On abelian points of varieties intersecting subgroups in a torus

Abstract

We show, under some natural conditions, that the set of abelian points on the non-anomalous dense 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.