论文标题
$ j = 0 $和mordell的椭圆形曲线 - 韦尔组
Hashing to elliptic curves of $j=0$ and Mordell--Weil groups
论文作者
论文摘要
考虑一个普通的椭圆曲线$ e_b \！：y^2 = x^3 -b $（of $ j $ -invariant $ 0 $）在有限字段$ \ mathbb {f} _ {\！q} $上本文试图解决构建有理$ \ mathbb {f} _ {\！q} $的问题 - 在直接产品的Kummer表面上曲线$ e_b \！\ times \！ e_b^\ prime $，其中$ e_b^\ prime $是二次$ \ mathbb {f} _ {\！q} $ - $ e_b $的扭曲。更确切地说，我们建议在无限顺序$ \ mathbb {f} _ {\！q} $之间搜索这样的曲线 - $ j = 0 $的某些椭圆表面的部分，分析其mordell-weel group。不幸的是，我们证明它只是同构至$ \ m athbb {z}/3 $。
Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 - b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_{\!q}$ such that $\sqrt[3]{b} \notin \mathbb{F}_{\!q}$. This article tries to resolve the problem of constructing a rational $\mathbb{F}_{\!q}$-curve on the Kummer surface of the direct product $E_b \!\times\! E_b^\prime$, where $E_b^\prime$ is the quadratic $\mathbb{F}_{\!q}$-twist of $E_b$. More precisely, we propose to search such a curve among infinite order $\mathbb{F}_{\!q}$-sections of some elliptic surface of $j=0$, analyzing its Mordell--Weil group. Unfortunately, we prove that it is just isomorphic to $\mathbb{Z}/3$.
