Essential Dimension in Mixed Characteristic

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 \operatorname{ed}_K(G) \geq \operatorname{ed}_k(G) . Here \operatorname{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 \operatorname{ed}_{L} G \geq \operatorname{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 \operatorname{ed}_k \mathbb{Z}/p^n \mathbb{Z}) = n for a field k of characteristic p>0 implies that \operatorname{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.

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