Effective multiplicative independence of three singular moduli

Bilu, Yuri; Gun, Sanoli; Tron, Emanuele

doi:10.2140/ant.2026.20.1073

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Effective multiplicative independence of three singular moduli

Yuri Bilu, Sanoli Gun and Emanuele Tron

Vol. 20 (2026), No. 6, 1073–1123

DOI: 10.2140/ant.2026.20.1073

Abstract

Pila and Tsimerman proved in 2017 that for every $k$ there exist at most finitely many $k$ -tuples $(x_{1}, \dots, x_{k})$ of nonzero singular moduli such that $x_{1}, \dots, x_{k}$ are multiplicatively dependent, but any proper subset of them is multiplicatively independent. The proof was noneffective, using Siegel’s lower bound for the class numbers. In 2019 Riffaut obtained an effective version of this result for $k = 2$ . Moreover, he determined all the instances of $x^{m} y^{n} \in ℚ^{\times}$ , where $x, y$ are distinct singular moduli and $m, n$ are nonzero integers. In this article we obtain a similar result for $k = 3$ . We show that $x^{m} y^{n} z^{r} \in ℚ^{\times}$ (where $x, y, z$ are distinct singular moduli and $m, n, r$ nonzero integers) implies that the discriminants of $x, y, z$ do not exceed $1 0^{10}$ .

Keywords

singular moduli

Mathematical Subject Classification

Primary: 11G15

Secondary: 11G18

Milestones

Received: 3 August 2022

Revised: 5 July 2024

Accepted: 2 May 2025

Published: 26 May 2026

Authors

Yuri Bilu
Institut de Mathématiques de Bordeaux Université de Bordeaux & CNRS Talence France

Sanoli Gun
Institute of Mathematical Sciences CIT Campus, Tharamani Chennai India

Emanuele Tron
Institut de Mathématiques de Bordeaux Université de Bordeaux & CNRS Talence France

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Vol. 20, No. 6, 2026