$a$-Numbers, higher Cartier ranks and maximality of van der Geer-van der Vlugt curves

Let $p$ be an odd prime and let $C_R$ be the van der Geer-van der Vlugt curve $y^p-y=xR(x)$ over an algebraically closed field of characteristic $p$, where $R(x)={\sum }_{i=0}^h{a}_i{x}^{p^i}$ is an additive polynomial of degree $p^h$. We determine the image and the rank of every power of the Cartier operator $\text{Cart}$ on $H^0(C_R,\text{Omg})$. For $1\le r\le h-1$ the image of ${\text{Cart}}^r$ is spanned by an explicit set of monomial differentials that does not depend on $R$, and $\text{rank}({\text{Cart}}^r)=p^{h-r-1}\left(\genfrac{}{}{0px}{}{p+r}{r+2}\right)$. The last iterate detects the linear coefficient: $\text{rank}({\text{Cart}}^h)=0$ if $a_0=0$ and $\text{rank}({\text{Cart}}^h)=\left(\genfrac{}{}{0px}{}{(p-1)\mathrm{/}2+h}{h+1}\right)$ if $a_0\mathrm{\ne }0$, so the nilpotency index of $\text{Cart}$ is $h$ or $h+1$. Hence, for $h\ge 2$, the loci $a_0=0$ and $a_0\mathrm{\ne }0$ have the same $a$-number, which is the largest $a$-number of a one-point Artin-Schreier curve with pole order $p^h+1$, but they share no Ekedahl-Oort type. Over finite fields we deduce that a curve over ${\text{F}}_{p^{2n}}$ that is geometrically isomorphic to $C_R$ can be ${\text{F}}_{p^{2n}}$-maximal or ${\text{F}}_{p^{2n}}$-minimal only if $n\ge h$, and only if $n\ge h+1$ when $a_0\mathrm{\ne }0$. Explicit examples show that the second bound is attained for $(p,h)=(3,1)$, $(3,2)$ and $(5,2)$.