Triples of singular moduli with rational product.

We show that all triples (x 1 , x 2 , x 3) of singular moduli satisfying x 1 x 2 x 3 ∈ ℚ × are "trivial". That is, either x 1 , x 2 , x 3 ∈ ℚ ; some x i ∈ ℚ and the remaining x j , x k are distinct, of degree 2 , and conjugate over ℚ ; or x 1 , x 2 , x 3 are pairwise distinct, of degree 3 , and conjugate over ℚ. 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 V α ⊂ ℂ 3 defined by an equation x 1 x 2 x 3 = α ∈ ℚ. [ABSTRACT FROM AUTHOR]