Toroidal Compactifications  of Integral Models of Shimura  Varieties of Hodge Type

Madapusi Pera, Keerthi

doi:10.24033/asens.2391

Geodesic

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Toroidal Compactifications of Integral Models of Shimura Varieties of Hodge Type
[Compactifications toroïdales des modèles entiers de variétés de Shimura de type de Hodge]

Madapusi Pera, Keerthi

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 2, pp. 393-514.

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Abstract (VO)
Résumé

We construct toroidal compactifications for integral models of Shimura varieties of Hodge type. We also construct integral models of the minimal (Satake-Baily-Borel) compactification. Our results essentially reduce the problem to understanding the integral models themselves. As such, they cover all previously known cases of PEL type. At primes where the level is hyperspecial, we show that our compactifications are canonical in a precise sense. We also provide a new proof of Y. Morita's conjecture on the everywhere good reduction of abelian varieties whose Mumford-Tate group is anisotropic modulo center. Along the way, we demonstrate an interesting rationality property of Hodge cycles on abelian varieties with respect to $p$ -adic analytic uniformizations.

Nous construisons des compactifications toroïdales pour les modèles entiers de variétés de Shimura de type de Hodge. Nous construisons également la compactification minimale (ou de Satake-Baily-Borel) pour ces modèles entiers. Nos résultats réduisent le problème à la compréhension des modèles entiers eux-mêmes. Donc ils recouvrent tous les cas déjá connus de type PEL. Quand le niveau est hyperspécial, nous montrons que nos compactifications sont canoniques dans un sens précis. Nous fournissons une nouvelle preuve de la conjecture de Y. Morita sur la bonne réduction de variétés abéliennes dont le groupe de Mumford-Tate est anisotrope modulo son centre. Sur le chemin, nous démontrons une propriété de rationalité intéressante de cycles de Hodge sur les variétés abéliennes par rapport aux uniformisations analytiques $p$ -adiques.

MR Zbl

DOI : 10.24033/asens.2391

Classification : 11G18, 14G35
Keywords: Variétés de Shimura, compactifications, variétés abéliennes, theorie de Dieudonné logarithmique.
Mots-clés : Shimura varieties, compactifications, abelian varieties, logarithmic Dieudonné theory.

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     title = {Toroidal {Compactifications}  of {Integral} {Models} of {Shimura}  {Varieties} of {Hodge} {Type}},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
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     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Madapusi Pera, Keerthi. Toroidal Compactifications  of Integral Models of Shimura  Varieties of Hodge Type. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 2, pp. 393-514. doi : 10.24033/asens.2391. http://geodesic.mathdoc.fr/articles/10.24033/asens.2391/

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