logo Automorphic Galois representations and the Sato-Tate conjecture

I will review what is expected and what is known regarding the conjectural Langlands correspondence between automorphic representations of GL(n) of a number field F and n-dimensional representations of the absolute Galois group of F. 
The principal application will be to the potential automorphy of even-dimensional symmetric powers of the Tate module of an elliptic curve over Q with non-integral j-invariant. 
The Sato-Tate conjecture for such elliptic curves is an immediate corollary.
The latter result is proved in joint work with Taylor, Clozel, and Shepherd-Barron.