{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:24P6ISCRMQZ6V4QHQUSXOBNGVU","short_pith_number":"pith:24P6ISCR","canonical_record":{"source":{"id":"1001.5302","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-01-29T01:32:23Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"b5860c40858e92bbbb0fb121814da6d84892ff12ac3f4fb99318a351bdb773e6","abstract_canon_sha256":"dd61270e9a7fe70b2409fe2f28ca4eca94006024790f0154d837bfec3f963415"},"schema_version":"1.0"},"canonical_sha256":"d71fe448516433eaf20785257705a6ad18ea25d7da80de7314a043e00d1f055f","source":{"kind":"arxiv","id":"1001.5302","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.5302","created_at":"2026-05-18T04:10:10Z"},{"alias_kind":"arxiv_version","alias_value":"1001.5302v1","created_at":"2026-05-18T04:10:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.5302","created_at":"2026-05-18T04:10:10Z"},{"alias_kind":"pith_short_12","alias_value":"24P6ISCRMQZ6","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"24P6ISCRMQZ6V4QH","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"24P6ISCR","created_at":"2026-05-18T12:26:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:24P6ISCRMQZ6V4QHQUSXOBNGVU","target":"record","payload":{"canonical_record":{"source":{"id":"1001.5302","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-01-29T01:32:23Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"b5860c40858e92bbbb0fb121814da6d84892ff12ac3f4fb99318a351bdb773e6","abstract_canon_sha256":"dd61270e9a7fe70b2409fe2f28ca4eca94006024790f0154d837bfec3f963415"},"schema_version":"1.0"},"canonical_sha256":"d71fe448516433eaf20785257705a6ad18ea25d7da80de7314a043e00d1f055f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:10.712484Z","signature_b64":"WAC9B0ngGQi9ixarmosFvmGzxqSiGKBcscXhCGnzUq6IjcUjeEzE6AJ4fOkiKO91hSDx+y0XrnT/PtbUFAS/Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d71fe448516433eaf20785257705a6ad18ea25d7da80de7314a043e00d1f055f","last_reissued_at":"2026-05-18T04:10:10.711732Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:10.711732Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1001.5302","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":"2026-05-18T04:10:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kpx83zbMbU/uVPCTJxjmgvOeyu9G7KjAg8UUywJ8zSSONZ4pNHdp8LWd4N+sXDs8rsOgkBQFJtO2fRIMhC4RDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T15:36:52.711165Z"},"content_sha256":"7e831db0ca56ca2d3862d7792c917d6e6b4623c8813a497f13074d4e8bd190bb","schema_version":"1.0","event_id":"sha256:7e831db0ca56ca2d3862d7792c917d6e6b4623c8813a497f13074d4e8bd190bb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:24P6ISCRMQZ6V4QHQUSXOBNGVU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Visualizing elements of Sha[3] in genus 2 jacobians","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Nils Bruin, Sander R. Dahmen","submitted_at":"2010-01-29T01:32:23Z","abstract_excerpt":"Mazur proved that any element xi of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that xi lies in the kernel of the natural homomorphism between the cohomology groups H^1(k,E) -> H^1(k,A). However, the abelian surface in Mazur's construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.5302","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":"2026-05-18T04:10:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CgdeXdK5SWq3e9nmTHRxJ5l4whmyXLYalttfufd1+v3uNi91kG5p8ha6ATANusAg5LH0uK3QHEYlnR1XOr6eDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T15:36:52.711537Z"},"content_sha256":"8b4d075498544b65b6a9d27fe4136bf4aae06f15ffc93715de52ce38386e182d","schema_version":"1.0","event_id":"sha256:8b4d075498544b65b6a9d27fe4136bf4aae06f15ffc93715de52ce38386e182d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/24P6ISCRMQZ6V4QHQUSXOBNGVU/bundle.json","state_url":"https://pith.science/pith/24P6ISCRMQZ6V4QHQUSXOBNGVU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/24P6ISCRMQZ6V4QHQUSXOBNGVU/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":"2026-06-25T15:36:52Z","links":{"resolver":"https://pith.science/pith/24P6ISCRMQZ6V4QHQUSXOBNGVU","bundle":"https://pith.science/pith/24P6ISCRMQZ6V4QHQUSXOBNGVU/bundle.json","state":"https://pith.science/pith/24P6ISCRMQZ6V4QHQUSXOBNGVU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/24P6ISCRMQZ6V4QHQUSXOBNGVU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:24P6ISCRMQZ6V4QHQUSXOBNGVU","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":"dd61270e9a7fe70b2409fe2f28ca4eca94006024790f0154d837bfec3f963415","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-01-29T01:32:23Z","title_canon_sha256":"b5860c40858e92bbbb0fb121814da6d84892ff12ac3f4fb99318a351bdb773e6"},"schema_version":"1.0","source":{"id":"1001.5302","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.5302","created_at":"2026-05-18T04:10:10Z"},{"alias_kind":"arxiv_version","alias_value":"1001.5302v1","created_at":"2026-05-18T04:10:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.5302","created_at":"2026-05-18T04:10:10Z"},{"alias_kind":"pith_short_12","alias_value":"24P6ISCRMQZ6","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"24P6ISCRMQZ6V4QH","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"24P6ISCR","created_at":"2026-05-18T12:26:03Z"}],"graph_snapshots":[{"event_id":"sha256:8b4d075498544b65b6a9d27fe4136bf4aae06f15ffc93715de52ce38386e182d","target":"graph","created_at":"2026-05-18T04:10:10Z","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":"Mazur proved that any element xi of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that xi lies in the kernel of the natural homomorphism between the cohomology groups H^1(k,E) -> H^1(k,A). However, the abelian surface in Mazur's construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get ","authors_text":"Nils Bruin, Sander R. Dahmen","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-01-29T01:32:23Z","title":"Visualizing elements of Sha[3] in genus 2 jacobians"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.5302","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":"sha256:7e831db0ca56ca2d3862d7792c917d6e6b4623c8813a497f13074d4e8bd190bb","target":"record","created_at":"2026-05-18T04:10:10Z","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":"dd61270e9a7fe70b2409fe2f28ca4eca94006024790f0154d837bfec3f963415","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-01-29T01:32:23Z","title_canon_sha256":"b5860c40858e92bbbb0fb121814da6d84892ff12ac3f4fb99318a351bdb773e6"},"schema_version":"1.0","source":{"id":"1001.5302","kind":"arxiv","version":1}},"canonical_sha256":"d71fe448516433eaf20785257705a6ad18ea25d7da80de7314a043e00d1f055f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d71fe448516433eaf20785257705a6ad18ea25d7da80de7314a043e00d1f055f","first_computed_at":"2026-05-18T04:10:10.711732Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:10:10.711732Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WAC9B0ngGQi9ixarmosFvmGzxqSiGKBcscXhCGnzUq6IjcUjeEzE6AJ4fOkiKO91hSDx+y0XrnT/PtbUFAS/Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:10:10.712484Z","signed_message":"canonical_sha256_bytes"},"source_id":"1001.5302","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7e831db0ca56ca2d3862d7792c917d6e6b4623c8813a497f13074d4e8bd190bb","sha256:8b4d075498544b65b6a9d27fe4136bf4aae06f15ffc93715de52ce38386e182d"],"state_sha256":"1cf3f50edcd2f73e9738119a5035d27acc2a57f1c278c80af2b6317362d5028c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BYoLC2aC103EhjVcw/yk2Yx2t4rI+vzI9UGbYdslaFVY5IPf5TzcEEtGwblxbbwaah9aKi/SoFg6ZxAcRsUaDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T15:36:52.713860Z","bundle_sha256":"ad57b255e3d3813a7e9b969b42e817cc651548bd986dd40a1ce07a04a25acab6"}}