Equations in three singular moduli: The equal exponent case

Let a∈Z>0 and ε1, ε2, ε3 ∈ {±1}. We classify explicitly all singular moduli x1, x2, x3 satisfying either ε1 x1 a + ε2 x2 a+ ε3 x3 a ∈ Q or (x1 ε1 x2 ε2 x3 ε3) a ∈ Q×. In particular, we show that all the solutions in singular moduli x1, x2, x3 to the Fermat equations x1 a +x2 a+ x3 a = 0 and x1 a+ x2 a − x3 a = 0 satisfy x1 x2 x3 = 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.