Galois representations,automorphic forms,and the Sato-Tate Conjecture

The present text consists of notes of several lectures on the proof of the Sato-Tate Conjecture given up through 2008. The goal of the lectures was to explain the statement and the main ideas of the proof. The notes are somewhat dated; shortly after they were written, the author, together with Bernet-Lamb, Geraghty, and Taylor, were able to prove the analogue of the Sato-Tate conjecture for all elliptic modular forms. In particular, Theorems 2.4 and 2.5 are not conditional, and the condition on the j-invariant in Theorem 1.1 is superfluous. Moreover, the methods of proof outlined in sections 3 and 4 have been generalized and extended in a number of ways, notably in a series of articles by Barnet-Lamb, Gee, Geraghty, and Taylor, by Thorne, and by Calegari and Geraghty.