= Solution
<Galois cohomology> records the obstruction to choosing Galois-invariant division points. Let $G_K=\operatorname{Gal}(\overline K/K)$ be the <absolute Galois group> and give its algebraic-point modules the discrete topology. A continuous $1$-cocycle with values in a <Galois module> $M$ is a function $c:G_K\to M$ satisfying
$$
c_{\sigma\tau}=c_\sigma+\sigma c_\tau.
$$
A <group coboundary> has the form $c_\sigma=\sigma a-a$. Quotienting cocycles by coboundaries defines $H^1(K,M)$. For the finite module $E[m]$ these cocycles have finite image and factor through finite data; continuity is essential because $G_K$ is a <profinite group>.
In characteristic zero, multiplication by $m\geq2$ on the algebraic points of an <elliptic curve> is surjective, with kernel $E[m]\cong(\mathbb Z/m\mathbb Z)^2$. The short exact sequence of Galois modules
$$
0\longrightarrow E[m]\longrightarrow E(\overline K)\xrightarrow{[m]}E(\overline K)\longrightarrow0
$$
gives the <Kummer exact sequence of an elliptic curve>
$$
\boxed{0\longrightarrow E(K)/mE(K)\xrightarrow{\delta}H^1(K,E[m])\longrightarrow H^1(K,E)[m]\longrightarrow0.}
$$
Explicitly, choose $Q$ with $mQ=P$ and set $\delta(P)_\sigma=\sigma Q-Q$. Changing $Q$ by an $m$-torsion point changes the cocycle by a coboundary. The class vanishes exactly when a suitable choice of $Q$ is Galois-fixed, that is, when $P\in mE(K)$. This proves the injection directly and identifies its arithmetic meaning.
The entire group $H^1(K,E[m])$ need not be finite. For example, <Kummer theory> gives $H^1(K,\mu_m)=K^*/K^{*m}$, which has classes supported on arbitrarily many different primes. The finite part needed for the <Weak Mordell-Weil theorem> comes from a ramification restriction on the image of $\delta$.
Choose a finite set $S$ of places containing the archimedean places, the primes dividing $m$ and all primes of <bad reduction of an elliptic curve>. At a finite place $v\notin S$, the curve has good reduction and $m$ is invertible in its valuation ring. The elliptic curve extends to a smooth proper group scheme, and $[m]$ on that model is finite etale. A point $P\in E(K_v)$ extends to an integral section by properness. Its division-point fibre is consequently finite etale over the valuation ring. Over the maximal unramified extension it has a point, so the Kummer cocycle restricts trivially to the <inertia group>. The module $E[m]$ itself is unramified there for the same reason. Thus
$$
\delta(E(K)/mE(K))\subseteq H^1_S(K,E[m]),
$$
where $H^1_S$ denotes classes unramified outside $S$.
Here is a proof that this restricted group is finite. Take a finite Galois extension $L/K$ containing all coordinates of $E[m]$ and the $m$th roots of unity, and enlarge $S$ by its ramified primes. Over $L$ the module is trivial and, after choosing a basis, is isomorphic to $\mu_m^2$. <Hilbert theorem 90> and the multiplicative Kummer sequence therefore give
$$
H^1(L,E[m])\cong(L^*/L^{*m})^2.
$$
For <unramified Kummer classes with bounded prime support>, a class unramified outside $S_L$ has valuations divisible by $m$ at every prime outside $S_L$: the valuation of an $m$th root in an unramified extension is integral. Consequently both coordinates lie in
$$
L(S_L,m)=\{[a]\in L^*/L^{*m}:v_{\mathfrak p}(a)\equiv0\pmod m\text{ for }\mathfrak p\notin S_L\}.
$$
To show this set finite, write the outside-$S_L$ divisor of $a$ as $mD$. The ideal class of $D$ belongs to the $m$-torsion of the <ideal class group> of the ring of $S_L$-integers. This gives the exact sequence
$$
0\longrightarrow\mathcal O_{L,S_L}^{\times}/(\mathcal O_{L,S_L}^{\times})^m
\longrightarrow L(S_L,m)
\longrightarrow\operatorname{Cl}(\mathcal O_{L,S_L})[m]
\longrightarrow0.
$$
The last map is onto: if $mD$ is principal in the $S_L$-ideal group, a generator represents a class with outside valuations divisible by $m$. The kernel consists exactly of <S-units> modulo $m$th powers. The <Dirichlet unit theorem>, with the finitely many inverted primes adjoined, makes the <S-unit group> finitely generated; the <ideal class group> is finite, and localization only quotients it. Both ends of the sequence are therefore finite.
Finally, the <inflation-restriction exact sequence> bounds the kernel of restriction from $H^1(K,E[m])$ to $H^1(L,E[m])$ by the finite group $H^1(\operatorname{Gal}(L/K),E[m])$. The unramified subgroup has finite image, contained in $L(S_L,m)^2$, and finite kernel, so it is finite. The Kummer injection now proves
$$
\boxed{E(K)/mE(K)\text{ is finite for every }m\geq2.}
$$
This is the <Weak Mordell-Weil theorem>, proved without first assuming finite generation of $E(K)$.
For computation one refines the unramified group by local solvability. The <Selmer group of an elliptic curve> consists of classes whose restriction at every completion lies in the corresponding local Kummer image. It is finite and sits in
$$
0\longrightarrow E(K)/mE(K)\longrightarrow\operatorname{Sel}_m(E/K)\longrightarrow\operatorname{Sha}(E/K)[m]\longrightarrow0,
$$
where the <Tate–Shafarevich group> $\operatorname{Sha}$ measures classes in $H^1(K,E)$ that become trivial at every completion. Thus locally soluble descent equations can give an upper bound without every class coming from a rational point. No finiteness assumption on the whole Tate-Shafarevich group is needed for the weak theorem. Combining the finite quotient with the <height descent lemma> from the height essay yields the full <Mordell-Weil theorem>. Replacing multiplication by $m$ with a smaller isogeny gives the same cohomological framework for <two-isogeny descent>.
Back to article page
