On cyclotomic multiplicatively dependent points on a curve in a torus

We show, under some natural conditions, that the set of cyclotomic (or abelian) multiplicatively dependent points in 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.