CLASS FIELD THEORY FOR GLOBAL FIELDS OF POSITIVE CHARACTERISTIC XEVI GUITART Abstract. We give the main statement of class field theory for global fields, paying more atten- tion to the case of positive characteristic. We basically follow Poonen's excellent notes [1], and x1.3 of Kerz's article [2]. 1. Introduction Definition 1.1. A global field K is either • a finite extension of Q (i.e., a number field), or • a finite extension of Fp(t); equivalently, the function field of a projective non-singular curve X over a finite field Fq = Fpr (in this case Fq is called the constant field of K). The aim of class field theory is to understand abelian extensions of K in terms of information contained in K. More precisely, it gives a description of Gal(Kab=K) in terms of K× (here Kab is the maximal abelian extension of K; that is, the union of all finite abelian extensions of K contained in a fixed separable closure Ks of K). In this notes we state the main results of class field theory, mainly focusing on the function field case. We will give an approach that passes through local class field theory first. In order to explain what local class field theory is about, recall that a place v of K is an equivalence class1 of valuations. These are well understood in the two situations: (1) If K is a number field: 1:1 fplaces of Kg ! fembeddings K,! Cg= ∼ [ fprimes p ⊂ OK g: Here ∼ means that we identify an embedding and its complex conjugate; the places corre- sponding to embeddings are arquimedean, and the ones corresponding to primes are discrete non-archimedean. (2) If K = Fq(X): fplaces of Kg 1:1! fclosed points of Xg; these are all discrete non-archimedean. Now the completion of K at v, denoted Kv, is a local field. Local class field theory describes ab × Gal(Kv =Kv) in terms of Kv . All these local descriptions are then combined in the so-called idelic formulation of class field theory for K. Date: October 24, 2014. 2010 Mathematics Subject Classification. 11G40 (11F41, 11Y99). 1a valuation determines an absolute value; two valuations are equivalent if they induce the same topology on K 1 CFT FOR FIELDS OF POSITIVE CHARACTERISTIC 2 ab × Remark 1.2. Observe that Gal(Kv =Kv) and Kv are of a different nature: the former is profinite (hence compact) but the later is not. In order to compare them, it will be useful to recall the following construction. If G is a topological group the profinite completion of G is G = lim G=U (U ranges over finite index open normal subgroups); b − U Observe that Gb is profinite, and that if G is already profinite then Gb ' G. In general there is a natural continuous map G ! Gb through which any other map from G to a profinite group factors. It not need be injective nor surjective. However, it sets up a bijection between the finite index open subgroups of G and Gb (which is what matters for Galois theory). 2. Local class field theory Definition 2.1. A field K is a (non-archimedean) local field if it is complete with respect to a 2 discrete valuation v : K× ! Z and its residue field is finite . Recall some standard terminology: •O v = fx 2 K : v(x) ≥ 0g the valuation ring (it is a DVR); • pv = fx 2 K : v(x) > 0g the maximal ideal; • k(v) = Ov=pv the residue field, which we assume is a finite field. Local fields are classified. Any such K is isomorphic to: (1) a finite extension of Qp; these arise as localizations of number fields at non-archimedean places. (2) Fq((t)) where, in fact, Fq = k(v); these arise as localizations of function fields. 2.1. Extensions. Let L=K be a finite Galois extension. Then v extends uniquely to w : L× ! Z e that makes L a local field as well. Since Ow ⊃ Ov we have that pvOw = pw for some e ≥ 1 (the ramification index), and k(w) is a finite extension of k(v). Reduction modulo pw gives a natural map (2.1) Gal(L=K) −! Gal(k(w)=k(v)); which turns out to be surjective. It is not injective in general, and the kernel IL=K is called the inertia subgroup of Gal(L=K). If L=K is unramified (i.e., if e = 1) the map (2.1) is an isomorphism. In particular, L=K is abelian. The group Gal(k(w)=k(v)) has a canonical generator, the Frobenius x 7! x#k(v); its preimage under (2.1) is the so-called Frobenius automorphism Frv 2 Gal(L=K). This can be passed to the limit: there is a one-to-one correspondence between finite unramified Galois extensions of K and finite extensions of k(v), so that Gal(Kur=K) ' Gal(k(v)=k(v)) ' Zb (here Kur denotes the union of all unramified extensions of K). We have an exact sequence ab ur ab ur 1 −! Gal(K =K ) −! Gal(K =K) −! Gal(K =K) ' Zb −! 