{"paper":{"title":"Andr\\'{e}'s theorem and weakly bounded height","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Guy Fowler","submitted_at":"2026-06-28T12:20:50Z","abstract_excerpt":"Let $V \\subset \\mathbb{A}^2(\\mathbb{C})$ be an algebraic curve such that $\\mathrm{deg} X \\neq \\mathrm{deg} Y$, where $X, Y$ denote the coordinate functions on $\\mathbb{A}^2(\\mathbb{C})$ restricted to $V$. We prove there exists an effectively computable constant $c$, that depends linearly on the height of $V$, such that $\\max \\{h(x), h(y)\\} \\leq c$ for every $(x, y) \\in V$ with $x$ and $y$ both CM $j$-invariants. This establishes, for such curves, an effective version of the Andr\\'{e}--Oort conjecture that has a better dependence on the height of $V$ than previous effective results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29369","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.29369/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}