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 ${\rho }_E:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \mathrm{GL}(2,\hat{\mathbb{Z}})$ be the adelic Galois representation attached to $E$. Much work has been done in recent years to study the image of ${\rho }_E$ (up to conjugation) as part of Mazur's so called ``Program B.'' In this paper, we describe and implement an efficient algorithm to compute the image of ${\rho }_E$ in $\mathrm{GL}(2,\hat{\mathbb{Z}})$ (up to conjugation) for an elliptic curve $E/\mathbb{Q}$ with complex multiplication (CM) and $j$-invariant not $0$ or $1728$.

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