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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

The Sato-Tate conjecture for Hilbert modular forms


Authors: Thomas Barnet-Lamb, Toby Gee and David Geraghty
Journal: J. Amer. Math. Soc. 24 (2011), 411-469
MSC (2010): Primary 11F33
Published electronically: December 28, 2010
MathSciNet review: 2748398
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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

DOI: http://dx.doi.org/10.1090/S0894-0347-2010-00689-3
PII: S 0894-0347(2010)00689-3
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.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.