The groups of points on abelian surfaces over finite fields

TL;DR

This paper classifies the possible groups of rational points on abelian surfaces over finite fields based on their Weil polynomials, providing a detailed understanding of their structure.

Contribution

It offers a complete classification of the groups of rational points on abelian surfaces over finite fields using Weil polynomials, a novel approach in this context.

Findings

Classification of groups based on Weil polynomials

Explicit descriptions of possible group structures

Connection between Weil polynomial and rational point groups

Abstract

Let $A$ be an abelian surface over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by a Weil polynomial $f_A$ of degree 4. We give a classification of the groups of $k$-rational points on varieties from this class in terms of $f_A$.