Some effective estimates for André-Oort in Y(1) ⁿ

Let X C Y (1)n be a subvariety defined over a number field F and let (P1, ⋯, Pn) ϵ X be a special point not contained in a positive-dimensional special subvariety of X. We show that if a coordinate Pi corresponds to an order not contained in a single exceptional Siegel-Tatuzawa imaginary quadratic field K∗, then the associated discriminant Δ ? (Pi) is bounded by an effective constant depending only on deg ?X and [F:Q]. We derive analogous effective results for the positive-dimensional maximal special subvarieties. From the main theorem we deduce various effective results of André-Oort type. In particular, we define a genericity condition on the leading homogeneous part of a polynomial, and give a fully effective André-Oort statement for hypersurfaces defined by polynomials satisfying this condition.