{"paper":{"title":"Computing points on modular curves over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jinxiang Zeng","submitted_at":"2013-05-20T11:53:38Z","abstract_excerpt":"In this paper, we present a probabilistic algorithm to compute the number of $\\mathbb{F}_p$-points of modular curve $X_1(n)$. Under the Generalized Riemann Hypothesis(GRH), the algorithm takes $\\textrm{O}(n^{56+\\delta+\\epsilon}\\log^{9+\\epsilon} p)$ bit operations, where $\\delta$ is an absolute constant and $\\epsilon$ is any positive real number. As an application, we can compute $#X_1(17)(\\mathbb{F}_p)\\textrm{mod} 17$ for huge primes $p$. For example, we have $#X_1(17)(\\mathbb{F}_{10^{1000}+1357})\\textrm{mod} 17=3$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4505","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}