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An isogeny class $\mathcal{A}$ of abelian varieties defined over finitefields is said to be "cyclic" if every variety in $\mathcal{A}$ has a cyclicgroup of rational points. In this paper we study the local cyclicity ofWeil-central isogeny classes of abelian varieties, i.e. those with Weilpolynomials of the form $f_\mathcal{A}(t)=t^{2g}+at^g+q^g$, as well as thelocal growth of the groups of rational points of the varieties in $\mathcal{A}$after finite field extensions. We exploit the criterion: an isogeny class$\mathcal{A}$ with Weil polynomial $f$ is cyclic if and only if $f'(1)$ iscoprime with $f(1)$ divided by its radical.

Some arithmetic properties of Weil polynomials of the form $t^{2g}+at^g+q^g$