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Unformatted text preview: CHAPTER 9: APPLICATIONS OF LOCAL FIELDS PETE L. CLARK The topological groups GL n ( R ) and GL n ( C ) have an inexhaustibly rich structure and importance in all parts of modern mathematics: analysis, geometry, topology, representation theory, number theory....The serious study of these groups was al- ready begun in the 19th century by Lie and his contemporaries. Somewhat more recently (say, about 1950) it has been realized that for a non- Archimedean locally compact field K , the groups GL n ( K ) also have a rich and useful structure. We will give some of this structure theory here: namely, we will classify the maxi- mal compact subgroups of GL n ( K ) for K a nondiscrete locally compact field. This has immediate applications to the structure of finite subgroups of GL n ( 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 GL n ( K ) is virtually torsionfree: i.e., has a finite index subgroup without any nontrivial elements of finite order. 1. General linear groups over locally compact fields 1.1. GL n ( K ) is a locally compact group. Let K be a nondiscrete locally com- pact field, and let n be a positive integer. We consider the group GL n ( K ) of invertible n n matrices with coefficients in K . We wish to endow GL n ( K ) with a natural locally compact topology. There are in fact two natural ways to do this, which, happily, lead to the same result. For any n Z + , we endow the Cartesian product K n with the product topology, which of course makes it a locally compact topological group. We will sometimes refer to this topology on K n and other topologies induced from it as the analytic topology , to distinguish it from the Zariski topology. (However, the reader need not know what the Zariski topology is in order to read these notes.) Let M n ( K ) be the ring of n n matrices with entries in K . As a K-vector space, M n ( K ) = K n 2 , and we give it the topology pulled back from the analytic topology on K n 2 via the isomorphism. (Easy exercise: the topology we get on M n ( K ) is independent of the chosen basis.) Now GL n ( K ) is a subset of M n ( K ). We claim that in the induced (subspace) topology it is locally compact, and indeed this is foisted off on the reader in the 1 2 PETE L. CLARK form of the following straightforward exercises. Exercise 9.1: Let P ( t 1 ,...,t n ) K [ t 1 ,...,t n ] be a polynomial, thought of as an algebraic object. Then P induces a function P : K n K in the usual way: ( x 1 ,...,x n ) 7 P ( x 1 ,...,x n ). Show that P is continuous for the analytic topolo- gies on K n and K ....
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