Separable isogeny

A separable isogeny is an isogeny whose induced extension of function fields is separable. Its degree equals the number of geometric points in its kernel.

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.

For an elliptic curve $E$ and a finite subgroup $G$ of its geometric points, Jacques Vélu's formulas construct the separable isogeny with kernel $G$. On a Short Weierstrass form away from characteristics two and three, its normalized coordinates areThese rational functions extend to a morphism of the smooth projective curves, sending $G$ to the new identity. Its degree is $\#G$. If $G$ is stable under the absolute Galois group, the summed expressions and quotient curve are defined over the base field, even when the individual points of $G$ are not rational.^[ref]

For $E:y^2=x^3-x$ and $G=\{O,(0,0)\}$, they give $x^{\prime }=x-1/x$ and $y^{\prime }=y(1+1/x^2)$, with quotient $y^{\prime 2}=x^{\prime 3}+4x^{\prime }$. The apparent pole at $(0,0)$ represents its image at infinity, not a missing value of the projective isogeny.