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. |