Finite translation orbits on double families of abelian varieties (with an appendix by E. Amerik)

Abstract

We study two families of $g$-dimensional abelian varieties, induced by distinct rational maps defined on a common variety $\overline{\mathcal A}$ and mapping to two bases $\overline{S}_1$ and $\overline{S}_2$. Two non-torsion sections induce birational fiberwise translations on $\overline{\mathcal A}$. We consider the action of a specific subset of the group generated by these translations. Under the assumption that $\operatorname{dim} \overline{S}_1 (= \operatorname{dim} \overline{S}_2) \leq g$, we prove that the points with finite orbit are contained in a proper Zariski closed subset. This subset is explicitly described to a certain extent. Our results generalize a theorem of Corvaja, Tsimermann, and Zannier to higher dimensions.