Linear independence of powers of singular moduli of degree 3

TL;DR

This paper proves that for degree 3 singular moduli, if certain powers are linearly dependent over rationals, then they generate the same quadratic or lower degree field, completing previous partial results.

Contribution

It extends Riffaut's theorem by resolving the remaining cases for degree 3 singular moduli, confirming the field degree bound in these special cases.

Findings

Remaining degree 3 cases are resolved.

Singular moduli with linear dependence generate at most quadratic fields.

Completes the classification of such dependencies for degree 3 singular moduli.

Abstract

We show that two distinct singular moduli $j(\tau),j(\tau')$, such that for some positive integers $m, n$ the numbers $1,j(\tau)^m$ and $j(\tau')^n$ 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.