Some arithmetic properties of Weil polynomials of the form t²ᵍ+atᵍ+qᵍ

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 fA(t)=t²ᵍ+atᵍ+qᵍ, 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.