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

Using \(p\)-adic local Langlands correspondence for \(\operatorname{GL}_2(\mathbb{Q}_p)\), 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 ne...

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Published in:arXiv.org 2021-03
Main Author: Shen-Ning, Tung
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Deformation
Dimensional stability
Number theory
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description Using \(p\)-adic local Langlands correspondence for \(\operatorname{GL}_2(\mathbb{Q}_p)\), 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.
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subjects Deformation
Dimensional stability
Number theory
title On the automorphy of 2-dimensional potentially semi-stable deformation rings of \(G_{\mathbb{Q}_p}\)
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