Local Sylow theory of totally disconnected, locally compact groups

We define a local Sylow subgroup of a totally disconnected, locally compact (t.d.l.c.) group G to be a maximal pro-p subgroup of an open compact subgroup of G. We use these subgroups to define the p-localization of G, a locally virtually pro-p group which maps continuously and injectively to G with dense image, and describe the relationship between the scale and modular functions of G and those of its p-localization. In the case of locally virtually prosoluble groups, we consider all primes simultaneously using local Sylow bases.