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

Consider two families of $g$-dimensional abelian varieties induced by two distinct rational maps on the same variety $\overline{\mathcal A}$ onto two bases $\overline S_1$ and $\overline S_2$ and having big common domain of definition. Two non-torsion sections of these families induce two (birational) fiberwise translations on $\overline{\mathcal A}$, respectively. We show that if $\dim \overline S_1+\dim \overline S_2\le 2g$, the points with finite orbit under the action of a certain subset of the group generated by both translations lie in a proper Zariski closed subset that can be described to a certain extent. Our work is a higher dimensional generalization of a result of Corvaja, Tsimermann and Zannier.
Comment: 26 pages, 2 figures. Added an appendix by E. Amerik