Effective multiplicative independence of 3 singular moduli

TL;DR

This paper proves that for three distinct singular moduli, any multiplicative relation with non-zero integer exponents implies bounded discriminants, extending effective results from two to three moduli.

Contribution

It establishes an effective bound on discriminants for multiplicative relations among three singular moduli, advancing previous results from two to three moduli.

Findings

Discriminants of such singular moduli do not exceed 10^{10}.

Any multiplicative relation among three singular moduli with non-zero exponents is effectively bounded.

Extends effective multiplicative independence results from pairs to triples of singular moduli.

Abstract

Pila and Tsimerman proved in 2017 that for every $k$ there exists at most finitely many $k$-tuples $(x_1,\ldots, x_k)$ of distinct non-zero singular moduli with the property "$x_1, \ldots,x_k$ are multiplicatively dependent, but any proper subset of them is multiplicatively independent". The proof was non-effective, using Siegel's lower bound for the Class Number. In 2019 Riffaut obtained an effective version of this result for $k=2$. Moreover, he determined all the instances of $x^my^n\in \mathbb Q^\times$, where $x,y$ are distinct singular moduli and $m,n$ non-zero integers. In this article we obtain a similar result for $k=3$. We show that $x^my^nz^r\in \mathbb Q^\times$ (where $x,y,z$ are distinct singular moduli and $m,n,r$ non-zero integers) implies that the discriminants of $x,y,z$ do not exceed $10^{10}$.