Uniform bounds on prime levels of abelian division fields of elliptic curves over number fields without rationally defined CM

[2608.12709] Uniform bounds on prime levels of abelian division fields of elliptic curves over number fields without rationally defined CM Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2608.12709 (math) [Submitted on 13 Aug 2026 ( v1 ), last revised 15 Aug 2026 (this version, v2)] Title: Uniform bounds on prime levels of abelian division fields of elliptic curves over number fields without rationally defined CM Authors: Hansol Kim View a PDF of the paper titled Uniform bounds on prime levels of abelian division fields of elliptic curves over number fields without rationally defined CM, by Hansol Kim View PDF HTML (experimental) Abstract: Enrique González-Jiménez and Álvaro Lozano-Robledo proved that there exists a uniform bound on the levels $n$ for which the $n$-division field $\mathbb{Q}\left(E\left[n\right]\right) / \mathbb{Q}$ of an elliptic curve $E/\mathbb{Q}$ defined over $\mathbb{Q}$ is abelian. Assuming GRH (Generalized Riemann Hypothesis), Allen and Genao partially generalized this result by restricting to prime levels while allowing arbitrary number fields without rationally defined CM. In this paper, we remove the GRH assumption and prove that the property established by Allen and Genao is in fact equivalent to the absence of rationally defined CM. Comments: 6 pages. The e-mail address of Hansol Kim has been corrected to jawlang@ajou. this http URL Subjects: Number Theory (math.NT) MSC classes: Primary 11G05, Secondary 14H52, 11F80 Cite as: arXiv:2608.12709 [math.NT] (or arXiv:2608.12709v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2608.12709 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Hansol Kim [ view email ] [v1] Thu, 13 Aug 2026 01:45:30 UTC (7 KB) [v2] Sat, 15 Aug 2026 11:23:55 UTC (7 KB) Full-text links: Access Paper: View a PDF of the paper titled Uniform bounds on prime levels of abelian division fields of elliptic curves over number fields without rationally defined CM, by Hansol Kim View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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