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

TL;DR

This paper provides a method to compute the size of the Galois image for CM elliptic curves over number fields, extending Serre's open image theorem through classical complex multiplication theory.

Contribution

It introduces a closed formula for calculating the Galois image size of CM elliptic curves, linking classical complex multiplication theory with modern Galois representation analysis.

Findings

Derived a closed formula for (E/F)

Connected classical CM theory with Galois image size computation

Enabled explicit calculation of Galois representation images for CM elliptic curves

Abstract

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