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arXiv 2006.00637math.NTmath.AG
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有限域上阿贝尔簇的有理点群的结构

The Structure of the Group of Rational Points of an Abelian Variety over a Finite Field

Caleb Springer

备注：
12 pages. New subsection (3.1) gives more background information on invertible ideals
发表信息：
European Journal of Mathematics (2021)
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摘要：

设$A$是定义在具有Frobenius自同态$π$的有限域$\mathbb{F}_q$上的维数为$g$的单阿贝尔簇.本文在一定的技术条件下，给出了定义在$\mathbb{F}_q$上的自同态环$R$上的模的有理点群$A(\mathbb{F}_{q^n})$的结构.如果$[\mathbb{Q}(π) : \mathbb{Q}]=2g$和$R$是Gorenstein环，则${A(\mathbb{F}_{q^n}) \cong R/R(π^n-1)}$。这包括当$A$是普通的并且具有最大实数乘法时的情况。否则，如果$Z$是$R$的中心，并且$(π^n - 1)Z$是$Z$中可逆素理想的乘积，则$A(\mathbb{F}_{q^n})^d \cong R/R(π^n - 1)$，其中$d = 2g/[\mathbb{Q}(π):\mathbb{Q}]$。最后，在相似的条件下，我们推导出了$A(\overline{\mathbb{F}}_q)$作为$R$上的模的结构.这些结果推广了Lenstra关于椭圆曲线的结果。

英文摘要：

Let $A$ be a simple abelian variety of dimension $g$ defined over a finite field $\mathbb{F}_q$ with Frobenius endomorphism $π$. This paper describes the structure of the group of rational points $A(\mathbb{F}_{q^n})$, for all $n \geq 1$, as a module over the ring $R$ of endomorphisms which are defined over $\mathbb{F}_q$, under certain technical conditions. If $[\mathbb{Q}(π) : \mathbb{Q}]=2g$ and $R$ is a Gorenstein ring, then ${A(\mathbb{F}_{q^n}) \cong R/R(π^n-1)}$. This includes the case when $A$ is ordinary and has maximal real multiplication. Otherwise, if $Z$ is the center of $R$ and $(π^n - 1)Z$ is the product of invertible prime ideals in $Z$, then $A(\mathbb{F}_{q^n})^d \cong R/R(π^n - 1)$ where $d = 2g/[\mathbb{Q}(π):\mathbb{Q}]$. Finally, we deduce the structure of $A(\overline{\mathbb{F}}_q)$ as a module over $R$ under similar conditions. These results generalize results of Lenstra for elliptic curves.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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