Oriented supersingular elliptic curves and Eichler orders of prime level

Abstract

Let $p>3$ be a prime and E be a supersingular elliptic curve defined over $F_{p^2}$. Let c be a prime with $c<3p/16$ and G be a subgroup of $E\left[c\right]$ of order c. The pair $(E,G)$ is called a supersingular elliptic curve with level-c structure, and the endomorphism ring $\text{End}(E,G)$ is isomorphic to an Eichler order with level c. We construct two kinds of Eichler orders $O_c(q,r)$ and $O_c^{\prime }(q,r^{\prime })$ with level c. Interestingly, we prove that each $O_c(q,r)$ or $O_c^{\prime }(q,r^{\prime })$ can represent a primitive reduced binary quadratic form with discriminant $-16cp$ or $-cp$ respectively. If a curve E is $Z\left[\sqrt{-cp}\right]$-oriented or $Z\left[\frac{1+\sqrt{-cp}}{2}\right]$-oriented, then we prove that $\text{End}(E,G)$ is isomorphic to $O_c(q,r)$ or $O_c^{\prime }(q,r^{\prime })$ respectively. Due to the fact that $Z\left[\sqrt{-cp}\right]$-oriented isogenies between $Z\left[\sqrt{-cp}\right]$-oriented elliptic curves could be represented by quadratic forms, we show that these isogenies are reflected in the corresponding Eichler orders via the composition law for their corresponding quadratic forms.