On abelian multiplicatively dependent points on a curve in a torus

2017/04/15 by Ostafe, Alina, Sha, Min, Shparlinski, Igor E. +1
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1704.04694

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 \Gmⁿ 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.

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• Multiplicative and linear dependence in finite fields and on elliptic curves modulo primes
• Roots of unity and torsion points of abelian varieties
• Primitive algebraic points on curves
• Theoreme de Dobrowolski-Laurent pour les extensions abeliennes sur une courbe elliptique a multiplication complexe