Un calcul d'anneaux de d\'eformations potentiellement Barsotti--Tate

TL;DR

This paper develops a local method to compute deformation rings of potentially Barsotti-Tate Galois representations over unramified extensions of Qp, and applies it to degree 2 cases, confirming a conjecture of Kisin in most instances.

Contribution

It introduces a new local approach for calculating deformation rings with tame Galois type and verifies Kisin's conjecture for degree 2 unramified extensions.

Findings

Computed almost all deformation rings in degree 2 cases

Established a link between deformation ring structure and Kisin variety geometry

Proved Kisin's conjecture in most degree 2 cases

Abstract

Let F be an unramified extension of Qp. The first aim of this work is to develop a purely local method to compute the potentially Barsotti-Tate deformations rings with tame Galois type of irreducible two-dimensional representations of the absolute Galois group of F. We then apply our method in the particular case where F has degree 2 over Q_p and determine this way almost all these deformations rings. In this particular case, we observe a close relationship between the structure of these deformations rings and the geometry of the associated Kisin variety. As a corollary and still assuming that F has degree 2 over Q_p, we prove, except in two very particular cases, a conjecture of Kisin which predicts that intrinsic Galois multiplicities are all equal to 0 or 1.