Greatest Common Divisor results on semiabelian varieties and a Conjecture of Silverman

TL;DR

This paper extends results on the growth of gcd in divisibility sequences to semiabelian varieties over function fields, supporting Silverman's conjecture using tools from unlikely intersections and Betti maps.

Contribution

It proves an analogue of Silverman's conjecture for abelian and split semiabelian varieties over function fields, generalizing previous elliptic curve results.

Findings

GCD of divisibility sequences behaves predictably in semiabelian varieties.

Silverman's conjecture holds over function fields for certain algebraic groups.

Utilizes unlikely intersections and Betti maps in the proof.

Abstract

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.