Eichler-Shimura relations and semisimplicity of étale cohomology of quaternionic Shimura varieties

Nekovář, Jan

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 5, pp. 1179-1252.

On montre que l'action galoisienne sur la partie sans multiplication complexe de la cohomologie étale d'un faisceau $\ell$-adique lisse automorphe sur une variété de Shimura quaternionique compacte est semi-simple. Si le poids du faisceau s'écrit $k=(k_1,...,k_d)$, où les $k_i$ ont la même parité, toute la cohomologie étale est semi-simple. Les mêmes résultats sont montrés pour la cohomologie d'intersection $\ell$-adique de la compactification de Baily-Borel des variétés modulaires de Hilbert. La preuve utilise un critère abstrait de semi-simplicité et les relations d'Eichler-Shimura pour les morphismes de Frobenius partiels.

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 $k=(k_1,...,k_d)$ 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.

Classification : 11G18, 11F41, 11F80, 14F20.
Keywords: $l$--adic cohomology, Shimura varieties, Galois representations, semisimplicity.
Mot clés : Cohomologie $l$--adique, variétés de Shimura, représentations galoisiennes, semi-simplicité.

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