# Binary quadratic forms attached to supersingular elliptic curves over F_p?

The question that I have is a more precise version of an earlier one (1), posted by myself on MO a little bit ago. Sorry for repeating myself.

Let $p$ be prime number which for simplicity shall be assumed $\equiv 1$ mod $4$ (so that $Z[\sqrt{-p}]=:R$ is the maximal order of the imaginary quadratic field $Q[\sqrt{-p}]=:K$, which we will see inside the complex numbers $C$).

Consider the following:

$Q$ is the set of binary quadratic forms $aX^2+bXY+cY^2$, in the variables $X$ and $Y$, with integer coefficients, with $a>0$, and with discriminant equal to $-4p$ (these forms are positive definite and necessarily primitive);

$I$ is the group of fractional ideals $I\subset K$, i.e., finitely generated, nonzero $R$ submodules of $K$, multiplication is given by $\otimes_R$;

$S$ is the "set(?)" of supersingular elliptic curves over $F_p$, which consists in fact of a unique $F_p$-isogeny class (that given by all the curves $E$ whose Frobenius endomorphisms $Frob_E:E\rightarrow E$ satisfies the polynomial $T^2+p$).

These three objects are naturally related, I will briefly recall how.

Given a quadratic $q(X,Y)$ form in $\mathcal{Q}$, take $\tau$ to be the root of $q(X,1)$ in the upper half plane and consider the $R$ submodule of $K$ given by $A_q=Z+Z\tau$.

Moreover, let me add that the unimodular group $\Gamma$ acts naturally, say to the left, of $Q$ and the map $q(X,Y)\rightarrow A_q$ induces an identification between $\Gamma\backslash Q$ and $Cl_K$, the ideal class group of $K$, namely the quotient of $I$ by the subgroup of principal fractional ideals. (The details of this are in a beautiful, elementary paper: J.-P. Serre, $\Delta=b^2-4ac$, OEUVRES, Collected papers. vol IV no 140)

The set $S$ is also related to $I$. Following Waterhouse thesis (W.C. Waterhouse, {\it Abelian varieties over finite fields}, Ann. Sc. Ec. Norm. Sup. 4.Ser, 2 (1969), 521-560), I shall recall how.

Choose an element $E_0$ of $S$ once for all. Let $\varphi:E_0\rightarrow E$ be an $F_p$-isogeny with source $E_0$ and target $E$, also an element of $S$. Define $A(\varphi)$ as the subset of $End_{F_p}(E_0)$ given by those elements $r$ that can be factored as $r'\varphi$, for some morphism $r':E\rightarrow E_0$. Then $A(\varphi)$ is an ideal of $R$. Waterhouse proves that $\varphi$ can be recovered from the ideal $A(\varphi)$ and that the map $\varphi\rightarrow A(\varphi)$ induces a bijection between $F_p$-isomorphism classes of elements of $E$ and $Cl_K$.

Questions:

1) Can we make this last bijection more canonical, and not just natural? Meaning is there a natural choice for the origin $E_0$?

2) Related to 1). Can we attach to any $E$ in $S$ and element $q_E$ of $Q$ in a natural way so as to recover a bijection between $S/\sim$ with $\Gamma\backslash Q=CL_K$?

3) Can we do the same thing as in 2) in such a way that the $j$-invariant of the lattice $A_{q_E}$ reduces mod $p$ to the $j$-invariant of $E$? (This however would involve (a priori) the choice of a place of the algebraic numbers above $p$..)

I guess I am only looking at these questions from a naive point of view, I would love to hear comments and learn more. In the literature there is a lot, but I could not exactly find what I am looking for.

Thanks.

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For #1, no natural base pt. Better formulation is not a bijection but rather a principal homogeneous space structure. For any invertible $R$-module $M$ and $E$ in $S$, the functor on $\mathbf{F}_p$-algebras defined by $A \rightsquigarrow M \otimes_R E(A)$ is represented by an elliptic curve denoted $M \otimes E$, and the natural map $M \otimes_E {\rm{Hom}}_R(E,E') \rightarrow {\rm{Hom}}_R(E,M \otimes E')$ between invertible $R$-modules is an isomorphism (pf: use Tate isogeny theorem at all primes). This shows that the tensoring operation makes $S$ a principal homogenous space for Pic($R$). – BCnrd Sep 28 '10 at 13:16
BCnrd, thanks for your comment. Can we explain why the number of supersingular $j$-invariants lying in the prime field is actually half as much as the number of $F_p$-isom. classes of objects of S? – Tommaso Centeleghe Sep 28 '10 at 13:48
The tensoring method works over $\overline{\mathbf{F}}_p$ using maximal orders in the quaternion algebra (and invertible modules over such orders), and you then need an "integral" refinement of Skolem-Noether applied to the subfield $K$. The 2-to-1 will come from the fact that the CM-structure by $R$ amounts to extra data, namely a choice of isomorphism of $R$ with the endomorphism ring over %\mathbf{F}_p\$, and these are two such choices. – BCnrd Sep 28 '10 at 16:00