We propose a simple criterion to know if an abelian variety A defined over a finite field Fq 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 Fq -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 Fq -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.
An isogeny class A of abelian varieties defined over finite fields is said to be cyclic if every variety in A has a cyclic group of rational points. In this paper we study the local cyclicity of Weil-central isogeny classes of abelian varieties, i.e. those with Weil polynomials of the form f A (t) = t 2g + at g + q g , as well as the local growth of the groups of rational points of the varieties in A after finite field extensions. We exploit the criterion: an isogeny class A with Weil polynomial f is cyclic if and only if f ′ (1) is coprime with f (1) divided by its radical.
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