Divisibility of torsion subgroups of abelian surfaces over number fields

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.