{"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/on-the-arithmetic-and-geometry-of-spaces-l-m","title":"On the arithmetic and geometry of spaces $L_{m+1,n}$","arxiv_id":"2503.19533","date":"2025-03-25","proceeding":null,"authors":["Michel Matignon","Guillaume Pagot","Daniele Turchetti"],"abstract":"Let $p$ be a prime number. Motivated by the local lifting problem for $(\\mathbb{Z}/p\\mathbb{Z})^n$ with $n>1$, we prove several new results on certain $\\mathbb{F}_p$-vector spaces of logarithmic differential forms on the projective line in characteristic $p$, called \"spaces $L_{m+1,n}$\". Expanding the previous work by the first two authors, we prove positive and negative results for the existence of spaces $L_{m+1,n}$ in many situations. Moreover, we classify all spaces $L_{4p,2}$ for any $p$, and all spaces $L_{15,2}$ for $p=3$. Among the novel tools we use, Moore determinants and computational algebra play a prominent role.","url_abs":"https://arxiv.org/abs/2503.19533v1","url_pdf":"https://arxiv.org/pdf/2503.19533v1.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":"on-the-arithmetic-and-geometry-of-spaces-l-m","repo_url":"https://github.com/danieleturchetti/equidistant","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}