Definition (Selmer-group). We define the $\varphi$-Selmer group

$$\begin{array}{llll}\hfill S^{(\varphi )}(E∕K)& =\ker \left(H^1(K,E[\varphi ])\to \prod _v{H}^1(K_v,E)\right)& \hfill & \\ \hfill & =\{\alpha \in H^1(K,E[\varphi ])|{\mathrm{res}}_v(\alpha )\in \mathrm{Im}({\delta }_v) \forall v\}& \hfill & \end{array}$$

(the map $H^1(K,E[\varphi ])\to {\prod }_v{H}^1(K_v,E)$ is as in the commutative diagram above).