{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:DSVQPIO37ADDFWKMO5AO2QIGGL","short_pith_number":"pith:DSVQPIO3","canonical_record":{"source":{"id":"1102.2980","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-02-15T07:08:10Z","cross_cats_sorted":[],"title_canon_sha256":"bc4e9a837bfae79964d141d06a285d599c497d01086035ba3cd7bdbfeb7a678a","abstract_canon_sha256":"ff8b8cc75c3a51ccb50f4a88077eb41db34ecef1bbd030b8327cdff54c082508"},"schema_version":"1.0"},"canonical_sha256":"1cab07a1dbf80632d94c7740ed410632c6d60645c1d6f274354dc0ddad1b4b81","source":{"kind":"arxiv","id":"1102.2980","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.2980","created_at":"2026-05-18T04:27:37Z"},{"alias_kind":"arxiv_version","alias_value":"1102.2980v3","created_at":"2026-05-18T04:27:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.2980","created_at":"2026-05-18T04:27:37Z"},{"alias_kind":"pith_short_12","alias_value":"DSVQPIO37ADD","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"DSVQPIO37ADDFWKM","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"DSVQPIO3","created_at":"2026-05-18T12:26:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:DSVQPIO37ADDFWKMO5AO2QIGGL","target":"record","payload":{"canonical_record":{"source":{"id":"1102.2980","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-02-15T07:08:10Z","cross_cats_sorted":[],"title_canon_sha256":"bc4e9a837bfae79964d141d06a285d599c497d01086035ba3cd7bdbfeb7a678a","abstract_canon_sha256":"ff8b8cc75c3a51ccb50f4a88077eb41db34ecef1bbd030b8327cdff54c082508"},"schema_version":"1.0"},"canonical_sha256":"1cab07a1dbf80632d94c7740ed410632c6d60645c1d6f274354dc0ddad1b4b81","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:27:37.629259Z","signature_b64":"Q0MI5Kkqf8F4/68P+lSVxXAMi/tW2z6oBi2UWlMyIux9NB6Ug+FwHoNt5/pFOlWypR9yr7mFqMv+uCdcePE3DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1cab07a1dbf80632d94c7740ed410632c6d60645c1d6f274354dc0ddad1b4b81","last_reissued_at":"2026-05-18T04:27:37.628347Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:27:37.628347Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1102.2980","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:27:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4rchl+zQYn4IJ4x9Nu2C0MNMTVoFpWQMDgVKbonADpWZLxXDMra9enIn0Ut9Azo72Fg+SRRNiPmEo0vZ4TqsDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T07:07:17.308670Z"},"content_sha256":"c43204b4ae68db6a945a21955fd19639f33464a574e850ddecaf202d0feb74d4","schema_version":"1.0","event_id":"sha256:c43204b4ae68db6a945a21955fd19639f33464a574e850ddecaf202d0feb74d4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:DSVQPIO37ADDFWKMO5AO2QIGGL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Visualizing elements of order four in the Shafarevich-Tate group of an elliptic curve","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mohammad Sadek","submitted_at":"2011-02-15T07:08:10Z","abstract_excerpt":"Let E be an elliptic curve defined over a number field K. Let h be an element of order 4 in the Shafarevich-Tate group of E. We prove that h is visible in infinitely many abelian surfaces up to isomorphism. This is to say that there are infinitely many abelian surfaces J such that E\\hookrightarrow J and h lies in the kernel of the natural map H^1(K,E)\\rightarrow H^1(K,J)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.2980","kind":"arxiv","version":3},"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":"2026-05-18T04:27:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ccvA4Ez7Chwcr0l2/9lzv2yZSYGlKNkjCOWrbzo8HfDGB4kkUBbRMWM/G8BaGVVm5if5vQ9xm60g4CzcoMR2DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T07:07:17.309201Z"},"content_sha256":"c3d11e0eb82f0a1dec1c95b4b715d8100ab9991799b1c2526b2fca6ea32fa4ad","schema_version":"1.0","event_id":"sha256:c3d11e0eb82f0a1dec1c95b4b715d8100ab9991799b1c2526b2fca6ea32fa4ad"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DSVQPIO37ADDFWKMO5AO2QIGGL/bundle.json","state_url":"https://pith.science