On the automorphy of 2-dimensional potentially semistable deformation rings of Gℚp

Abstract

Using the $p$-adic local Langlands correspondence for ${\text{GL}}_2\left({\mathbb{Q}}_p\right)$, we prove that the support of the patched modules $M_{\infty }\left(\sigma \right)\left[1∕p\right]$ constructed by Caraiani et al. (Compos. Math. 154:3 (2018), 503–548) meets every irreducible component of the potentially semistable deformation ring $R_{\overline{r}}^{\square }(\sigma )[1∕p]$. This gives a new proof of the Breuil–Mézard conjecture for 2-dimensional representations of the absolute Galois group of ${\mathbb{Q}}_p$ when , which is new for $p=3$ and $\overline{r}$ 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 the Fontaine–Mazur conjecture by Kisin (J. Amer. Math. Soc. 22:3 (2009), 641–690) is removed.