{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:LWCC4MLN5BNCRZ6GP5D2QECRY2","short_pith_number":"pith:LWCC4MLN","schema_version":"1.0","canonical_sha256":"5d842e316de85a28e7c67f47a81051c68632791d196e49aa1f25a8fc3d25edb5","source":{"kind":"arxiv","id":"1601.04415","version":1},"attestation_state":"computed","paper":{"title":"Heegner Points on Modular Curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Li Cai, Yihua Chen, Yu Liu","submitted_at":"2016-01-18T07:21:18Z","abstract_excerpt":"In this paper, we study the Heegner points on more general modular curves other than $X_0(N)$, which generalizes Gross' work \"Heegner points on $X_0(N)$\". The explicit Gross-Zagier formula and the Euler system property are stated in this case. Using such kind of Heegner points, we construct certain families of quadratic twists of $X_0(36)$, with the ranks of Mordell-Weil groups being zero and one respectively, and show that the $2$-part of their BSD conjectures hold."},"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":"1601.04415","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-18T07:21:18Z","cross_cats_sorted":[],"title_canon_sha256":"33239aaf3af496af943bd7e8aa6a3421e1bd58232f5002d9cad07b35a602b1ea","abstract_canon_sha256":"89d9b396e27bcc0de7ea8a37be718d8d303f618b977b0be4cb3cea45a47a0a3f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:45.101119Z","signature_b64":"W4tCXNlWckRaqqU7ez1AEE4FGK9c+AYe6G9cPIecKkh3pLjE240EAPTzPUpYDzdgEMfSbdj5VrJkamaz7/0HCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d842e316de85a28e7c67f47a81051c68632791d196e49aa1f25a8fc3d25edb5","last_reissued_at":"2026-05-18T01:22:45.100600Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:45.100600Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Heegner Points on Modular Curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Li Cai, Yihua Chen, Yu Liu","submitted_at":"2016-01-18T07:21:18Z","abstract_excerpt":"In this paper, we study the Heegner points on more general modular curves other than $X_0(N)$, which generalizes Gross' work \"Heegner points on $X_0(N)$\". The explicit Gross-Zagier formula and the Euler system property are stated in this case. Using such kind of Heegner points, we construct certain families of quadratic twists of $X_0(36)$, with the ranks of Mordell-Weil groups being zero and one respectively, and show that the $2$-part of their BSD conjectures hold."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04415","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":"1601.04415","created_at":"2026-05-18T01:22:45.100675+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.04415v1","created_at":"2026-05-18T01:22:45.100675+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.04415","created_at":"2026-05-18T01:22:45.100675+00:00"},{"alias_kind":"pith_short_12","alias_value":"LWCC4MLN5BNC","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"LWCC4MLN5BNCRZ6G","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"LWCC4MLN","created_at":"2026-05-18T12:30:29.479603+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/LWCC4MLN5BNCRZ6GP5D2QECRY2","json":"https://pith.science/pith/LWCC4MLN5BNCRZ6GP5D2QECRY2.json","graph_json":"https://pith.science/api/pith-number/LWCC4MLN5BNCRZ6GP5D2QECRY2/graph.json","events_json":"https://pith.science/api/pith-number/LWCC4MLN5BNCRZ6GP5D2QECRY2/events.json","paper":"https://pith.science/paper/LWCC4MLN"},"agent_actions":{"view_html":"https://pith.science/pith/LWCC4MLN5BNCRZ6GP5D2QECRY2","download_json":"https://pith.science/pith/LWCC4MLN5BNCRZ6GP5D2QECRY2.json","view_paper":"https://pith.science/paper/LWCC4MLN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.04415&json=true","fetch_graph":"https://pith.science/api/pith-number/LWCC4MLN5BNCRZ6GP5D2QECRY2/graph.json","fetch_events":"https://pith.science/api/pith-number/LWCC4MLN5BNCRZ6GP5D2QECRY2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LWCC4MLN5BNCRZ6GP5D2QECRY2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LWCC4MLN5BNCRZ6GP5D2QECRY2/action/storage_attestation","attest_author":"https://pith.science/pith/LWCC4MLN5BNCRZ6GP5D2QECRY2/action/author_attestation","sign_citation":"https://pith.science/pith/LWCC4MLN5BNCRZ6GP5D2QECRY2/action/citation_signature","submit_replication":"https://pith.science/pith/LWCC4MLN5BNCRZ6GP5D2QECRY2/action/replication_record"}},"created_at":"2026-05-18T01:22:45.100675+00:00","updated_at":"2026-05-18T01:22:45.100675+00:00"}