Author:
Barroero Fabrizio,Capuano Laura,Turchet Amos
Abstract
AbstractA divisibility sequence is a sequence of integers $$\{d_n\}$$
{
d
n
}
such that $$d_m$$
d
m
divides $$d_n$$
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.
Funder
Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni
Università degli Studi Roma Tre
Publisher
Springer Science and Business Media LLC