Equations in three singular moduli: The equal exponent case

Abstract

Let a∈Z>0 and ϵ1,ϵ2,ϵ3∈{±1}. We classify explicitly all singular moduli x1,x2,x3 satisfying either ϵ1x1a+ϵ2x2a+ϵ3x3a∈Q or (x1ϵ1x2ϵ2x3ϵ3)a∈Q×. In particular, we show that all the solutions in singular moduli x1,x2,x3 to the Fermat equations x1a+x2a+x3a=0 and x1a+x2a−x3a=0 satisfy x1x2x3=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.