Potentially semi-stable deformation rings for discrete series extended types

Rozensztajn, Sandra

Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 179-211.

Nous définissons des anneaux de déformations pour les déformations potentiellement semi-stables ayant un type étendu de la série discrète fixé en dimension $2$. Dans le cas des représentations du groupe de Galois de ${\mathbb{Q}}_p$, nous prouvons un analogue de la conjecture de Breuil-Mézard pour ces anneaux. Nous donnons comme application de ceci des résultats sur l’existence de congruences modulo $p$ pour les formes nouvelles dans $S_k\left({\Gamma }_0\left(p\right)\right)$.

We define deformation rings for potentially semi-stable deformations of fixed discrete series extended type in dimension $2$. In the case of representations of the Galois group of ${\mathbb{Q}}_p$, we prove an analogue of the Breuil-Mézard conjecture for these rings. As an application, we give some results on the existence of congruences modulo $p$ for newforms in $S_k\left({\Gamma }_0\left(p\right)\right)$.

Classification : 11F80, 11F33
Keywords: Galois representations, deformation rings, Breuil-Mézard conjecture
Mot clés : Représentations galoisiennes, anneaux de déformations, conjecture de Breuil-Mézard

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