Group Structure of Abelian Varieties

TL;DR

This paper extends previous results to classify the group structures of all abelian varieties over finite fields, linking their finite group points to polynomial invariants and geometric polygons.

Contribution

It generalizes Rybakov's classification to encompass all abelian varieties, providing a comprehensive framework for their group structures over finite fields.

Findings

Extended Rybakov's theorem to all abelian varieties

Established criteria for group structures based on polygons

Unified classification framework for abelian varieties' groups

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 $G = A'(\mathbb{F}_q)$ for some $A'$ 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.