Vari\'et\'es de Kisin stratifi\'ees et d\'eformations potentiellement Barsotti-Tate

TL;DR

This paper explicitly computes the Kisin variety for certain potentially Barsotti-Tate deformation families of a Galois representation, revealing its structure as a union of products of projective lines and its stratification.

Contribution

It provides an explicit description of the Kisin variety as a union of products of P^1 and introduces a stratification to aid in understanding Barsotti-Tate deformation rings.

Findings

Kisin variety is a finite union of products of P^1

Explicit embedding of the Kisin variety into P^1^[F:Qp]

Stratification helps in understanding deformation rings

Abstract

Let F be a unramified finite extension of Qp and rhobar 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 rhobar. We prove that this variety is a finite union of products of P^1. Moreover, it appears as an explicit closed subvariety of P^1^[F:\Qp]. 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 rhobar.