The Elliptic Curve Kummer Sequence

Let $E$ be an elliptic curve over a number field $K$. Consider the abelian group $E(\overline{\mathbb{Q}})$ of all points on $E$ defined over a fixed choice $\overline{\mathbb{Q}}$ of algebraic closure of $\mathbb{Q}$. Then $A$ is a module over $\Gal (\overline{\mathbb{Q}}/K)$, and we may consider the Galois cohomology groups

$$\H^n(K, E),\qquad\text{for $n=0,1,2,\ldots$}$$

which are of great interest in the study of elliptic curves, especially for $n=0,1$.

If $L$ is a finite Galois extension of $K$, then the inf-res sequence, written in terms of Galois chomology, is

$$0 \to \H^1(L/K, E(L)) \to \H^1(K, E) \to \H^1(L, E).$$

For any positive integer $n$ consider the homomorphism

$$[n]: E(\overline{\mathbb{Q}}) \to E(\overline{\mathbb{Q}}).$$

This is a surjective homomorphism of abelian groups, so we have an exact sequence

$$0 \to E[n] \to E \xrightarrow{[n]} E \to 0.$$

The associated long exact sequence of Galois cohomology is

$$0 \to E(K)[n] \to E(K) \xrightarrow{[n]} E(K) \to \H^1(K, E[n]) \to \H^1(K, E)\xrightarrow{[n]} \H^1(K, E)\to \cdots.$$

An interesting way to rewrite the begining part of this sequence is as

$$0 \to E(K)/n E(K) \to \H^1(K,E[n]) \to H^1(K,E)[n]\to 0.$$ (2.2.1)

The sequence (2.2.1) is called the Kummer sequence associated to the elliptic curve.