{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:62Q2JYC65A5E2W7JU2GROWMSZK","short_pith_number":"pith:62Q2JYC6","schema_version":"1.0","canonical_sha256":"f6a1a4e05ee83a4d5be9a68d175992ca8587b249934861e5b2cc2e28adbda5c8","source":{"kind":"arxiv","id":"1112.6250","version":2},"attestation_state":"computed","paper":{"title":"Lifts of projective congruence groups, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.NT","authors_text":"Ian Kiming","submitted_at":"2011-12-29T08:29:07Z","abstract_excerpt":"We continue and complete our previous paper `Lifts of projective congruence groups' [2] concerning the question of whether there exist noncongruence subgroups of $\\SL_2(\\Z)$ that are projectively equivalent to one of the groups $\\Gamma_0(N)$ or $\\Gamma_1(N)$. A complete answer to this question is obtained: In case of $\\Gamma_0(N)$ such noncongruence subgroups exist precisely if $N\\not\\in {3,4,8}$ and we additionally have either that $4\\mid N$ or that $N$ is divisible by an odd prime congruent to 3 modulo 4. In case of $\\Gamma_1(N)$ these noncongruence subgroups exist precisely if $N>4$.\n  As i"},"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":"1112.6250","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-12-29T08:29:07Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"e1106772fdccbd23be7c165ef785f33c21208173f5578198a04aab55aaecd568","abstract_canon_sha256":"6aee41231e297a7edc28dca49ed3f0599e460e301f894f0e1393c03feee03e80"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:02.918960Z","signature_b64":"pxP/ACDw6PbA4zDP62xDZwShsKBpW8XAyQHdE696UhwXcVNUxCbMHJP5S6ncwaa3V5MpedK3LgIZWfIbjq8TBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f6a1a4e05ee83a4d5be9a68d175992ca8587b249934861e5b2cc2e28adbda5c8","last_reissued_at":"2026-05-18T03:38:02.918457Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:02.918457Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lifts of projective congruence groups, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.NT","authors_text":"Ian Kiming","submitted_at":"2011-12-29T08:29:07Z","abstract_excerpt":"We continue and complete our previous paper `Lifts of projective congruence groups' [2] concerning the question of whether there exist noncongruence subgroups of $\\SL_2(\\Z)$ that are projectively equivalent to one of the groups $\\Gamma_0(N)$ or $\\Gamma_1(N)$. A complete answer to this question is obtained: In case of $\\Gamma_0(N)$ such noncongruence subgroups exist precisely if $N\\not\\in {3,4,8}$ and we additionally have either that $4\\mid N$ or that $N$ is divisible by an odd prime congruent to 3 modulo 4. In case of $\\Gamma_1(N)$ these noncongruence subgroups exist precisely if $N>4$.\n  As i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.6250","kind":"arxiv","version":2},"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":"1112.6250","created_at":"2026-05-18T03:38:02.918538+00:00"},{"alias_kind":"arxiv_version","alias_value":"1112.6250v2","created_at":"2026-05-18T03:38:02.918538+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.6250","created_at":"2026-05-18T03:38:02.918538+00:00"},{"alias_kind":"pith_short_12","alias_value":"62Q2JYC65A5E","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_16","alias_value":"62Q2JYC65A5E2W7J","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_8","alias_value":"62Q2JYC6","created_at":"2026-05-18T12:26:22.705136+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/62Q2JYC65A5E2W7JU2GROWMSZK","json":"https://pith.science/pith/62Q2JYC65A5E2W7JU2GROWMSZK.json","graph_json":"https://pith.science/api/pith-number/62Q2JYC65A5E2W7JU2GROWMSZK/graph.json","events_json":"https://pith.science/api/pith-number/62Q2JYC65A5E2W7JU2GROWMSZK/events.json","paper":"https://pith.science/paper/62Q2JYC6"},"agent_actions":{"view_html":"https://pith.science/pith/62Q2JYC65A5E2W7JU2GROWMSZK","download_json":"https://pith.science/pith/62Q2JYC65A5E2W7JU2GROWMSZK.json","view_paper":"https://pith.science/paper/62Q2JYC6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1112.6250&json=true","fetch_graph":"https://pith.science/api/pith-number/62Q2JYC65A5E2W7JU2GROWMSZK/graph.json","fetch_events":"https://pith.science/api/pith-number/62Q2JYC65A5E2W7JU2GROWMSZK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/62Q2JYC65A5E2W7JU2GROWMSZK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/62Q2JYC65A5E2W7JU2GROWMSZK/action/storage_attestation","attest_author":"https://pith.science/pith/62Q2JYC65A5E2W7JU2GROWMSZK/action/author_attestation","sign_citation":"https://pith.science/pith/62Q2JYC65A5E2W7JU2GROWMSZK/action/citation_signature","submit_replication":"https://pith.science/pith/62Q2JYC65A5E2W7JU2GROWMSZK/action/replication_record"}},"created_at":"2026-05-18T03:38:02.918538+00:00","updated_at":"2026-05-18T03:38:02.918538+00:00"}