{"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"}