{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:QCWSTM7SA3BTZU76TSIUXJFRPE","short_pith_number":"pith:QCWSTM7S","canonical_record":{"source":{"id":"1204.5803","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-04-26T01:35:56Z","cross_cats_sorted":[],"title_canon_sha256":"124405b1a0610da1a487cf9011056a85f838aadb8bb2b0b3c3772aa904538f2e","abstract_canon_sha256":"63a4c4b9b547f4c599c315e2a9f532d76d308870f31198d888007c59cb706835"},"schema_version":"1.0"},"canonical_sha256":"80ad29b3f206c33cd3fe9c914ba4b1790de6dbe714c2ed8b3ca161e12f70a53b","source":{"kind":"arxiv","id":"1204.5803","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.5803","created_at":"2026-05-18T02:56:32Z"},{"alias_kind":"arxiv_version","alias_value":"1204.5803v2","created_at":"2026-05-18T02:56:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.5803","created_at":"2026-05-18T02:56:32Z"},{"alias_kind":"pith_short_12","alias_value":"QCWSTM7SA3BT","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QCWSTM7SA3BTZU76","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QCWSTM7S","created_at":"2026-05-18T12:27:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:QCWSTM7SA3BTZU76TSIUXJFRPE","target":"record","payload":{"canonical_record":{"source":{"id":"1204.5803","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-04-26T01:35:56Z","cross_cats_sorted":[],"title_canon_sha256":"124405b1a0610da1a487cf9011056a85f838aadb8bb2b0b3c3772aa904538f2e","abstract_canon_sha256":"63a4c4b9b547f4c599c315e2a9f532d76d308870f31198d888007c59cb706835"},"schema_version":"1.0"},"canonical_sha256":"80ad29b3f206c33cd3fe9c914ba4b1790de6dbe714c2ed8b3ca161e12f70a53b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:56:32.994127Z","signature_b64":"CwSlM7LqIfXEEMoJRasD4hiFQVSKGgWB7n5WP2k5Qa4KmMZNSnV4+uEz/d69iSsAYqU1XtjYwn+2jcyytNICBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80ad29b3f206c33cd3fe9c914ba4b1790de6dbe714c2ed8b3ca161e12f70a53b","last_reissued_at":"2026-05-18T02:56:32.993292Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:56:32.993292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1204.5803","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:56:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s1e8Y9yB2patT2mQV20JCJVJT7mkkqmoijCX8wkspVSyjucd+IwCtl87WTk/a5T+JkoH9Cl3qVheCIXJeLxqBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T11:51:20.110629Z"},"content_sha256":"539d061c63f24b4eb676e23fa805615423a8bebec8270761c2fb89a19964c7d2","schema_version":"1.0","event_id":"sha256:539d061c63f24b4eb676e23fa805615423a8bebec8270761c2fb89a19964c7d2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:QCWSTM7SA3BTZU76TSIUXJFRPE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Explicit descent in the Picard group of a cyclic cover of the projective line","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Brendan Creutz","submitted_at":"2012-04-26T01:35:56Z","abstract_excerpt":"Given a curve X of the form y^p = h(x) over a number field, one can use descents to obtain explicit bounds on the Mordell-Weil rank of the Jacobian or to prove that the curve has no rational points. We show how, having performed such a descent, one can easily obtain additional information which may rule out the existence of rational divisors on X of degree prime to p. This can yield sharper bounds on the Mordell-Weil rank by demonstrating the existence of nontrivial elements in the Shafarevich-Tate group. As an example we compute the Mordell-Weil rank of the Jacobian of a genus 4 curve over Q "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.5803","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:56:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kHzcX0GTz3yntzpqX+fa9w86zNUDvnML2qRib72TsODWLNYY+OIIBics2V8dFFvZfac8AGkV37Gx53Ymz1/1Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T11:51:20.110981Z"},"content_sha256":"2c31e6de6fb8630f9220a7037c06d365c64ca6880e571e6ee9b901a6b275d297","schema_version":"1.0","event_id":"sha256:2c31e6de6fb8630f9220a7037c06d365c64ca6880e571e6ee9b901a6b275d297"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QCWSTM7SA3BTZU76TSIUXJFRPE/bundle.json","state_url":"https://pith.science/pith/QCWSTM7SA3BTZU76TSIUXJFRPE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QCWSTM7SA3BTZU76TSIUXJFRPE/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":"2026-05-27T11:51:20Z","links":{"resolver":"https://pith.science/pith/QCWSTM7SA3BTZU76TSIUXJFRPE","bundle":"https://pith.science/pith/QCWSTM7SA3BTZU76TSIUXJFRPE/bundle.json","state":"https://pith.science/pith/QCWSTM7SA3BTZU76TSIUXJFRPE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QCWSTM7SA3BTZU76TSIUXJFRPE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:QCWSTM7SA3BTZU76TSIUXJFRPE","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":"63a4c4b9b547f4c599c315e2a9f532d76d308870f31198d888007c59cb706835","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-04-26T01:35:56Z","title_canon_sha256":"124405b1a0610da1a487cf9011056a85f838aadb8bb2b0b3c3772aa904538f2e"},"schema_version":"1.0","source":{"id":"1204.5803","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.5803","created_at":"2026-05-18T02:56:32Z"},{"alias_kind":"arxiv_version","alias_value":"1204.5803v2","created_at":"2026-05-18T02:56:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.5803","created_at":"2026-05-18T02:56:32Z"},{"alias_kind":"pith_short_12","alias_value":"QCWSTM7SA3BT","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QCWSTM7SA3BTZU76","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QCWSTM7S","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:2c31e6de6fb8630f9220a7037c06d365c64ca6880e571e6ee9b901a6b275d297","target":"graph","created_at":"2026-05-18T02:56:32Z","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":"Given a curve X of the form y^p = h(x) over a number field, one can use descents to obtain explicit bounds on the Mordell-Weil rank of the Jacobian or to prove that the curve has no rational points. We show how, having performed such a descent, one can easily obtain additional information which may rule out the existence of rational divisors on X of degree prime to p. This can yield sharper bounds on the Mordell-Weil rank by demonstrating the existence of nontrivial elements in the Shafarevich-Tate group. As an example we compute the Mordell-Weil rank of the Jacobian of a genus 4 curve over Q ","authors_text":"Brendan Creutz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-04-26T01:35:56Z","title":"Explicit descent in the Picard group of a cyclic cover of the projective line"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.5803","kind":"arxiv","version":2},"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":"sha256:539d061c63f24b4eb676e23fa805615423a8bebec8270761c2fb89a19964c7d2","target":"record","created_at":"2026-05-18T02:56:32Z","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":"63a4c4b9b547f4c599c315e2a9f532d76d308870f31198d888007c59cb706835","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-04-26T01:35:56Z","title_canon_sha256":"124405b1a0610da1a487cf9011056a85f838aadb8bb2b0b3c3772aa904538f2e"},"schema_version":"1.0","source":{"id":"1204.5803","kind":"arxiv","version":2}},"canonical_sha256":"80ad29b3f206c33cd3fe9c914ba4b1790de6dbe714c2ed8b3ca161e12f70a53b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"80ad29b3f206c33cd3fe9c914ba4b1790de6dbe714c2ed8b3ca161e12f70a53b","first_computed_at":"2026-05-18T02:56:32.993292Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:56:32.993292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CwSlM7LqIfXEEMoJRasD4hiFQVSKGgWB7n5WP2k5Qa4KmMZNSnV4+uEz/d69iSsAYqU1XtjYwn+2jcyytNICBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:56:32.994127Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.5803","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:539d061c63f24b4eb676e23fa805615423a8bebec8270761c2fb89a19964c7d2","sha256:2c31e6de6fb8630f9220a7037c06d365c64ca6880e571e6ee9b901a6b275d297"],"state_sha256":"410b79d131fb764185bf76e4987daf1cdfeb8abb16b6956fbab2586b0a52f47b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w9oqi1i2nUy1uW3ZfzpxIUpQjkk2nwWXUIDTqep5qHrSx2HHNeaKSiDr3J+qi7PanVDyBPhn+ueOlRQ9NvlCBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T11:51:20.113052Z","bundle_sha256":"cf3c02096d05c9722bba201e2fe52deb03544a99c1a8102c0e2ab56203dfbb95"}}