{"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/refined-selmer-equations-for-the-thrice","title":"Refined Selmer equations for the thrice-punctured line in depth two","arxiv_id":"2106.10145","date":"2021-06-18","proceeding":null,"authors":["Alex J. Best","L. Alexander Betts","Theresa Kumpitsch","Martin Lüdtke","Angus W. McAndrew","Lie Qian","Elie Studnia","Yujie Xu"],"abstract":"In [Kim05], Kim gave a new proof of Siegel's Theorem that there are only finitely many $S$-integral points on $\\mathbb P^1_{\\mathbb Z}\\setminus\\{0,1,\\infty\\}$. One advantage of Kim's method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of $S$ increases. In this paper, we implement a refinement of Kim's method to explicitly compute various examples where $S$ has size $2$ which has been introduced in [BD19]. In so doing, we exhibit new examples of a natural generalisation of a conjecture of Kim.","url_abs":"https://arxiv.org/abs/2106.10145v2","url_pdf":"https://arxiv.org/pdf/2106.10145v2.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":"refined-selmer-equations-for-the-thrice","repo_url":"https://github.com/martinluedtke/dcw_coefficients","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}