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Some special families of hyperelliptic curves

T. Shaska

Abstract

Let $Ł_g^G$ denote the locus of hyperelliptic curves of genus $g$ whose automorphism group contains a subgroup isomorphic to $G$. We study spaces $Ł_g^G$ for $G \iso \Z_n, \Z_2ø\Z_n, \Z_2øA_4$, or $SL_2(3)$. We show that for $G \iso \Z_n, \Z_2ø\Z_n$, the space $Ł_g^G$ is a rational variety and find generators of its function field. For $G\iso \Z_2øA_4, SL_2(3)$ we find a necessary condition in terms of the coefficients, whether or not the curve belongs to $Ł_g^G$. 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.

Some special families of hyperelliptic curves

Abstract

Let denote the locus of hyperelliptic curves of genus whose automorphism group contains a subgroup isomorphic to . We study spaces for , or . We show that for , the space is a rational variety and find generators of its function field. For we find a necessary condition in terms of the coefficients, whether or not the curve belongs to . Further, we describe algebraically the loci of such curves for and show that for all curves in these loci the field of moduli is a field of definition.

Paper Structure

This paper contains 9 sections, 10 theorems, 46 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperelliptic curves and their automorphisms
  4. Hurwitz spaces of covers $\psi: \mathcal{X}_g \to \mathbb P^1$
  5. Invariants of Binary Forms
  6. The reduced automorphism group is cyclic
  7. The reduced automorphism group is isomorphic to $A_4$
  8. Applications, subvarieties $\mathcal{L}_g^G$ of dimension $\delta \leq 1$
  9. Field of moduli versus field of definition

Key Result

Lemma 2.1

Let $\mathcal{X}_g$ be a genus $g\geq 2$ hyperelliptic curve with $\overline G:= \mathbb Z_n, A_4$. Then, $G:=\hbox{Aut}(\mathcal{X}_g)$, the dimension $\delta$ of $\mathcal{L}_G^\sigma$, the signature $\sigma$, and the number of involutions $i(G)$ of $G$ are as follows:

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Example 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • Theorem 3.1
  • proof
  • Remark 3.1
  • ...and 17 more