Finite translation orbits on double families of abelian varieties

Paolo Dolce, Francesco Tropeano

Published 2024-01-13Version 1

Consider two distinct families of $g$-dimensional abelian varieties with the same domain $\mathcal A$ and codomain $S$, and both endowed with a non-torsion section. Such sections induce two fiberwise translations on $\mathcal A$. We show that if $\dim S\le g$, 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.