The characteristic polynomials of abelian varieties of higher dimension over finite fields

TL;DR

This paper advances understanding of the characteristic polynomials of abelian varieties over finite fields, especially in higher dimensions, by establishing new relations and explicit descriptions for dimensions up to five.

Contribution

It introduces new relations between parameters and provides explicit formulas for characteristic polynomials in higher dimensions, extending prior knowledge beyond dimension 4.

Findings

Relation between dimension and multiplicity for simple abelian varieties

Explicit description of characteristic polynomials for dimension g when e=g

Explicit coefficients for dimension 5 abelian varieties

Abstract

The characteristic polynomials of abelian varieties over the finite field $\mathbb{F}_q$ with $q=p^n$ elements have a lot of arithmetic and geometric information. They have been explicitly described for abelian varieties up to dimension 4, but little is known in higher dimension. In this paper, among other things, we obtain the following three results on the characteristic polynomial of abelian varieties. First, we prove a relation between $n$ and $e$, where $e$ is a certain multiplicity associated with a simple abelian variety of arbitrary dimension over $\mathbb{F}_q$. Second, we explicitly describe the characteristic polynomials of simple abelian varieties of arbitrary dimension $g$, when $e=g$. Finally, we explicitly describe the coefficients of characteristic polynomials of abelian varieties of dimension 5 over $\mathbb{F}_q$.