Endomorphism Rings of Supersingular Elliptic Curves and Ternary Quadratic Forms
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911053157236736 |
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| author | Xiao, Guanju Zhou, Zijian Qu, Longjiang |
| author_facet | Xiao, Guanju Zhou, Zijian Qu, Longjiang |
| contents | 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 $\text{End}(E,G)=\{ θ\in \text{End}(E): θ(G) \subseteq G \}$, 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 $B_{p,\infty}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_11025 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Endomorphism Rings of Supersingular Elliptic Curves and Ternary Quadratic Forms Xiao, Guanju Zhou, Zijian Qu, Longjiang Number Theory 14H52, 14K22, 11R52 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 $\text{End}(E,G)=\{ θ\in \text{End}(E): θ(G) \subseteq G \}$, 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 $B_{p,\infty}$. |
| title | Endomorphism Rings of Supersingular Elliptic Curves and Ternary Quadratic Forms |
| topic | Number Theory 14H52, 14K22, 11R52 |
| url | https://arxiv.org/abs/2409.11025 |