# Abelian surfaces with prescribed groups

## Abstract

Let A be an abelian surface over Fq, the field of q elements. The rational points on A/Fq form an abelian group A(Fq)≃Z/n1Z×Z/n1n2Z×Z/n1n2n3Z×Z/n1n2n3n4Z. 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 on certain combinations of coefficients of the corresponding Weil polynomials. We use Rybakov's criterion to show that groups Z/n1Z×Z/n1n2Z×Z/n1n2n3Z×Z/n1n2n3n4Z do not occur if n1 is very large with respect to n2,n3,n4 (Theorem 1.1), and occur with density zero in a wider range of the variables (Theorem 1.2).

## Repository Citation

David, C., D. Garton, Z. Scherr, A. Shankar, E. Smith, and L. Thompson. 2014. "Abelian surfaces with prescribed groups." Bulletin of the London Mathematical Society 46: 779-792.

## Publisher

London Mathematical Society

## Publication Date

1-1-2014

## Publication Title

Bulletin of the London Mathematical Society

## Department

Mathematics

## Document Type

Article

## DOI

https://dx.doi.org/10.1112/blms/bdu033

## Language

English

## Format

text