How big is the image of the Galois representations attached to 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.

13 pages. Accepted for publication in the Proceedings of the 18th Conference on Arithmetic, Geometry, Cryptography, and Coding Theory

References in corpus (1)

• Torsion points and Galois representations on CM elliptic curves