The Sato-Tate conjecture

Abstract

In this thesis, we explore the connection between Galois representations and the Sato-Tate group. We give an explicit formulation of the Sato-Tate con-jecture for smooth and projective varieties over Q and we see what it con-cretely implies. We present an approach to this subject following Serre's notes Lectures on N X (p). We then discuss an example to give the reader a taste of how the proof of the Sato-Tate conjecture works in the case of elliptic curve with complex multiplication. Acknowledgments