Endomorphism rings of supersingular elliptic curves over $\mathbb{F}_p$

Abstract

Let $p>3$ be a fixed prime. For a supersingular elliptic curve $E$ over $\mathbb{F}_p$ with $j$-invariant $j(E)\in \mathbb{F}_p\backslash\{0, 1728\}$, it is well known that the Frobenius map $\pi=((x,y)\mapsto (x^p, y^p))\in \mathrm{End}(E)$ satisfies ${\pi}^2=-p$. A result of Ibukiyama tells us that $\mathrm{End}(E)$ is a maximal order in $\mathrm{End}(E)\otimes \mathbb{Q}$ associated to a (minimal) prime $q$ satisfying $q\equiv 3 \bmod 8$ and the quadratic residue $\bigl(\frac{p}{q}\bigr)=-1$ according to $\frac{1+\pi}{2}\notin \mathrm{End}(E)$ or $\frac{1+\pi}{2}\in \mathrm{End}(E)$. Let $q_j$ denote the minimal $q$ for $E$ with $j=j(E)$. Firstly, we determine the neighborhood of the vertex $[E]$ in the supersingular $\ell$-isogeny graph if $\frac{1+\pi}{2}\notin \mathrm{End}(E)$ and $p>q\ell^2$ or $\frac{1+\pi}{2}\in \mathrm{End}(E)$ and $p>4q\ell^2$. In particular, under our assumption, we show that there are at most two vertices defined over $\mathbb{F}_p$ adjacent to $[E]$. Next, under GRH, we obtain the bound $M(p)$ of $q_j$ for all $j$ and estimate the number of supersingular elliptic curves with $q_j<c\sqrt{p}$. We also computer the upper bound $M(p)$ for all $p<2000$ numerically and show that $M(p)>\sqrt{p}$ except $p=11,23$ and $M(p)<p\log^2 p$ for all $p$.