Equations with powers of singular moduli

TL;DR

This paper investigates equations involving powers of singular moduli, establishing degree bounds for certain linear dependencies and demonstrating that products of powers are rarely rational, with explicit exceptions.

Contribution

It provides new results on the algebraic relations of singular moduli powers, including degree bounds and non-rationality of their products, extending understanding of their algebraic properties.

Findings

Two singular moduli with linearly dependent powers have degree at most 2.

Products of powers of singular moduli are generally not rational, except in specific cases.

Explicit exceptions are identified for both types of equations.

Abstract

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli j(\tau),j(\tau') such that the numbers 1, j(\tau)^m and j(\tau')^n are linearly dependent over $\mathbb{Q}$ for some positive integers m,n, must be of degree at most 2. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number.