Kummer-theoretic proof of the weak Mordell-Weil theorem

The Kummer map of an elliptic curve embeds $E(K)/nE(K)$ into $H^1(K,E[n])$. Outside a finite set containing the primes over $n$ and the primes of bad reduction, the resulting classes are unramified. Class-group and unit finiteness implies that only finitely many such classes exist, so $E(K)/nE(K)$ is finite.

The exact sequence of Galois modulesinduces the Kummer map of an elliptic curve from $E(K)/nE(K)$ into $H^1(K,E[n])$.

The $n$-Selmer group consists of the classes in $H^1(K,E[n])$ whose restriction at every place lies in the image of the local Kummer map of an elliptic curve. It is finite and contains $E(K)/nE(K)$.

The Tate–Shafarevich group consists of the principal homogeneous spaces for $E$ over $K$ that have a point over every completion of $K$. Its $n$-torsion is the quotient of the $n$-Selmer group by $E(K)/nE(K)$.