{"paper":{"title":"Impossible intersections in a Weierstrass family of elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Niki Myrto Mavraki","submitted_at":"2015-07-25T01:07:09Z","abstract_excerpt":"Consider the Weierstrass family of elliptic curves $E_{\\lambda}:y^2=x^3+\\lambda$ parametrized by nonzero $\\lambda\\in\\overline{\\mathbb{Q}_2}$, and let $P_{\\lambda}(x)=(x,\\sqrt{x^3+\\lambda})\\in E_{\\lambda}$. In this article, given $\\alpha,\\beta\\in\\overline{\\mathbb{Q}_2}$ such that $\\frac{\\alpha}{\\beta}\\in\\mathbb{Q}$, we provide an explicit description for the set of parameters $\\lambda$ such that $P_{\\lambda}(\\alpha)$ and $P_{\\lambda}(\\beta)$ are simultaneously torsion for $E_{\\lambda}$. In particular we prove that the aforementioned set is empty unless $\\frac{\\alpha}{\\beta}\\in\\{-2,-\\frac{1}{2}\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07047","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"}