= Prime-to-characteristic torsion of an abelian variety
{title2=$A[N]\cong(\mathbb Z/N\mathbb Z)^{2\dim A}$}
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 $[N]^*L\cong L^{N^2}$ 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.
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