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/pn-torsor becomes at most one-dimensional over a field extension of degree prime to p, and Ledet conjectured edk(Z/pn)=n. Using the finite-degree profile edd dk(τ) and its jump degrees introduced in \cite{PA}, we study the generic Z/pn-torsor tgen, whose profile records how many Artin--Schreier--Witt layers an extension of degree ≤ 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/pn-curve has degree ≥ pn−1. Hence edd dk(tgen)≥2 for d<pn−1 and d1(tgen)=pn−1 for all p and n≥2; in particular the truncation tower is optimal in degree pn−2. We prove a quantitative Reichstein--Vistoli theorem, d1(p')(tgen)≤ ln(p)=1+(p−1)∑j=1n−1p2j−1, and show that it is attained: d1(p')(tgen)=ln(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 edk(Z/8)∈{2,3}, that is, Ledet's conjecture at level $3$. We show that the subgroup formula eddpk(τn)=n−1 is equivalent to Ledet's conjecture at level n−1 together with a statement about extensions of degree ≤ p disjoint from L, so that the profile relocates, but does not decide, Ledet's conjecture.
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BibTeX
@misc{shukla2026finitedegree,
title = {{The finite-degree profile of essential dimension II: cyclic $p$-groups in characteristic $p$}},
author = {Abhishek Shukla},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09298},
eprint = {2610.09298},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09298},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The finite-degree profile of essential dimension II: cyclic p-groups in characteristic p
AU - Abhishek Shukla
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.09298
UR - https://arxiv.org/abs/2610.09298
AB - 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(τ) 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 ≤ 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 ≥ p^n−1. Hence edd d_k(tgen)≥2 for d<p^n−1 and d_1(tgen)=p^n−1 for all p and n≥2; in particular the truncation tower is optimal in degree p^n−2. We prove a quantitative Reichstein--Vistoli theorem, d_1^(p')(tgen)≤ l_n(p)=1+(p−1)∑_j=1^n−1p^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)∈{2,3}, that is, Ledet's conjecture at level $3$. We show that the subgroup formula eddp_k(τ_n)=n−1 is equivalent to Ledet's conjecture at level n−1 together with a statement about extensions of degree ≤ p disjoint from L, so that the profile relocates, but does not decide, Ledet's conjecture.
ER -