Chapter 9: Applications of Local Fields

We will give some of this structure theory here: namely, we will classify the maximal compact subgroups of GLn(K) for K a nondiscrete locally compact field. This has immediate applications to the structure of finite subgroups of GLn(Q), which are of intrinsic interest and are quite useful in areas like representation theory and modular and automorphic forms. Moreover, this material (actually, a small piece of it suffices) can be combined with a beautiful embedding theorem of J.W.S. Cassels to deduce a celebrated 1960 theorem of A. Selberg: for any field K of characteristic 0, a finitely generated subgroup of GLn(K) is virtually torsionfree: i.e., has a finite index subgroup without any nontrivial elements of finite order.