Hyperelliptic curves of genus 3 with prescribed automorphism group
Abstract
We study genus 3 hyperelliptic curves which have an extra involution. The locus 3 of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli 3. We find a birational parametrization of this locus by affine 3-space. For every moduli point ∈ 3 such that | ()|>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 ∈ 3 such that |()|>4. This is the first time that such a rational model of these curves appears in the literature.
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