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

2022/05/11 by Fabrizio Barroero, Barroero, Fabrizio, Laura Capuano +3 · 1 citation
Computer Science · Mathematics · #11G10 #14G05 #14K15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2205.05562

openalex publication_date 2022/05/11 · openalex created_date 2022/05/22 · openalex updated_date 2026/07/28

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.

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• Singular intersections in families of abelian varieties

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