Greatest common divisor results on semiabelian varieties and a conjecture of Silverman

A divisibility sequence is a sequence of integers { d n } such that d m divides d n if m divides n . Results of Bugeaud, Corvaja, Zannier, among others, have shown that the gcd of two divisibility sequences corresponding to subgroups of the multiplicative group grows in a controlled way. Silverman c...

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Bibliographic Details
Published in:Research in number theory 2024-03, Vol.10 (1), Article 17
Main Authors: Barroero, Fabrizio, Capuano, Laura, Turchet, Amos
Format: Article
Language:English
Subjects:
Curves
Group theory
Mathematics
Mathematics and Statistics
Number Theory
Subgroups
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Summary:A divisibility sequence is a sequence of integers { d n } such that d m divides d n if m divides n . Results of Bugeaud, Corvaja, Zannier, among others, have shown that the gcd of two divisibility sequences corresponding to subgroups of the multiplicative group grows in a controlled way. Silverman conjectured that a similar behaviour should appear in many algebraic groups. We extend results by Ghioca–Hsia–Tucker and Silverman for elliptic curves and prove an analogue of Silverman’s conjecture over function fields for abelian and split semiabelian varieties and some generalizations of this result. We employ tools coming from the theory of unlikely intersections as well as properties of the so-called Betti map associated to a section of an abelian scheme.
ISSN:2522-0160
2363-9555
DOI:10.1007/s40993-023-00494-2