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

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve and let $ρ_E \colon \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{GL}(2, \widehat{\mathbb{Z}})$ be the adelic Galois representation attached to $E$. We describe and implement an algorithm to compute the image of $ρ_E$ 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.

Figures (3)

• Figure 1: Diagram of groups and maps that appear in Prop. \ref{['prop-normalizerCRT']}
• Figure 2: A diagram for the groups and maps for the proof of Prop. \ref{['prop-levelofdefngcd']}.
• Figure 3: Field diagram of division fields from Theorem \ref{['thm-cartan-image']}

Theorems & Definitions (103)

• Theorem 1.1
• Example 1.2
• Example 1.3
• Example 1.4
• Remark 2.1
• Theorem 2.2: lozano-galoiscm, Theorems 1.1, 1.2, and 4.1
• Theorem 2.3: lozano-galoiscm, Theorem 1.2
• Theorem 2.4: lozano-galoiscm, Theorem 1.5
• Remark 2.5
• Theorem 2.6: lozano-galoiscm, Theorem 1.6