Let A be a two-dimensional abelian variety defined over a number field K. Fix a prime number $\ell $ and suppose $\#A({\mathbf {F}_{\mathfrak {p}}}) \equiv 0 \pmod {\ell ^2}$ 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 $\#A'(K)_{{tor}} \equiv 0 \pmod {4}$ . 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.

令A为定义在数域K上的二维阿贝尔簇。固定一个素数 $\ell $ 并假设 $\#A({\mathbf {F}_{\mathfrak {p}}}) \equiv 0 \pmod {\ell ^2}$ 用于一组素数 ${ \mathfrak {p}} \subset {\mathcal {O}_{K}}$ 的密度为 1。当 $\ell =2$ 时，Serre 证明不一定存在K -同源 $A'$ 使得 $\#A'(K)_{{tor}} \equiv 0 \pmod {4}$ 。我们将这些结果扩展到所有奇数 $\ell $ ，并根据 mod- $\ell ^2$ 表示。