A uniform effective André--Oort result

We prove the André–Oort conjecture for hypersurfaces $V\subset Y(1{)}^n\cong {\mathbb{A}}_{\mathbb{C}}^n$ defined by an equation $a_1{x}_1^m+\dots +a_n{x}_n^m=b$, where $a_1,\dots ,a_n,b\in \bar{\mathbb{Q}}$ and $m\in {\mathbb{Z}}_{>0}$. Unlike previous proofs, our result is both effective and uniform in the height of the coefficients $a_1,\dots ,a_n,b$. 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 $V\subset Y(1{)}^n\times {\mathbb{G}}_m^l$.