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 $\dim {\overline{S}}_1(=\dim {\overline{S}}_2)\le g$, 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.

26 pages, 2 figures. Substantially rewritten. Corrected proof of the second case in the main theorem