Torsion points on isogenous abelian varieties

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-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.

arXiv comment31 pages, to appear in Compositio

References in corpus (12)

• Torsion points and Galois representations on CM elliptic curves
• Families of abelian varieties with many isogenous fibres
• Equidistribution in Families of Abelian Varieties and Uniformity
• Modular invariants and isogenies
• Explicit Kummer Theory for Elliptic Curves
• The Uniform Mordell-Lang Conjecture
• Point counting for foliations over number fields
• Unlikely intersections between isogeny orbits and curves
• Unlikely intersections with isogeny orbits in a product of elliptic schemes
• Some uniform bounds for elliptic curves over $\mathbb{Q}$
• On the Zilber-Pink conjecture for complex abelian varieties
• Homothéties explicites des représentations $\ell$-adiques

Cited by in corpus (1)

• The Uniform Mordell-Lang Conjecture