Chabauty--Kim, finite descent, and the Section Conjecture for locally geometric sections

[2305.09462] Chabauty--Kim, finite descent, and the Section Conjecture for locally geometric sections Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2305.09462 (math) [Submitted on 16 May 2023 ( v1 ), last revised 11 Apr 2026 (this version, v2)] Title: Chabauty--Kim, finite descent, and the Section Conjecture for locally geometric sections Authors: L. Alexander Betts , Theresa Kumpitsch , Martin Lüdtke View a PDF of the paper titled Chabauty--Kim, finite descent, and the Section Conjecture for locally geometric sections, by L. Alexander Betts and 1 other authors View PDF HTML (experimental) Abstract: Let $X$ be a smooth projective curve of genus $\geq2$ over a number field. A natural variant of Grothendieck's Section Conjecture postulates that every section of the fundamental exact sequence for $X$ which everywhere locally comes from a point of $X$ in fact globally comes from a point of $X$. We show that $X/\mathbb{Q}$ satisfies this version of the Section Conjecture if it satisfies Kim's Conjecture for almost all choices of auxiliary prime $p$, and give the appropriate generalisation to $S$-integral points on hyperbolic curves. This gives a new "computational" strategy for proving instances of this variant of the Section Conjecture, which we carry out for the thrice-punctured line over $\mathbb{Z}[1/2]$. Comments: 51 pages; minor revision; appendix rewritten, 'finite descent' added to the title Subjects: Number Theory (math.NT) ; Algebraic Geometry (math.AG) MSC classes: Primary: 14H30. Secondary: 11G30, 14H25 Cite as: arXiv:2305.09462 [math.NT] (or arXiv:2305.09462v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2305.09462 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Martin Lüdtke [ view email ] [v1] Tue, 16 May 2023 14:21:28 UTC (56 KB) [v2] Sat, 11 Apr 2026 17:46:17 UTC (53 KB) Full-text links: Access Paper: View a PDF of the paper titled Chabauty--Kim, finite descent, and the Section Conjecture for locally geometric sections, by L. Alexander Betts and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2023-05 Change to browse by: math math.AG 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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