= Galois cohomology and weak Mordell-Weil
{c}
For a discrete module $M$ over the absolute Galois group $G_K$, the first <Galois cohomology> group is
$$
H^1(K,M)=Z^1(G_K,M)/B^1(G_K,M),
$$
where 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 $n\geq2$, the <Kummer exact sequence of an elliptic curve>
$$
0\longrightarrow E[n]\longrightarrow E(\overline K)
\xrightarrow{[n]}E(\overline K)\longrightarrow0
$$
produces the injective <Kummer map of an elliptic curve>
$$
\delta:E(K)/nE(K)\hookrightarrow H^1(K,E[n]).
$$
Explicitly, 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> is
$$
\operatorname{Sel}^{(n)}(E/K)
=\{c\in H^1(K,E[n]):c_v\in\operatorname{im}\delta_v\text{ for every }v\}.
$$
It fits into
$$
0\longrightarrow E(K)/nE(K)
\longrightarrow\operatorname{Sel}^{(n)}(E/K)
\longrightarrow\operatorname{Sha}(E/K)[n]
\longrightarrow0,
$$
where $\operatorname{Sha}(E/K)$ 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 $\operatorname{Sel}^{(n)}(E/K)$ 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>.
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