# On abelian multiplicatively dependent points on a curve in a torus

## TL;DR

This paper proves finiteness results for abelian and cyclotomic multiplicatively dependent points on algebraic curves in tori, characterizing their structure via roots of unity and primitive characters, with extensions to Bogomolov fields.

## Contribution

It introduces the concept of primitive multiplicative dependence and establishes finiteness theorems for such points over specific number field extensions.

## Key findings

- Finiteness of abelian multiplicatively dependent points on curves in tori.
- Characterization of these points as preimages of roots of unity via primitive characters.
- Extension of finiteness results to Bogomolov extensions.

## Abstract

We show, under some natural conditions, that the set of abelian (and thus also cyclotomic) multiplicatively dependent points on an irreducible curve over a number field is a finite union of preimages of roots of unity by a certain finite set of primitive characters from $\Gm^n$ to $\Gm$ restricted to the curve, and a finite set. We also introduce the notion of primitive multiplicative dependence and obtain a finiteness result for primitively multiplicatively dependent points defined over a so-called Bogomolov extension of a number field.