= Abel-Jacobi map of a genus-one curve
{title2=$P\mapsto[P-O]$}
Let $E$ be a <genus one curve> over an algebraically closed field and choose $O\in E$. The map
$$
E\longrightarrow\operatorname{Cl}^0(E),
\qquad P\longmapsto[P-O]
$$
is an isomorphism. For a degree-zero divisor $D$, <Riemann-Roch theorem>[Riemann--Roch] gives a nonzero section of $D+O$, so $D\sim P-O$ for some point $P$. If $P-O\sim P'-O$ with $P\ne P'$, a rational function with divisor $P-P'$ would give a degree-one map to the <projective line>, contradicting that $E$ has genus one. The Abel--Jacobi map transports addition in the divisor class group to the elliptic-curve group law.
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