The invariant of $PGU(3,q)$ in the Hermitian function field

Abstract

Let $F=F|\mathbb{K}$ a be function field over an algebraically closed constant field $\mathbb{K}$ of positive characteristic $p$. For a $\mathbb{K}$-automorphism group $G$ of $F$, the invariant of $G$ is the fixed field $F^G$ of $G$. If $F$ has transendency degree $1$ (i.e. $F$ is the function field of an irreducible curve) and $F^G$ is rational, then each generator of $F^G$ uniquely determines $F^G$ and it makes sense to call each of them the invariant of $G$. In this paper, $F$ is the Hermitian function field $\mathbb{K}({\mathcal{H}}_q)=\mathbb{K}(x,y)$ with $y^q+y-x^{q+1}=0$ and $q=p^r$. We determine the invariant of $Aut(\mathbb{K}({\mathcal{H}}_q))\cong PGU(3,q)$, and discuss some related questions on Galois subcovers of maximal curves over finite fields.