On abelian multiplicatively dependent points on a curve in a torus

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 \(\Gm\) restrict...

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Bibliographic Details
Published inarXiv.org
Main Authors Ostafe, Alina, Sha, Min, Shparlinski, Igor E, Zannier, Umberto
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 31.01.2018
Subjects
Dependence
Number theory
Toruses
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Summary: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.
ISSN:2331-8422