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

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 $n \geq 1$,asamoduleoverthering$R$ofendomorphismswhicharedefinedover$$\mathbb{F}$q$,undercertaintechnicalconditions\mathrm{.}If$[$\mathbb{Q}$(\pi) : $\mathbb{Q}$]=2g$and$R$isaGorensteinring,then${A($\mathbb{F}${q^n}) \cong R/R(\pi^n-1)}$\mathrm{.}Thisincludesthecasewhen$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 $A({\bar{\mathbb{F}}}_q)$ as a module over $R$ under similar conditions. These results generalize results of Lenstra for elliptic curves.