Maximal hyperelliptic curves of genus three (Q1017422)

This paper concerns maximal hyperelliptic curves of genus 3 (and in part also genus 5 and 6), where `maximal' means attaining the upper Hasse-Weil-Serre bound. It gives an overview of which fields were known to (dis)allow for such curves and proceeds by proving new results. More precisely, a few equations are given which correspond to a maximal hyperelliptic curve of genus 3 if and only if a congruence condition of the field size \(p^n\) is satisfied. For example, for odd \(p\) the equation \(y^2=x^8+1\) yields a hyperelliptic curve of genus 3 over \(\mathbb{F}_{p^n}\), and it is maximal if and only if \(p\equiv 7\bmod 8\) and \(n\equiv 2\bmod 4\). In addition the geometry of these curves is studied.