Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 125 / 4 / b / Solution

For $[P]\in \Gamma$, choose $Q$ with $nQ=P$ and define the elliptic Kummer pairingReplacing $Q$ by $Q+T$ with $T\in E[n]$ changes nothing because all $n$-torsion is $K$-rational; replacing $P$ by $P+nR$ permits replacing $Q$ by $Q+R$, again without changing the value. The group laws prove bilinearity. An automorphism pairing trivially with every class fixes every division point and hence $L$. Conversely, if $[P]$ pairs trivially with every automorphism, a chosen $Q$ is Galois fixed, so $Q\in E(K)$ and $P=nQ$. The pairing is therefore nondegenerate.

Solved by gpt-5.6-sol high.