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We give some sufficient conditions for the local-global divisibility by $p$ in ${\\mathcal{A}}$ and the triviality of $Sha (k,{\\mathcal{A}}[p])$. When ${\\mathcal{A}}$ is an abelian variety principally polarized, those conditions imply that the elements of the Tate-Shafarevich group $Sha(k,{\\mathcal{A}})$ are divisible by $p$ in the Weil-Ch\\^atelet group $H^1(k,{\\mathcal{A}})$ and the local-globa"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1603.05857","kind":"arxiv","version":8},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-18T12:36:01Z","cross_cats_sorted":[],"title_canon_sha256":"4f715a6d78fdd8597af88da89095e8923f2d8f87a876d4b060f40bf6067549b1","abstract_canon_sha256":"1003e17b7916b4d4bf7fcb45b8b53f943f10b28a758db92f1e235b5a22d4b1ce"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:12.327725Z","signature_b64":"6Nn1iAo6HbRrvRezrcKHYo8dDzqHkS6i0jiDhtLRbEF9wOYwKRY8cMC6JTK9IoLYyyGJ1PbLy6Tg2NSLoomwDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e45d7976b470b6d6a29757af36422e9a90864133865633fb096f8b2ffc39e977","last_reissued_at":"2026-05-17T23:49:12.327033Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:12.327033Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Divisibility questions in commutative algebraic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Laura Paladino","submitted_at":"2016-03-18T12:36:01Z","abstract_excerpt":"Let $k$ be a number field, let ${\\mathcal{A}}$ be a commutative algebraic group defined over $k$ and let $p$ be a prime number. Let ${\\mathcal{A}}[p]$ denote the $p$-torsion subgroup of ${\\mathcal{A}}$. We give some sufficient conditions for the local-global divisibility by $p$ in ${\\mathcal{A}}$ and the triviality of $Sha (k,{\\mathcal{A}}[p])$. When ${\\mathcal{A}}$ is an abelian variety principally polarized, those conditions imply that the elements of the Tate-Shafarevich group $Sha(k,{\\mathcal{A}})$ are divisible by $p$ in the Weil-Ch\\^atelet group $H^1(k,{\\mathcal{A}})$ and the local-globa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.05857","kind":"arxiv","version":8},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1603.05857","created_at":"2026-05-17T23:49:12.327142+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.05857v8","created_at":"2026-05-17T23:49:12.327142+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.05857","created_at":"2026-05-17T23:49:12.327142+00:00"},{"alias_kind":"pith_short_12","alias_value":"4ROXS5VUOC3N","created_at":"2026-05-18T12:29:58.707656+00:00"},{"alias_kind":"pith_short_16","alias_value":"4ROXS5VUOC3NNIUX","created_at":"2026-05-18T12:29:58.707656+00:00"},{"alias_kind":"pith_short_8","alias_value":"4ROXS5VU","created_at":"2026-05-18T12:29:58.707656+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4ROXS5VUOC3NNIUXK6XTMQROTK","json":"https://pith.science/pith/4ROXS5VUOC3NNIUXK6XTMQROTK.json","graph_json":"https://pith.science/api/pith-number/4ROXS5VUOC3NNIUXK6XTMQROTK/graph.json","events_json":"https://pith.science/api/pith-number/4ROXS5VUOC3NNIUXK6XTMQROTK/events.json","paper":"https://pith.science/paper/4ROXS5VU"},"agent_actions":{"view_html":"https://pith.science/pith/4ROXS5VUOC3NNIUXK6XTMQROTK","download_json":"https://pith.science/pith/4ROXS5VUOC3NNIUXK6XTMQROTK.json","view_paper":"https://pith.science/paper/4ROXS5VU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.05857&json=true","fetch_graph":"https://pith.science/api/pith-number/4ROXS5VUOC3NNIUXK6XTMQROTK/graph.json","fetch_events":"https://pith.science/api/pith-number/4ROXS5VUOC3NNIUXK6XTMQROTK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4ROXS5VUOC3NNIUXK6XTMQROTK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4ROXS5VUOC3NNIUXK6XTMQROTK/action/storage_attestation","attest_author":"https://pith.science/pith/4ROXS5VUOC3NNIUXK6XTMQROTK/action/author_attestation","sign_citation":"https://pith.science/pith/4ROXS5VUOC3NNIUXK6XTMQROTK/action/citation_signature","submit_replication":"https://pith.science/pith/4ROXS5VUOC3NNIUXK6XTMQROTK/action/replication_record"}},"created_at":"2026-05-17T23:49:12.327142+00:00","updated_at":"2026-05-17T23:49:12.327142+00:00"}