{"paper":{"title":"An explicit Andr\\'e-Oort type result for P^1(C) x G_m(C) based on logarithmic forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Roland Paulin","submitted_at":"2014-03-12T14:33:41Z","abstract_excerpt":"Using linear forms in logarithms we prove an explicit result of Andr\\'e-Oort type for $\\mathbb{P}^1(\\mathbb{C}) \\times \\mathbb{G}_m(\\mathbb{C})$. In this variation the special points of $\\mathbb{P}^1(\\mathbb{C}) \\times \\mathbb{G}_m(\\mathbb{C})$ are of the form $(\\alpha, \\lambda)$, with $\\alpha$ a singular modulus and $\\lambda$ a root of unity. The qualitative version of our result states that if $\\mathcal{C}$ is a closed algebraic curve in $\\mathbb{P}^1(\\mathbb{C}) \\times \\mathbb{G}_m(\\mathbb{C})$, defined over a number field, not containing a horizontal or vertical line, then $\\mathcal{C}$ co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.2949","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"}