Un calcul d'anneaux de déformations potentiellement Barsotti--Tate

Xavier Caruso (1) , Agnès David (2) , Ariane Mézard (3)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Uni.lu - Université du Luxembourg
3 DMA - Département de Mathématiques et Applications - ENS Paris

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.

Mots clés: représentations galoisiennes p-adiques; variétés de Kisin; conjecture de Breuil-Mézard; espaces de déformations

Domaines: Théorie des nombres [math.NT]

Xavier Caruso, Agnès David, Ariane Mézard. Un calcul d'anneaux de déformations potentiellement Barsotti--Tate. Transactions of the American Mathematical Society, 2018, 370 (9), pp.6041-6096. ⟨10.1090/tran/6973⟩. ⟨hal-00944666⟩