{"paper":{"title":"Explicit points on the Legendre curve II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chris Hall, Douglas Ulmer, Ricardo Concei\\c{c}\\~ao","submitted_at":"2013-07-16T11:52:44Z","abstract_excerpt":"Let $E$ be the elliptic curve $y^2=x(x+1)(x+t)$ over the field $\\Fp(t)$ where $p$ is an odd prime. We study the arithmetic of $E$ over extensions $\\Fq(t^{1/d})$ where $q$ is a power of $p$ and $d$ is an integer prime to $p$. The rank of $E$ is given in terms of an elementary property of the subgroup of $(\\Z/d\\Z)^\\times$ generated by $p$. We show that for many values of $d$ the rank is large. For example, if $d$ divides $2(p^f-1)$ and $2(p^f-1)/d$ is odd, then the rank is at least $d/2$. When $d=2(p^f-1)$, we exhibit explicit points generating a subgroup of $E(\\Fq(t^{1/d}))$ of finite index in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.4251","kind":"arxiv","version":2},"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"}