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

Abstract

For a prime number $\ell$, an isogeny class $\mathcal{A}$ of abelian varieties is called $\ell$-cyclic if every variety in $\mathcal{A}$ have a cyclic $\ell$-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 $\ell$-cyclic for every $\ell\in\mathcal{S}$. 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.