The finite-degree profile of essential dimension II: cyclic $p$-groups in characteristic $p$

Let $k$ be algebraically closed of characteristic $p$. Reichstein and Vistoli showed that every $\mathbb{Z}\mathrm{/}{p}^n$-torsor becomes at most one-dimensional over a field extension of degree prime to $p$, and Ledet conjectured ${\text{ed}}_k(\mathbb{Z}\mathrm{/}{p}^n)=n$. Using the finite-degree profile $\text{edd}{d}_k(\tau )$ and its jump degrees introduced in \cite{PA}, we study the generic $\mathbb{Z}\mathrm{/}{p}^n$-torsor $\text{tgen}$, whose profile records how many Artin–Schreier–Witt layers an extension of degree $\le d$ can absorb. An Abel–Jacobi rigidity argument, in which the purely transcendental total space of $\text{tgen}$ forces divisor classes on twisted curves to be constant, shows that a $G$-stable linear system on a faithful $\mathbb{Z}\mathrm{/}{p}^n$-curve has degree $\ge p^{n-1}$. Hence $\text{edd}{d}_k(\text{tgen})\ge 2$ for $d<p^{n-1}$ and $d_1(\text{tgen})=p^{n-1}$ for all $p$ and $n\ge 2$; in particular the truncation tower is optimal in degree $p^{n-2}$. We prove a quantitative Reichstein–Vistoli theorem, $d_1^{(p^{\mathrm{\prime }})}(\text{tgen})\le l_n(p)=1+(p-1){\sum }_{j=1}^{n-1}{p}^{2j-1}$, and show that it is attained: $d_1^{(p^{\mathrm{\prime }})}(\text{tgen})=l_n(p)$ for all $p$ and $n$. We determine the profile at $(p,n)=(2,2)$ and every slot but the first at $(2,3)$, where the remaining slot is ${\text{ed}}_k(\mathbb{Z}\mathrm{/}8)\in \{2,3\}$, that is, Ledet’s conjecture at level $3$. We show that the subgroup formula $\text{edd}{p}_k({\tau }_n)=n-1$ is equivalent to Ledet’s conjecture at level $n-1$ together with a statement about extensions of degree $\le p$ disjoint from $L$, so that the profile relocates, but does not decide, Ledet’s conjecture.