{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:V2CZZDND7ZYBI7HLRRRRKJFSNB","short_pith_number":"pith:V2CZZDND","canonical_record":{"source":{"id":"1005.4105","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-05-22T05:49:06Z","cross_cats_sorted":[],"title_canon_sha256":"a65951cadb16bd85def6e2e4cb3724226afe0c8618c510c8e641b3e3357a9bfa","abstract_canon_sha256":"1b9f63cbc3df3217cd54f0a0d0cd78381a60c2f2e595dbb41ce6caa7d1017dfa"},"schema_version":"1.0"},"canonical_sha256":"ae859c8da3fe70147ceb8c631524b2684f32a9489c4bcb9b199dbd783a2159d5","source":{"kind":"arxiv","id":"1005.4105","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.4105","created_at":"2026-05-18T04:14:38Z"},{"alias_kind":"arxiv_version","alias_value":"1005.4105v4","created_at":"2026-05-18T04:14:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.4105","created_at":"2026-05-18T04:14:38Z"},{"alias_kind":"pith_short_12","alias_value":"V2CZZDND7ZYB","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"V2CZZDND7ZYBI7HL","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"V2CZZDND","created_at":"2026-05-18T12:26:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:V2CZZDND7ZYBI7HLRRRRKJFSNB","target":"record","payload":{"canonical_record":{"source":{"id":"1005.4105","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-05-22T05:49:06Z","cross_cats_sorted":[],"title_canon_sha256":"a65951cadb16bd85def6e2e4cb3724226afe0c8618c510c8e641b3e3357a9bfa","abstract_canon_sha256":"1b9f63cbc3df3217cd54f0a0d0cd78381a60c2f2e595dbb41ce6caa7d1017dfa"},"schema_version":"1.0"},"canonical_sha256":"ae859c8da3fe70147ceb8c631524b2684f32a9489c4bcb9b199dbd783a2159d5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:38.223823Z","signature_b64":"tA+GWEvEUOo4oY0msIvAGZpSXgW4SMxUoOSbERjLU36QYcVtINzih7rzpcnhI5X8hetfj9rLiS364kY25vrJAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ae859c8da3fe70147ceb8c631524b2684f32a9489c4bcb9b199dbd783a2159d5","last_reissued_at":"2026-05-18T04:14:38.223165Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:38.223165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1005.4105","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:14:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yzhp5TWqtoO3LL+Yhym++3LRlBNmQiEKXrW128+urUpNeQ+fk6mBBSWeQpnioZK2/jcEyuR8f7Bv/x0FytsFCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T20:40:43.068886Z"},"content_sha256":"be56839bbb2555ceaf697294c647a35ad6ca541365effab0429231247a031692","schema_version":"1.0","event_id":"sha256:be56839bbb2555ceaf697294c647a35ad6ca541365effab0429231247a031692"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:V2CZZDND7ZYBI7HLRRRRKJFSNB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"A.O.L. Atkin, Ling Long, Tong Liu, Wen-Ching Winnie Li","submitted_at":"2010-05-22T05:49:06Z","abstract_excerpt":"In this paper we study the compatible family of degree-4 Scholl representations $\\rho_{\\ell}$ associated with a space $S$ of weight $\\kappa> 2$ noncongruence cusp forms satisfying Quaternion Multiplications over a biquadratic field $K$. It is shown that when either $K$ is totally real or $\\kappa$ is odd, $\\rho_\\ell$ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $GL_4(\\mathbb Q)$. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in $S$ and those of certain automorphic forms via"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.4105","kind":"arxiv","version":4},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:14:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jkZ0vjlcP/n5fnsqRw72aq+V2cr3h+QnBSa08pue+W0U/ubhl6fsWFtVN53Vn2DkPDh86onxEzGPFoMhDedUBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T20:40:43.069433Z"},"content_sha256":"61d8e3ddfe58e988283ccab8b7d74408638bafb833fc441502d966b4e17e2f7f","schema_version":"1.0","event_id":"sha256:61d8e3ddfe58e988283ccab8b7d74408638bafb833fc441502d966b4e17e2f7f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V2CZZDND7ZYBI7HLRRRRKJFSNB/bundle.json","state_url":"https://pith.science