On algebraic curves with many automorphisms in characteristic p

Abstract

Let $\mathcal{X}$ be an irreducible, non-singular, algebraic curve defined over a field of odd characteristic $p$. Let $g$ and $\gamma$ be the genus and $p$-rank of $\mathcal{X}$, respectively. The influence of $g$ and $\gamma$ on the automorphism group $Aut(\mathcal{X})$ of $\mathcal{X}$ is well-known in the literature. If $g \geq 2$ then $Aut(\mathcal{X})$ is a finite group, and unless $\mathcal{X}$ is the so-called Hermitian curve, its order is upper bounded by a polynomial in $g$ of degree four (Stichtenoth). In 1978 Henn proposed a refinement of Stichtenoth's bound of cube order in $g$ up to few exceptions, all having $p$-rank zero. In this paper a further refinement of Henn's result is proposed. First, we prove that if an algebraic curve of genus $g \geq 2$ has more than $336g^2$ automorphisms then its automorphism group has exactly two short orbits, one tame and one non-tame. Then we show that if $|Aut(\mathcal{X})| \geq 900g^2$, the quotient curve $\mathcal{X}/Aut(\mathcal{X})_P^{(1)}$ where $P$ is contained in the non-tame short orbit is rational, and the stabilizer of 2 points is either a $p$-group or a prime-to-$p$ group, then the $p$-rank of $\mathcal{X}$ is equal to zero.