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

摘要

We propose a simple criterion to know if an abelian variety A defined over a finite field F-q is cyclic, i.e., it has a cyclic group of rational points; this criterion is based on the endomorphism ring End(Fq) (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 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 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. (C) 2019 Elsevier Inc. All rights reserved.