# On the modularity of 2-adic potentially semi-stable deformation rings

## TL;DR

This paper proves the Breuil-Mézard conjecture for certain 2-dimensional Galois representations at p=2 using p-adic Langlands correspondence and R=T theorems, revealing the modularity structure of deformation rings.

## Contribution

It introduces a new proof of the Breuil-Mézard conjecture for 2-adic potentially semi-stable deformation rings, especially for twists of extensions of trivial characters.

## Key findings

- Support of patched modules meets all irreducible components.
- Provides a new proof of the Breuil-Mézard conjecture at p=2.
- Removes a local restriction in the Fontaine-Mazur conjecture proof.

## Abstract

Using $p$-adic local Langlands correspondence for $\operatorname{GL}_2(\mathbb{Q}_2)$ and an ordinary $R = \mathbb{T}$ theorem, we prove that the support of patched modules for quaternionic forms meet every irreducible component of the potentially semi-stable deformation ring. This gives a new proof of the Breuil-M\'{e}zard conjecture for 2-dimensional representations of the absolute Galois group of $\mathbb{Q}_2$, which is new in the case $\overline{r}$ a twist of an extension of the trivial character by itself. As a consequence, a local restriction in Pa\v{s}k\=unas' proof of Fontaine-Mazur conjecture is removed.