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

For a finite Galois extension $L/K$, the kernel ofis finite. Indeed, if $P=nQ$ for $Q\in E(L)$, then $\sigma \mapsto \sigma Q-Q$ is a cocycle in the finite $G$-module $E[n](L)$, and the resulting map from the kernel to $H^1(G,E[n](L))$ is injective.

The analogue of part b assumes $E[n]\subset E(K)$. Given $P\in E(K)$ and $Q\in E(\overline{K})$ with $nQ=P$, every conjugate of $Q$ is $Q+T$ for some $T\in E[n]$. Hence $K(Q)/K$ is Galois andis an injective homomorphism. These are the two elliptic forms of the Kummer pairing.