{"paper":{"title":"Variations on twists of tuples of hyperelliptic curves and related results","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Maciej Ulas, Tomasz J\\k{e}drzejak","submitted_at":"2014-01-03T08:07:34Z","abstract_excerpt":"Let $f\\in\\Q[x]$ be a square-free polynomial of degree $\\geq 3$ and $m\\geq 3$ be an odd positive integer. Based on our earlier investigations we prove that there exists a function $D_{1}\\in\\Q(u,v,w)$ such that the Jacobians of the curves \\begin{equation*} C_{1}:\\;D_{1}y^2=f(x),\\quad C_{2}:\\;y^2=D_{1}x^m+b,\\quad C_{3}:\\;y^2=D_{1}x^m+c, \\end{equation*} have all positive ranks over $\\Q(u,v,w)$. Similarly, we prove that there exists a function $D_{2}\\in\\Q(u,v,w)$ such that the Jacobians of the curves \\begin{equation*} C_{1}:\\;D_{2}y^2=h(x),\\quad C_{2}:\\;y^2=D_{2}x^m+b,\\quad C_{3}:\\;y^2=x^m+cD_{2}, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.0602","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"}