Are all supersingular elliptic curves defined over $\mathbb{F}_{p^2}$?

Question

In the paper introducing supersingular isogeny Diffie-Hellman, they write

Every supersingular elliptic curve in characteristic $p$ is defined over either $F_p$ or $F_{p^2}$ (see Silverman's AEC)

The most I am able to conclude from Silverman's book is that any supersingular elliptic curve is isomorphic to a curve defined over $\mathbb{F}_{p^2}$

V.3.1 Says that supersingular elliptic curves have $j$-invariant in $\mathbb{F}_{p^2}$

III.1.4 Says that elliptic curves with equal $j$-invariants are isomorphic and given some $j_0\in \bar{K}$, there exists a curve defined over $K(j_0)$ with $j$-invariant $j_0$.

Answer 1

Number theorists sometimes use "defined over F" about a morphism or variety over a large field $k$ containing $F$ to mean "base change of a morphism (or variety) over F".

In clearer language, you might say: if $L$ is a field containing $\mathbb{F}_{p^2}$ and $E$ is a supersingular elliptic curve over $L$, then there exists an elliptic curve $E'$ over $\mathbb{F}_{p^2}$ such that $E'_L$ is isomorphic to $E$.

Answer 2

In the case of (abstract) elliptic curves, “being isomorphic to something defined over $\mathbb{F}_{p^2}$” is what is meant by “being defined over $\mathbb{F}_{p^2}$”.