= Solution
Embed $E$ as a smooth plane cubic with $O_E$ an <inflection point of a plane cubic>[inflection point]. A line through $P,Q$, using the tangent when $P=Q$, has a third intersection $S$ counted with multiplicity. The <chord-and-tangent group law> defines $P+Q=-S$, where $-S$ is the third point on the line through $S$ and $O_E$.
The divisor cut out by the first line is
$$
(P)+(Q)+(S)-3(O_E),
$$
and the line through $S$ and $O_E$ gives $(S)+(-S)-2(O_E)$. Their quotient therefore shows
$$
[(P)-(O_E)]+[(Q)-(O_E)]=[(P+Q)-(O_E)].
$$
Under the bijection from part (a), the chord-and-tangent operation is exactly addition in the <abelian group> $\operatorname{Pic}^0(E)$. It is consequently associative and commutative, has identity $O_E$, and has the geometrically defined point $-P$ as inverse. Thus it makes $E$ an abelian group.
Back to article page
