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Isogeny orbits in a family of abelian varieties. (English) Zbl 1382.11045
Summary: 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 D. Bertrand [Duke Math. J. 80, No. 1, 223–250 (1995; Zbl 0847.11036)].

11G18 Arithmetic aspects of modular and Shimura varieties
11G50 Heights
14K10 Algebraic moduli of abelian varieties, classification
Full Text: DOI arXiv
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