The Applied Mathematics of the Nottingham Group

10/28/2008 - 12:15pm
10/28/2008 - 1:10pm
Jonathan Lubin (Brown University - emeritus)

If k is a finite field, say with pn elements, then we may form the group of all formal power series u(x) ∈ k[[t]] for which u(0) = 0, u'(0) = 1, the group operation being substitution (composition). This group is often called the Nottingham group over k. It's a pro-p-group, i.e. the projective limit of finite p-groups, simple enough in definition, but in many ways, very mysterious in behavior. Camina has shown that every finite p-group can be embedded in Nottingham, and Klopsch has classified all the conjugacy classes of elements of order p. They remarked a while back that they did not know of any explicitly given elements of order even as low as p2. In this talk I will apply old mathematics to give a description of how to construct all elements of the Nottingham group of p-power order, and tell a classification up to conjugacy. But a characterization of the conjugacy classes that's as satisfactory as Klopsch's seems elusive.

ML 211