How big is the image of the Galois representations attached to CM elliptic curves?

Abstract

A. Using an analogue of Serre's open image theorem for elliptic curves with complex multiplication, one can associate to each CM elliptic curve 𝐸 de ned over a number eld 𝐹 a natural number I (𝐸/𝐹 ) which describes how big the image of the Galois representation associated to 𝐸 is. We show how one can compute I (𝐸/𝐹 ), using a closed formula that we obtain from the classical theory of complex multiplication.