{"paper":{"title":"Explicit points on the Legendre curve","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Douglas Ulmer","submitted_at":"2010-02-17T17:04:05Z","abstract_excerpt":"We study the elliptic curve E given by y^2=x(x+1)(x+t) over the rational function field k(t) and its extensions K_d=k(\\mu_d,t^{1/d}). When k is finite of characteristic p and d=p^f+1, we write down explicit points on E and show by elementary arguments that they generate a subgroup V_d of rank d-2 and of finite index in E(K_d). Using more sophisticated methods, we then show that the Birch and Swinnerton-Dyer conjecture holds for E over K_d, and we relate the index of V_d in E(K_d) to the order of the Tate-Shafarevich group \\sha(E/K_d). When k has characteristic 0, we show that E has rank 0 over"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.3313","kind":"arxiv","version":3},"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"}