A ‘Non-Additive’ Characterization of $${\wp}$$ -Adic Norms

Abstract

For F a \({\wp}\)-adic field together with a \({\wp}\)-adic valuation, we present a new characterization for a map \({p: F^{n} \rightarrow {\mathbb R}\cup\{-\infty}\}\) to be a \({\wp}\)-adic norm on the vector space Fn. This characterization was motivated by the concept of tight maps — maps that naturally arise within the theory of valuated matroids and tight spans. As an immediate consequence, we show that the two descriptions of the affine building of SLn(F) in terms of (i) \({\wp}\)-adic norms given by Bruhat and Tits and (ii) tight maps given by Terhalle essentially coincide. The result suggests that similar characterizations of affine buildings of other classical groups should exist, and that the theory of affine buildings may turn out as a particular case of a yet to be developed geometric theory of valuated (and δ-valuated) matroids and their tight spans providing simply-connected G-spaces for large classes of appropriately specified groups G that could serve as a basis for an affine variant of Gromov’s theory.