This question was previously asked on Math.Stackexchange.
It is about a proposition from R. Schoof’s article Nonsingular Plane Cubic Curves over Finite Fields:
Proposition 3.7: Let $ E$ be an elliptic curve over the finite field $ {\mathbb F}_q$ of characteristic $ p$ and $ n\in {\mathbb N}$ such that $ p\nmid n$ . Further, let $ t$ be the trace of the Frobenius $ \phi$ , that is, $ t := q + 1 – |E({\mathbb F}_q)|$ . Then the following are equivalent:
- $ E(\overline{{\mathbb F}_q})[n]\subset E({\mathbb F}_q)$ , i.e. every $ n$ -torsion point is defined over $ {\mathbb F}_q$ .
- $ n | q-1$ , $ n^2 | q + 1 – t$ , and either $ \phi\in{\mathbb Z}$ or $ {\mathcal O}\left(\frac{t^2-4q}{n^2}\right)\subset\text{End}_{{\mathbb F}_q}(E)$ .
Here $ {\mathcal O}(\Delta)$ is introduced earlier as the complex quadratic order with discriminant $ \Delta$ .
I’m having trouble understanding the precise meaning of the second statement if $ E$ is not ordinary. First, under the assumption that $ n | q – 1$ and $ n^2 | q + 1 – t$ , one has $ n^2 | t^2 – 4q$ , so $ {\mathcal O}\left(\frac{t^2-4q}{n^2}\right)$ is defined. But:
Q: Is $ {\mathcal O}\left(\frac{t^2-4q}{n^2}\right)\subset\text{End}_{{\mathbb F}_q}(E)$ really only supposed to mean that there’s some embedding, even if $ \text{End}_{{\mathbb F}_q}(E)$ is not assumed to be commutative?
As far as I understand, for a commutative domain $ R$ the image of an embedding $ {\mathcal O}\left(\Delta\right)\to R$ is unique if it exists, namely the subring of $ R$ spanned by all solutions of quadratic equations with discriminant $ \Delta$ – and in fact, a single such would suffice. However, over the non-commutative domain $ \text{End}_{{\mathbb F}_q}(E)$ this canonicity seems to fail – e.g. in the ring of integral quaternions $ {\mathbb Z}[i,j,k]$ we have $ X^2 + 1 = (X-i)(X+i) = (X-j)(X+j) = (X-k)(X+k)$ corresponding to different embeddings $ {\mathcal O}(-4)={\mathbb Z}[i]\hookrightarrow {\mathbb Z}[i,j,k]$ . Given that such quaternion algebras actually arise as endomorphism algebras of supersingular elliptic curves, this example doesn’t seem to be too far-fetched. In fact, this is the case I am most interested in.
Q: Considering that the proposition is specifically about describing $ \frac{\phi-1}{n}$ as opposed to any quadratic element of $ \text{End}_{{\mathbb F}_q}(E)$ of discriminant $ \frac{t^2-4q}{n^2}$ , perhaps one should restrict the discussion to the intersection of $ \text{End}_{{\mathbb F}_q}(E)$ with the $ {\mathbb Q}$ -subalgebra of $ \text{End}_{\overline{{\mathbb F}_q}}(E)\otimes_{\mathbb Z}{\mathbb Q}$ generated by $ \phi$ ?
Looking at the proof doesn’t help me, unfortunately, because it also uses notation I’m lacking a precise definition for.
I’d be happy if somebody could clarify. Thanks alot!
