Linear independence of powers of singular moduli of degree 3

Florian Luca, Antonin Riffaut

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 $\mathbb{Q}$ ge…