Scale functions on p-adic Lie groups

Let G be a p-adic Lie group. Then G is a locally compact, totally disconnected group, to which Willis [14] associates its scale function s: G --> N. We show that s can be computed on the Lie algebra level. The image of s consists of powers of p. If G is a linear algebraic group over Q(p), s(x) = s(h) is determined by the semisimple part h of x is an element of G. For every finite extension K of Q(p), the scale functions of G and H := G(K) are related by s(H \ G) = s(G)([K:QP]). More generally, we clarify the relations between the scale function of a p-adic Lie group and the scale functions of its closed subgroups and Hausdorff quotients.