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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 Sato-Tate conjecture for Hilbert modular forms
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by Thomas Barnet-Lamb, Toby Gee and David Geraghty PDF
J. Amer. Math. Soc. 24 (2011), 411-469 Request permission

Abstract:

We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of $\operatorname {GL}_2(\mathbb {A}_F)$, $F$ a totally real field, which are not of CM type. The argument is based on the potential automorphy techniques developed by Taylor et al., but makes use of automorphy lifting theorems over ramified fields, together with a “topological” argument with local deformation rings. In particular, we give a new proof of the conjecture for modular forms, which does not make use of potential automorphy theorems for non-ordinary $n$-dimensional Galois representations.
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Additional Information
  • Thomas Barnet-Lamb
  • Affiliation: Department of Mathematics, Brandeis University, 415 South Street MS 050, Waltham, Massachusetts 02138
  • Email: tbl@brandeis.edu
  • Toby Gee
  • Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
  • Address at time of publication: Department of Mathematics, Northwestern University, 2033 Sheridan Road Evanston, Ilinois 60208-2730
  • Email: tgee@math.harvard.edu, gee@math.northwestern.edu
  • David Geraghty
  • Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
  • Address at time of publication: Princeton University and Institute for Advanced Study, Princeton, New Jersey 08540
  • Email: geraghty@math.harvard.edu, geraghty@math.ias.edu
  • Received by editor(s): December 17, 2009
  • Received by editor(s) in revised form: November 4, 2010
  • Published electronically: December 28, 2010
  • Additional Notes: The second author was partially supported by NSF grant DMS-0841491.
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 24 (2011), 411-469
  • MSC (2010): Primary 11F33
  • DOI: https://doi.org/10.1090/S0894-0347-2010-00689-3
  • MathSciNet review: 2748398