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}$are\mathrm{"}trivial\mathrm{"}\mathrm{.}Thatis,either$x_1, x_2, x_3 \in $\mathbb{Q}$$;some$x_i \in $\mathbb{Q}$$andtheremaining$x_j, x_k$ 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}$$.