{"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/the-complete-classification-of-rational","title":"The Complete Classification of Rational Preperiodic Points of Quadratic Polynomials over Q: A Refined Conjecture","arxiv_id":"math/9512217","date":"1995-12-11","proceeding":null,"authors":["Bjorn Poonen"],"abstract":"We classify the graphs that can occur as the graph of rational preperiodic points of a quadratic polynomial over $\\bold Q$, assuming the conjecture that it is impossible to have rational points of period $4$ or higher. In particular, we show under this assumption that the number of preperiodic points is at most~$9$. Elliptic curves of small conductor and the genus~$2$ modular curves $X_1(13)$, $X_1(16)$, and $X_1(18)$ all arise as curves classifying quadratic polynomials with various combinations of preperiodic points. To complete the classification, we compute the rational points on a non-modular genus~$2$ curve by performing a $2$-descent on its Jacobian and afterwards applying a variant of the method of Chabauty and Coleman.","url_abs":"https://arxiv.org/abs/math/9512217v1","url_pdf":"https://arxiv.org/pdf/math/9512217v1.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":"the-complete-classification-of-rational","repo_url":"https://github.com/safaama1/Find-all-preperiodic-points-of-Dynamics-System-over-function-fields","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"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}