Group Structure of Abelian Varieties

Abstract

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 $|G|=h_A(1)$, then $G=A^{\prime }({\mathbb{F}}_q)$ for some $A^{\prime }$ isogenous to $A$ if and only if the ${\ell }^{th}$ Hodge polygon of $G$ is under the ${\ell }^{th}$ Newton polygon of $h_A(1-t)$ for all primes $\ell$. In this paper, we will extend this result to get a classification theorem for the group structure of all abelian varieties.

Withdrawn to rework Theorem 1.2