Explicit valuation of elliptic nets for elliptic curves with complex multiplication

[2512.09601] Explicit valuation of elliptic nets for elliptic curves with complex multiplication Skip to main content Search Submit Donate Log in Mathematics > Number Theory arXiv:2512.09601 (math) [Submitted on 10 Dec 2025 ( v1 ), last revised 26 Jun 2026 (this version, v2)] Title: Explicit valuation of elliptic nets for elliptic curves with complex multiplication Authors: Edison H L Au-Yeung View PDF HTML (experimental) Abstract: Division polynomials associated to an elliptic curve $E/K$ are polynomials $\phi_n, \psi_n^2$ that arise from the sequence of points $\{nP\}_{n \in \N}$ on this curve. If one wishes to study $\Z$--linear combination of points on $E(K)$, we can use net polynomials $\Phi_{\bm{v}}, \Psi_{\bm{v}}^2$ which are higher--dimensional \textcolor{red}{analogues} of division polynomials. It turns out they are also elliptic nets, an $n$--dimensional array with values in $K$ satisfying the same nonlinear recurrence relation that division polynomials do as well. Now further assume the elliptic curve $E/K$ has complex multiplication by an order of a quadratic imaginary field $F \subseteq K$, we will prove a formula for the common valuation of $\Phi_{\bm{v}}$ and $\Psi_{\bm{v}}^2$ associated to multiples of points by elements of an order in $F$. As an application, we will use the formula to show that elliptic divisibility sequences associated to multiples of points indexed by elements of an order also satisfy a recurrence relation when indexed by elements of an order, subject to certain conditions on the indices. Subjects: Number Theory (math.NT) MSC classes: 11G05, 11B37, 11B39, 14H52 Cite as: arXiv:2512.09601 [math.NT] (or arXiv:2512.09601v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2512.09601 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Canadian Mathematical Communications. 2026;1:e6 Related DOI : https://doi.org/10.4153/S2976859426100058 Focus to learn more DOI(s) linking to related resources Submission history From: Hang Lok Edison Au-Yeung [ view email ] [v1] Wed, 10 Dec 2025 12:48:20 UTC (42 KB) [v2] Fri, 26 Jun 2026 12:17:37 UTC (42 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2025-12 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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)

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