Selmer group

Definition

Let $K$ be an algebraic number field with absolute Galois group $G_K$. Let $\phi \colon E \to E'$ be an isogeny of elliptic curves over $K$. That is, $\phi$ is a nonconstant morphism over $K$ that is also a group homomorphism. Let $E[\phi] = \ker(\phi \colon E(\overline{K}) \to E'(\overline{K}))$, a finite $G_K$-module. For each place $v$ of $K$, let $\delta_v \colon E'(K_v) \to H^1(K_v, E[\phi])$ be the connecting map of the Galois cohomology sequence of $0 \to E[\phi] \to E \xrightarrow{\phi} E' \to 0$ over the completion $K_v$. The $\phi$-Selmer group of $E$ over $K$ is $$\mathrm{Sel}^{(\phi)}(E/K) = \set{\xi \in H^1(K, E[\phi]) : \mathrm{res}_v(\xi) \in \delta_v(E'(K_v)) \text{ for every place } v \text{ of } K},$$ where $\mathrm{res}_v \colon H^1(K, E[\phi]) \to H^1(K_v, E[\phi])$ is restriction. For an integer $n \ge 2$, the $n$-Selmer group $\mathrm{Sel}^{(n)}(E/K)$ is the case $\phi = [n] \colon E \to E$.