42 M. FLACH 3. The idele class group and idelic class field theory We have seen that for a global field K the modulus m appears as a parameter for the tower of ray class fields K(m). The content of class field theory is that the K(m) have an explicit Galois group and that their union is the maximal abelian extension of K (the union of all abelian extensions in a given algebraic closure). This same information can be repackaged in a more canonical construction which at the same time makes the relation to local class field theory much more transparent and allows to compute the Brauer group of the global field K. This is the role of the idele class group. It is a more elegant but slightly more abstract approach to class field theory since one has to operate with profinite or more general topological groups. Definition 3.0.2. Let K be a global field. Define the idele group of K as × × × AK = AK,S =: Kp S p with the direct limit topology where A× × × O× K,S := Kp Kp p∈S p∈/S has the product topology and S runs through all finite sets of places containing all archimedean ones. This is a locally compact group by Tychonoff’s theorem. × Remark 3.0.4. As is suggested by the notation AK is the unit group of the similarly defined adele ring AK for which we have no use in this course, except for Lemma 3.1 below. Since any element α ∈ K is integral at all but finitely many places we have a × × natural embedding K ⊆ AK . We also have an absolute value map |(αp)| := |αp|p p and K× lies in its kernel by the product formula. Definition 3.0.3. The idele class group CK of the global field K is the quotient group × × CK := AK /K . It is a (split) extension || 1 log 1 → CK → CK −−−→ R → 0 if K is a number field and ||/ q 1 log log( ) 1 → CK → CK −−−−−−−−→ Z → 0 1 if K has characteristic p and Fq is the algebraic closure of Fp in K. Here CK = ker(||). Proposition 3.1. The topology on K× induced from the locally compact topology × of AK is the discrete topology. Therefore CK is a locally compact abelian group. 1 Moreover, the group CK is actually compact. COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 43 Proof. It suffices to find a neighborhood of 1 containing no other element of K. O× × Take U = p∞ Kp p|∞ B(1,) where B(1,)={x ∈ Kp||x − 1|p <} is the ball of radius <1. Now look at 1 = α ∈ U ∩ K.Then0= α − 1 ∈OK and |α − 1|p <for all archimedean places. But this contradicts the product formula. Lemma 3.1. There is a constant C>0 with the following property: If a =(ap) ∈ × × AK with |a| >C then there exists α ∈ K such that |α|p ≤|ap|p. Proof. This is reminiscent of the Minkowski argument in the classical geometry of numbers. We use without proof that K is a discrete subgroup of AK (similar × argument to the one just given for AK ) and that AK /K is compact. Let c0 be the Haar measure of AK /K and and c1 that of the set W = {ξ ∈ AK ||ξp|p ≤ p} where p = 1 for non-archimedean p and p =1/10 for archimedean p.ForC = c0/c1 the set T = {ξ ∈ AK ||ξp|p ≤ p|ap|p} has measure c1 |ap|p >c1C = c0 p and therefore there must be a pair of distinct points t1,t2 of T with the same image × in AK /K.Thenα := t1 − t2 ∈ K and |α|p = |t1 − t2|p ≤|ap|p by the triangle inequality and its ultrametric sharpening. For an idele a with |a| >Cconsider the set × W = {ξ ∈ AK ||ξp|p ≤|ap|p} × which is compact since it is a product of compact sets. Now for b ∈ AK with |b| =1 −1 × −1 we also have |b a| >C, hence the Lemma gives α ∈ K with |α|p ≤|bp ap|p, i.e. αb ∈ W . 