Some effective estimates for André–Oort in Y(1)n
AbstractLet {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 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 {\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 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.