TY - JOUR

T1 - On abelian multiplicatively dependent points on a curve in a torus

AU - Ostafe, Alina

AU - Sha, Min

AU - Shparlinski, Igor E.

AU - Zannier, Umberto

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85048628979&partnerID=8YFLogxK

U2 - 10.1093/qmath/hax045

DO - 10.1093/qmath/hax045

M3 - Article

VL - 69

SP - 391

EP - 401

JO - Quarterly Journal of Mathematics

JF - Quarterly Journal of Mathematics

SN - 0033-5606

IS - 2

ER -