The Sato-Tate conjecture for Hilbert modular forms

Barnet-Lamb, Thomas; Gee, Toby; Geraghty, David

doi:10.1090/S0894-0347-2010-00689-3

The Sato-Tate conjecture for Hilbert modular forms
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by Thomas Barnet-Lamb, Toby Gee and David Geraghty

J. Amer. Math. Soc. 24 (2011), 411-469

DOI: https://doi.org/10.1090/S0894-0347-2010-00689-3

Published electronically: December 28, 2010

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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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