Hyperelliptic curves with extra involutions
J. Gutierrez and T. Shaska
Abstract: The purpose of this paper is to study hyperelliptic
curves with extra involutions. The locus Lg of such genus-g hyperelliptic
curves is a g-dimensional subvariety of the moduli space of hyperelliptic curves
Hg. The authors present a birational parameterization of
Lg via 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 ∈ Hg with |Aut(p)| > 2, the field of moduli is a
field of definition, and they prove this conjecture for any point p ∈
Lg such that the Klein 4-group is embedded in the reduced automorphism group of
p. Further, for g = 3, they show that for every moduli point p ∈
H3 such 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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