Hyperelliptic Curves with Extra Involutions

The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus L g of such genus- g hyperelliptic curves is a g -dimensional subvariety of the moduli space of hyperelliptic curves H g . The authors present a birational parameterization of L g 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 ∈ H g with |Aut( p )| > 2, the field of moduli is a field of definition, and they prove this conjecture for any point p ∈ L g 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 ∈ H 3 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.