Equations in three singular moduli: the equal exponent case

Abstract

Let $a\in {\mathbb{Z}}_{>0}$ and ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3\in \{\pm 1\}$. We classify explicitly all singular moduli $x_1,x_2,x_3$ satisfying either ${\epsilon }_1{x}_1^a+{\epsilon }_2{x}_2^a+{\epsilon }_3{x}_3^a\in \mathbb{Q}$ 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.

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