Large $p$-groups of automorphisms of algebraic curves in characteristic $p$

Abstract

Let $S$ be a $p$-subgroup of the $\mathbb{K}$-automorphism group $Aut(\mathcal{X})$ of an algebraic curve $\mathcal{X}$ of genus $g\ge 2$ and $p$-rank $\gamma$ defined over an algebraically closed field $\mathbb{K}$ of characteristic $p\ge 3$. Nakajima proved that if $\gamma \ge 2$ then $|S|\le \frac{p}{p-2}(g-1)$. If equality holds, $\mathcal{X}$ is a Nakajima extremal curve. We prove that if $$|S|>\frac{p^2}{p^2-p-1}(g-1)$$ then one of the following cases occurs: (i) $\gamma =0$ and the extension $\mathbb{K}(\mathcal{X})|\mathbb{K}(\mathcal{X}{)}^S$ completely ramifies at a unique place, and does not ramify elsewhere. (ii) $|S|=p$, and $\mathcal{X}$ is an ordinary curve of genus $g=p-1$. (iii) $\mathcal{X}$ is an ordinary, Nakajima extremal curve, and $\mathbb{K}(\mathcal{X})$ is an unramified Galois extension of a function field of a curve given in (ii). There are exactly $p-1$ such Galois extensions. Moreover, if some of them is an abelian extension then $S$ has maximal nilpotency class. The full $\mathbb{K}$-automorphism group of any Nakajima extremal curve is determined, and several infinite families of Nakajima extremal curves are constructed by using their pro-$p$ fundamental groups.