{"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/computing-l-polynomials-of-picard-curves-from","title":"Computing L-Polynomials of Picard curves from Cartier-Manin matrices","arxiv_id":"2010.07247","date":"2020-10-14","proceeding":null,"authors":["Sualeh Asif","Francesc Fité","Dylan Pentland"],"abstract":"We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve $C:y^3=f(x)$ defined over $\\mathbb{Q}$ at primes $p$ of good reduction for $C$. We define a degree 9 polynomial $\\psi_f\\in \\mathbb{Q}[x]$ such that the splitting field of $\\psi_f(x^3/2)$ is the $2$-torsion field of the Jacobian of $C$. We prove that, for all but a density zero subset of primes, the zeta function $Z(C_p,T)$ is uniquely determined by the Cartier-Manin matrix $A_p$ of $C$ modulo $p$ and the splitting behavior modulo $p$ of $f$ and $\\psi_f$; we also show that for primes $\\equiv 1 \\pmod{3}$ the matrix $A_p$ suffices and that for primes $\\equiv 2 \\pmod{3}$ the genericity assumption on $C$ is unnecessary. An element of the proof, which may be of independent interest, is the determination of the density of the set of primes of ordinary reduction for a generic Picard curve. By combining this with recent work of Sutherland, we obtain a practical deterministic algorithm that computes $Z(C_p,T)$ for almost all primes $p \\le N$ using $N\\log(N)^{3+o(1)}$ bit operations. This is the first practical result of this type for curves of genus greater than 2.","url_abs":"https://arxiv.org/abs/2010.07247v3","url_pdf":"https://arxiv.org/pdf/2010.07247v3.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":"computing-l-polynomials-of-picard-curves-from","repo_url":"https://github.com/sualehasif/computingPicard","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}