1; and the group Gal(Kab=Kur) is called the inertia subgroup of Gal(Kab=K). 2 an archimedean local field is, by definition, either R or C; in this case the Galois theory is very easy so in this section we will not mention them, and whenever we say local field we mean non-archimedean local field CFT FOR FIELDS OF POSITIVE CHARACTERISTIC 3 2.2. The local Artin map. Let K be a non-archimedean local field. The following is the main theorem of local class field theory3. Theorem 2.2. There exists a homomorphism, which is called the local Artin map4, θ : K× ! Gal(Kab=K) which induces an isomorphism Kd× ' Gal(Kab=K). × × Z × Observe that K 'Ov · π 'Ov × Z, where π is a uniformizer. Since Ov is already profinite, × × we have that Kd 'Ov × Zb. So the above theorem is giving the abstract structure of the group Gal(Kab=K), namely: ab × Gal(K =K) 'Ov × Zb: But in fact class field theory also gives much more precise information about θ. We also know the following: • θ is injective5. • θ maps any uniformizer to a Frobenius automorphism in Gal(Kab=K) (i.e., an element which induces the Frobenius in Gal(k(v)=k(v)) × ab un • θ(Ov ) ' Gal(K =K ); that is, θ maps the local units isomorphically to the inertia. We also have the following functoriality property: if L=K is a finite extension then the following diagram commutes: θ L× L/ Gal(Lab=L) NL=K res θ K K/ Gal(Kab=K) From this, and the fact that θ induces an isomorphism Kd× ' Gal(Kab=K) one can deduce the following correspondence: ffinite index subgroups of K×g 1:1! ffinite abelian extensions of Kg N −! (Kab)θ(N) × NL=K (L ) − L: Remark 2.3. Let L=K be a finite abelian extension. Then the composition K× −!θ Gal(Kab=K) −!res Gal(L=K) × × maps Ov to the inertia group IL=K . In particular, if L=K is unramified it maps Ov to 1. 3. Global class field theory From now on K denotes a global field. For any place v the completion Kv is a local field and by LCFT we have the local Artin map at v: × ab θv : Kv −! Gal(Kv =Kv): Now we would like to combine all these local maps in order to say something about Kab. One might Q × think that we want to consider then the product v Kv , where v runs through all the places of K. 3there is an analogous result for archimedean local fields 4or local reciprocity map 5this would not be true for an archimedean local field CFT FOR FIELDS OF POSITIVE CHARACTERISTIC 4 It turns out that this is not quite the right object to consider, but we need to consider a subgroup of this: the group of ideles. We denote by Ov the valuation ring of Kv if v is non-archimedean and Kv if v is archimedean. × Definition 3.1. The idele group of K is the restricted product of the Kv with respect to Ov; that is to say × Y 0 × Y × AK := Kv := f(av) 2 Kv : av 2 Ov for almost all vg: v v × × Observe that there is a natural injection K ,! AK , given by the diagonal embedding. We will × × think of it as an inclusion K ⊂ AK . Let L be a finite abelian extension of K. Then we can consider the composition × θv ab res ab res Kv −! Gal(Kv =Kv) −! Gal(K =K) −! Gal(L=K); which (by abusing the notation) we also denote by θv. Then define a map × A −! Gal(L=K) K Q (av) 7−! v θv(av): × This map is well defined. Indeed, if L=K is unramified at v and av 2 Ov (which happens for all but finitely many v's) then θv(av) = 1 (cf. Remark 2.3). Now taking the projective limit with respect to L we obtain a map × ab (3.1) AK −! Gal(K =K): Theorem 3.2 (Reciprocity law, idelic form). K× lies in the kernel of the above homomorphism. × × One defines the idele class group of K as CK := K nAK . Then (3.1) induces the so-called global Artin map ab θK : CK −! Gal(K =K): ab Theorem 3.3. θK induces an isomorphism CdK ' Gal(K =K). As in the local case, the Artin homomorphism also behaves well under finite extensions L=K.