Abelian surfaces over finite fields with prescribed groups

TL;DR

This paper investigates which finite abelian groups can be realized as the group of rational points on abelian surfaces over finite fields, providing criteria and density results for their occurrence based on Rybakov's characterization.

Contribution

It applies Rybakov's criterion to determine conditions under which certain group structures do or do not occur, including bounds and density results for these groups.

Findings

Groups with large n_1 relative to n_2, n_3, n_4 do not occur.

Such groups occur with density zero in broader parameter ranges.

The paper establishes explicit congruence conditions for realizability.

Abstract

Let A be an abelian surface over F_q, the field of q elements. The rational points on A/\F_q form an abelian group A(\F_q) \simeq \Z/n_1\Z \times \Z/n_1 n_2 \Z \times \Z/n_1 n_2 n_3\Z \times\Z/n_1 n_2 n_3 n_4\Z. We are interested in knowing which groups of this shape actually arise as the group of points on some abelian surface over some finite field. For a fixed prime power q, a characterization of the abelian groups that occur was recently found by Rybakov. One can use this characterization to obtain a set of congruences modulo the integers $n_1, n_2, n_3, n_4$ on certain combinations of coefficients of the corresponding Weil polynomials. We use Rybakov's criterion to show that groups \Z/n_1\Z \times \Z/n_1 n_2 \Z \times \Z/n_1 n_2 n_3\Z \times\Z/n_1 n_2 n_3 n_4\Z do not occur if n_1 is very large with respect to n_2, n_2, n_4 (Theorem \ref{splitbound}), and occur with density zero in a wider range of the variables (Theorem \ref{splitbound-average}).