{"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/counting-intersection-numbers-of-closed","title":"Counting intersection numbers of closed geodesics on Shimura curves","arxiv_id":"2104.01968","date":"2021-04-05","proceeding":null,"authors":["James Rickards"],"abstract":"Let $\\Gamma\\subseteq\\text{PSL}(2, \\mathbb{R})$ correspond to the group of units of norm $1$ in an Eichler order $\\mathrm{O}$ of an indefinite quaternion algebra over $\\mathbb{Q}$. Closed geodesics on $\\Gamma\\backslash\\mathbb{H}$ correspond to optimal embeddings of real quadratic orders into $\\mathrm{O}$. The weighted intersection numbers of pairs of these closed geodesics conjecturally relates to the work of Darmon-Vonk on a real quadratic analogue to the difference of singular moduli. In this paper, we study the total intersection number over all embeddings of a given pair of discriminants. We precisely describe the arithmetic of each intersection, and produce a formula for the total intersection. This formula is a real quadratic analogue of the work of Gross and Zagier on factorizing the difference of singular moduli. The results are fairly general, allowing for a large class of non-maximal Eichler orders, and non-fundamental/non-coprime discriminants. The paper ends with some explicit examples illustrating the results of the paper.","url_abs":"https://arxiv.org/abs/2104.01968v2","url_pdf":"https://arxiv.org/pdf/2104.01968v2.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":"counting-intersection-numbers-of-closed","repo_url":"https://github.com/JamesRickards-Canada/Q-Quadratic","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}