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VARIÉTÉS DE KISIN STRATIFIÉES ET DÉFORMATIONS POTENTIELLEMENT BARSOTTI-TATE

Published online by Cambridge University Press:  08 September 2016

Xavier Caruso
Affiliation:
IRMAR, Université de Rennes 1, UMR 6625, Campus de Beaulieu, 35042 Rennes Cedex, France (xavier.caruso@normalesup.org)
Agnès David
Affiliation:
Laboratoire de Mathématiques de Besançon, UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France (Agnes.David@math.cnrs.fr)
Ariane Mézard
Affiliation:
Institut de Mathématiques de Jussieu Paris Rive-Gauche, UMR 7586, LabEx SMP Université Pierre et Marie Curie, 75005 Paris, France (ariane.mezard@upmc.fr)

Abstract

Let $F$ be a unramified finite extension of $\mathbb{Q}_{p}$ and $\overline{\unicode[STIX]{x1D70C}}$ be an irreducible mod $p$ two-dimensional representation of the absolute Galois group of $F$. The aim of this article is the explicit computation of the Kisin variety parameterizing the Breuil–Kisin modules associated to certain families of potentially Barsotti–Tate deformations of $\overline{\unicode[STIX]{x1D70C}}$. We prove that this variety is a finite union of products of $\mathbb{P}^{1}$. Moreover, it appears as an explicit closed connected subvariety of $(\mathbb{P}^{1})^{[F:\mathbb{Q}_{p}]}$. We define a stratification of the Kisin variety by locally closed subschemes and explain how the Kisin variety equipped with its stratification may help in determining the ring of Barsotti–Tate deformations of $\overline{\unicode[STIX]{x1D70C}}$.

Soient $F$ une extension finie non ramifiée de $\mathbb{Q}_{p}$ et $\overline{\unicode[STIX]{x1D70C}}$ une représentation modulo $p$ irréductible de dimension 2 du groupe de Galois absolu de $F$. L’objet de ce travail est la détermination de la variété de Kisin qui paramètre les modules de Breuil-Kisin associés à certaines familles de déformations potentiellement Barsotti-Tate de $\overline{\unicode[STIX]{x1D70C}}$. Nous démontrons que cette variété est une réunion finie de produits de $\mathbb{P}^{1}$ qui s’identifie à une sous-variété explicite connexe de $(\mathbb{P}^{1})^{[F:\mathbb{Q}_{p}]}$. Nous définissons une stratification de la variété de Kisin en sous-schémas localement fermés et expliquons enfin comment la variété de Kisin ainsi stratifiée peut aider à déterminer l’anneau des déformations potentiellement Barsotti-Tate de $\overline{\unicode[STIX]{x1D70C}}$.

Type
Research Article
Copyright
© Cambridge University Press 2016 

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VARIÉTÉS DE KISIN STRATIFIÉES ET DÉFORMATIONS POTENTIELLEMENT BARSOTTI-TATE
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