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arXiv · 1803.07451

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

Shen-Ning Tung
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Abstract

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-M\'ezard 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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Shen-Ning Tung. 2018-03-19. On the automorphy of 2-dimensional potentially semi-stable deformation rings of $G_{\mathbb{Q}_p}$. https://arxiv.org/abs/1803.07451

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