{"ID":23003497,"CreatedAt":"2026-09-17T04:20:54.235930923Z","UpdatedAt":"2026-09-17T04:20:54.235930923Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18934","arxiv_id":"2609.18934","title":"A uniform effective André--Oort result","abstract":"We prove the André--Oort conjecture for hypersurfaces $V \\subset Y(1)^n \\cong \\mathbb{A}^n_\\mathbb{C}$ defined by an equation $a_1 x_1^m + \\ldots + a_n x_n^m = b$, where $a_1, \\ldots, a_n, b \\in \\overline{\\mathbb{Q}}$ and $m \\in \\mathbb{Z}_{\u003e0}$. Unlike previous proofs, our result is both effective and uniform in the height of the coefficients $a_1, \\ldots, a_n, b$. This is the first effective proof of a uniform André--Oort statement for a class of subvarieties with arbitrary dimension and non-empty special locus. We also prove an analogous result for hypersurfaces $V \\subset Y(1)^n \\times \\mathbb{G}_m^l$.","short_abstract":"We prove the André--Oort conjecture for hypersurfaces $V \\subset Y(1)^n \\cong \\mathbb{A}^n_\\mathbb{C}$ defined by an equation $a_1 x_1^m + \\ldots + a_n x_n^m = b$, where $a_1, \\ldots, a_n, b \\in \\overline{\\mathbb{Q}}$ and $m \\in \\mathbb{Z}_{\u003e0}$. Unlike previous proofs, our result is both effective and uniform in the h...","url_abs":"https://arxiv.org/abs/2609.18934","url_pdf":"https://arxiv.org/pdf/2609.18934v1","authors":"[\"Guy Fowler\"]","published":"2026-09-16T17:09:04Z","proceeding":"math.NT","tasks":"[\"math.NT\",\"math.AG\"]","methods":"[]","has_code":false}