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

We characterise abelian surfaces defined over finite fields containing no curves 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 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 2, containing no abelian surface with a polarisation of degree 4. Finally, we describe absolutely irreducible genus 3 curves lying on abelian surfaces containing no curves of genus less than or equal to 2, and show that their number of rational points is far from the Serre-Weil bound.