The maximal abelian extension contained in a division field of an elliptic curve over $\mathbb{Q}$ with complex multiplication
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929478268092416 |
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| author | Hamakiotes, Asimina S. |
| author_facet | Hamakiotes, Asimina S. |
| contents | Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j_{K,f})$, where $j_{K,f}=j(E)$. It has been shown by the author and Lozano-Robledo that $\operatorname{Gal}(\mathbb{Q}(j_{K,f},E[N])/\mathbb{Q}(j_{K,f}))$ is only abelian for $N=2,3$, and $4$. Let $p$ be a prime and let $n\geq 1$ be an integer. In this article, we bound the commutator subgroups of $\operatorname{Gal}(\mathbb{Q}(E[p^n])/\mathbb{Q})$ and classify the maximal abelian extensions contained in $\mathbb{Q}(E[p^n])/\mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16164 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The maximal abelian extension contained in a division field of an elliptic curve over $\mathbb{Q}$ with complex multiplication Hamakiotes, Asimina S. Number Theory Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j_{K,f})$, where $j_{K,f}=j(E)$. It has been shown by the author and Lozano-Robledo that $\operatorname{Gal}(\mathbb{Q}(j_{K,f},E[N])/\mathbb{Q}(j_{K,f}))$ is only abelian for $N=2,3$, and $4$. Let $p$ be a prime and let $n\geq 1$ be an integer. In this article, we bound the commutator subgroups of $\operatorname{Gal}(\mathbb{Q}(E[p^n])/\mathbb{Q})$ and classify the maximal abelian extensions contained in $\mathbb{Q}(E[p^n])/\mathbb{Q}$. |
| title | The maximal abelian extension contained in a division field of an elliptic curve over $\mathbb{Q}$ with complex multiplication |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.16164 |