Triples of singular moduli with rational product

We show that all triples $(x_1,x_2,x_3)$ of singular moduli satisfying $x_1 x_2 x_3 \in \mathbb{Q}^{\times}$ are "trivial". That is, either $x_1, x_2, x_3 \in \mathbb{Q}$; some $x_i \in \mathbb{Q}$ and the remaining $x_j, x_k$ are distinct, of degree $2$, and conjugate over $\mathbb{Q}$; or $x_1, x_2, x_3$ are pairwise distinct, of degree $3$, and conjugate over $\mathbb{Q}$. 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 Andr\'e--Oort conjecture for the family of subvarieties $V_{\alpha} \subset \mathbb{C}^3$ defined by an equation $x_1 x_2 x_3 = \alpha \in \mathbb{Q}$.