Kisin modules with descent data and parahoric local models

Caraiani, Ana; Levin, Brandon

doi:10.24033/asens.2354

Kisin modules with descent data and parahoric local models
[Modules de Kisin avec données de descente et modèles parahoriques locaux]

Caraiani, Ana ; Levin, Brandon

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 1, pp. 181-213.

Résumé
Abstract

Nous construisons un espace de modules $Y^{μ, τ}$ de modules de Kisin avec donnée de descente modérée $τ$ et type de Hodge $p$ -adique $μ$ , pour une extension finie $K / ℚ_{p}$ . Nous démontrons une équivalence lisse entre $Y^{μ, τ}$ et le modèle local pour la restriction de scalaires ${Res}_{K / ℚ_{p}} {GL}_{n}$ , co-caractère ${μ}$ et structure de niveau parahorique. Cette équivalence est ensuite utilisée pour construire l'analogue de la stratification de Kottwitz-Rapoport sur la fibre spéciale de $Y^{μ, τ}$ , paramétrée par l'ensemble des éléments $μ$ -admissibles. Nous décrivons aussi la relation entre $Y^{μ, τ}$ et l'espace de déformations galoisiennes potentiellement cristallines.

We construct a moduli space $Y^{μ, τ}$ of Kisin modules with tame descent datum $τ$ and with $p$ -adic Hodge type $\leq μ$ , for some finite extension $K / ℚ_{p}$ . We show that this space is smoothly equivalent to the local model for ${Res}_{K / ℚ_{p}} {GL}_{n}$ , cocharacter ${μ}$ , and parahoric level structure. We use this to construct the analog of Kottwitz-Rapoport strata on the special fiber $Y^{μ, τ}$ indexed by the $μ$ -admissible set. We also relate $Y^{μ, τ}$ to potentially crystalline Galois deformation rings.

Publié le : 2018-11-12

MR Zbl

DOI : 10.24033/asens.2354

Classification : 11F80, 14M15, 14D20.
Keywords: Galois deformations, integral $p$-adic Hodge theory, Kisin modules, local models of Shimura varieties, affine flag varieties.
Mot clés : Déformations Galoisiennes, théorie de Hodge $p$-adique entière, modules de Kisin, modèles locaux des variétés de Shimura, variété de drapeaux affine.

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     author = {Caraiani, Ana and Levin, Brandon},
     title = {Kisin modules with descent data and parahoric local models},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {181--213},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {1},
     year = {2018},
     doi = {10.24033/asens.2354},
     mrnumber = {3764041},
     zbl = {1450.11049},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2354/}
}

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Caraiani, Ana; Levin, Brandon. Kisin modules with descent data and parahoric local models. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 1, pp. 181-213. doi : 10.24033/asens.2354. http://www.numdam.org/articles/10.24033/asens.2354/

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