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 , 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.
Vélu's formulas construct the quotient of a Weierstrass elliptic curve by a specified finite subgroup and give rational functions for the resulting separable isogeny.

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  1. Isogeny of elliptic curves
  2. Elliptic curve
  3. Genus one curve
  4. Geometric genus
  5. Normalization of an algebraic curve
  6. Algebraic geometry
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