Some special families of hyperelliptic curves

Abstract

Let $\L_g^G$ denote the locus of hyperelliptic curves of genus $g$ whose automorphism group contains a subgroup isomorphic to $G$. We study spaces $\L_g^G$ for $G \iso \Z_n, \Z_2{\o}\Z_n, \Z_2{\o}A_4$, or $SL_2(3)$. We show that for $G \iso \Z_n, \Z_2{\o}\Z_n$, the space $\L_g^G$ is a rational variety and find generators of its function field. For $G\iso \Z_2{\o}A_4, SL_2(3)$ we find a necessary condition in terms of the coefficients, whether or not the curve belongs to $\L_g^G$. Further, we describe algebraically the loci of such curves for $g\leq 12$ and show that for all curves in these loci the field of moduli is a field of definition.