Torsion in the Nottingham group

Abstract

From local class-field theory and higher ramification theory one gets a classification up to conjugacy of the torsion elements of arbitrary order in the Nottingham group over a finite field, in terms of continuous characters on the multiplicative group of principal units in the ring of formal power series over the field. An essential part of this classification is to partition the torsion elements according to the depths of their successive p-power iterates. The classification described here is good enough to permit an independent proof of Klopsch's theorem on torsion elements of order p, but not good enough to give a full description of the finite set of classes of torsion elements with a given depth-sequence. The final section exhibits an efficient method of calculating the first few hundred terms of a series of order pn, limited only by the capabilities of the computation package used, but gives no idea of any formula for describing the coefficients.