Hyperelliptic curves of genus 3 with prescribed automorphism group

Jaime Gutierrez, David Sevilla, Tanush Shaska

I-AIMS paper 2005-3 · Published in Lecture Notes Ser. Comput. 13, World Sci. (2005), 109–123 · 2005

We study genus 3 hyperelliptic curves which have an extra involution. The locus ℒ₃ of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli ℋ₃. We find a birational parametrization of this locus by affine 3-space. For every moduli point 𝔭 ∈ ℋ₃ 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 ∈ ℋ₃ such that |Aut(𝔭)| > 4. This is the first time that such a rational model of these curves appears in the literature.

MSC classes: Primary 14H37; Secondary 14H25, 14H10

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