Hyperelliptic curves with extra involutions
Gutierrez, J. · Shaska, T.
Original · EN
The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus Łg of such genus g hyperelliptic curves is a g-dimensional subvariety of the moduli space of hyperelliptic curves. We discover a birational parametrization of Łg via dihedral invariants and show how these invariants can be used to determine the field of moduli of points ∈ Łg. We conjecture that for ∈ with |()| > 2 the field of moduli is a field of definition and prove this conjecture for any point ∈ Łg such that the Klein 4-group is embedded in the reduced automorphism group of. Further, for g=3 we show that for every moduli point ∈ ₃ such that | () | > 4, the field of moduli is a field of definition and provide a rational model of the curve over its field of moduli.
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