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Unformatted text preview: A BRIEF SUMMARY OF THE STATEMENTS OF CLASS FIELD THEORY BJORN POONEN 0. Profinite completions of topological groups Let G be a topological group. The profinite completion of G is G := lim - U G U , where U ranges over the finite-index open normal subgroups of G . If G is discrete, this gives the usual notion of profinite completion. If G is profinite already, then the natural homomorphism G G is an isomorphism. In general, G G need not be injective or surjective. Nevertheless, we think of G as being almost isomorphic to G : The finite-index open subgroups of G are in bijection with those of G . And finite-index open subgroups of certain Galois groups are what we are interested in. . . 1. Local class field theory 1.1. Notation associated to a discrete valuation ring. O : a complete discrete valuation ring K := Frac( O ) v : the valuation K Z p : the maximal ideal of O k : the residue field O / p K s : a fixed separable closure of K K ab : the maximal abelian extension of K in K s K unr : the maximal unramified extension of K in K s k s : the residue field of K unr , so k s is a separable closure of k . Equip K and its subsets with the topology coming from the absolute value | x | := exp(- v ( x )). Date : September 23, 2006. I thank David Zywina for comments. The writing of this article was supported by NSF grant DMS- 0301280. 1 1.2. Local fields. Definition 1.1. A nonarchimedean local field is a complete discrete-valued field K as in Section 1.1 such that the residue field k is finite. An archimedean local field is R or C . Facts: A nonarchimedean local field of characteristic 0 is isomorphic to a finite extension of Q p . A (nonarchimedean) local field of characteristic p > 0 is isomorphic to F q (( t )) for some power q of p . 1.3. The local Artin homomorphism. Let K be a local field. Local class field theory says that there is a homomorphism : K Gal( K ab /K ) that is almost an isomorphism. The homomorphism is called the local Artin homomorphism . It cannot be literally an isomorphism, because Gal( K ab /K ) is a profinite group, hence com- pact, while K is not. What is true is that induces an isomorphism of topological groups d K Gal( K ab /K ). If K is archimedean, then : K Gal( K ab /K ) is surjective and its kernel is the con- nected component of the identity in K . For the rest of Section 1.3, we assume that K is nonarchimedean. Then is injective: The choice of a uniformizer O lets us write K = O Z ' O Z , and O is already profinite, so d K ' O Z . Thus local class field theory says that there is an isomorphism O Z Gal( K ab /K ) ....
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