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

Author(s): Mark Kisin
Journal: J. Amer. Math. Soc. 22 (2009), 641-690.
MSC (2000): Primary 11F80
Posted: 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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Additional Information:

Mark Kisin
Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Email: kisin@math.uchicago.edu

DOI: 10.1090/S0894-0347-09-00628-6
PII: S 0894-0347(09)00628-6
Received by editor(s): June 25, 2007
Posted: January 21, 2009
Additional Notes: The author was partially supported by NSF grant DMS-0400666 and a Sloan Research Fellowship.
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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