#### Vol. 15, No. 9, 2021

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

### Shen-Ning Tung

Vol. 15 (2021), No. 9, 2173–2194
##### Abstract

Using the $p$-adic local Langlands correspondence for ${GL}_{2}\left({ℚ}_{p}\right)$, 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 ${R}_{\stackrel{̄}{r}}^{\square }\left(\sigma \right)\left[1∕p\right]$. This gives a new proof of the Breuil–Mézard conjecture for 2-dimensional representations of the absolute Galois group of ${ℚ}_{p}$ when $p>2$, which is new for $p=3$ and $\stackrel{̄}{r}$ 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.

##### Keywords
$p$-adic Langlands, Fontaine–Mazur, modularity lifting
Primary: 11F80
Secondary: 11F33
##### Milestones
Received: 9 November 2018
Revised: 24 January 2021
Accepted: 28 February 2021
Published: 23 December 2021
##### Authors
 Shen-Ning Tung Fakultät für Mathematik Universität Duisburg–Essen Essen, Germany Department of Mathematics University of British Columbia Vancouver, BC Canada