# The endomorphismen ring of a supersingular elliptic curve is a (maximal) order in a quaternion algebra (ramified at $p$ and $\infty$)

I'm trying to prove that the endomorphism ring of a supersingular curve is indeed an order in a quaternion algebra over $Q$. I have been following in the lines of Silverman's proof which goes more or less in the following way:

Proof: Suppose by contradiction that $End(E)$ is not an order in a quaternion algebra over $Q$. As a consequence $$\mathcal{K}:= End(E)\otimes Q$$ is either $Q$ or an imaginary quadratic extension of $Q$.

Because $E$ is a supersingular elliptic curve there are only finitely many elliptic curves (up to $\overline{F}_{p^{n}}$-isomorphism) isogenous to $E$.

Let $\ell Z$ be a prime with $\ell \neq p$ such that $\ell$ is a prime element in $End(E')$ (the order of an imaginary quadratic extension or $Q$) for every elliptic curve $E'$ isogenous to $E$. Because there are only finitely many such subrings $End(E')\subset \mathcal{K}$, it is clear such a prime exists.

Because $E[\ell^{i}]\subset E(\overline{K})$ and $$E[\ell^{i}] \simeq Z_{\ell^{i}}\times Z_{\ell^{i}}$$ there exists a sequence of subgroups $$S_{1}\subset S_{2} \subset \cdots \subset E(\overline{K}), \quad \text{ where } S_{i}\simeq Z_{\ell^{i}}.$$ we know there exists a (unique up to $\overline{F}_{p^{n}}$-isomorphism) isogeny $\varphi_{i} \colon E \to E/S_{i}$ with kernel $S_{i}$. But since there exists an infinite number of isogenies $\varphi_{i}$ and only a finite number of elliptic curves isogenous to $E$ there must exist integers $m,k>0$ such that $E_{k+m}$ and $E_{k}$ are $F_{p^{n}}$-isomorphic. Let $\rho \colon E_{k} \to E_{k+m}$ be the natural projection with cyclic kernel $$\ker(\rho)=S_{k+m}/S_{k}$$ of order $\ell^{m}$ and $\iota \colon E_{k+m}\to E_{k}$ the isomorphism, then the composition $$\rho \circ \iota \colon E_{k} \to E_{k}$$ yields an endomorphism of $E_{k}$.

Now as for the question, because $\ell$ is a prime in $End(E_m)$, Silverman claims that if we compare degrees, then we must have $\rho=u \circ [\ell^{n/2}]$ for some automorphism $u$ of $E_m$? While this is not the case in general is guess this has something to do with $\ell$ been a prime in $End(E_m)$, am I missing something here or are there more than meets the eye?

I tried having a look a Deuring's original article which apparently is what Silverman used, though my germen is a bit rusty and translating engines are of no help what so ever.

Because $S_{k+m}/S_k$ is a subgroup of $E[\ell^{k+m}]$ there must exists another isogeny $\psi$ such that $\rho \circ \psi = [\ell^{k+m}]$. Now composing both isogenies with appropiate isomorphismens and enabeling the fact that $\ell$ is prime in $End(E_k)$ we find that $\iota \circ \rho = u \circ [\ell^x]$ for some integer $x$, the rest follows from comparing degrees, as stated by Silverman.