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

TL;DR

This paper investigates the proportion of isogeny classes of abelian varieties over finite fields that have cyclic groups of rational points for specified primes, providing bounds as the field size grows.

Contribution

It establishes bounds on the fraction of $ ext{S}$-cyclic isogeny classes of abelian varieties over finite fields, extending understanding of their distribution.

Findings

Provides lower and upper bounds on the fraction of $ ext{S}$-cyclic classes

Analyzes behavior as the size of the finite field tends to infinity

Focuses on abelian varieties of fixed dimension 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.