Abstract
AbstractThe purpose of this paper is to study hyperelliptic curves with extra involutions. The locusLgof such genus-ghyperelliptic curves is ag-dimensional subvariety of the moduli space of hyperelliptic curvesHg. The authors present a birational parameterization ofLgvia dihedral invariants, and show how these invariants can be used to determine the field of moduli of points p ∈ Lg. They conjecture that for p ∈Hgwith |Aut(p)| > 2, the field of moduli is a field of definition, and they prove this conjecture for any point p ∈Lgsuch that the Klein 4-group is embedded in the reduced automorphism group ofp. Further, forg= 3, they show that for every moduli point p ∈H3such that |Aut(p)| > 4, the field of moduli is a field of definition. A rational model of the curve over its field of moduli is provided.
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