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 GL2(p), we prove that the support of the patched modules M(σ)[1p] constructed by Caraiani et al. (Compos. Math. 154:3 (2018), 503–548) meets every irreducible component of the potentially semistable deformation ring Rr̄(σ)[1p]. This gives a new proof of the Breuil–Mézard conjecture for 2-dimensional representations of the absolute Galois group of p when p > 2, which is new for p = 3 and r̄ 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
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