Multiplicative relations among differences of singular moduli

TL;DR

This paper proves that multiplicative relations among differences of singular moduli are confined to a finite set and explicitly describable algebraic curves, revealing structure in these special algebraic numbers.

Contribution

It establishes a finiteness result for multiplicative relations among differences of singular moduli and provides explicit descriptions of the associated algebraic curves.

Findings

Finite set of solutions for multiplicative relations among singular moduli

Explicit determination of algebraic curves for given n

Structural insight into differences of singular moduli

Abstract

Let $n \in \mathbb{Z}_{>0}$. We prove that there exist a finite set $V$ and finitely many algebraic curves $T_1, \ldots, T_k$ with the following property: if $(x_1, \ldots, x_n, y)$ is an $(n+1)$-tuple of pairwise distinct singular moduli such that $\prod_{i=1}^n (x_i - y)^{a_i}=1$ for some $a_1, \ldots, a_n \in \mathbb{Z} \setminus \{0\}$, then $(x_1, \ldots, x_n, y) \in V \cup T_1 \cup \ldots \cup T_k$. Further, the curves $T_1, \ldots, T_k$ may be determined explicitly for a given $n$.