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

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.
• For $C$ a finite subgroup of an elliptic curve $E$ then $E/C$ is an elliptic curve isogeneous to $E$. The elliptic curve is $(x,y) \in \overline{\mathbf{F}_p},f(x,y) = 0$ plus the point at $\infty$, but in cryptography you restrict to the subgroups $E(\mathbf{F}_{p^k})= \{ (x,y) \in \mathbf{F}_{p^k}, f(x,y)=0\}\cup \{\infty\}$ Commented Dec 27, 2017 at 8:40
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).