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

Nekovář, Jan

doi:10.24033/asens.2374

Eichler-Shimura relations and semisimplicity of étale cohomology of quaternionic Shimura varieties
[Les relations d'Eichler-Shimura et la semi-simplicité de la cohomologie étale des variétés de Shimura quaternioniques]

Nekovář, Jan

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

Résumé
Abstract

On montre que l'action galoisienne sur la partie sans multiplication complexe de la cohomologie étale d'un faisceau $ℓ$ -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 $ℓ$ -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 $ℓ$ -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 $ℓ$ -adic étale cohomology is semisimple. For Hilbert modular varieties, analogous results are proved for $ℓ$ -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.

Publié le : 2018-11-12

MR Zbl

DOI : 10.24033/asens.2374

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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     author = {Nekov\'a\v{r}, Jan},
     title = {Eichler-Shimura relations and semisimplicity of \'etale cohomology  of quaternionic {Shimura} varieties},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1179--1252},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
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     mrnumber = {3942040},
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     url = {http://www.numdam.org/articles/10.24033/asens.2374/}
}

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Nekovář, Jan. Eichler-Shimura relations and semisimplicity of étale cohomology  of quaternionic Shimura varieties. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 5, pp. 1179-1252. doi : 10.24033/asens.2374. http://www.numdam.org/articles/10.24033/asens.2374/

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