Characteristic Polynomials of Simple Ordinary Abelian Varieties over Finite Fields

Abstract

We provide an easy method for the construction of characteristic polynomials of simple ordinary abelian varieties $\mathcal{A}$ of dimension $g$ over a finite field ${\mathbb{F}}_q$, when $q\ge 4$ and $2g={\rho }^{b-1}(\rho -1)$ for some prime $\rho \ge 5$, with $b\ge 1$. Moreover, we show that $\mathcal{A}$ is absolutely simple if $b=1$ and $\rho =5$, but $\mathcal{A}$ is not absolutely simple for arbitrary $\rho$ with $b>1$.