The Selmer Group

Let $E$ be an elliptic curve over a number field $K$. Let $v$ be either a prime $\wp$ of $K$ or a real Archimedian place (i.e., embedding $K\to \mathbb{R}$). As in Section 2.2.1 we also obtain a local Kummer sequence

$$0 \to E(K_v)/n E(K_v) \to \H^1(K_v,E[n]) \to H^1(K_v,E)[n]\to 0.$$

Putting these together for all $v$ we obtain a commutative diagram:

$$\xymatrix @=1.2em{ 0 \ar[r]& {E(K)/n E(K)} \ar[r]\ar[d]& {\... ... \H^1(K_v,E[n])} \ar[r]& {\prod_v H^1(K_v,E)[n]} \ar[r]& 0. }$$ (2.2.2)

Definition 2.7   The $n$-Selmer group of an elliptic curve $E$ over a number field $K$ is

$$\Sel ^{(n)}(E/K) = \ker\left(\H^1(K,E[n]) \to \prod_v \H^1(K_v,E)[n]\right).$$