Galois representations attached to elliptic curves with complex multiplication

The goal of this article is to give an explicit classification of the possible p-adic Galois representations that are attached to elliptic curves E with CM defined over Q(j(E)). More precisely, let K be an imaginary quadratic field, and let OK,f be an order in K of conductor f≥ 1. Let E be an elliptic curve with CM by OK,f, such that E is defined by a model over Q(j(E)). Let p≥ 2 be a prime, let GQ(j(E)) be the absolute Galois group of Q(j(E)), and let ρE,p^∞ GQ(j(E))→ GL(2,Zₚ) be the Galois representation associated to the Galois action on the Tate module Tₚ(E). The goal is then to describe, explicitly, the groups of GL(2,Zₚ) that can occur as images of ρE,p^∞, up to conjugation, for an arbitrary order OK,f.