2018/01/07 by Patrick Brosnan, Zinovy Reichstein, Brosnan, Patrick +3 · 1 citation
Mathematics · Social Sciences · #Algebraic Geometry and Number Theory #Advanced Differential Equations and Dynamical Systems #Communism, Protests, Social Movements
paper · pdf · doi:10.48550/arxiv.1801.02245
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 étale 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 \mathbbFp, 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, edk(ℤ/pnℤ), 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 edℂ(G) ≥ 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.