# Torsion points with multiplicatively dependent coordinates on elliptic curves

## TL;DR

This paper investigates the finiteness of torsion points on elliptic curves with multiplicatively dependent coordinates, providing effective results over certain fields.

## Contribution

It proves finiteness of such torsion points on elliptic curves over number fields and offers effective bounds for curves over rationals or with complex multiplication.

## Key findings

- Finiteness of torsion points with multiplicative dependence over number fields
- Effective bounds for rational and CM elliptic curves
- Extension of known results to broader classes of elliptic curves

## Abstract

In this paper, we study the finiteness problem of torsion points on an elliptic curve whose coordinates satisfy some multiplicative dependence relations. In particular, we prove that on an elliptic curve defined over a number field there are only finitely many torsion points whose coordinates are multiplicatively dependent. Moreover, we produce an effective result when the elliptic curve is defined over the rational numbers or has complex multiplication.