Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 125 / 5 / a / Solution

Let $T=(0,0)$, the rational point of order two. Definebywhere the right sides denote square classes.

To prove multiplicativity, let a nonvertical line $y=mx+c$ meet $E$ in points with $x$-coordinates $x_1,x_2,x_3$. Substitution gives the monic cubicso Vieta formulas give $x_1{x}_2{x}_3=c^2$. If the first two intersection points are $P,Q$, the third is $-(P+Q)$ and has the same $x$-coordinate as $P+Q$. Thereforein the square-class group. Tangencies follow by taking a repeated root, and the vertical and exceptional cases give the stated values at $O$ and $T$. Thus $\alpha$ is a group homomorphism.