Triples of singular moduli with rational product

We show that all triples (x₁, x₂, x₃) of singular moduli satisfying x₁x₂x₃ ∈ Q^× are "trivial". That is, either x₁, x₂, x₃ ∈ Q; some x_i ∈ Q and the remaining x_j, x_k are distinct, of degree 2, and conjugate over Q; or x₁, x₂, x₃ 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 ⊂ C³ defined by an equation x₁x₂x₃ = α ∈ Q.