Isogeny orbits in a family of abelian varieties
Volume 170 / 2015
Acta Arithmetica 170 (2015), 161-173 MSC: Primary 11G18; Secondary 14K12, 11G50. DOI: 10.4064/aa170-2-4
We prove that if a curve of a nonisotrivial family of abelian varieties over a curve contains infinitely many isogeny orbits of a finitely generated subgroup of a simple abelian variety, then it is either torsion or contained in a fiber. This result fits into the context of the Zilber–Pink conjecture. Moreover, by using the polyhedral reduction theory we give a new proof of a result of Bertrand.