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

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 $p\geq 3$. Nakajima proved that if $\gamma \ge 2$ then $|S|\leq \textstyle\frac{p}{p-2}(g-1)$. If equality holds, $\mathcal X$ is a Nakajima extremal curve. We prove that if $$|S|>\textstyle\frac{p^2}{p^2-p-1}(g-1)$$ 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.