/pith/V2CZZDND7ZYBI7HLRRRRKJFSNB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V2CZZDND7ZYBI7HLRRRRKJFSNB/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-26T20:40:43Z","links":{"resolver":"https://pith.science/pith/V2CZZDND7ZYBI7HLRRRRKJFSNB","bundle":"https://pith.science/pith/V2CZZDND7ZYBI7HLRRRRKJFSNB/bundle.json","state":"https://pith.science/pith/V2CZZDND7ZYBI7HLRRRRKJFSNB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V2CZZDND7ZYBI7HLRRRRKJFSNB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:V2CZZDND7ZYBI7HLRRRRKJFSNB","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1b9f63cbc3df3217cd54f0a0d0cd78381a60c2f2e595dbb41ce6caa7d1017dfa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-05-22T05:49:06Z","title_canon_sha256":"a65951cadb16bd85def6e2e4cb3724226afe0c8618c510c8e641b3e3357a9bfa"},"schema_version":"1.0","source":{"id":"1005.4105","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.4105","created_at":"2026-05-18T04:14:38Z"},{"alias_kind":"arxiv_version","alias_value":"1005.4105v4","created_at":"2026-05-18T04:14:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.4105","created_at":"2026-05-18T04:14:38Z"},{"alias_kind":"pith_short_12","alias_value":"V2CZZDND7ZYB","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"V2CZZDND7ZYBI7HL","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"V2CZZDND","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:61d8e3ddfe58e988283ccab8b7d74408638bafb833fc441502d966b4e17e2f7f","target":"graph","created_at":"2026-05-18T04:14:38Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper we study the compatible family of degree-4 Scholl representations $\\rho_{\\ell}$ associated with a space $S$ of weight $\\kappa> 2$ noncongruence cusp forms satisfying Quaternion Multiplications over a biquadratic field $K$. It is shown that when either $K$ is totally real or $\\kappa$ is odd, $\\rho_\\ell$ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $GL_4(\\mathbb Q)$. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in $S$ and those of certain automorphic forms via","authors_text":"A.O.L. Atkin, Ling Long, Tong Liu, Wen-Ching Winnie Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-05-22T05:49:06Z","title":"Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.4105","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:be56839bbb2555ceaf697294c647a35ad6ca541365effab0429231247a031692","target":"record","created_at":"2026-05-18T04:14:38Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1b9f63cbc3df3217cd54f0a0d0cd78381a60c2f2e595dbb41ce6caa7d1017dfa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-05-22T05:49:06Z","title_canon_sha256":"a65951cadb16bd85def6e2e4cb3724226afe0c8618c510c8e641b3e3357a9bfa"},"schema_version":"1.0","source":{"id":"1005.4105","kind":"arxiv","version":4}},"canonical_sha256":"ae859c8da3fe70147ceb8c631524b2684f32a9489c4bcb9b199dbd783a2159d5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ae859c8da3fe70147ceb8c631524b2684f32a9489c4bcb9b199dbd783a2159d5","first_computed_at":"2026-05-18T04:14:38.223165Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:14:38.223165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tA+GWEvEUOo4oY0msIvAGZpSXgW4SMxUoOSbERjLU36QYcVtINzih7rzpcnhI5X8hetfj9rLiS364kY25vrJAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:14:38.223823Z","signed_message":"canonical_sha256_bytes"},"source_id":"1005.4105","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:be56839bbb2555ceaf697294c647a35ad6ca541365effab0429231247a031692","sha256:61d8e3ddfe58e988283ccab8b7d74408638bafb833fc441502d966b4e17e2f7f"],"state_sha256":"e1ccc4dbf6c3349aeab8c94bfd08d6e9093d3c6ecb7b7ed3c346aaf037349d64"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yRaWoG4ZKucbgAhshTh3OD7oJv78W1cJpnu5Pq+s0PT6fjrv6HS6ImDaXJBfctJJaxvDa/3VNjtXn4T8THH6DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T20:40:43.072882Z","bundle_sha256":"91b35790dedce7f59a311f8828fa82373a25f8f35f015f57c662e4722ae3a57c"}}