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Stratified Kisin varieties and potentially Barsotti-Tate deformations. (Variétés de Kisin stratifiées et déformations potentiellement Barsotti-Tate.) (French. English summary) Zbl 1450.11050
Summary: Let \(F\) be a unramified finite extension of \(\mathbb{Q}_p\) and \(\overline{\rho}\) 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 \(\overline{\rho}\). 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 \(\overline{\rho}\).

MSC:
11F80 Galois representations
11S20 Galois theory
14L15 Group schemes
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