Oriented Supersingular Elliptic Curves and Eichler Orders

TL;DR

This paper explores the relationship between supersingular elliptic curves with level structures and Eichler orders, establishing a correspondence with quadratic forms and analyzing the implications for oriented isogenies.

Contribution

It constructs specific Eichler orders associated with supersingular elliptic curves and demonstrates their representation of quadratic forms, linking endomorphism rings to orientations and isogenies.

Findings

Eichler orders $ ext{End}(E,G)$ are linked to quadratic forms with specific discriminants.

Orientation of elliptic curves determines the isomorphism class of their endomorphism rings.

Isogenies between oriented elliptic curves correspond to compositions of quadratic forms.

Abstract

Let $p>3$ be a prime and $E$ be a supersingular elliptic curve defined over $\mathbb{F}_{p^2}$. Let $c$ be a prime with $c < 3p/16$ and $G$ be a subgroup of $E[c]$ 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 $\mathcal{O}_c(q,r)$ and $\mathcal{O}'_c(q,r')$ with level $c$. Interestingly, we prove that each $\mathcal{O}_c(q,r)$ or $\mathcal{O}'_c(q,r')$ can represent a primitive reduced binary quadratic form with discriminant $-16cp$ or $-cp$ respectively. If a curve $E$ is $\mathbb{Z}[\sqrt{-cp}]$-oriented or $\mathbb{Z}[\frac{1+\sqrt{-cp}}{2}]$-oriented, then we prove that $\text{End}(E,G)$ is isomorphic to $\mathcal{O}_c(q,r)$ or $\mathcal{O}'_c(q,r')$ respectively. Due to the fact that $\mathbb{Z}[\sqrt{-cp}]$-oriented isogenies between $\mathbb{Z}[\sqrt{-cp}]$-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.