{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:F7D6T2CZ4Q5NZAYQL76SKAQ6MK","short_pith_number":"pith:F7D6T2CZ","canonical_record":{"source":{"id":"1203.1030","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-03-05T20:36:43Z","cross_cats_sorted":[],"title_canon_sha256":"95049520f02ddb009e5fa5fbf11414bd5a7160b254010f15a6ce846fe6609161","abstract_canon_sha256":"41bca00c7538bae98107aa592ebbb9c5b1016d2072fe9ec1f0d1725deb665cc4"},"schema_version":"1.0"},"canonical_sha256":"2fc7e9e859e43adc83105ffd25021e6289ea9aa2f59c885c02b743a6553e2324","source":{"kind":"arxiv","id":"1203.1030","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.1030","created_at":"2026-05-18T04:00:13Z"},{"alias_kind":"arxiv_version","alias_value":"1203.1030v2","created_at":"2026-05-18T04:00:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.1030","created_at":"2026-05-18T04:00:13Z"},{"alias_kind":"pith_short_12","alias_value":"F7D6T2CZ4Q5N","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"F7D6T2CZ4Q5NZAYQ","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"F7D6T2CZ","created_at":"2026-05-18T12:27:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:F7D6T2CZ4Q5NZAYQL76SKAQ6MK","target":"record","payload":{"canonical_record":{"source":{"id":"1203.1030","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-03-05T20:36:43Z","cross_cats_sorted":[],"title_canon_sha256":"95049520f02ddb009e5fa5fbf11414bd5a7160b254010f15a6ce846fe6609161","abstract_canon_sha256":"41bca00c7538bae98107aa592ebbb9c5b1016d2072fe9ec1f0d1725deb665cc4"},"schema_version":"1.0"},"canonical_sha256":"2fc7e9e859e43adc83105ffd25021e6289ea9aa2f59c885c02b743a6553e2324","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:13.143060Z","signature_b64":"c+DEfH7Z6pm+cC8hzC+/kzoRnMbgAdHs4scLJvKeWe9Byj1s8vnfQX0wVhuvWiwPgbraDqzUvare6oneha8ZDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2fc7e9e859e43adc83105ffd25021e6289ea9aa2f59c885c02b743a6553e2324","last_reissued_at":"2026-05-18T04:00:13.142361Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:13.142361Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1203.1030","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":"2026-05-18T04:00:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JprX9YeVKO/XYXA1DiLpJIsZLUly1FDxO9kkpisSeHkO3/WfSkPxpH6QwrJ+JyWN5pJx20Gqrw+OvKw5G54xAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T22:17:17.449769Z"},"content_sha256":"24d83a1c19e24c5845fe90fc806c4120c2a86d409930a2e051490adf280ab216","schema_version":"1.0","event_id":"sha256:24d83a1c19e24c5845fe90fc806c4120c2a86d409930a2e051490adf280ab216"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:F7D6T2CZ4Q5NZAYQL76SKAQ6MK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Distribution of 2-Selmer Ranks within Quadratic Twist Families of Elliptic Curves with Partial Rational Two-Torsion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zev Klagsbrun","submitted_at":"2012-03-05T20:36:43Z","abstract_excerpt":"This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1030","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":"2026-05-18T04:00:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mqfJtg6MgnDs+dlPcq0QeOWRHvfmZO8Pof5E+zKPTu2gUTde5nrjDtsko8ArRZEqaRW87j7hZRRs8YQB1BIbAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T22:17:17.450108Z"},"content_sha256":"6d9fb37453acc4013695202a8cffedf69c42d45b16a47a347f632484610a6731","schema_version":"1.0","event_id":"sha256:6d9fb37453acc4013695202a8cffedf69c42d45b16a47a347f632484610a6731"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F7D6T2CZ4Q5NZAYQL76SKAQ6MK/bundle.json","state_url":"https://pith.science/pith/F7D6T2CZ4Q5NZAYQL76SKAQ6MK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F7D6T2CZ4Q5NZAYQL76SKAQ6MK/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":"2026-06-08T22:17:17Z","links":{"resolver":"https://pith.science/pith/F7D6T2CZ4Q5NZAYQL76SKAQ6MK","bundle":"https://pith.science/pith/F7D6T2CZ4Q5NZAYQL76SKAQ6MK/bundle.json","state":"https://pith.science/pith/F7D6T2CZ4Q5NZAYQL76SKAQ6MK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F7D6T2CZ4Q5NZAYQL76SKAQ6MK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:F7D6T2CZ4Q5NZAYQL76SKAQ6MK","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":"41bca00c7538bae98107aa592ebbb9c5b1016d2072fe9ec1f0d1725deb665cc4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-03-05T20:36:43Z","title_canon_sha256":"95049520f02ddb009e5fa5fbf11414bd5a7160b254010f15a6ce846fe6609161"},"schema_version":"1.0","source":{"id":"1203.1030","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.1030","created_at":"2026-05-18T04:00:13Z"},{"alias_kind":"arxiv_version","alias_value":"1203.1030v2","created_at":"2026-05-18T04:00:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.1030","created_at":"2026-05-18T04:00:13Z"},{"alias_kind":"pith_short_12","alias_value":"F7D6T2CZ4Q5N","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"F7D6T2CZ4Q5NZAYQ","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"F7D6T2CZ","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:6d9fb37453acc4013695202a8cffedf69c42d45b16a47a347f632484610a6731","target":"graph","created_at":"2026-05-18T04:00:13Z","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":"This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion o","authors_text":"Zev Klagsbrun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-03-05T20:36:43Z","title":"On the Distribution of 2-Selmer Ranks within Quadratic Twist Families of Elliptic Curves with Partial Rational Two-Torsion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1030","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":"sha256:24d83a1c19e24c5845fe90fc806c4120c2a86d409930a2e051490adf280ab216","target":"record","created_at":"2026-05-18T04:00:13Z","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":"41bca00c7538bae98107aa592ebbb9c5b1016d2072fe9ec1f0d1725deb665cc4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-03-05T20:36:43Z","title_canon_sha256":"95049520f02ddb009e5fa5fbf11414bd5a7160b254010f15a6ce846fe6609161"},"schema_version":"1.0","source":{"id":"1203.1030","kind":"arxiv","version":2}},"canonical_sha256":"2fc7e9e859e43adc83105ffd25021e6289ea9aa2f59c885c02b743a6553e2324","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2fc7e9e859e43adc83105ffd25021e6289ea9aa2f59c885c02b743a6553e2324","first_computed_at":"2026-05-18T04:00:13.142361Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:13.142361Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"c+DEfH7Z6pm+cC8hzC+/kzoRnMbgAdHs4scLJvKeWe9Byj1s8vnfQX0wVhuvWiwPgbraDqzUvare6oneha8ZDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:13.143060Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.1030","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:24d83a1c19e24c5845fe90fc806c4120c2a86d409930a2e051490adf280ab216","sha256:6d9fb37453acc4013695202a8cffedf69c42d45b16a47a347f632484610a6731"],"state_sha256":"db4d7998d9412361126683a59b35bc5d83d14e7eb5294af152d3896172a131d4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wTIVGPcnts3M+4t5I4q5XNiUaG7CJqTnIkoo74o+MwJlF+z1GXRu8j/5kudZUkaO+jV0X9dwXQNx97xlrtwRCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T22:17:17.452127Z","bundle_sha256":"18d7aaa90fe369bc2ad85f06033ab803eb51b679ccdecdc79b19132ad35218b6"}}