Kummer pairing

When ${\mu }_n\subset K$, the Kummer pairing sends $\sigma \in \mathrm{Gal}(L/K)$ and $[x]\in K^{\ast }/(K^{\ast }{)}^n$ to $\sigma (\sqrt[n]{x})/\sqrt[n]{x}\in {\mu }_n$. Its elliptic analogue sends $\sigma$ and $[P]\in E(K)/nE(K)$ to $\sigma (Q)-Q\in E[n]$ for $nQ=P$.

For a number field $K$ and a finite set $S$ of finite primes,