Isogeny rigidity under algebraic group actions and finite translation orbits
Paolo Dolce, Francesco Tropeano, Ekaterina Amerik
Abstract
We study finite orbits generated by fibrewise translations on varieties carrying two abelian fibrations. Under natural dominance and generation hypotheses, non-isotriviality of the first family forces points with finite orbit to lie over proper closed subsets of both bases. This extends results of Corvaja, Tsimerman and Zannier to complex families of arbitrary, possibly different relative dimensions, including torsors. The proof combines relative Manin-Mumford with a general isogeny rigidity theorem for algebraic group actions.
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