Endomorphism Rings of Supersingular Elliptic Curves and Ternary Quadratic Forms

Abstract

Let $c<3p/16$ be a prime or $c=1$. Let $E$ be a $\mathbb{Z}[\sqrt{-cp}]$-oriented supersingular elliptic curve defined over $\mathbb{F}_{p^2}$. There exists a $c$-isogeny from $E$ to $E^p$ with kernel $G \subset E[c]$. Given an Eichler order corresponding to the endomorphism ring $\text{End}(E,G)=\{ \theta \in \text{End}(E): \theta(G) \subseteq G \}$, we can compute a ternary quadratic form with discriminant $p$ by solving two square roots in $\mathbb{F}_c$, and the ternary quadratic form corresponds to a maximal order $\mathcal{O} \cong \text{End}(E)$ in $B_{p,\infty}$ by Brandt--Sohn correspondence. Let $D$ be a prime with $D<p$ (resp. $4D<p$). If an imaginary quadratic order with discriminant $-D$ (resp. $-4D$) can be embedded into $\text{End}(E)$, then we can compute a maximal order in $B_{p,\infty}$ corresponding to $\text{End}(E)$ by solving one square root in $\mathbb{F}_D$ and two square roots in $\mathbb{F}_c$. As we know, any isogeny between supersingular elliptic curves can be translated into a kernel ideal of the endomorphism ring. We study the action of the kernel ideal and give a basis of its right order. In general, we propose an efficient algorithm for computing a maximal order from an Eichler order in $B_{p,\infty}$.