Starting from a geometric object, can we say something about the relation of its geometry and its arithmetic? One such example is the circle S^1. If we think of it as the solutions of x^2 + y^2 = 1, then the coordinates of its torsion points generate interesting extensions of Q. Moreover, the so called Galois representation of S^1, namely the action of the Galois group on the torsion points, actually ‘comes from geometry’. In this talk I will describe an analogue of this phenomenon in the case of elliptic curves with complex multiplication (CM elliptic curves) and some of the consequences.