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

TL;DR

This paper characterizes the structure of the group of rational points of a simple abelian variety over finite fields as a module over its endomorphism ring, generalizing known results for elliptic curves.

Contribution

It provides a detailed description of the rational points' structure for abelian varieties over finite fields, extending Lenstra's results to higher dimensions and more general cases.

Findings

Structure of $A(F_{q^n})$ as an $R$-module under certain conditions

Explicit isomorphisms involving the endomorphism ring and Frobenius endomorphism

Generalization of elliptic curve results to higher-dimensional abelian varieties

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 $n \geq 1$, as a module over the ring $R$ of endomorphisms which are defined over $\mathbb{F}_q$, under certain technical conditions. If $[\mathbb{Q}(\pi) : \mathbb{Q}]=2g$ 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.