{"paper":{"title":"Explicit points on $y^2 + xy - t^d y = x^3$ and related character sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Christopher Davis, Tommy Occhipinti","submitted_at":"2014-09-26T09:54:38Z","abstract_excerpt":"Let $\\mathbb{F}_q$ denote a finite field of characteristic $p \\geq 5$ and let $d = q+1$. Let $E_d$ denote the elliptic curve over the function field $\\mathbb{F}_{q^2}(t)$ defined by the equation $y^2 + xy - t^d y = x^3$. Its rank is $q$ when $q \\equiv 1 \\bmod 3$ and its rank is $q-2$ when $q \\equiv 2 \\bmod 3$. We describe an explicit method for producing points on this elliptic curve. In case $q \\not\\equiv 11 \\bmod 12$, our method produces points which generate a full-rank subgroup. Our strategy for producing rational points on $E_d$ makes use of a dominant map from the degree $d$ Fermat surfa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.7519","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"}