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The Fontaine-Mazur conjecture for $ {GL}_2$

Author: Mark Kisin
Journal: J. Amer. Math. Soc. 22 (2009), 641-690
MSC (2000): Primary 11F80
Published electronically: January 21, 2009
MathSciNet review: 2505297
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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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Additional Information

Mark Kisin
Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637

Received by editor(s): June 25, 2007
Published electronically: January 21, 2009
Additional Notes: The author was partially supported by NSF grant DMS-0400666 and a Sloan Research Fellowship.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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