Multiplicative relations among differences of singular moduli

TL;DR

This work studies multiplicative relations among differences of singular moduli, proving that for any fixed $n$ every $(n+1)$-tuple $(x_1,\dots,x_n,y)$ of pairwise distinct singular moduli with $\prod_{i=1}^n (x_i - y)^{a_i}=1$ (for some nonzero integers $a_i$) lies either in a finite set $V$ or on finitely many explicitly describable multiplicative special curves $T_j$, with these curves effectively computable. The main technique combines model-theoretic o-minimal point-counting (Pila–Wilkie, Habegger–Pila) with functional transcendence (Ax–Lindemann) and the Zilber–Pink philosophy, reducing potential infinite families to MSCs built from modular polynomials $\Phi_N$ via $F_N(X)=\Phi_N(X,X)$ and the identity $F_N(j(z))=\prod_{g\in C(N)} (j(z)-j(gz))$. The paper provides an explicit structural description of all MSCs, proves finiteness of MSCs for each $n$ (and shows none exist for $n\le 5$), and establishes a conditional link to Zilber–Pink that would yield the main finiteness result unconditionally given ZP. Collectively, these results give a concrete, effective framework for classifying multiplicative relations among singular moduli, with potential applications to atypical intersections and CM theory.

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 $n$.

Figures (1)

• Figure 1: The fundamental domain $\mathfrak{F}_j$

Theorems & Definitions (70)

• Theorem 1.1
• Definition 1.2: BiluLucaMasser17
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Example 1.7
• Proposition 2.1
• proof
• Lemma 2.2