Eichler-Shimura relations and semisimplicity of étale cohomology of quaternionic Shimura varieties. (English. French summary) Zbl 1458.11100
Summary: We show that the non CM part of \(\ell\)-adic étale cohomology of any compact quaternionic Shimura variety with coefficients in any automorphic local system is a semisimple Galois representation. If the local system has weight \(k = (k_1,\ldots,k_d)\) with all \(k_i\) of the same parity, the full \(\ell\)-adic étale cohomology is semisimple. For Hilbert modular varieties, analogous results are proved for \(\ell\)-adic intersection cohomology of the Baily-Borel compactification. The proof combines a representation-theoretical criterion of semisimplicity with Eichler-Shimura relations for partial Frobenius morphisms.
MSC:
11G18 | Arithmetic aspects of modular and Shimura varieties |
11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
11F80 | Galois representations |
14F20 | Étale and other Grothendieck topologies and (co)homologies |