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Torsion points on isogenous abelian varieties

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry Abelian varieties and schemes

Published online by Cambridge University Press:  20 July 2022

Gabriel A. Dill
Gabriel A. Dill*
Affiliation:
Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Welfengarten 1, 30167 Hannover, Germany dill@math.uni-hannover.de
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Abstract

Investigating a conjecture of Zannier, we study irreducible subvarieties of abelian schemes that dominate the base and contain a Zariski dense set of torsion points that lie on pairwise isogenous fibers. If everything is defined over the algebraic numbers and the abelian scheme has maximal variation, we prove that the geometric generic fiber of such a subvariety is a union of torsion cosets. We go on to prove fully or partially explicit versions of this result in fibered powers of the Legendre family of elliptic curves. Finally, we apply a recent result of Galateau and Martínez to obtain uniform bounds on the number of maximal torsion cosets in the Manin–Mumford problem across a given isogeny class. For the proofs, we adapt the strategy, due to Lang, Serre, Tate, and Hindry, of using Galois automorphisms that act on the torsion as homotheties to the family setting.

Keywords

isogenyabelian schemeManin–Mumfordeffectivityuniformityhomothety

MSC classification

Primary: 11G05: Elliptic curves over global fields 14K02: Isogeny 14K12: Subvarieties
Secondary: 11G50: Heights 14G40: Arithmetic varieties and schemes; Arakelov theory; heights
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 5 , May 2022 , pp. 1020 - 1051
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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