Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 125 / 4 / d / Solution

Pass to the finite extension $L=K(E[n])$. Part c shows that the kernel of $E(K)/nE(K)\to E(L)/nE(L)$ is finite, so it is enough to control the image after $n$-torsion becomes rational. The Kummer map of an elliptic curveassociates to $P$ the finite extension generated by one $n$-division point of $P$.

Let $S$ contain the places over $n$, all archimedean places, and all places of bad reduction. The local theory of elliptic curves shows that these Kummer classes are unramified outside $S$. Because $E[n]$ is finite and constant over $L$, such classes are controlled by finitely many $S$-unramified power classes. Finiteness of the class group and finite generation of the unit group make that power-class group finite. The Kummer image is therefore finite, proving the Weak Mordell-Weil theorem that $E(K)/nE(K)$ is finite; this is the Kummer-theoretic proof of the weak Mordell-Weil theorem.