about the Hodge-De Rham filtration on \(H^ d_{DR}(X_ \Gamma,{\mathcal M})\). This conjecture is proved for \(d\leq 2\). In the last section the author analyses the case where \(M\) comes from a \(K\)-rational representation of \(SL_{d+1}(K)\).
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14F40
zbMATH DE Number: 126351
zbMATH Keywords: rigid analytic spaces
zbMATH Keywords: de Rham cohomology groups
reviewed by: Guido Van Steen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 126351