Endomorphism Rings of Supersingular Elliptic Curves and Quadratic Forms

Abstract: Brandt--Sohn correspondence shows a connection between maximal orders in quaternion algebra $B_{p,\infty}$ and ternary quadratic forms with discriminant $p$. In this paper, we use this correspondence to compute endomorphism rings of supersingular elliptic curves. In particular, we study relations between orientations of supersingular elliptic curves and coefficients of corresponding ternary quadratic forms. 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 $G$. Given the endomorphism ring $\text{End}(E,G)$, we can compute the endomorphism ring $\text{End}(E)$ by solving two square roots in $\mathbb{F}_c$. 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 $\text{End}(E)$ by solving one square root in $\mathbb{F}_D$ and two square roots in $\mathbb{F}_c$. As we know, isogenies between supersingular elliptic curves can be translated into kernel ideals of endomorphism rings. We study the action of kernel ideals and represent their right orders by ternary quadratic forms. In general, given an Eichler order in $B{p,\infty}$, we can compute the maximal orders containing it in polynomial time.