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Unformatted text preview: ADELES AND IDELES PETE L. CLARK Contents 6.1. What should the adeles do? 1 6.2. The adele ring 2 6.3. Basic results on the topology of the adeles 4 6.4. Ideles 6 6.1. What should the adeles do? We have now come to what is probably the most important topic in our course: the systematic study of global fields using their locally compact completions. Let K be a global field – we call that this means that K is either a finite ex tension of Q (a number field) or a finite separable extension of F q ( t ) (a function field). In the case of a locally compact field K , the additive group ( K, +) and the mul tiplicative group ( K × , · ) play key roles in the theory. Each is a locally compact abelian group, hence amenable to the methods of Fourier analysis. Moreover, the additive group is selfPontrjagin dual, and the multiplicative group K × is a target group for class field theory on K : that is, there is a bijective correspondence be tween the finite abelian extensions L/K and the finite index open subgroups H L of K × such that K × /H L ∼ = Gal( L/K ). We seek global analogues of all of these facts. That is, for K a global field, we will define a commutative topological ring A K , the adele ring , which is locally compact and selfPontrjagin dual. (This allows us to do harmonic analysis on global fields, as was first done in John Tate’s 1950 Princeton thesis. We will not actually do this in our course, but it is an allimportant topic in modern number theory, and I wish to be aware of it and be prepared to learn it!) Moreover, the group of units, suitably topologized, is called the idele group I K . It is again a locally compact abelian group. There are further important topological properties that we do not see in the local case: namely, we will have canonical embeddings K , → A K and K × , → I K . In the additive case, K is discrete (hence closed) as a subgroup of the adele ring, and the quotient A K is compact. In the multiplicative case, K × is again a discrete 1 2 PETE L. CLARK subgroup of I K ; the quotient group is denoted C ( K ). C ( K ) need not be compact, but it is again a target group for class field theory on K . 6.2. The adele ring. For each place v of K , the completion K v is a locally compact field, so it seems natural not to proceed merely by analogy but actually to use the fields K v in the construction of our putative A K . The first idea is simply to take the product of all the completions: Q v K v . However, this will not work: Exercise 6.1: Let { X i } i ∈ I be an indexed family of nonempty topological spaces. Show that TFAE: (i) X = Q i ∈ I X i is locally compact. (ii) Each X i is locally compact, and { i ∈ I  X i is not compact } is finite....
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This note was uploaded on 10/26/2011 for the course MATH 8410 taught by Professor Staff during the Fall '11 term at UGA.
 Fall '11
 Staff
 Number Theory, Topology

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