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 $q\equiv 3\mathrm{mod}8$ and the quadratic residue $\left(\frac{p}{q}\right)=-1$ 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 $p$.