{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:WOOQGHINFV4DEDC4YR3TC53OTC","short_pith_number":"pith:WOOQGHIN","schema_version":"1.0","canonical_sha256":"b39d031d0d2d78320c5cc47731776e988bb00e09fa5b32673a515cf7cc89afef","source":{"kind":"arxiv","id":"1504.07364","version":1},"attestation_state":"computed","paper":{"title":"Generators of the ring of weakly holomorphic modular functions for $\\Gamma_1(N)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dong Sung Yoon, Ja Kyung Koo","submitted_at":"2015-04-28T07:14:00Z","abstract_excerpt":"For a positive integer $N$ divisible by $4,5,6,7$ or $9$, let $\\mathcal{O}_{1,N}(\\mathbb{Q})$ be the ring of weakly holomorphic modular functions for the congruence subgroup $\\Gamma_1(N)$ with rational Fourier coefficients. We present explicit generators of the ring $\\mathcal{O}_{1,N}(\\mathbb{Q})$ over $\\mathbb{Q}$ by making use of modular units which have infinite product expansions."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1504.07364","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-04-28T07:14:00Z","cross_cats_sorted":[],"title_canon_sha256":"7ed867ce32b05821d95caa0e17e951ff38fd15cb7727da5fc50463c82127d097","abstract_canon_sha256":"30fa2b6aa510f652109da01ac516c4148d5a28d940d3440d0b382af8a4c57b39"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:40.137482Z","signature_b64":"2JAyiuYxoVIqHkeRS1KnIb01tNW2MCz/stZElzXQaCIr4VyPXxM/rh/USF6YYT6YJnKNc8DPkF8M91X8IZFzDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b39d031d0d2d78320c5cc47731776e988bb00e09fa5b32673a515cf7cc89afef","last_reissued_at":"2026-05-18T00:24:40.136811Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:40.136811Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generators of the ring of weakly holomorphic modular functions for $\\Gamma_1(N)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dong Sung Yoon, Ja Kyung Koo","submitted_at":"2015-04-28T07:14:00Z","abstract_excerpt":"For a positive integer $N$ divisible by $4,5,6,7$ or $9$, let $\\mathcal{O}_{1,N}(\\mathbb{Q})$ be the ring of weakly holomorphic modular functions for the congruence subgroup $\\Gamma_1(N)$ with rational Fourier coefficients. We present explicit generators of the ring $\\mathcal{O}_{1,N}(\\mathbb{Q})$ over $\\mathbb{Q}$ by making use of modular units which have infinite product expansions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07364","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1504.07364","created_at":"2026-05-18T00:24:40.136920+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.07364v1","created_at":"2026-05-18T00:24:40.136920+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.07364","created_at":"2026-05-18T00:24:40.136920+00:00"},{"alias_kind":"pith_short_12","alias_value":"WOOQGHINFV4D","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_16","alias_value":"WOOQGHINFV4DEDC4","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_8","alias_value":"WOOQGHIN","created_at":"2026-05-18T12:29:47.479230+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WOOQGHINFV4DEDC4YR3TC53OTC","json":"https://pith.science/pith/WOOQGHINFV4DEDC4YR3TC53OTC.json","graph_json":"https://pith.science/api/pith-number/WOOQGHINFV4DEDC4YR3TC53OTC/graph.json","events_json":"https://pith.science/api/pith-number/WOOQGHINFV4DEDC4YR3TC53OTC/events.json","paper":"https://pith.science/paper/WOOQGHIN"},"agent_actions":{"view_html":"https://pith.science/pith/WOOQGHINFV4DEDC4YR3TC53OTC","download_json":"https://pith.science/pith/WOOQGHINFV4DEDC4YR3TC53OTC.json","view_paper":"https://pith.science/paper/WOOQGHIN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.07364&json=true","fetch_graph":"https://pith.science/api/pith-number/WOOQGHINFV4DEDC4YR3TC53OTC/graph.json","fetch_events":"https://pith.science/api/pith-number/WOOQGHINFV4DEDC4YR3TC53OTC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WOOQGHINFV4DEDC4YR3TC53OTC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WOOQGHINFV4DEDC4YR3TC53OTC/action/storage_attestation","attest_author":"https://pith.science/pith/WOOQGHINFV4DEDC4YR3TC53OTC/action/author_attestation","sign_citation":"https://pith.science/pith/WOOQGHINFV4DEDC4YR3TC53OTC/action/citation_signature","submit_replication":"https://pith.science/pith/WOOQGHINFV4DEDC4YR3TC53OTC/action/replication_record"}},"created_at":"2026-05-18T00:24:40.136920+00:00","updated_at":"2026-05-18T00:24:40.136920+00:00"}