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 $m \in \mathbb{Z}_{>0}$. 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 $V \subset Y(1)^n \times \mathbb{G}_m^l$.