Equations in three singular moduli: the equal exponent case

Let $a\in {\mathbb{Z}}_{>0}$ and $\epsilon_1, \epsilon_2, \epsilon_3 \in {\pm 1}$\mathrm{.}Weclassifyexplicitlyallsingularmoduli$x_1, x_2, x_3$ satisfying either ${ϵ}_1{x}_1^a+{ϵ}_2{x}_2^a+{ϵ}_3{x}_3^a\in \mathbb{Q}$ or $(x_1^{\epsilon_1} x_2^{\epsilon_2} x_3^{\epsilon_3})^{a} \in $\mathbb{Q}$^{\times}$. In particular, we show that all the solutions in singular moduli $x_1,x_2,x_3$ to the Fermat equations $x_1^a+x_2^a+x_3^a=0$ and $x_1^a+x_2^a-x_3^a=0$ satisfy $x_1{x}_2{x}_3=0$. Our proofs use a generalisation of a result of Faye and Riffaut on the fields generated by sums and products of two singular moduli, which we also establish.