Potentially semi-stable deformation rings for discrete series extended types

Abstract

Abstract. 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 Qp, we prove an analogue of the BreuilMézard conjecture for these rings. As an application, we give some results on the existence of congruences modulo p for newforms in S k (Γ 0 (p)).

On the locus of $2$-dimensional crystalline representations with a given reduction modulo $p$

“…Then Kisin in [Kis08] defines deformation rings R ψ (k, τ, ρ) that are quotients of R ψ (ρ). We will use a refinement of these rings introduced in [Roz15], which are better for our purposes in view of Theorem 5.3.1. If the Galois type τ is an inertial type, we denote by R ψ (k, τ, ρ) the ring classifying potentially crystalline representations with Hodge-Tate weights (0, k−1), inertial type τ , determinant ψ with reduction isomorphic to ρ, as defined by Kisin in [Kis08].…”

“…If the Galois type τ is an inertial type, we denote by R ψ (k, τ, ρ) the ring classifying potentially crystalline representations with Hodge-Tate weights (0, k−1), inertial type τ , determinant ψ with reduction isomorphic to ρ, as defined by Kisin in [Kis08]. If the Galois type τ is a discrete series extended type, we denote by R ψ (k, τ, ρ) the complete local noetherian O E -algebra which is a quotient of R ψ (ρ), classifying potentially semi-stable representations with Hodge-Tate weights (0, k − 1), extended type τ , determinant ψ with reduction isomorphic to ρ defined in [Roz15,2.3.3].…”

“…The Breuil-Mézard conjecture gives us some information about these rings ([BM02], proved in [Kis09], [Paš15], [Paš16]; and [Roz15] for the cases of discrete series extended type):…”

A remark on non-integral p-adic slopes for modular forms

“…Then, R k,± r is defined as the maximal reduced quotient of R k,st r supported on the union of components in Spec(R k,st r ) that contain at least one semi-stable lift r of r such that the associated Weil-Deligne representation WD(r) has non-trivial monodromy and Frobenius acting by τ ± . See [63,Section 2.3.3].…”