{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:S5NGIZUEUVGF7LIB72LYSSEGMK","short_pith_number":"pith:S5NGIZUE","canonical_record":{"source":{"id":"1509.07817","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-09-25T18:11:15Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"374d4ad5195a04f19ba6e1031d0459a492b9f986207ae4b4143599bf33900570","abstract_canon_sha256":"190a18bff84714336f757081b6f5dc3f38b8f3fd93a4277e84ef95d6240fc7f2"},"schema_version":"1.0"},"canonical_sha256":"975a646684a54c5fad01fe9789488662af18acfedac534c5416d841ef9aa2bdd","source":{"kind":"arxiv","id":"1509.07817","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.07817","created_at":"2026-05-18T01:32:01Z"},{"alias_kind":"arxiv_version","alias_value":"1509.07817v1","created_at":"2026-05-18T01:32:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07817","created_at":"2026-05-18T01:32:01Z"},{"alias_kind":"pith_short_12","alias_value":"S5NGIZUEUVGF","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"S5NGIZUEUVGF7LIB","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"S5NGIZUE","created_at":"2026-05-18T12:29:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:S5NGIZUEUVGF7LIB72LYSSEGMK","target":"record","payload":{"canonical_record":{"source":{"id":"1509.07817","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-09-25T18:11:15Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"374d4ad5195a04f19ba6e1031d0459a492b9f986207ae4b4143599bf33900570","abstract_canon_sha256":"190a18bff84714336f757081b6f5dc3f38b8f3fd93a4277e84ef95d6240fc7f2"},"schema_version":"1.0"},"canonical_sha256":"975a646684a54c5fad01fe9789488662af18acfedac534c5416d841ef9aa2bdd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:01.877689Z","signature_b64":"4foYx897gAXhbRxYOTSRrrLNDFDs+eKzUo0op7gp74muwKbBmb1fAq9YmqY4z4eqg4PrKkrCJtdUuMDds7NmDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"975a646684a54c5fad01fe9789488662af18acfedac534c5416d841ef9aa2bdd","last_reissued_at":"2026-05-18T01:32:01.877288Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:01.877288Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.07817","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":"2026-05-18T01:32:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hssq+QRZSxSXqa6MFgmmBi3reLv6c9TdVRvsNIUrBkvcM+UKanaj6Lyob4YcCMFqTKs63m5SBeRXpcV1shMgDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T21:06:33.427741Z"},"content_sha256":"88fda638380bcbeb552e6d0583d260f25edfe2174c6a1e266dd2ddd7f1789095","schema_version":"1.0","event_id":"sha256:88fda638380bcbeb552e6d0583d260f25edfe2174c6a1e266dd2ddd7f1789095"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:S5NGIZUEUVGF7LIB72LYSSEGMK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Vari\\'et\\'es ab\\'eliennes sur les corps de fonctions de courbes sur des corps locaux sup\\'erieurs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Diego Izquierdo","submitted_at":"2015-09-25T18:11:15Z","abstract_excerpt":"Let $k$ be a higher-dimensional local field and $X$ be a smooth projective geometrically integral curve over $k$. Let $K$ be the function field of $X$. We define Tate-Shafarevich groups of an abelian variety via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We prove local duality theorems for abelian varieties over $k$, as well as global duality theorems for Tate-Shafarevich groups of abelian varieties over $K$.\n  Soient $k$ un corps local sup\\'erieur et $X$ une courbe projective lisse g\\'eom\\'etriquement int\\`egre de corps de fonctions $K$. On"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07817","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":"2026-05-18T01:32:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k+N5qKes3BsZNURZo8OtvRo8TNT0omR45aG7Eu3QZo/rC5OfNkL8NqV08I5IGPO0SRfAS9SJa3ig7ZAXoz9FBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T21:06:33.428113Z"},"content_sha256":"ef9840206abe36330e6a0d35dc46da19cd6e9156cb9a9ce1076b0fd7614c934f","schema_version":"1.0","event_id":"sha256:ef9840206abe36330e6a0d35dc46da19cd6e9156cb9a9ce1076b0fd7614c934f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/S5NGIZUEUVGF7LIB72LYSSEGMK/bundle.json","state_url":"https://pith