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

Abstract

Let $k$ be algebraically closed of characteristic $p$. Reichstein and Vistoli showed that every $\Z/p^n$-torsor becomes at most one-dimensional over a field extension of degree prime to $p$, and Ledet conjectured $\ed_k(\Z/p^n)=n$. Using the finite-degree profile $\edd d_k(\tau)$ and its jump degrees introduced in \cite{PA}, we study the generic $\Z/p^n$-torsor $\tgen$, 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 $\tgen$ forces divisor classes on twisted curves to be constant, shows that a $G$-stable linear system on a faithful $\Z/p^n$-curve has degree $\ge p^{n-1}$. Hence $\edd d_k(\tgen)\ge2$ for $d<p^{n-1}$ and $d_1(\tgen)=p^{n-1}$ for all $p$ and $n\ge2$; in particular the truncation tower is optimal in degree $p^{n-2}$. We prove a quantitative Reichstein--Vistoli theorem, $d_1^{(p')}(\tgen)\le l_n(p)=1+(p-1)\sum_{j=1}^{n-1}p^{2j-1}$, and show that it is attained: $d_1^{(p')}(\tgen)=l_n(p)$ 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 $\ed_k(\Z/8)\in\{2,3\}$, that is, Ledet's conjecture at level $3$. We show that the subgroup formula $\edd{p}_k(\tau_n)=n-1$ 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.