p^k-torsion of genus two curves over F_{p^m}

TL;DR

This paper classifies abelian surfaces and genus-2 Jacobians over finite fields based on the divisibility of their rational point groups by q^2, providing a detailed understanding of their isogeny classes.

Contribution

It offers a complete classification of isogeny classes of these abelian varieties with specific divisibility properties over finite fields.

Findings

Identifies all isogeny classes with group order divisible by q^2

Provides explicit criteria for Jacobians of genus-2 curves

Enhances understanding of the structure of rational points on abelian surfaces

Abstract

We determine the isogeny classes of abelian surfaces over F_q whose group of F_q-rational points has order divisible by q^2. We also solve the same problem for Jacobians of genus-2 curves.