The image of the adelic Galois representation of an elliptic curve with complex multiplication

TL;DR

This paper develops an algorithm to compute the adelic Galois representation image of CM elliptic curves over  with specific j-invariants, revealing entanglement properties of their division fields.

Contribution

It introduces a new algorithm for explicitly determining the Galois representation image of CM elliptic curves and proves related entanglement results between their division fields.

Findings

Successful implementation of the algorithm for CM elliptic curves.

Identification of entanglement phenomena in division fields.

Explicit descriptions of Galois image structures.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve and let $\rho_E \colon \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{GL}(2, \widehat{\mathbb{Z}})$ be the adelic Galois representation attached to $E$. We describe and implement an algorithm to compute the image of $\rho_E$ in $\operatorname{GL}(2, \widehat{\mathbb{Z}})$ (up to conjugation) for an elliptic curve $E/\mathbb{Q}$ with complex multiplication (CM) and $j$-invariant not $0$ or $1728$. In the process, we prove certain entanglement results between division fields of elliptic curves over $\mathbb{Q}$ with CM.