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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 pn 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 pn. 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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