Mordell--Lang and disparate Selmer ranks of odd twists of some superelliptic curves over global function fields

[2504.20594] Mordell--Lang and disparate Selmer ranks of odd twists of some superelliptic curves over global function fields Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2504.20594 (math) [Submitted on 29 Apr 2025 ( v1 ), last revised 15 Aug 2026 (this version, v3)] Title: Mordell--Lang and disparate Selmer ranks of odd twists of some superelliptic curves over global function fields Authors: Sun Woo Park View a PDF of the paper titled Mordell--Lang and disparate Selmer ranks of odd twists of some superelliptic curves over global function fields, by Sun Woo Park View PDF HTML (experimental) Abstract: Fix a prime number $\ell \geq 5$. Let $K = \mathbb{F}_q(t)$ be a global function field of characteristic $p$ coprime to $2,3$, and $q \equiv 1 \text{ mod } \ell$. Let $C:y^\ell = F(x)$ be a non-isotrivial superelliptic curve over $K$ such that $F$ is a degree $3$ polynomial over $\mathbb{F}_q(t)$. Denote by $C_f: fy^\ell = F(x)$ the twist of $C$ by a polynomial $f$ over $\mathbb{F}_q$. Assuming some conditions on $C$, we show that the expected number of $K$-rational points of $C_f$ is bounded, and at least $99\%$ of such curves $C_f$ have at most $(3p)^{4\ell} \cdot \ell!$ many $K$-rational points, as $f$ ranges over the set of polynomials of sufficiently large degree over $\mathbb{F}_q$. To achieve this, we compute the distribution of dimensions of $1-\zeta_\ell$ Selmer groups of Jacobians of such superelliptic curves. This is done by generalizing the technique of constructing a governing Markov operator, as developed from previous studies by Swinnerton-Dyer, Klagsbrun--Mazur--Rubin, Yu, and the author. Comments: Version 3: 42 pages, 1 figure. Substantial revisions made, pertaining to the fact that the Weil pairing over $1-ζ_\ell$ torsion module of the Jacobian of $C_f$ is symmetric. We devise a new Markov model which governs the $1-ζ_\ell$ Selmer groups of Jacobians of $C_f$. Comments Welcome Subjects: Number Theory (math.NT) MSC classes: 11G10, 11G30, 11M50, 60J10, Report number: MPIM-Bonn-2025 Cite as: arXiv:2504.20594 [math.NT] (or arXiv:2504.20594v3 [math.NT] for this version) https://doi.org/10.48550/arXiv.2504.20594 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Sun Woo Park [ view email ] [v1] Tue, 29 Apr 2025 09:47:43 UTC (21 KB) [v2] Thu, 26 Feb 2026 15:25:48 UTC (21 KB) [v3] Sat, 15 Aug 2026 09:22:18 UTC (88 KB) Full-text links: Access Paper: View a PDF of the paper titled Mordell--Lang and disparate Selmer ranks of odd twists of some superelliptic curves over global function fields, by Sun Woo Park View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2025-04 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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