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. Using this definition, we show that if G is weakly tame at char(k), then edK(G) ≥ edk(G). Here edF (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 X over R. As a corollary, we show that if G is weakly tame at p, then edL(G) ≥ edk(G) for any field L of characteristic 0 and any field k of characteristic p, provided that k contains Fp. We also show that a conjecture of A. Ledet, asserting that edk(Z/pnZ) = n for a field k of characteristic p > 0 implies that edC(G) ≥ n for any finite group G which is weakly tame at p and contains an element of order pn. We give a number of examples, where an unconditional proof of the last inequality is out of the reach of all presently known techniques.