Multiplicative relations among singular moduli

TL;DR

This paper investigates the finiteness of multiplicative relations among singular moduli, contributing to the understanding of Diophantine problems related to the Zilber-Pink conjecture in the context of complex multiplication.

Contribution

It proves a finiteness result for multiplicative relations among singular moduli, advancing the study of modular-multiplicative Diophantine problems.

Findings

Finiteness of multiplicative relations among singular moduli established

Supports aspects of the Zilber-Pink conjecture in this setting

Provides new insights into complex multiplication and modular relations

Abstract

We consider some Diophantine problems of mixed modular-multiplicative type associated with the Zilber-Pink conjecture. In particular, we prove a finiteness statement for the number of multiplicative relations between singular moduli (j-invariants of elliptic curves with complex multiplication.)