Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 125 / 1 / b

Embed $E$ as a smooth plane cubic with $O_E$ an 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 isand the line through $S$ and $O_E$ gives $(S)+(-S)-2(O_E)$. Their quotient therefore showsUnder the bijection from part (a), the chord-and-tangent operation is exactly addition in the abelian group ${\mathrm{Pic}}^0(E)$. 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.