Multiplicative relations among differences of singular moduli

authored by

Vahagn Aslanyan, Sebastian Eterović, Guy Fowler

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 \(n\).

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Preprint

Publication date

23.08.2023

Publication status

E-pub ahead of print

Electronic version(s)

https://doi.org/10.48550/arXiv.2308.12244 (Access: Open)