Translation invariance of a Weierstrass differential (${\tau }_Q^{\ast }\omega =\omega$)

On a nonsingular equation $y^2=x^3+a_2{x}^2+a_{4}x+a_6$ 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 ${\mu }^{\ast }\omega ={\mathrm{pr}}_1^{\ast }\omega +{\mathrm{pr}}_2^{\ast }\omega$, and $[n{]}^{\ast }\omega =n\omega$.