 Galois cohomology and weak Mordell-Weil

For a discrete module $M$ over the absolute Galois group $G_K$, the first Galois cohomology group iswhere a one-cocycle satisfies $c(\sigma \tau )=c(\sigma )+\sigma c(\tau )$ and a coboundary has the form $c(\sigma )=\sigma m-m$.

For an integer $n\ge 2$, the Kummer exact sequence of an elliptic curveproduces the injective Kummer map of an elliptic curveExplicitly, if $nQ=P$, then $\delta (P)$ is represented by $\sigma \mapsto \sigma Q-Q$.

For every completion $K_v$ there is a local Kummer map. The n-Selmer group isIt fits intowhere $\mathrm{Sha}(E/K)$ is the Tate–Shafarevich group.

Only finitely many places divide $n$, are places of bad reduction, or are Archimedean. Outside this finite set $S$, every Selmer class is unramified. Since the finite Galois module $E[n]$ has finite order, there are only finitely many $E[n]$-valued cohomology classes unramified outside $S$; equivalently, the relevant finite extensions have bounded degree and ramification, and their number is finite by the Hermite–Minkowski theorem. Hence ${\mathrm{Sel}}^{(n)}(E/K)$ is finite, and its subgroup $E(K)/nE(K)$ is finite. This is the Weak Mordell-Weil theorem. Combined with height descent, which chooses representatives of bounded height in the finitely many cosets modulo $nE(K)$, it yields the finite generation asserted by the Mordell-Weil theorem.