{"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-points-on-superelliptic-curves-in","title":"Counting points on superelliptic curves in average polynomial time","arxiv_id":"2004.10189","date":"2020-04-21","proceeding":null,"authors":["Andrew V. Sutherland"],"abstract":"We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over $\\mathbb Q$ that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic curves $y^m=f(x)$ with $m\\ge 2$ and $f\\in \\mathbb Z[x]$ any squarefree polynomial of degree $d\\ge 3$, along with a positive integer $N$. It can compute $\\#X(\\mathbb F_p)$ for all $p\\le N$ not dividing $m\\mathrm{lc}(f)\\mathrm{disc}(f)$ in time $O(md^3 N\\log^3 N\\log\\log N)$. It achieves this by computing the trace of the Cartier--Manin matrix of reductions of $X$. We can also compute the Cartier--Manin matrix itself, which determines the $p$-rank of the Jacobian of $X$ and the numerator of its zeta function modulo~$p$.","url_abs":"https://arxiv.org/abs/2004.10189v5","url_pdf":"https://arxiv.org/pdf/2004.10189v5.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-points-on-superelliptic-curves-in","repo_url":"https://github.com/sualehasif/computingPicard","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}