Computation of potentially Barsotti-Tate deformation rings. (Un calcul d’anneaux de déformations potentiellement Barsotti-Tate.) (French. English summary) Zbl 1442.11087
Summary: Let \( F\) be an unramified extension of \( \mathbb{Q}_{p}\). The first aim of this work is to develop a purely local method to compute the potentially Barsotti-Tate deformation rings with tame Galois type of level \( [F : \mathbb{Q}_{p}] \) 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 \( \mathbb{Q}_{p}\) and determine in this way almost all these deformation rings. In this particular case, we observe a close relationship between the structure of these deformation rings and the geometry of the associated Kisin variety. As a corollary and still assuming that \( F\) has degree \( 2\) over \( \mathbb{Q}_{p}\), we prove, except in two very particular cases, a conjecture of Kisin which predicts that intrinsic Galois multiplicities are all equal to 0 or \( 1\).
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