论文标题
在有限场上，阿贝尔品种的一组理性点的结构
The Structure of the Group of Rational Points of an Abelian Variety over a Finite Field
论文作者
论文摘要
让$ a $是通过有限字段$ \ mathbb {f} _q $与Frobenius内态$π$定义的简单的Abelian dimension $ g $。本文将所有$ n \ geq 1 $ a（\ mathbb {f} _ {q^n}）$的一组理性点的结构描述为所有$ n \ geq 1 $，作为在某些技术条件下定义在$ \ mathbb {f} _q $的$ \ rathbb {f} _q $的圈$ r $ r $的模块。如果$ [\ mathbb {q}（π）：\ mathbb {q}] = 2g $，$ r $是gorenstein戒指，则$ {a（\ mathbb {f} _ {q^n}）这包括$ a $是普通且具有最大真实乘法的情况。否则，如果$ z $是$ r $和$（π^n -1）的中心，则z $是$ z $中的可演变理想的产物，然后是$ a（\ mathbb {f} _ {q^n} _ {q^n}）^d \ cong r/r/r/r/r（π^n -1）$ d = 2g/d = 2g/d = 2g/d = 2g/[\ mathbbbb {q y}最后，我们将$ a（\ overline {\ mathbb {f}} _ q）的结构推定为在类似条件下$ r $的模块。这些结果概括了椭圆曲线的伦斯特拉的结果。
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.
