Group Structure of Abelian Varieties

Let $A$ be an abelian variety over ${\mathbb{F}}_q$. Let $h_A(t)$ be the characteristic polynomial of $A$. Rybakov showed that if $h_A(t)$ is squarefree and $G$ is any finite group with $\mathrm{\mid }G\mathrm{\mid }=h_A(1)$, then $G=A^{\mathrm{\prime }}({\mathbb{F}}_q)$ for some $A^{\mathrm{\prime }}$ isogenous to $A$ if and only if the ${\mathrm{\ell }}^{th}$ Hodge polygon of $G$ is under the ${\mathrm{\ell }}^{th}$ Newton polygon of $h_A(1-t)$ for all primes $\mathrm{\ell }$. In this paper, we will extend this result to get a classification theorem for the group structure of all abelian varieties.