{"paper":{"title":"The Cassels-Tate pairing on polarized abelian varieties","license":"","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Bjorn Poonen, Michael Stoll","submitted_at":"1999-11-01T00:00:00Z","abstract_excerpt":"Let (A,\\lambda) be a principally polarized abelian variety defined over a global field k, and let \\Sha(A) be its Shafarevich-Tate group. Let \\Sha(A)_\\nd denote the quotient of \\Sha(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing \\Sha(A)_\\nd \\times \\Sha(A)_\\nd \\rightarrow \\Q/\\Z. If A is an elliptic curve, then by a result of Cassels the pairing is alternating. But in general it is only antisymmetric.\n  Using some new but equivalent definitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9911267","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"}