Some effective estimates for Andr\'e-Oort in $Y(1)^n$

Abstract

Let $X\subset Y(1)^n$ be a subvariety defined over a number field $\mathbb F$ and let $(P_1,\ldots,P_n)\in X$ be a special point not contained in a positive-dimensional special subvariety of $X$. We show that the if a coordinate $P_i$ corresponds to an order not contained in a single exceptional Siegel-Tatuzawa imaginary quadratic field $K_*$ then the associated discriminant $|\Delta(P_i)|$ is bounded by an effective constant depending only on $\operatorname{deg} X$ and $[{\mathbb F}:{\mathbb Q}]$. We derive analogous effective results for the positive-dimensional maximal special subvarieties. From the main theorem we deduce various effective results of Andr\'e-Oort type. In particular we define a genericity condition on the leading homogeneous part of a polynomial, and give a fully effective Andr\'e-Oort statement for hypersurfaces defined by polynomials satisfying this condition.