Linear independence of powers of singular moduli of degree 3

Abstract

We show that two distinct singular moduli $j(\tau ),j({\tau }^{\prime })$, such that for some positive integers $m,n$ the numbers $1,j(\tau {)}^m$ and $j({\tau }^{\prime }{)}^n$ are linearly dependent over $\mathbb{Q}$ generate the same number field of degree at most $2$. This completes a result of Riffaut, who proved the above theorem except for two explicit pair of exceptions consisting of numbers of degree $3$. The purpose of this article is to treat these two remaining cases.