2 extraneous sentence

This largely repeats

(EDIT: I've rewritten my argument in terms of the inverse functor, i.e., base extension, since it is clearer and more natural this way.)

Much of what is below is simply a reorganization of what Robin Chapman wrote, but let me write it anyway.

There is a

Theorem: For each prime $j$-invariant-preserving p$, the base extension functor from the category of elliptic curves over$\overline{\mathbf{F}}_p$to the category \mathcal{C}_{-p}$ of elliptic curves over $\mathbf{F}_{p^2}$ on which the $p^2$-Frobenius endomorphism acts as $-p$. There -p$to the category of supersingular elliptic curves over $\overline{\mathbf{F}}_p$ is another functor an equivalence of categories. Proof: To show that does the same but for Frobenius acting as$+p$. Moreover, each functor defines is an equivalence of categories, it suffices to show that the functor is full, faithful, and essentially surjective. These results hold for every$p$, including$2$It is faithful (trivially), and$3$. A uniform proof of the existence full (because homomorphisms between base extensions of the elliptic curves in $\mathbf{F}_{p^2}$-model with \mathcal{C}_{-p}$ automatically respect the Frobenius $-p$ is given in on each side). Essential surjectivity follows from Lemma 3.2.1 of in

though it sounds as if from Robin Chapman's comments, it may also be somewhere in Lang,Elliptic functions. The idea

which is to construct proved by constructing a model for one curve and to get getting models for the others via separable isogenies. Once a model for each elliptic curve over $\overline{\mathbf{F}}_p$ is fixed, each homomorphism over $\overline{\mathbf{F}}_p$ descends to $\mathbf{F}_{p^2}$ because it respects \square$The same holds for the Frobenius acting on each side; that is why one gets a functor. In particular, if two elliptic curves over category$\mathbf{F}_{p^2}$\mathcal{C}_p$ defined analogously, but with Frobenius $-p$ have the same $j$-invariant, any isomorphism between their base extensions to $\overline{\mathbf{F}}_{p^2}$ comes from an acting as $\mathbf{F}_{p^2}$-isomorphism.+p$.Here are two approaches for constructing the second functor, proving essential surjectivity for Frobenius$+p$:\mathcal{C}_p$:

2) Use Honda-Tate theory (actually, it goes back to Deuring in this case) to find one supersingular elliptic curve over $\mathbf{F}_{p^2}$ with Frobenius $+p$, and then repeat the proof of Lemma 3.2.1 to construct the models of all other supersingular elliptic curves via separable isogenies.

1

This largely repeats what Robin Chapman wrote, but let me write it anyway.

There is a $j$-invariant-preserving functor from the category of elliptic curves over $\overline{\mathbf{F}}_p$ to the category of elliptic curves over $\mathbf{F}_{p^2}$ on which the $p^2$-Frobenius endomorphism acts as $-p$. There is another functor that does the same but for Frobenius acting as $+p$. Moreover, each functor defines an equivalence of categories.

These results hold for every $p$, including $2$ and $3$. A uniform proof of the existence of the $\mathbf{F}_{p^2}$-model with Frobenius $-p$ is given in Lemma 3.2.1 of

though it sounds as if from Robin Chapman's comments, it may also be somewhere in Lang, Elliptic functions. The idea is to construct a model for one curve and to get the others via isogenies. Once a model for each elliptic curve over $\overline{\mathbf{F}}_p$ is fixed, each homomorphism over $\overline{\mathbf{F}}_p$ descends to $\mathbf{F}_{p^2}$ because it respects the Frobenius acting on each side; that is why one gets a functor. In particular, if two elliptic curves over $\mathbf{F}_{p^2}$ with Frobenius $-p$ have the same $j$-invariant, any isomorphism between their base extensions to $\overline{\mathbf{F}}_{p^2}$ comes from an $\mathbf{F}_{p^2}$-isomorphism.

Here are two approaches for constructing the second functor, for Frobenius $+p$:

1) If $G:=\operatorname{Gal}(\overline{\mathbf{F}}_p/\mathbf{F}_{p^2})$ and $E$ is an elliptic curve over $\mathbf{F}_{p^2}$, and $\overline{E}$ is its base extension to $\overline{\mathbf{F}}_{p^2}$, then the image of the nontrivial element under $H^1(G,\{\pm 1\}) \to H^1(G,\operatorname{Aut} \overline{E})$ gives the quadratic twist of $E$ (even when $p$ is $2$ or $3$, and even when $j$ is $0$ or $1728$). Applying this to each $E$ with Frobenius $-p$ gives the corresponding elliptic curve with Frobenius $+p$.

2) Use Honda-Tate theory (actually, it goes back to Deuring in this case) to find one supersingular elliptic curve over $\mathbf{F}_{p^2}$ with Frobenius $+p$, and then repeat the proof of Lemma 3.2.1 to construct the models of all other supersingular elliptic curves via isogenies.