The cohomology of local systems on \(p\)-adically uniformized varieties
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 \(SL_{d+1}(K)\).
scientific article; zbMATH DE number 126351
scientific article