Models of CM elliptic curves with a prescribed $\ell$-adic Galois image

TL;DR

This paper classifies all possible -adic Galois representations for elliptic curves with complex multiplication by certain imaginary quadratic orders, and identifies all elliptic curves with a given -adic Galois image.

Contribution

It provides a complete classification of -adic Galois images for CM elliptic curves over their minimal fields for orders with class number one or two.

Findings

All -adic Galois representations for these CM elliptic curves are determined.

Explicit isomorphism classes of elliptic curves with prescribed -adic Galois images are identified.

The results extend understanding of Galois representations in CM theory.

Abstract

For each prime number $\ell$ and for each imaginary quadratic order of class number one or two, we determine all the possible $\ell$-adic Galois representations that occur for any elliptic curve with complex multiplication by such an order over its minimal field of definition, and then we determine all the isomorphism classes of elliptic curves that have a prescribed $\ell$-adic Galois representation.