Characteristic Polynomials of Simple Ordinary Abelian Varieties over Finite Fields

TL;DR

This paper introduces a straightforward method to construct characteristic polynomials of simple ordinary abelian varieties over finite fields, revealing conditions for their absolute simplicity based on parameters like dimension and prime factors.

Contribution

It provides an easy construction method for characteristic polynomials and characterizes when these abelian varieties are absolutely simple based on specific parameter conditions.

Findings

Constructed characteristic polynomials for certain abelian varieties.

Identified conditions for absolute simplicity when b=1 and ρ=5.

Showed non-absolute simplicity for other parameter values.

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 $b>1$.