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Isogeny orbits in a family of abelian varieties

Volume 170 / 2015

Qian Lin, Ming-Xi Wang Acta Arithmetica 170 (2015), 161-173 MSC: Primary 11G18; Secondary 14K12, 11G50. DOI: 10.4064/aa170-2-4

Abstract

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.

Authors

  • Qian LinCenter of Mathematical Sciences and Applications
    Harvard University
    Cambridge, MA 02138 U.S.A.
    e-mail
  • Ming-Xi WangDepartment of Mathematics
    University of Salzburg
    Hellbrunnerstr. 34/I
    5020 Salzburg, Austria
    e-mail

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