/pith/DSVQPIO37ADDFWKMO5AO2QIGGL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DSVQPIO37ADDFWKMO5AO2QIGGL/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":"2026-06-07T07:07:17Z","links":{"resolver":"https://pith.science/pith/DSVQPIO37ADDFWKMO5AO2QIGGL","bundle":"https://pith.science/pith/DSVQPIO37ADDFWKMO5AO2QIGGL/bundle.json","state":"https://pith.science/pith/DSVQPIO37ADDFWKMO5AO2QIGGL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DSVQPIO37ADDFWKMO5AO2QIGGL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:DSVQPIO37ADDFWKMO5AO2QIGGL","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":"ff8b8cc75c3a51ccb50f4a88077eb41db34ecef1bbd030b8327cdff54c082508","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-02-15T07:08:10Z","title_canon_sha256":"bc4e9a837bfae79964d141d06a285d599c497d01086035ba3cd7bdbfeb7a678a"},"schema_version":"1.0","source":{"id":"1102.2980","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.2980","created_at":"2026-05-18T04:27:37Z"},{"alias_kind":"arxiv_version","alias_value":"1102.2980v3","created_at":"2026-05-18T04:27:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.2980","created_at":"2026-05-18T04:27:37Z"},{"alias_kind":"pith_short_12","alias_value":"DSVQPIO37ADD","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"DSVQPIO37ADDFWKM","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"DSVQPIO3","created_at":"2026-05-18T12:26:26Z"}],"graph_snapshots":[{"event_id":"sha256:c3d11e0eb82f0a1dec1c95b4b715d8100ab9991799b1c2526b2fca6ea32fa4ad","target":"graph","created_at":"2026-05-18T04:27:37Z","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":"Let E be an elliptic curve defined over a number field K. Let h be an element of order 4 in the Shafarevich-Tate group of E. We prove that h is visible in infinitely many abelian surfaces up to isomorphism. This is to say that there are infinitely many abelian surfaces J such that E\\hookrightarrow J and h lies in the kernel of the natural map H^1(K,E)\\rightarrow H^1(K,J).","authors_text":"Mohammad Sadek","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-02-15T07:08:10Z","title":"Visualizing elements of order four in the Shafarevich-Tate group of an elliptic curve"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.2980","kind":"arxiv","version":3},"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:c43204b4ae68db6a945a21955fd19639f33464a574e850ddecaf202d0feb74d4","target":"record","created_at":"2026-05-18T04:27:37Z","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":"ff8b8cc75c3a51ccb50f4a88077eb41db34ecef1bbd030b8327cdff54c082508","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-02-15T07:08:10Z","title_canon_sha256":"bc4e9a837bfae79964d141d06a285d599c497d01086035ba3cd7bdbfeb7a678a"},"schema_version":"1.0","source":{"id":"1102.2980","kind":"arxiv","version":3}},"canonical_sha256":"1cab07a1dbf80632d94c7740ed410632c6d60645c1d6f274354dc0ddad1b4b81","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1cab07a1dbf80632d94c7740ed410632c6d60645c1d6f274354dc0ddad1b4b81","first_computed_at":"2026-05-18T04:27:37.628347Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:37.628347Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Q0MI5Kkqf8F4/68P+lSVxXAMi/tW2z6oBi2UWlMyIux9NB6Ug+FwHoNt5/pFOlWypR9yr7mFqMv+uCdcePE3DA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:37.629259Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.2980","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c43204b4ae68db6a945a21955fd19639f33464a574e850ddecaf202d0feb74d4","sha256:c3d11e0eb82f0a1dec1c95b4b715d8100ab9991799b1c2526b2fca6ea32fa4ad"],"state_sha256":"9a891f370aca372db750ed45cd5687ea1dd808cc9eb14a9c17c843b02acfcd93"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o22U7foshq1gttRQtZTTh5+PzDLalMPwtLZCv/tTUzo+pEnSDGDjzKGhu1a57BQYRsLJfAMh8ZT/J6vXaOsZBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T07:07:17.311776Z","bundle_sha256":"169ab0dec6d1f039aafcbbe46cf5f15fd203b7c054e3f91b27eae9ba7f9009d4"}}