  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.
