Prime-to-characteristic geometric torsion
= Prime-to-characteristic geometric torsion
{title2=$E[\ell]\cong(\mathbb Z/\ell\mathbb Z)^2$}
If $\ell$ is a prime different from the characteristic, multiplication by $\ell$ on an <elliptic curve> has degree $\ell^2$ and nonzero differential. It is a <separable isogeny>, so its geometric kernel has $\ell^2$ points, all killed by $\ell$. This gives the displayed group over an algebraic closure; the rational subgroup over the base field may be smaller.