Galois 1-cocycle ($z_{\sigma \tau }=z_{\sigma }+\sigma z_{\tau }$)

A continuous one-cocycle for a Galois group acting on an abelian module $M$ is a map satisfying the displayed identity. Cocycles modulo maps $\sigma \mapsto \sigma u-u$ form $H^1(K,M)$ in Galois cohomology. For an elliptic curve, choosing $mQ=P$ gives the cocycle $\sigma \mapsto \sigma Q-Q$ in $E[m]$, which represents the Kummer map of an elliptic curve.