{"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/quadratic-chabauty-for-bi-elliptic-curves-and","title":"Quadratic Chabauty for (bi)elliptic curves and Kim's conjecture","arxiv_id":"1904.04622","date":"2019-04-09","proceeding":null,"authors":["Francesca Bianchi"],"abstract":"We explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. We remove the assumption of semistability in the description of the quadratic Chabauty sets $\\mathcal{X}(\\mathbb{Z}_p)_2$ containing the integral points $\\mathcal{X}(\\mathbb{Z})$ of an elliptic curve of rank at most $1$. Motivated by a conjecture of Kim, we then investigate theoretically and computationally the set-theoretic difference $\\mathcal{X}(\\mathbb{Z}_p)_2\\setminus \\mathcal{X}(\\mathbb{Z})$. We also consider some algorithmic questions arising from Balakrishnan--Dogra's explicit quadratic Chabauty for the rational points of a genus-two bielliptic curve. As an example, we provide a new solution to a problem of Diophantus which was first solved by Wetherell. Computationally, the main difference from the previous approach to quadratic Chabauty is the use of the $p$-adic sigma function in place of a double Coleman integral.","url_abs":"http://arxiv.org/abs/1904.04622v2","url_pdf":"http://arxiv.org/pdf/1904.04622v2.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":"quadratic-chabauty-for-bi-elliptic-curves-and","repo_url":"https://github.com/bianchifrancesca/quadratic_chabauty","is_official":1,"mentioned_in_paper":1,"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}