.science/pith/S5NGIZUEUVGF7LIB72LYSSEGMK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/S5NGIZUEUVGF7LIB72LYSSEGMK/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":"2026-06-22T21:06:33Z","links":{"resolver":"https://pith.science/pith/S5NGIZUEUVGF7LIB72LYSSEGMK","bundle":"https://pith.science/pith/S5NGIZUEUVGF7LIB72LYSSEGMK/bundle.json","state":"https://pith.science/pith/S5NGIZUEUVGF7LIB72LYSSEGMK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/S5NGIZUEUVGF7LIB72LYSSEGMK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:S5NGIZUEUVGF7LIB72LYSSEGMK","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":"190a18bff84714336f757081b6f5dc3f38b8f3fd93a4277e84ef95d6240fc7f2","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-09-25T18:11:15Z","title_canon_sha256":"374d4ad5195a04f19ba6e1031d0459a492b9f986207ae4b4143599bf33900570"},"schema_version":"1.0","source":{"id":"1509.07817","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.07817","created_at":"2026-05-18T01:32:01Z"},{"alias_kind":"arxiv_version","alias_value":"1509.07817v1","created_at":"2026-05-18T01:32:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07817","created_at":"2026-05-18T01:32:01Z"},{"alias_kind":"pith_short_12","alias_value":"S5NGIZUEUVGF","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"S5NGIZUEUVGF7LIB","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"S5NGIZUE","created_at":"2026-05-18T12:29:39Z"}],"graph_snapshots":[{"event_id":"sha256:ef9840206abe36330e6a0d35dc46da19cd6e9156cb9a9ce1076b0fd7614c934f","target":"graph","created_at":"2026-05-18T01:32:01Z","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 $k$ be a higher-dimensional local field and $X$ be a smooth projective geometrically integral curve over $k$. Let $K$ be the function field of $X$. We define Tate-Shafarevich groups of an abelian variety via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We prove local duality theorems for abelian varieties over $k$, as well as global duality theorems for Tate-Shafarevich groups of abelian varieties over $K$.\n  Soient $k$ un corps local sup\\'erieur et $X$ une courbe projective lisse g\\'eom\\'etriquement int\\`egre de corps de fonctions $K$. On","authors_text":"Diego Izquierdo","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-09-25T18:11:15Z","title":"Vari\\'et\\'es ab\\'eliennes sur les corps de fonctions de courbes sur des corps locaux sup\\'erieurs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07817","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":"sha256:88fda638380bcbeb552e6d0583d260f25edfe2174c6a1e266dd2ddd7f1789095","target":"record","created_at":"2026-05-18T01:32:01Z","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":"190a18bff84714336f757081b6f5dc3f38b8f3fd93a4277e84ef95d6240fc7f2","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-09-25T18:11:15Z","title_canon_sha256":"374d4ad5195a04f19ba6e1031d0459a492b9f986207ae4b4143599bf33900570"},"schema_version":"1.0","source":{"id":"1509.07817","kind":"arxiv","version":1}},"canonical_sha256":"975a646684a54c5fad01fe9789488662af18acfedac534c5416d841ef9aa2bdd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"975a646684a54c5fad01fe9789488662af18acfedac534c5416d841ef9aa2bdd","first_computed_at":"2026-05-18T01:32:01.877288Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:01.877288Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4foYx897gAXhbRxYOTSRrrLNDFDs+eKzUo0op7gp74muwKbBmb1fAq9YmqY4z4eqg4PrKkrCJtdUuMDds7NmDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:01.877689Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.07817","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:88fda638380bcbeb552e6d0583d260f25edfe2174c6a1e266dd2ddd7f1789095","sha256:ef9840206abe36330e6a0d35dc46da19cd6e9156cb9a9ce1076b0fd7614c934f"],"state_sha256":"e7406e2b73cdcf1932422130286984f570070b6d8df59389414688a720c65fca"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZUZaG0YEViGnEim4Xj6ZKxDONlUnKkPQHfe7R+wb5Vp5JK7v0hpYiWKrLjT/oXZaek2FrKZw6A6wohz/r24vDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T21:06:33.430121Z","bundle_sha256":"307d0747838dd8c9e528f432bedf9b67b10f4d6448b5a53a239e2a75f61a50c4"}}