Hyperelliptic curves of genus 3 with prescribed automorphism group

TL;DR

This paper investigates genus 3 hyperelliptic curves with extra involutions, providing a birational parametrization of their moduli locus, and constructs explicit rational models over their fields of moduli for certain automorphism groups.

Contribution

It offers the first explicit rational models of these curves over their fields of moduli and characterizes the locus with automorphism groups larger than 2.

Findings

Birational parametrization of the locus in affine 3-space.

Field of moduli is a field of definition for curves with automorphism group size > 2.

Explicit rational models over the field of moduli for curves with automorphism group size > 4.

Abstract

We study genus 3 hyperelliptic curves which have an extra involution. The locus $\L_3$ of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli $\H_3$. We find a birational parametrization of this locus by affine 3-space. For every moduli point $\p \in \H_3$ such that $|\Aut (\p)|>2$, the field of moduli is a field of definition. We provide a rational model of the curve over its field of moduli for all moduli points $\p \in \H_3$ such that $|\Aut(\p)|>4$. This is the first time that such a rational model of these curves appears in the literature.