Essential Dimension in Mixed Characteristic

Abstract

Let $G$ be a finite group, and let $R$ be a discrete valuation ring with residue field $k$ and fraction field $K$. We say that $G$ is weakly tame at a prime $p$ if it has no non-trivial normal $p$-subgroups. By convention, every finite group is weakly tame at $0$. We show that if $G$ is weakly tame at char(k), then ${\mathrm{ed}}_K(G)\ge {\mathrm{ed}}_k(G)$. Here ${\mathrm{ed}}_F(G)$ denotes the essential dimension of $G$ over the field $F$. We also prove a more general statement of this type, for a class of étale gerbes $\mathcal{X}$ over $R$. As a corollary, we show that if $G$ is weakly tame at $p$, then ${\mathrm{ed}}_{L}G\ge {\mathrm{ed}}_{k}G$ for any field $L$ of characteristic $0$ and any field $k$ of characteristic $p$, provided that $k$ contains ${\overline{\mathbb{F}}}_p$. We also show that a conjecture of A. Ledet asserting that ${\mathrm{ed}}_k\mathbb{Z}/p^n\mathbb{Z})=n$ for a field $k$ of characteristic $p>0$ 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.