On abelian points of varieties intersecting subgroups in a torus

TL;DR

This paper proves finiteness and structural results for abelian points on certain subvarieties of tori, extending previous work to non-connected subgroups and applications in curves and dynamics.

Contribution

It generalizes existing theorems by removing connectivity assumptions and applies to broader contexts like curves and arithmetic dynamics.

Findings

Finiteness of abelian points under specified conditions

Generalization of structure theorem to non-connected subgroups

Application to curves and arithmetic dynamics

Abstract

We show, under some natural conditions, that the set of abelian points on the non-anomalous subset of a closed irreducible subvariety $X$ intersected with the union of connected algebraic subgroups of codimension at least $\dim X$ in a torus is finite, generalising results of Ostafe, Sha, Shparlinski and Zannier (2017). We also generalise their structure theorem for such sets when the algebraic subgroups are not necessarily connected, and obtain a related result in the context of curves and arithmetic dynamics.