3.1. Class field theory using the idele group. In section we reformulate the main theorems of section 1 in terms of the idele group. We start by describing ray class groups Hm in terms of the idele class group. Given a modulus np m = mf m∞ = p p we define an open subgroup ⎧ ⎪O× p ⎪ Kp finite, np =0 ⎨⎪ pnp O p 1+ Kp finite, np > 0 U(np)= ⎪ 2 p ⎪Kp real and np =1 ⎩ × Kp p infinite, np =0 × of Kp for each place p, and then we define the open subgroup m U = U(np) p 44 M. FLACH × of AK . To a finite Galois extension L/K we can also associate an open subgroup A× × NL/K ( L )= NLP/Kp LP p where we choose a P | p and note that the local norm subgroup is independent of this choice. We remark that this is just the image of the norm map for the finite × × flat ring extension AL/AK and this map sends L to K . Hence there is also an induced map NL/K : CL → CK . m m Proposition 3.2. Denoting by U¯ the image of U in CK we have ¯ m × × m ∼ m m m CK /U = AK /K · U −→ J /P = H and × × × ∼ m m m CK /NL/K (CL)=AK /K · NL/K (AL ) = J /P · NL/K JL . Proof. We have × m ∼ m × AK /U = J × Kp /U(np) p∈S where S is the union of the set of primes dividing m and the infinite places. By weak approximation there is an exact sequence ×,m × × 1 → K → K → Kp /U(np) → 1 p∈S where K×,m = {α ∈ K×|α ≡ 1modm} is as in Lemma 1.6. Then ¯ m ∼ × m × ∼ m × × CK /U = AK /U · K = (J × Kp /U(np))/ im(K ) p∈S ∼ m ×,m = (J × 1)/ im(K ) = J m/P m = Hm. For a finite extension L/K we can choose a modulus m so that ⊆ × U(np) NLP/Kp LP m × for all places p and hence that U ⊆ NL/K (AL ). Weak approximation gives us the isomorphism (16) × × · ×,m −∼→ × × K /NL/K L K Kp /NLP/Kp (LP) p∈S COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 45 as in the proof of Lemma 1.6. Then we have CK /NL/K (CL) ∼ × × × =A /K · NL/K (A ) ⎛K L ⎞ ⎛ ⎞ ∼ ⎝ m × × ⎠ ⎝ m × × ⎠ · × = J Kp /U(np) / NL/K JL NLP/Kp (LP)/U(np) im(K ) p∈S p∈S ∼ m m ×,m × =(J × 1)/(NL/K JL × 1) · im(K · NL/K L ) m m m =J /(NL/K JL · P ). Hence we can view the Artin map for a finite abelian extension L/K as a map ρ˜ =˜ρL/K : CK → Gal(L/K) inducing an isomorphism ∼ CK /NL/K CL = Gal(L/K). × In particular, viewing Kp as a subgroup of CK , via the embedding × αp → (1,...,1,αp, 1,...) ∈ AK we get homomorphisms for all primes × ρ˜p : Kp → Gal(L/K) which clearly coincide with the local Artin map ρp defined in Ma160b if p is unrami- fied in L/K. It is not so clear whetherρ ˜p = ρp for all places p. One can now proceed in two ways. Either one shows thatρ ˜p takes values in Gal(LP/Kp) ⊆ Gal(L/K) and in fact induces an isomorphism × × ∼ Kp /NLP/Kp LP = Gal(LP/Kp) for any place p. This is done, for example, in Lang’s Algebraic Number Theory (Thm. 3 in Ch. XI,§4) and this is how local class field theory was discovered in the first place (by Hasse). However, the problem still remains to identify this map with ρp (and thereby show independence of the choice of K, for example). Perhaps, after one checks compatibilities with change of fields one can do this but I haven’t seen it anywhere in the literature. Alternatively, one can show that the global Artin map × ρ = ρp : AK → Gal(L/K), p defined as the product of local Artin maps, is trivial on K×. Since it is clearly × trivial on NL/K (AL ) and coincides withρ ˜ on the unramified primes we must have ρ =˜ρ and thereforeρ ˜p = ρp for all primes p. Theorem 3.1. Let K be a global field. Then one has: a) (Reciprocity) For every abelian extension L/K the map ρ is trivial on K×. b) (Isomorphism) For every abelian extension L/K the map ρ is surjective × × with kernel K NL/K (AL ). c) (Existence) For every open subgroup U ⊆ CK of finite index there is a unique abelian extension L/K with norm subgroup U. d) (Classification) There is an inclusion reversing bijection between abelian extensions L/K and open subgroups U of CK of finite index. 46 M. FLACH Proof. The first statement is unfortunately somewhat lengthy to prove, even though we already have the reciprocity theorem in its classical formulation. However, Theorem 1.6 only implies that ρ is trivial on K×,m. We shall give a proof in conjunction with the computation of the cohomology of the idele class group in × the next section. For b) note that ρ is trivial on NL/K (AL ) by local class field × × theory and hence trivial on K NL/K (AL ) by a). By Prop. 3.2 this subgroup has index [L : K]andρ is surjective since it restricts to the classical Artin map on the unramified primes.