A Family of Calabi-Yau Varieties and Potential Automorphy II

We prove new potential modularity theorems for n-dimensional essentially self-dual l-adic representations of the absolute Galois group of a totally real field. Most notably, in the ordinary case we prove quite a general result. Our results suffice to show that all the symmetric powers of any non-CM, holomorphic, cuspidal; elliptic modular newform of weight greater than one are potentially cuspidal automorphic. This in turns proves the Sato-Tate conjecture for such forms. (In passing we also note that the Sato-Tate conjecture can now be proved for any elliptic curve over a totally real field.)