What does the phrase “For supersingular elliptic curves, isogenies are equivalently defined by points inside their kernel” mean?

Question

On the wikipedia page for supersingular isogenies, it says:

The supersingular isogeny Diffie-Hellman method works with the set of supersingular elliptic curves $E$ over $F_{p^2}$, where the number of points on any such curve will be $(p ± 1)^2$. An isogeny of an elliptic curve $E$ is a rational map from $E$ to another elliptic curve $E'$ such that the number of points on both curves is the same. For supersingular elliptic curves, isogenies are equivalently defined by points inside their kernel.

What does the bold statement mean?

Answer 1

This is false, and so is the claim about the number of points being $(p \pm 1)^2$, as described here. It's true for separable isogenies, where "points" means $\overline{\mathbb{F}_p}$-points, but not in general, e.g. if $E$ is supersingular then $[p] : E \to E$, by definition, has trivial kernel in the sense of $\overline{\mathbb{F}_p}$-points but isn't isomorphic as an isogeny out of $E$ to the identity (because they can be distinguished by their group scheme kernels).

The bolded statement has now been corrected; I think I'll just remove the claim about the number of points since I don't think it's used anywhere else in the text anyway.