A geometric approach to the uniform boundedness of $\ell$-primary torsion points

TL;DR

This paper introduces a geometric method leveraging Betti foliations and equidistribution to prove the uniform boundedness of $\,\ell$-primary torsion points on abelian schemes over curves, providing new proofs and resolving existing conjectures.

Contribution

It offers a novel geometric approach to the uniform boundedness of torsion points, extending previous results and resolving conjectures without extra assumptions.

Findings

Proved that the genus of multi-sections with small heights tends to infinity.

Provided a new proof of the uniform boundedness of $\,\ell$-primary torsion points.

Resolved a conjecture of Cadoret and Tamagawa using geometric methods.

Abstract

We prove that for a non-isotrivial abelian scheme over a smooth curve, the genus of a generic sequence of multi-sections with small heights tends to infinity. As an application, we give a new proof of the uniform boundedness of $\ell$-primary torsion points on fibers of an abelian scheme over a smooth curve, a result originally proved by Cadoret and Tamagawa. Furthermore, our approach allows us to resolve a conjecture of Cadoret and Tamagawa without additional assumptions. Our approach is based on the theory of Betti foliations and the arithmetic equidistribution theorem.