Witt-Burnside functor attached to $\mathbf{Z}_p^2$ and $p$-adic Lipschitz continuous functions

TL;DR

This paper interprets Witt-Burnside rings for $ extbf{Z}_p^2$ as rings of Lipschitz continuous functions on the $p$-adic upper half plane, revealing their infinite Krull dimension and a basis analogous to van der Put functions.

Contribution

It provides a concrete interpretation of $ extbf{W}_{ extbf{Z}_p^2}(k)$ rings in terms of Lipschitz functions and analyzes their algebraic properties, including Krull dimension and basis structure.

Findings

Krull dimension of $ extbf{W}_{ extbf{Z}_p^d}(k)$ is infinite for $d \u003e= 2

Teichmüller representatives form an analogue of the van der Put basis

Rings $ extbf{W}_{ extbf{Z}_p^2}(k)$ are interpreted as 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 a concrete interpretation of the rings $\mathbf{W}_{\mathbf{Z}_p^2}(k)$ where $k$ is a field of characteristic $p > 0$ in terms of rings of Lipschitz continuous functions on the $p$-adic upper half plane $\mathbf{P}^1(\mathbf{Q}_p)$. As a consequence we show that the Krull dimensions of the rings $\mathbf{W}_{\mathbf{Z}_p^d}(k)$ are infinite for $d \geq 2$ and we show the Teichm\"uller representatives form an analogue of the van der Put basis for continuous functions on $\mathbf{Z}_p$.