Divisibility of torsion subgroups of abelian surfaces over number fields

TL;DR

This paper investigates the divisibility properties of torsion subgroups of abelian surfaces over number fields, extending previous results for prime 2 to all odd primes and classifying exceptions via Galois representations.

Contribution

It generalizes Serre's divisibility results from prime 2 to all odd primes and characterizes abelian varieties that do not satisfy the divisibility principle using mod-$oldsymbol{\ell^2}$ Galois representations.

Findings

Extends divisibility results to all odd primes.

Classifies abelian varieties failing the divisibility principle.

Connects failures to the image of mod-$oldsymbol{\ell^2}$ Galois representations.

Abstract

Let $A$ be a 2-dimensional abelian variety defined over a number field $K$. Fix a prime number $\ell$ and suppose $\#A(\mathbb{F}_p) \equiv 0 \pmod{\ell^2}$ for a set of primes $\mathfrak{p} \subset \mathcal{O}_K$ of density 1. When $\ell=2$ Serre has shown that there does not necessarily exist a $K$-isogenous $A'$ such that $\#A'(K)_{\mathrm{tors}} \equiv 0 \pmod{4}$. We extend those results to all odd $\ell$ and classify the abelian varieties that fail this divisibility principle for torsion in terms of the image of the mod-$\ell^2$ representation.