= Halving cocycle with rational two-torsion
{title2=$c_P(\sigma)=\sigma(Q)-Q,\quad 2Q=P$}
Suppose an <elliptic curve> $E/K$ has all four geometric <2-torsion> points rational. For $P\in E(K)$ choose $Q\in E(\overline K)$ with $2Q=P$. The function $c_P(\sigma)=\sigma(Q)-Q$ takes values in $E[2]$ and is a <group homomorphism> from the <absolute Galois group> to $E[2]$: its usual cocycle law becomes additivity because the action on $E[2]$ is trivial. Replacing $Q$ by another half, or $P$ by $P+2R$ with $R\in E(K)$, leaves it unchanged. Adding chosen halves proves additivity in $P$. Its kernel in $E(K)/2E(K)$ is zero, since $c_P=0$ precisely when the chosen half is rational. This constructs the injective map directly, without assuming any theorem of <two-descent on an elliptic curve> or finite generation of the <Mordell-Weil group>.
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