The Sato-Tate conjecture for Hilbert modular forms

Details The Sato-Tate conjecture for Hilbert modular forms The Sato-Tate conjecture for Hilbert modular forms

2009-12-05

arxiv.org/abs/0912.1054v2

Mathematics

Number Theory

59 pages. Essentially final version, to appear in Journal of the AMS. This version does not incorporate any minor changes (e.g

Scientific paper

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 $\GL_2(\A_F)$, $F$ 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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