{"paper":{"title":"Generators of graded rings of modular forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Nadim Rustom","submitted_at":"2012-09-18T07:55:22Z","abstract_excerpt":"We study graded rings of modular forms over congruence subgroups, with coefficients in a subring $A$ of $\\mathbb{C}$, and specifically the highest weight needed to generate these rings as $A$-algebras. In particular, we determine upper bounds, independent of $N$, for the highest needed weight that generates the $\\mathbb{C}$-algebras of modular forms over $\\Gamma(N)$, $\\Gamma_1(N)$ and $\\Gamma_0(N)$ with some conditions on $N$. For $N \\geq 5$, we prove that the $\\mathbb{Z}[1/N]$-algebra of modular forms over $\\Gamma_1(N)$ with coefficients in $\\mathbb{Z}[1/N]$ is generated in weight at most 3. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3864","kind":"arxiv","version":3},"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"}