The Sato-Tate conjecture for Hilbert modular forms

TL;DR

This paper proves the Sato-Tate conjecture for Hilbert modular forms over totally real fields, extending previous results and introducing new automorphy lifting techniques that do not rely on potential automorphy theorems for non-ordinary Galois representations.

Contribution

It provides a new proof of the Sato-Tate conjecture for modular forms using automorphy lifting over ramified fields, avoiding previous reliance on potential automorphy theorems.

Findings

Proves the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of GL(2) over totally real fields.

Develops automorphy lifting theorems over ramified fields with a topological argument.

Offers a new proof for the conjecture in the classical modular form case without potential automorphy theorems.

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