In "The Arithmetic of Elliptic Curves" by Silverman, the following result about the endomorphism rings of elliptic curves is proven:
Next a similar fact is mentioned in Schoof's "Nonsingular Plane Cubic Curves over Finite Fields":
The difference is, now the curve is over $\mathbb{F}_q$, so he narrows the possible quaternion algebras down to $\mathbb{Q}_{\infty,p}$. I tried to find a proof for this specific claim, but so far i couldn't find it.
Can someone provide a proof or a link to the proof?
Best Answer
The endomorphism ring of $E$ acts on the $\ell$-adic Tate module of $E$. If we tensor the endomorphism ring with $\mathbb Q$ we get this quaternion algebra, so it follows that the quaternion algebra acts on the $\ell$-adic Tate module tensored with $\mathbb Q$, which is $\mathbb Q_\ell^2$ as long as $\ell\neq p$.
If the quaternion algebra is ramified at $\ell$, then it cannot act on $\mathbb Q_\ell^{2}$, as tensoring with $\mathbb Q_\ell$ it is a rank $4$ division algebra, and thus cannot act on anything of dimension less than $4$.
Thus, the quaternion algebra is not ramified at any prime but $p$ and $\infty$. Because every quaternion algebra is ramified at at least two primes, it is ramified at those two primes.