Potentially semi-stable deformation rings for discrete series extended types

TL;DR

This paper introduces deformation rings for potentially semi-stable Galois representations with fixed discrete series types and proves an analogue of the Breuil-Mézard conjecture, impacting the understanding of modular form congruences.

Contribution

It defines new deformation rings for specific Galois representations and establishes a Breuil-Mézard type conjecture for these rings in the case of al representations.

Findings

Proof of the Breuil-Mézard conjecture analogue for these deformation rings

Results on congruences modulo p for newforms in S_k(b0(p))

Construction of deformation rings for potentially semi-stable representations

Abstract

We define deformation rings for potentially semi-stable deformations of fixed discrete series inertial type in dimension $2$. In the case of representations of the Galois group of $\mathbf{Q}_p$, we prove an analogue of the Breuil-M\'ezard conjecture for these rings. As an application, we give some results on the existence of congruences modulo $p$ for newforms in $S_k(\Gamma_0(p))$.