Weil polynomials of small degree

TL;DR

This paper corrects previous inaccuracies in the classification of Weil polynomials of small degree, specifically for dimensions 3, 4, and 5, providing accurate descriptions and proofs for these cases.

Contribution

It provides corrected descriptions and proofs for Weil polynomials of degrees 6, 8, and 10, clarifying the classification of isogeny classes of abelian varieties over finite fields.

Findings

Corrected descriptions of Weil polynomials for g=3, 4, 5.

Established a criterion for real polynomials with only real roots.

Built on a new root-determination criterion for polynomial analysis.

Abstract

Honda and Tate showed that the isogeny classes of abelian varieties of dimension $g$ over a finite field $\mathbb{F}_q$ are classified in terms of $q$-Weil polynomials of degree $2g$, that is, monic integer polynomials whose set of complex roots consists of $g$ conjugate pairs of absolute value $\sqrt{q}$. There are descriptions of the space of such polynomials for $g \leq 5$, but for $g=3$, $4$ and $5$, these results contain mistakes. We correct these statements. Our proofs build on a criterion that determines when a real polynomial has only real roots in terms of the non-necessarily distinct roots of its first derivative.