Kummer map of an elliptic curve

The multiplication-by-$n$ sequence gives an injective connecting mapLocal conditions restrict its image to the finite $n$-Selmer group.

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,

For $E:y^2=x^3+a{x}^2+bx$ and its two-isogenous curve $E^{\prime }:y^2=x^3-2a{x}^2+(a^2-4b)x$, square-class maps on $E(\mathbb{Q})$ and $E^{\prime }(\mathbb{Q})$ determine the Mordell-Weil rank.