On the cyclicity of the rational points group of abelian varieties over finite fields

TL;DR

This paper introduces criteria based on endomorphism rings and characteristic polynomials to determine when abelian varieties over finite fields are cyclic, providing asymptotic bounds and specific results for surfaces.

Contribution

It offers new criteria for cyclicity of abelian varieties and isogeny classes over finite fields, along with asymptotic bounds and special cases for surfaces.

Findings

Criteria for cyclicity based on endomorphism rings and characteristic polynomials.

Asymptotic lower bounds on the fraction of cyclic isogeny classes as q grows.

Proof that certain maximal rational point varieties are cyclic and ordinary for even q.

Abstract

We propose a simple criterion to know if an abelian variety $A$ defined over a finite field $\mathbb{F}_q$ is cyclic, i.e., it has a cyclic group of rational points; this criterion is based on the endomorphism ring End$_{\mathbb{F}_q}(A)$. We also provide a criterion to know if an isogeny class is cyclic, i.e., all its varieties are cyclic; this criterion is based on the characteristic polynomial of the isogeny class. We find some asymptotic lower bounds on the fraction of cyclic $\mathbb{F}_q$-isogeny classes among certain families of them, when $q$ tends to infinity. Some of these bounds require an additional hypothesis. In the case of surfaces, we prove that this hypothesis is achieved and, over all $\mathbb{F}_q$-isogeny classes with endomorphism algebra being a field and where $q$ is an even power of a prime, we prove that the one with maximal number of rational points is cyclic and ordinary.