{"paper":{"title":"Structure of the module of vector-valued modular forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.NT","authors_text":"Christopher Marks, Geoffrey Mason","submitted_at":"2009-01-27T22:31:53Z","abstract_excerpt":"Let $V$ be a representation of the modular group $\\Gamma$ of dimension $p$. We show that the $\\mathbb{Z}$-graded space $\\mathcal{H}(V)$ of holomorphic vector-valued modular forms associated to $V$ is a free module of rank $p$ over the algebra $\\mathcal{M}$ of classical holomorphic modular forms. We study the nature of $\\mathcal{H}$ considered as a functor from $\\Gamma$-modules to graded $\\mathcal{M}$-lattices and give some applications, including the calculation of the Hilbert-Poincar\\'{e} of $\\mathcal{H}(V)$ in some cases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.4367","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"}