Effective multiplicative independence of 3 singular moduli

TL;DR

The paper proves an explicit bound for triples of singular moduli: if x,y,z are distinct nonzero singular moduli and m,n,r ≠ 0 with x^m y^n z^r ∈ Q^×, then max{|Δ_x|,|Δ_y|,|Δ_z|} < 10^10. Building on Pila–Tsimerman and Riffaut, it develops an effective framework by linking multiplicative relations to linear relations among exponents via quantities f(x)/a(x) tied to Gauss denominators, and then bounds the exponents using Masser-type height arguments and isogeny theory. A detailed, case-based analysis using ring class fields, 2-elementary discriminants, and Galois-theoretic lemmas rules out all large-discriminant configurations, yielding the explicit bound and enabling potential complete classifications of such triples. The results advance the understanding of multiplicative independence among CM values and provide a template for effective finiteness results for higher-k cases in the arithmetic 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 $10^{10}$.

Theorems & Definitions (84)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Remark 2.2
• proof
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• proof