{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:FC5PW5VYBGKTLR3SLZUHXDPYYX","short_pith_number":"pith:FC5PW5VY","canonical_record":{"source":{"id":"1306.1410","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-06-06T13:58:42Z","cross_cats_sorted":[],"title_canon_sha256":"d476a29ca4f18080c368b1da2ef37fc62a0c0a9c86223f652d383987bcc94389","abstract_canon_sha256":"24e5272491e530a08663b4f7de6e3a1d3041b1f44f8d090da7a5302e7e8a893a"},"schema_version":"1.0"},"canonical_sha256":"28bafb76b8099535c7725e687b8df8c5ee493ea5fa906a59949800221ef415ed","source":{"kind":"arxiv","id":"1306.1410","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.1410","created_at":"2026-05-18T03:21:34Z"},{"alias_kind":"arxiv_version","alias_value":"1306.1410v1","created_at":"2026-05-18T03:21:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.1410","created_at":"2026-05-18T03:21:34Z"},{"alias_kind":"pith_short_12","alias_value":"FC5PW5VYBGKT","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"FC5PW5VYBGKTLR3S","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"FC5PW5VY","created_at":"2026-05-18T12:27:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:FC5PW5VYBGKTLR3SLZUHXDPYYX","target":"record","payload":{"canonical_record":{"source":{"id":"1306.1410","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-06-06T13:58:42Z","cross_cats_sorted":[],"title_canon_sha256":"d476a29ca4f18080c368b1da2ef37fc62a0c0a9c86223f652d383987bcc94389","abstract_canon_sha256":"24e5272491e530a08663b4f7de6e3a1d3041b1f44f8d090da7a5302e7e8a893a"},"schema_version":"1.0"},"canonical_sha256":"28bafb76b8099535c7725e687b8df8c5ee493ea5fa906a59949800221ef415ed","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:34.292416Z","signature_b64":"4UmHCmUpBRmdQ2vn/uoQdD/+fhKlh2okp/pCwztlDNU5HT6cXPcWcomGXp6XtAaIv+tQp/6dtY4AZn6P3hPMBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28bafb76b8099535c7725e687b8df8c5ee493ea5fa906a59949800221ef415ed","last_reissued_at":"2026-05-18T03:21:34.291652Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:34.291652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1306.1410","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:21:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6q15EOuKKUl1y8x7M2eJ9RUulUC9n/yAoVHvFWr4RQ+BkCnc9TPttpe4yuQYlfnoalbnc5EXf7BZMNhiCwXZBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T01:15:04.999045Z"},"content_sha256":"2c816fc1f64a28ac99ae3a6e4e1ef0933d315cbdd2d084edc5ce46ecd141eb8c","schema_version":"1.0","event_id":"sha256:2c816fc1f64a28ac99ae3a6e4e1ef0933d315cbdd2d084edc5ce46ecd141eb8c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:FC5PW5VYBGKTLR3SLZUHXDPYYX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Computing the Cassels-Tate pairing on the 3-Selmer group of an elliptic curve","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Rachel Newton, Tom Fisher","submitted_at":"2013-06-06T13:58:42Z","abstract_excerpt":"We extend the method of Cassels for computing the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the calculation. Our method is practical in sufficiently small examples, and can be used to improve the upper bound for the rank of an elliptic curve obtained by 3-descent."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.1410","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:21:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rk8CR6ONpV2unx7ARN+zj+J63fLbH0JUSyPZkguGt+XcI7OjaLDMmUh+q6xwR8gZdeze9z1HDGZAjUq1EEj8Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T01:15:04.999772Z"},"content_sha256":"d92c20180c291b75929ba84a0e547b7e7481aab5771ce9c91992bc903266c56f","schema_version":"1.0","event_id":"sha256:d92c20180c291b75929ba84a0e547b7e7481aab5771ce9c91992bc903266c56f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FC5PW5VYBGKTLR3SLZUHXDPYYX/bundle.json","state_url":"https://pith.science/pith/FC5PW5VYBGKTLR3SLZUHXDPYYX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FC5PW5VYBGKTLR3SLZUHXDPYYX/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":"2026-06-10T01:15:05Z","links":{"resolver":"https://pith.science/pith/FC5PW5VYBGKTLR3SLZUHXDPYYX","bundle":"https://pith.science/pith/FC5PW5VYBGKTLR3SLZUHXDPYYX/bundle.json","state":"https://pith.science/pith/FC5PW5VYBGKTLR3SLZUHXDPYYX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FC5PW5VYBGKTLR3SLZUHXDPYYX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:FC5PW5VYBGKTLR3SLZUHXDPYYX","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":"24e5272491e530a08663b4f7de6e3a1d3041b1f44f8d090da7a5302e7e8a893a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-06-06T13:58:42Z","title_canon_sha256":"d476a29ca4f18080c368b1da2ef37fc62a0c0a9c86223f652d383987bcc94389"},"schema_version":"1.0","source":{"id":"1306.1410","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.1410","created_at":"2026-05-18T03:21:34Z"},{"alias_kind":"arxiv_version","alias_value":"1306.1410v1","created_at":"2026-05-18T03:21:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.1410","created_at":"2026-05-18T03:21:34Z"},{"alias_kind":"pith_short_12","alias_value":"FC5PW5VYBGKT","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"FC5PW5VYBGKTLR3S","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"FC5PW5VY","created_at":"2026-05-18T12:27:45Z"}],"graph_snapshots":[{"event_id":"sha256:d92c20180c291b75929ba84a0e547b7e7481aab5771ce9c91992bc903266c56f","target":"graph","created_at":"2026-05-18T03:21:34Z","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":"We extend the method of Cassels for computing the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the calculation. Our method is practical in sufficiently small examples, and can be used to improve the upper bound for the rank of an elliptic curve obtained by 3-descent.","authors_text":"Rachel Newton, Tom Fisher","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-06-06T13:58:42Z","title":"Computing the Cassels-Tate pairing on the 3-Selmer group of an elliptic curve"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.1410","kind":"arxiv","version":1},"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":"sha256:2c816fc1f64a28ac99ae3a6e4e1ef0933d315cbdd2d084edc5ce46ecd141eb8c","target":"record","created_at":"2026-05-18T03:21:34Z","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":"24e5272491e530a08663b4f7de6e3a1d3041b1f44f8d090da7a5302e7e8a893a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-06-06T13:58:42Z","title_canon_sha256":"d476a29ca4f18080c368b1da2ef37fc62a0c0a9c86223f652d383987bcc94389"},"schema_version":"1.0","source":{"id":"1306.1410","kind":"arxiv","version":1}},"canonical_sha256":"28bafb76b8099535c7725e687b8df8c5ee493ea5fa906a59949800221ef415ed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"28bafb76b8099535c7725e687b8df8c5ee493ea5fa906a59949800221ef415ed","first_computed_at":"2026-05-18T03:21:34.291652Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:21:34.291652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4UmHCmUpBRmdQ2vn/uoQdD/+fhKlh2okp/pCwztlDNU5HT6cXPcWcomGXp6XtAaIv+tQp/6dtY4AZn6P3hPMBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:21:34.292416Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.1410","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2c816fc1f64a28ac99ae3a6e4e1ef0933d315cbdd2d084edc5ce46ecd141eb8c","sha256:d92c20180c291b75929ba84a0e547b7e7481aab5771ce9c91992bc903266c56f"],"state_sha256":"40c62db11c98f74cc51d8b5f4368ceea439518228b9cc910b533d13421fc871d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w7KmaMgX0xrB94zOAxzK4wVa6CoUBi21IIrQggG6V2X0p+hsYBXtiDuhBYTxMHtfUin1NdZAGbZn+3FXPWgmDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T01:15:05.003863Z","bundle_sha256":"7b466a84fc56cebceefe9cd607524bafe9bca3e3036a6f9173347a40863c17ac"}}