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

Abstract

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

12 pages. New subsection (3.1) gives more background information on invertible ideals

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