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 \'etale 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 $\mathrm{ed}_{\mathbb{C}}(G) \geq n$ 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.