Essential dimension in mixed characteristic

Abstract

Suppose $G$ is a finite group and $p$ is either a prime number or $0$. For $p$ positive, we say that $G$ is weakly tame at $p$ if $G$ has no non-trivial normal $p$-subgroups. By convention we say that every finite group is weakly tame at $0$. Now suppose that $G$ is a finite group which is weakly tame at the residue characteristic of a discrete valuation ring $R$. Our main result shows that the essential dimension of $G$ over the fraction field $K$ of $R$ is at least as large as the essential dimension of $G$ over the residue field $k$. We also prove a more general statement of this type for a class of étale gerbes over $R$. As a corollary, we show that, if $G$ is weakly tame at $p$ and $k$ is any field of characteristic $p>0$ containing the algebraic closure of ${\mathbb{F}}_p$, then the essential dimension of $G$ over $k$ is less than or equal to the essential dimension of $G$ over any characteristic $0$ field. A conjecture of A. Ledet asserts that the essential dimension, ${\mathrm{ed}}_k(\mathbb{Z}/p^n\mathbb{Z})$, of the cyclic group of order $p^n$ over a field $k$ is equal to $n$ whenever $k$ is a field of characteristic $p$. We show that this conjecture implies that ${\mathrm{ed}}_{\mathbb{C}}(G)\ge n$ for any finite group $G$ which is weakly tame at $p$ and contains an element of order $p^n$. To the best of our knowledge, an unconditional proof of the last inequality is out of the reach of all presently known techniques.

16 pages. Corrected some minor mistakes, improved the exposition, and added some additional examples. To appear in Documenta Mathematica