{"paper":{"title":"Explicit points on the Legendre curve III","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Douglas Ulmer","submitted_at":"2014-06-25T19:26:55Z","abstract_excerpt":"We continue our study of the Legendre elliptic curve $y^2=x(x+1)(x+t)$ over function fields $K_d=\\mathbf{F}_p(\\mu_d,t^{1/d})$. When $d=p^f+1$, we have previously exhibited explicit points generating a subgroup $V_d$ of $E(K_d)$ of rank $d-2$ and of finite, $p$-power index. We also proved the finiteness of $III(E/K_d)$ and a class number formula: $[E(K_d):V_d]^2=|III(E/K_d)|$. In this paper, we compute $E(K_d)/V_d$ and $III(E/K_d)$ explicitly as modules over $\\mathbf{Z}_p[\\mathrm{Gal}(K_d/F_p(t))]$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.6674","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"}