{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-maximal-abelian-extension-contained-in-a","title":"The maximal abelian extension contained in a division field of an elliptic curve over $\\mathbb{Q}$ with complex multiplication","arxiv_id":"2408.16164","date":"2024-08-28","proceeding":null,"authors":["Asimina S. Hamakiotes"],"abstract":"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 $\\mathbb{Q}(E[p^n])/\\mathbb{Q}$.","url_abs":"https://arxiv.org/abs/2408.16164v1","url_pdf":"https://arxiv.org/pdf/2408.16164v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-maximal-abelian-extension-contained-in-a","repo_url":"https://github.com/asiminah/max-ab-extn-contained-in-div-flds","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}