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 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 W Z p 2 ( k ) , 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 P 1 ( Q p ) . As a consequence we show that the Krull dimensions of the rings W Z p d ( k ) are infinite for d ≥ 2 and we show the Teichmuller representatives form an analogue of the van der Put basis for continuous functions on Z p .