Abelian surfaces over finite fields containing no curves of genus $3$ or less

TL;DR

This paper characterizes abelian surfaces over finite fields that contain no low-genus curves, expanding previous classifications and linking the presence of genus 3 curves to polarisation degrees, with implications for identifying specific isogeny classes.

Contribution

It completes the classification of isogeny classes of abelian surfaces with no low-genus curves and relates genus 3 curves to polarisation degrees, extending prior work.

Findings

Characterization of isogeny classes with no genus ≤ 2 curves.

Equivalence between genus 3 curves and polarisation degree 4 in simple abelian surfaces.

Identification of isogeny classes with no genus ≤ 2 curves and no polarisation of degree 4.

Abstract

We study abelian surfaces defined over finite fields which do not contain any possibly singular curve of genus less than or equal to $3$. Firstly, we complete and expand the characterisation of isogeny classes of abelian surfaces with no curves of genus up to $2$ initiated by the first author \emph{et al.~}in previous work. Secondly, we show that, for simple abelian surfaces, containing a curve of genus $3$ is equivalent to admitting a polarisation of degree $4$. Thanks to this result, we can use existing algorithms to check which isomorphism classes in the isogeny classes containing no genus $2$ curves have a polarisation of degree $4$. Thirdly, we characterise isogeny classes of abelian surfaces with no curves of genus $\leq 2$, containing no abelian surface with a polarisation of degree $4$. Finally, we describe the absolutely irreducible genus $3$ curves lying on abelian surfaces containing no curves of genus less than or equal to $2$.