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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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The Fontaine-Mazur conjecture for ${GL}_2$
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by Mark Kisin PDF
J. Amer. Math. Soc. 22 (2009), 641-690 Request permission

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
  • MR Author ID: 352758
  • Email: kisin@math.uchicago.edu
  • 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.
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 22 (2009), 641-690
  • MSC (2000): Primary 11F80
  • DOI: https://doi.org/10.1090/S0894-0347-09-00628-6
  • MathSciNet review: 2505297