Oriented Supersingular Elliptic Curves and Eichler Orders
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| Format: | Preprint |
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2023
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| _version_ | 1866916385233305600 |
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| author | Xiao, Guanju Zhou, Zijian Qu, Longjiang |
| author_facet | Xiao, Guanju Zhou, Zijian Qu, Longjiang |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_08844 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Oriented Supersingular Elliptic Curves and Eichler Orders Xiao, Guanju Zhou, Zijian Qu, Longjiang Number Theory 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. |
| title | Oriented Supersingular Elliptic Curves and Eichler Orders |
| topic | Number Theory |
| url | https://arxiv.org/abs/2312.08844 |