Differential criterion for separability of an isogeny (${\varphi }^{\ast }\omega \ne 0$)

A nonzero isogeny of elliptic curves is separable exactly when its pullback of an invariant differential on an elliptic curve is nonzero. For a map of smooth curves in characteristic $p$, a zero differential means the function-field extension has a nontrivial inseparable part. The criterion turns sums of elliptic endomorphisms into computations of their tangent scalars.