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VARIÉTÉS DE KISIN STRATIFIÉES ET DÉFORMATIONS POTENTIELLEMENT BARSOTTI-TATE
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 17 / Issue 5 / November 2018
- Published online by Cambridge University Press:
- 08 September 2016, pp. 1019-1064
- Print publication:
- November 2018
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- Article
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- You have access
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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 
$\overline{\unicode[STIX]{x1D70C}}$.
