An $\mathbb{F}_{p^2}$-maximal Wiman's sextic and its automorphisms

TL;DR

This paper extends Wiman's classical results on the automorphism group of a specific genus 6 curve to fields of positive characteristic, analyzes its maximality over finite fields, and explores its relation to Hermitian curves.

Contribution

It proves that Wiman's automorphism group result holds over algebraically closed fields of characteristic p ≥ 7, and investigates the curve's maximality and Galois coverings over finite fields.

Findings

Automorphism group of Wiman's sextic extends to characteristic p ≥ 7.

Curve is rational and has automorphism group PGL(2,𝕂) in characteristic 5.

The curve is not Galois covered by the Hermitian curve over 𝔽_{19^2}.

Abstract

In 1895 Wiman introduced a Riemann surface $\mathcal{W}$ of genus $6$ over the complex field $\mathbb{C}$ defined by the homogeneous equation $\mathcal{W}:X^6+Y^6+Z^6+(X^2+Y^2+Z^2)(X^4+Y^4+Z^4)-12X^2 Y^2 Z^2=0$, and showed that its full automorphism group is isomorphic to the symmetric group $S_5$. The curve $\mathcal{W}$ was previously studied as a curve defined over a finite field $\mathbb{F}_{p^2}$ where $p$ is a prime, and necessary and sufficient conditions for its maximality over $\mathbb{F}_{p^2}$ were obtained. In this paper we first show that the result of Wiman concerning the automorphism group of $\mathcal{W}$ holds also over an algebraically closed field $\mathbb{K}$ of positive characteristic $p$, provided that $p \geq 7$. For $p=2,3$ the polynomial $X^6+Y^6+Z^6+(X^2+Y^2+Z^2)(X^4+Y^4+Z^4)-12X^2 Y^2 Z^2$ is not irreducible over $\mathbb{K}$, while for $p=5$ the curve $\mathcal{W}$ is rational and $Aut(\mathcal{W}) \cong PGL(2,\mathbb{K})$. We also show that the $\mathbb{F}_{19^2}$-maximal Wiman's sextic $\mathcal{W}$ is not Galois covered by the Hermitian curve $\mathcal{H}_{19}$ over $\mathbb{F}_{19^2}$.