The image of the adelic Galois representation of an elliptic curve with complex multiplication
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918380130271232 |
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| author | Lozano-Robledo, Álvaro York, Benjamin |
| author_facet | Lozano-Robledo, Álvaro York, Benjamin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08545 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The image of the adelic Galois representation of an elliptic curve with complex multiplication Lozano-Robledo, Álvaro York, Benjamin Number Theory Algebraic Geometry 11G05 (Primary) 11G15, 14H52, 14K22 (Secondary) 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. |
| title | The image of the adelic Galois representation of an elliptic curve with complex multiplication |
| topic | Number Theory Algebraic Geometry 11G05 (Primary) 11G15, 14H52, 14K22 (Secondary) |
| url | https://arxiv.org/abs/2603.08545 |