Large automorphism groups compared to the $p$-rank of algebraic curves in characteristic $p$

Abstract

Let $\cX$ be a (projective, geometrically irreducible, non-singular) algebraic curve of genus $\ge 2$ and positive $p$-rank $γ(\cX)$, defined over an algebraically closed field $\mathbb{K}$ of positive characteristic $p>0$. Contrary to what occurs for the genus, no function $h(\gamma )$ exists such that $|\aut(\cX)|\le h(γ)$ whenever $γ=γ(\cX)$. Thus, to have a bound on $|\aut(\cX)|$ only depending on $γ(\cX)$, some restrictions on $\cX$ and $\aut(\cX)$ are needed. In this context, the following theorem is proven. Let $\Gamma$ be a subgroup of $\aut(\cX)$. Assume that there is a point $P\in \cX$ such that the quotient curve $\cX/S_P$ is rational, where $S_P$ denotes the Sylow p-subgroup of the stabilizer ${\Gamma }_P$ of $P$ in $\Gamma$. Then the following $p$-rank analog of the Riemann-Hurwitz bound \begin{equation*} |Γ|\le 24 \left(\frac{p}{p-1}\right)^4 γ(\cX)^4 \end{equation*} holds, unless a subgroup of index $\le 2$ of $\Gamma$ fixes $P$. This bound is sharp up to a constant depending only on $p$.