Torsion points on isogenous abelian varieties

TL;DR

This paper explores the distribution of torsion points on isogenous abelian varieties, proving structural results about subvarieties with dense torsion points and applying these to uniform bounds in the Manin-Mumford problem.

Contribution

It extends the understanding of torsion point distributions to families of abelian varieties, providing explicit results and bounds in the context of isogeny classes.

Findings

The geometric generic fiber of certain subvarieties is a union of torsion cosets.

Explicit bounds on the number of maximal torsion cosets in the Manin-Mumford problem.

Application of Galois automorphisms to analyze torsion points in families.

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-Mart\'inez 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.