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arXiv · 2603.08545

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

Álvaro Lozano-Robledo·Benjamin York
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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 $E$. Much work has been done in recent years to study the image of $ρ_E$ (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 $ρ_E$ 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 $1728$.

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Álvaro Lozano-Robledo, Benjamin York. 2026-06-17. The image of the adelic Galois representation of an elliptic curve with complex multiplication. https://arxiv.org/abs/2603.08545

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