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Unformatted text preview: Math 676. A compactness theorem for the idele group 1. Introduction Let K be a global field, so K is naturally a discrete subgroup of the idele group A K and by the product formula it lies in the kernel ( A K ) 1 of the continuous idelic norm || || K : A K R > . We saw in class that if S is a finite non-empty set of places of K that contains all archimedean places, then the combined statement that O K,S has finite class group and finitely generated unit group with rank | S |- 1 is logically equivalent to the assertion (that does not involve S ) that the quotient ( A K ) 1 /K is compact . (Some aspects of that proof are addressed in 5 below.) In the case of number fields we saw how to directly prove that O K,S has finite class group and O K,S is finitely generated with rank | S | - 1 for any S containing the archimedean places. These methods rest on Minkowskis lemma, and so to carry them over to global function fields one needs a generalization of Minkowskis lemma. Our aim in this handout is to bypass the problem by giving a purely adelic proof of: Theorem 1.1. For any global field K , ( A K ) 1 /K is compact. The proof will be uniform across all global fields, and the key to the proof is an adelic replacement for the role of Minkowskis lemma in the classical argument for number fields. In particular, this gives a new proof of Dirichlets unit theorem and finiteness of class groups for rings of S-integers O K,S . However, if one strips away the adelic language in the case of number fields (especially when S is precisely the set of archimedean places) then one essentially recovers the classical argument. It must be emphasized that the power of the adelic approach is that it is more systematic across all global fields K and it is methodologically simpler . Also, this approach shows the close logical connection between the two fundamental finiteness theorems of algebraic number theory, a closeness that one cannot fully appreciate until one has come across their relation with Theorem 1.1. Our exposition of the proof of Theorem 1.1 follows Chapter 2 ( 14 17) in the book Algebraic number theory edited by Cassels and Frohlich. 2. The adelic Minkowski lemma The classical Minkowski lemma concerns compact quotients V/ with V a finite-dimensional R-vector space and a lattice in V : if is the Haar measure on V that is adapted to counting measure on and the volume-1 measure on V/ , then for X V is compact, convex, and symmetric about the origin with ( X ) > 2 dim V the intersection X is nonzero. As a special case, let V = K Q R Q v | K v for a number field K and let = O K . For any C > and any = ( v ) Q v | K v with Q v || v || v > C (where |||| v is the square of the standard absolute value for complex v ) we may consider the compact, convex, centrally symmetric region X V consisting of points ( x v ) such that || x v || v ||...
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