= Three-torsion points are flexes of a plane cubic
{title2=$3(P-O)=0$}
Let $E$ be a smooth plane cubic with an inflection point $O$ as identity. A point $P$ with $3(P-O)=0$ in <Picard group> has a line whose intersection divisor with $E$ is $3P$. Indeed, $\mathcal O_E(3P)\cong\mathcal O_E(3O)\cong\mathcal O_E(1)$. On this genus-one curve, the <Riemann-Roch theorem> gives $h^0(\mathcal O_E(1))=3$: the dual term has negative degree. Restriction of ambient linear forms is injective and its source also has dimension three, so it is an isomorphism. The section vanishing at $3P$ therefore comes from a line. Conversely, such a line gives $3P\sim3O$, so $P$ is a three-torsion point. This includes $O$; points of exact order three are the other such points.
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