Hyperelliptic curves with reduced automorphism group A5

TL;DR

This paper classifies hyperelliptic curves with automorphism group A_5, providing explicit equations over their field of moduli, and shows these curves form a rational variety with models over their moduli fields.

Contribution

It offers explicit equations for hyperelliptic curves with automorphism group A_5 over their field of moduli, extending previous real-case results to algebraically closed fields.

Findings

Locus of such curves is a rational variety.

Existence of rational models over the field of moduli.

Explicit equations with decomposable polynomials in x^2 or x^5.

Abstract

We study genus $g$ hyperelliptic curves with reduced automorphism group $A_5$ and give equations $y^2=f(x)$ for such curves in both cases where $f(x)$ is a decomposable polynomial in $x^2$ or $x^5$. For any fixed genus the locus of such curves is a rational variety. We show that for every point in this locus the field of moduli is a field of definition. Moreover, there exists a rational model $y^2=F(x)$ or $y^2=x F(x)$ of the curve over its field of moduli where $F(x)$ can be chosen to be decomposable in $x^2$ or $x^5$. While similar equations have been given in Bujalance, Cirre, Gamboa and Gromadzki (2001) over $\mathbb R$, this is the first time that these equations are given over the field of moduli of the curve.