Triples of singular moduli with rational product

Abstract

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 André--Oort conjecture for the family of subvarieties $V_{\alpha }\subset {\mathbb{C}}^3$ defined by an equation $x_1{x}_2{x}_3=\alpha \in \mathbb{Q}$.

14 pages. Version 2 contains a significant strengthening of the main result; we exclude all but the "trivial" cases of triples with rational product