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

Tung, Shen-Ning

doi:10.2140/ant.2021.15.2173

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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

DOI: 10.2140/ant.2021.15.2173

Abstract

Using the $p$ -adic local Langlands correspondence for ${GL}_{2} (ℚ_{p})$ , we prove that the support of the patched modules $M_{\infty} (σ) [1 ∕ p]$ constructed by Caraiani et al. (Compos. Math. 154:3 (2018), 503–548) meets every irreducible component of the potentially semistable deformation ring $R_{\bar{r}}^{□} (σ) [1 ∕ p]$ . 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 $\bar{r}$ 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 the Fontaine–Mazur conjecture by Kisin (J. Amer. Math. Soc. 22:3 (2009), 641–690) is removed.

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Keywords

$p$-adic Langlands, Fontaine–Mazur, modularity lifting

Mathematical Subject Classification 2010

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

Vol. 15, No. 9, 2021