{"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/gross-lattices-of-supersingular-elliptic","title":"Gross lattices of supersingular elliptic curves","arxiv_id":"2503.03478","date":"2025-03-05","proceeding":null,"authors":["Chenfeng He","Gaurish Korpal","Ha T. N. Tran","Christelle Vincent"],"abstract":"Let $p$ be a prime, $E$ be a supersingular elliptic curve defined over $\\bar{\\mathbb{F}}_p$, and $\\mathscr{O}$ be its (geometric) endomorphism ring. Earlier results of Chevyrev-Galbraith and Goren-Love have shown that the successive minima of the Gross lattice of $\\mathscr{O}$ characterize the isomorphism class of $\\mathscr{O}$. In this paper, we extend this work and show that the value of the third successive minimum $D_3$ of the Gross lattice gives necessary and sufficient conditions for the curve to have its $j$-invariant in the field $\\mathbb{F}_p$ or in the set $\\mathbb{F}_{p^2} \\setminus \\mathbb{F}_p$, as well as finer information about the endomorphism ring of $E$ when its $j$-invariant belongs to $\\mathbb{F}_p$ and $p \\equiv 3 \\pmod{4}$. We end our article with an investigation of the geometry of Gross lattices of supersingular elliptic curves.","url_abs":"https://arxiv.org/abs/2503.03478v1","url_pdf":"https://arxiv.org/pdf/2503.03478v1.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":"gross-lattices-of-supersingular-elliptic","repo_url":"https://github.com/gkorpal/minimal-gross","is_official":1,"mentioned_in_paper":1,"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}