= Galois cohomology of an elliptic curve
{c}
{title2=$H^i(K,E)$}
For an <elliptic curve> $E/K$, regard $E(K^{\rm sep})$ as a discrete <Galois module> for the <absolute Galois group> $G_K$. Its continuous <Galois cohomology> has $H^0(K,E)=E(K)$; $H^1(K,E)$ classifies torsors under $E$. For an integer $n$ invertible in $K$, the <Kummer exact sequence of an elliptic curve> gives $0\to E(K)/nE(K)\to H^1(K,E[n])\to H^1(K,E)[n]\to0$. If $[n]Q=P$, the first map sends $P$ to $\sigma\mapsto\sigma Q-Q$. Changing $Q$ changes the cocycle by a coboundary; this class vanishes exactly when $P$ has a division point over $K$.
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