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

Abstract

Using $p$-adic local Langlands correspondence for ${\mathrm{GL}}_2({\mathbb{Q}}_2)$ 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-Mézard 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 Paškūnas' proof of Fontaine-Mazur conjecture is removed.

Final version