On the automorphy of 2-dimensional potentially semi-stable deformation rings of $G_{{\mathbb{Q}}_p}$

Abstract

Using $p$-adic local Langlands correspondence for ${\mathrm{GL}}_2({\mathbb{Q}}_p)$, we prove that the support of patched modules constructed by Caraiani, Emerton, Gee, Geraghty, Paskunas, and Shin meet every irreducible component of the potentially semistable deformation ring. This gives a new proof of the Breuil-Mézard conjecture for 2-dimensional representations of the absolute Galois group of ${\mathbb{Q}}_p$ when $p>2$, which is new in the case $p=3$ and $\bar{r}$ a twist of an extension of the trivial character by the mod p cyclotomic character. As a consequence, a local restriction in the proof of Fontaine-Mazur conjecture by Kisin is removed.

Final version, accepted for publication in 'Algebra & Number Theory'