Prime-to-characteristic torsion of an abelian variety ($A[N]\cong (\mathbb{Z}/N\mathbb{Z}{)}^{2\dim A}$)

If $N$ is invertible in the base field, multiplication by $N$ on an abelian variety is finite étale: its differential is $N$ times the identity, and properness turns its zero-dimensional fibres into finite fibres. The Theorem of the Cube gives $[N{]}^{\ast }L\cong L^{N^2}$ for a symmetric ample line bundle. Top intersection products give degree $N^{2\dim A}$. For each prime power ${\ell }^r\mid N$, the kernels killed by ${\ell }^s$ have orders ${\ell }^{2s\dim A}$; the classification of finite abelian groups forces $2\dim A$ cyclic factors of length $r$. Combining the prime powers gives the displayed group.