Super-singular reduction at a given prime

Question

Given a prime $p$ should there always exist an elliptic curve over $\mathbb{Q}$ having super-singular reduction at $p$ ? I know examples with $p \equiv 2 \pmod 3,$ or $\equiv 3 \pmod 4$. But I am asking for all other $p$

Answer 1

This follows from Deuring's work on endomorphism rings (M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272). I couldn't find a free version online, but the content you need is summarized in sections 1-3 of T. Yang, Minimal CM liftings of supersingular elliptic curves.

Specifically, if $B_p$ is the unique quaternion algebra over $\mathbb{Q}$ ramified over exactly $p$ and $\infty$, then maximal orders in $B_p$ are in natural bijection with supersingular elliptic curves over $\overline{\mathbb{F}_p}$, by taking endomorphism rings. A maximal order corresponds to a curve over $\mathbb{F}_p$ if and only if the order contains an element $w$ satisfying $w^2 = -p$ (corresponding to the Frobenius auto-isogeny). Once we have a supersingular curve over $\mathbb{F}_p$, we may write down a Weierstrass form over $\mathbb{F}_p$, and choose a corresponding Weierstrass form over $\mathbb{Q}$ in the usual way.

It remains to show that there is always a maximal order with a suitable $w$. In fact, Deuring showed that when $p \equiv 3 \pmod{4}$, then the number of such orders is $\frac{1}{2}(h_p + h_{4p})$, and when $p \equiv 1 \pmod{4}$, then the number is $\frac{1}{2}h_{4p}$. These numbers are always strictly positive.