Equations in three singular moduli: the equal exponent case

TL;DR

This paper classifies all singular moduli solutions to certain algebraic equations involving equal exponents, including Fermat equations, showing that solutions must have a zero among the moduli.

Contribution

It provides an explicit classification of singular moduli solutions to specific algebraic equations and generalizes previous results on fields generated by sums and products of two singular moduli.

Findings

Solutions to the Fermat equations in singular moduli have a zero among the variables.

All solutions to the classified equations satisfy that the product of the three singular moduli is zero.

The paper extends known results on fields generated by sums and products of two singular moduli.

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.