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

We study two families of $g$-dimensional abelian varieties, induced by distinct rational maps defined on a common variety $\bar{\mathcal{A}}$ and mapping to two bases ${\bar{S}}_1$ and ${\bar{S}}_2$. Two non-torsion sections induce birational fiberwise translations on $\bar{\mathcal{A}}$. We consider the action of a specific subset of the group generated by these translations. Under the assumption that $\dim {\bar{S}}_1(=\dim {\bar{S}}_2)\le g$, 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.