In a letter to Tate from 1987, Serre describes a beautiful Theorem relating mod p modular forms to quaternions ("Two letters on quaternions and modular forms (mod p)", Israel J. Math. 95 (1996), 281--299). At the beginning of Remark (4) on page 284, Serre says that every supersingular elliptic curve E over \bar F\_p has a "canonical and functorial" structure E\_0 over the finite field F with p^2 elements. He further says (if I understand it correctly) that such F-structure is determined by the condition that the relative Frobenius endomorphism of E\_0 is equal to -p in the endomorphism ring of E\_0 over F.


My question is: does anyone have a detailed reference for the proof of this fact, please? Also, can one find an F-structure of a given supersingular elliptic curve E where the relative Frobenius acts as multiplication by p (instead of -p)? Thanks



Supersingular elliptic curves and their "functorial" structure over F_p^2
(**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.



> 
> **Theorem:** For each prime $p$, the base extension functor from the category $\mathcal{C}\_{-p}$ of elliptic curves over $\mathbf{F}\_{p^2}$ on which the $p^2$-Frobenius endomorphism acts as $-p$ to the category of supersingular elliptic curves over $\overline{\mathbf{F}}\_p$ is an [equivalence of categories](http://en.wikipedia.org/wiki/Equivalence_of_categories).
> 
> 
> 


**Proof:** To show that the functor is an equivalence of categories, it suffices to show that the functor is full, faithful, and essentially surjective. It is faithful (trivially), and full (because homomorphisms between base extensions of elliptic curves in $\mathcal{C}\_{-p}$ automatically respect the Frobenius on each side). Essential surjectivity follows from Lemma 3.2.1 in


[Baker, González-Jiménez, González, Poonen, "Finiteness theorems for modular curves of genus at least 2", *Amer. J. Math.* **127** (2005), 1325–1387](http://www-math.mit.edu/~poonen/papers/finiteness.pdf),


which is proved by constructing a model for one curve and getting models for the others via separable isogenies. $\square$




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The same holds for the category $\mathcal{C}\_p$ defined analogously, but with Frobenius acting as $+p$.
Here are two approaches for proving essential surjectivity for $\mathcal{C}\_p$:


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 separable isogenies.


