{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>.

Galois cohomology and weak Mordell-Weil

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 125 / 4 / Solution / Galois cohomology and weak Mordell-Weil

Pass to the finite extension $L=K(E[n])$. Part c shows that the kernel of $E(K)/nE(K)\to E(L)/nE(L)$ is finite, so it is enough to control the image after $n$-torsion becomes rational. The <Kummer map of an elliptic curve>
$$
E(L)/nE(L)\hookrightarrow H^1(L,E[n])
$$
associates to $P$ the finite extension generated by one $n$-division point of $P$.

Let $S$ contain the places over $n$, all archimedean places, and all places of bad reduction. The local theory of elliptic curves shows that these Kummer classes are unramified outside $S$. Because $E[n]$ is finite and constant over $L$, such classes are controlled by finitely many $S$-unramified power classes. Finiteness of the class group and finite generation of the unit group make that power-class group finite. The Kummer image is therefore finite, proving the <Weak Mordell-Weil theorem> that $E(K)/nE(K)$ is finite; this is the <Kummer-theoretic proof of the weak Mordell-Weil theorem>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 125 / 4 / d / Solution

{c}

For an elliptic curve over a number field and an integer $n\geq2$, the quotient $E(K)/nE(K)$ is finite. Combining this finite quotient with a height descent proves finite generation of $E(K)$.


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 Incoming links: Weak Mordell-Weil theorem