Some effective estimates for André-Oort in $Y(1{)}^n$

Abstract

Let $X\subset Y(1{)}^n$ be a subvariety defined over a number field $\mathbb{F}$ and let $(P_1,\dots ,P_n)\in X$ 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 $K_{\ast }$ then the associated discriminant $|\Delta (P_i)|$ is bounded by an effective constant depending only on $\mathrm{deg}X$ 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.

Contains an appendix by Emmanuel Kowalski

References in corpus (2)

• Intersections of Class Fields
• Linear Equations in Singular Moduli

Cited by in corpus (2)

• Effective estimates for the degrees of maximal special subvarieties
• No singular modulus is a unit