Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 125 / 4 / a

The natural maphas finite image by hypothesis. It remains to bound its kernel. If $P\in E(K)$ becomes $nQ$ for $Q\in E(L)$, thenis a one-cocycle for $G=\mathrm{Gal}(L/K)$. Changing $Q$ by an $n$-torsion point changes this cocycle by a coboundary, producing a well-defined map from the kernel toIf its cohomology class is zero, subtracting the corresponding torsion point from $Q$ makes $Q$ Galois fixed, so $P\in nE(K)$. The map is therefore injective. Both $G$ and $E(L)[n]$ are finite, so this group cohomology set is finite. A finite kernel and finite image give