= Translation invariance of a Weierstrass differential
{title2=$\tau_Q^*\omega=\omega$}
On a nonsingular equation $y^2=x^3+a_2x^2+a_4x+a_6$ in characteristic different from two, $\omega=dx/(2y)$ is regular and nowhere zero. For a chord of slope $h$ giving $P+Q=(X,Y)$, differentiating the line-intersection identity yields $dX/dx=Y/y$, so $dX/(2Y)=dx/(2y)$. Extension across the exceptional cases proves invariance under every <translation on an elliptic curve>. The addition map consequently satisfies $\mu^*\omega=\operatorname{pr}_1^*\omega+\operatorname{pr}_2^*\omega$, and $[n]^*\omega=n\omega$.
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