Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 125 / 5 / b / Solution

The principal divisor criterion on an elliptic curve says that $D=\sum n_P(P)$ is principal exactly when $\mathrm{deg}D=0$ and $\sum [n_P]P=O$.

On $E^{\prime }:y^2+dy=x^3$, takeThe line $y=0$ meets the cubic three times at $T$, while $y$ has a triple pole at the point at infinity. HenceWith $g=u/v\in \mathbb{Q}(E)$, part (a) givesTaking divisors and cancelling the factor three yieldsPullback on degree-zero divisor classes is the dual isogeny, so the pulled-back class is represented by $\hat{\varphi }(T)$. It is principal, and therefore