Triples of singular moduli with rational product

Abstract

We show that all triples [Formula: see text] of singular moduli satisfying [Formula: see text] are “trivial”. That is, either [Formula: see text]; some [Formula: see text] and the remaining [Formula: see text] are distinct, of degree [Formula: see text], and conjugate over [Formula: see text]; or [Formula: see text] are pairwise distinct, of degree [Formula: see text], and conjugate over [Formula: see text]. This theorem is best possible and is the natural three-dimensional analogue of a result of Bilu, Luca and Pizarro-Madariaga in two dimensions. It establishes an explicit version of the André–Oort conjecture for the family of subvarieties [Formula: see text] defined by an equation [Formula: see text].