Local cyclicity of isogeny classes of abelian varieties defined over finite fields

For a prime number $\mathrm{\ell }$, an isogeny class $\mathcal{A}$ of abelian varieties is called $\mathrm{\ell }$-cyclic if every variety in $\mathcal{A}$ have a cyclic $\mathrm{\ell }$-part of its group of rational points. More generally, for a finite set of prime numbers $\mathcal{S}$, $\mathcal{A}$ is said to be $\mathcal{S}$-cyclic if it is $\mathrm{\ell }$-cyclic for every $\mathrm{\ell }\in \mathcal{S}$. We give lower and upper bounds on the fraction of $\mathcal{S}$-cyclic $g$-dimensional isogeny classes of abelian varieties defined over the finite field ${\mathbb{F}}_q$, when $q$ tends to infinity.