The Fontaine-Mazur conjecture for 𝐺𝐿₂

Kisin, Mark

doi:10.1090/S0894-0347-09-00628-6

The Fontaine-Mazur conjecture for ${GL}_2$
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by Mark Kisin;

J. Amer. Math. Soc. 22 (2009), 641-690

DOI: https://doi.org/10.1090/S0894-0347-09-00628-6

Published electronically: January 21, 2009

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Abstract:

We prove new cases of the Fontaine-Mazur conjecture, that a $2$-dimensional $p$-adic representation $\rho$ of $G_{\mathbb {Q}, S}$ which is potentially semi-stable at $p$ with distinct Hodge-Tate weights arises from a twist of a modular eigenform of weight $k\geq 2$. Our approach is via the Breuil-Mézard conjecture, which we prove (many cases of) by combining a global argument with recent results of Colmez and Berger-Breuil on the $p$-adic local Langlands correspondence.

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