Some special families of hyperelliptic curves

TL;DR

This paper investigates special families of hyperelliptic curves with specific automorphism groups, establishing their rationality, describing their loci for genus up to 12, and analyzing their fields of moduli and definition.

Contribution

It characterizes the loci of hyperelliptic curves with certain automorphism groups, proving rationality for some and providing algebraic descriptions for low genus cases.

Findings

Loci are rational varieties for certain groups.

Necessary coefficient conditions for membership in loci.

Fields of moduli are fields of definition for studied 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.