{"paper":{"title":"Rational points on certain elliptic surfaces","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Maciej Ulas","submitted_at":"2007-05-21T12:10:43Z","abstract_excerpt":"Let $\\mathcal{E}_{f}:y^2=x^3+f(t)x$, where $f\\in\\Q[t]\\setminus\\Q$, and let us assume that $\\op{deg}f\\leq 4$. In this paper we prove that if $\\op{deg}f\\leq 3$, then there exists a rational base change $t\\mapsto\\phi(t)$ such that on the surface $\\cal{E}_{f\\circ\\phi}$ there is a non-torsion section. A similar theorem is valid in case when $\\op{deg}f=4$ and there exists $t_{0}\\in\\Q$ such that infinitely many rational points lie on the curve $E_{t_{0}}:y^2=x^3+f(t_{0})x$. In particular, we prove that if $\\op{deg}f=4$ and $f$ is not an even polynomial, then there is a rational point on $\\cal{E}_{f}$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0705.2955","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"}