Determining the automorphism group of a hyperelliptic curve

TL;DR

This paper presents new methods for determining the automorphism groups of hyperelliptic curves, including classical invariants and dihedral invariants, with results applicable across all genera and conditions for specific automorphism groups.

Contribution

It introduces a novel approach using dihedral invariants for hyperelliptic curves, applicable to all genera, and provides new conditions for automorphism groups like A4, S4, and A5.

Findings

Dihedral invariants are effective for all genera.

Conditions for automorphism groups A4, S4, A5 are established.

Normal form solutions involve nonlinear systems.

Abstract

In this note we discuss techniques for determining the automorphism group of a genus $g$ hyperelliptic curve $\X_g$ defined over an algebraically closed field $k$ of characteristic zero. The first technique uses the classical $GL_2 (k)$-invariants of binary forms. This is a practical method for curves of small genus, but has limitations as the genus increases, due to the fact that such invariants are not known for large genus. The second approach, which uses dihedral invariants of hyperelliptic curves, is a very convenient method and works well in all genera. First we define the normal decomposition of a hyperelliptic curve with extra automorphisms. Then dihedral invariants are defined in terms of the coefficients of this normal decomposition. We define such invariants independently of the automorphism group $\Aut (\X_g)$. However, to compute such invariants the curve is required to be in its normal form. This requires solving a nonlinear system of equations. We find conditions in terms of classical invariants of binary forms for a curve to have reduced automorphism group $A_4$, $S_4$, $A_5$. As far as we are aware, such results have not appeared before in the literature.