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

Let $S$ be a $p$-subgroup of the $\mathbb{K}$-automorphism group $Aut(\mathcal X)$ofanalgebraiccurve$\mathcal X$ofgenus$g\ge 2$and$p$-rank$\gamma$definedoveranalgebraicallyclosedfield$$\mathbb{K}$$ofcharacteristic$p\geq 3$\mathrm{.}Nakajimaprovedthatif$\gamma \ge 2$then$|S|\leq \textstyle\frac{p}{p-2}(g-1)$\mathrm{.}Ifequalityholds,$\mathcal X$ is a Nakajima extremal curve. We prove that if $$\mathrm{\mid }S\mathrm{\mid }>\frac{p^2}{p^2-p-1}(g-1)$$ 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) $\mathrm{\mid }S\mathrm{\mid }=p$, 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.