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.)