Cyclicity of some families of isogeny classes of abelian varieties over finite fields

Alejandro J. Giangreco-Maidana

Published 2020-01-04Version 1

An isogeny class $\mathcal{A}$ of abelian varieties defined over finite fields is said to be "cyclic" if every variety in $\mathcal{A}$ has a cyclic group of rational points. In these notes we study the cyclicity of isogeny classes of abelian varieties with Weil polynomials of the form $f_\mathcal{A}(t)=t^{2g}+at^g+q^g$. We exploit the criterion: an isogeny class $\mathcal{A}$ with Weil polynomial $f$ is cyclic if and only if $\widehat{f(1)}$ is coprime with $f'(1)$, where $\widehat{f(1)}$ is the ratio of $f(1)$ to its radical.