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