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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-2-4,
author = {Qian Lin and Ming-Xi Wang},
title = {Isogeny orbits in a family of abelian varieties},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {161-173},
zbl = {06471887},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-2-4}
}
Qian Lin; Ming-Xi Wang. Isogeny orbits in a family of abelian varieties. Acta Arithmetica, Tome 168 (2015) pp. 161-173. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-2-4/