pbrosnan@gmail.com, reichst@math.ubc.ca, angelo.vistoli@sns.it

Submission: 2018, Jan 10

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
the prime field, 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 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
the essential dimension of G over the complex numbers is at least 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.

2010 Mathematics Subject Classification: 14A20, 13A18, 13A50

Keywords and Phrases: Essential dimension, Ledet's conjecture, genericity theorem, gerbe, mixed characteristic

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