Hyperelliptic curves with extra involutions

Abstract

The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus ${Ł}_g$ of such genus $g$ hyperelliptic curves is a $g$-dimensional subvariety of the moduli space of hyperelliptic curves $\H_g$. We discover a birational parametrization of ${Ł}_g$ via dihedral invariants and show how these invariants can be used to determine the field of moduli of points $\p \in Ł_g$. We conjecture that for $\p\in \H_g$ with $|\Aut(\p)| > 2$ the field of moduli is a field of definition and prove this conjecture for any point $\p\in Ł_g$ such that the Klein 4-group is embedded in the reduced automorphism group of $\p$. Further, for $g=3$ we show that for every moduli point $\p \in \H_3$ such that $| \Aut (\p) | > 4$, the field of moduli is a field of definition and provide a rational model of the curve over its field of moduli.