Witt–Burnside functor attached to $\boldsymbol{Z}_{p}^{2}$ and $p$-adic Lipschitz continuous functions

Abstract

Dress and Siebeneicher gave a significant generalization of the construction of Witt vectors, by producing for any profinite group $G$, a ring-valued functor ${\mathbf{W}}_G$. This paper gives the first concrete interpretation of any Witt–Burnside rings outside the procyclic cases in terms of known rings. In particular, the rings ${\mathbf{W}}_{{\mathbf{Z}}_p^2}\left(k\right)$, where $k$ is a field of characteristic $p>0$ have a quotient realized as rings of Lipschitz continuous functions on the $p$-adic upper half plane ${\mathbf{P}}^1\left({\mathbf{Q}}_p\right)$. As a consequence we show that the Krull dimensions of the rings ${\mathbf{W}}_{{\mathbf{Z}}_p^d}\left(k\right)$ are infinite for $d\ge 2$ and we show the Teichmüller representatives form an analogue of the van der Put basis for continuous functions on ${\mathbf{Z}}_p$.