三次AGM图与有限域$\mathbb{F}_q$（奇特征且$q \equiv 2 \pmod3$）上的Hessian $3$-同源

中文摘要 AI 辅助

设$\mathbb{F}_q$为奇特征且$q \equiv 2 \pmod3$的有限域。我们研究由Borwein--Borwein三次算术-几何平均（AGM）在$\mathbb{F}_q$上定义的有向图。我们证明该图是不相交的有向环的并集。我们将Hessian曲线与该AGM相关联。我们还证明每条边对应于其初始顶点和终顶点所关联曲线之间定义在$\mathbb{F}_q$上的一个$3$-同源。然后我们利用Hessian曲线的计数公式推导出环数量的下界。

英文摘要

Let $\mathbb{F}_q$ be a finite field of odd characteristic with $q \equiv 2 \pmod3$. We study the directed graph defined by the Borwein--Borwein cubic arithmetic--geometric mean (AGM) over $\mathbb{F}_q$. We prove that this graph is a disjoint union of directed cycles. We associate Hessian curves with this AGM. We also show that each edge corresponds to a $3$-isogeny defined over $\mathbb{F}_q$ between the curves associated with its initial and terminal vertices. We then use a counting formula for Hessian curves to derive a lower bound for the number of cycles.

发表机构

• School of Science and Engineering, Tokyo Denki University（东京都立大学理工学部）
• Institute of Mathematics for Industry (IMI), Kyushu University（九州大学产业数学研究院）
• Cyber Physical Security Research Institute (CPSEC), National Institute of Advanced Industrial Science and Technology (AIST)（独立行政法人产业技术综合研究所网络物理安全研究所）

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