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 Paramet

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

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 unramified 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. For c) and d) it suffices to note that the groups U m, resp. U¯ m, × form a basis of open subgroups of AK , resp. a basis of open subgroups of finite index of CK and apply the classical existence theorem, Theorem 1.7.

Remark 3.1.1. If K is a global field of characteristic 0 we could replace ”open subgroup of finite index” by ”open subgroup” in statements c) and d) since any open subgroup of CK has finite index. Remark 3.1.2. Another advantage of the idelic approach is that it easily general- izes to infinite abelian extensions. The inverse limit of the reciprocity maps → ab ψK : lim←− CK /U ←lim− Gal(L/K) = Gal(K /K) U L is injective with image { ∈ ab |∃ ∈ Z | n} σ Gal(K /K n ,σ F¯p =Frob

0 if char(K)=p>0 and it is surjective with kernel the connected component CK of the identity in CK if char(K)=0. One has an isomorphism

0 ∼ 1 r2 r1+r2−1 CK = R × (S ) × V where V 1 ∼ R × Zˆ Z ∼ QD =← lim− S = ( )/ = n is the solenoid, the Pontryagin dual of the discrete group Q. Note that the connected ×,0 × component AK of the identity in AK easily computes to ×, 0 ∼ × ×,>0 ∼ 1 r2 r1+r2 AK = C × R = (S ) × R . p complex p real 3.2. Galois cohomology of the idele class group. In the next section on class formations we shall give the proofs of Theorem 3.1 a) and b) without using results from section 1, in particular without any use of ray class L-functions. We shall not reprove c) and d) since the proofs in the idelic setting are really identical to the classical ones. In order to verify that the idele class group satisﬁes the axioms of a class formation axioms we shall compute the Galois cohomology of the idele class group in this subsection, more precisely just the H0,H1 and H2. This computation goes hand in hand with the determination of the Brauer group of the global ﬁeld K about which the classical approach had nothing to say. It turns out that the Galois cohomology of the idele class group contains a lot of information which one does not see in the m m m m classical formulation. For a start, H or J /P ·NL/K JL are not naturally Galois cohomology groups whereas CK /NL/K CL is: COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 47

Lemma 3.2. For any Galois extension L/K with group G we have 0 H (G, CL)=CK and hence 0 Hˆ (G, CL)=CK /NL/K (CL).

Proof. The short exact sequence of G-modules × × (29) 1 → L → AL → CL → 1 induces an exact cohomology sequence × × 0 1 → K → AK → H (G, CL) → 1 since H1(G, L×) = 0 by Hilbert 90. The following result is in some sense the analogue of the norm index theorem in the idelic approach. On the one hand it implies general results on Galois cohomology of the idele class group which are needed in the proof of Theorem 3.1 a) and b). On the other hand it has a direct proof using purely algebraic techniques, i.e. avoiding the analytic theory of ray class L-functions. At this point we can deduce it rather quickly from Prop. 3.2 and the results of section 1. Theorem 3.2. If L/K is cyclic with group G = Gal(L/K) then

i [L : K]=|G| i =0 |Hˆ (G, CL)| = 1 i =1.

Proof. By Lemma 3.2 and Prop. 3.2 we get 0 ∼ m m m ∼ Hˆ (G, CL)=CK /NL/K (CL) = J /P · NL/K JL = G and so it suffices to show that q(CL)=|G|. Consider the open subgroup A× O× × × L,S = LP LP P∈/S P∈S × of AL , where S is a finite G-stable set of places of L containing the archimedean places and those ramified in L/K and large enough so that Cl(OL,S) = 0. Corollary 1.6 implies i × i × O p O (30) H (G, LP )=H (G , LP )=0 P|p for i =0, 1andp ∈/ S and since cohomology commutes with products we further obtain form Corollary 1.6 × × q(AL,S)= q(Gp,LP)= |Gp|. p∈S p∈S Then with m = P∈S P one has × × × ∼ m × ∼ AL /L · AL,S = JL / im(L ) = Cl(OL,S)=0 and ∼ × × ∼ × × × × × CL = AL /L = AL,S/L ∩ AL,S = AL,S/OL,S. 48 M. FLACH and therefore | | × × p∈S Gp q(CL)=q(AL,S)/q(OL,S)= × = |G| q(OL,S) by Lemma 1.7. For the rest of this section we will develop consequences of Theorem 3.2 without using any other results from class field theory. In other words, Part 1 will only enter in this section via Theorem 3.2. Later we will then give an independent proof of Theorem 3.2 making all results independent of Part 1. First, using Theorem 3.2 we can compute H1 of the idele class group. It satisfies an analogue of Hilbert’s theorem 90. Proposition 3.3. For any Galois extension L/K with group G we have 1 H (G, CL)=0.

Proof. Assume first that G is a p-group. Then G has a cyclic quotient of order p. If L/K denotes the corresponding Galois extension we have an inflation restriction sequence 1 1 1 0 → H (Gal(L /K),CL ) → H (G, CL) → H (Gal(L/L ),CL) obtained form the Hochschild-Serre spectral sequence for the group extension 1 → Gal(L/L) → Gal(L/K) → Gal(L/K) → 1. 1 By induction on |G| we deduce H (G, CL) = 0 from Thm. 3.2. In general let Gp ⊆ G be a p-Sylow subgroup. Then the composite map 1 res 1 cor 1 H (G, CL) −−→ H (Gp,CL)=0−−→ H (G, CL) is multiplication with [G : Gp], hence induces a bijection on the p-primary sub- 1 ∞ 1 group H (G, CL)[p ]ofH (G, CL) (recall that this is a |G|-torsion group). So 1 ∞ 1 H (G, CL)[p ] = 0 and since this holds for all p ||G| we get H (G, CL)=0. 2 Our next aim is to compute H (G, CL) but it is not easy to do this without pass- 2 2 ing to the limit and considering the full group H (K, CK¯ ) = lim−→L H (Gal(L/K),CL). Since the system of all finite extensions L/K is filtered, the direct limit → ¯ × → A× → → 1 K K¯ CK¯ 1 of the exact sequence (29) is an exact sequence of discrete GK -modules, and its cohomology in degree 2 is given by the following theorem. Theorem 3.3. One has a commutative diagram of short exact sequences 2 × 2 × 2 0 −−−−→ H (K, K¯ ) −−−−→ H (K, A ) −−−−→ H (K, C ¯ ) −−−−→ 0 ⏐ K¯ ⏐ K ∼⏐ ∼⏐ = =invK p invp 0 −−−−→ Br(K) −−−−→ p Br(Kp) −−−−−→ Q/Z −−−−→ 0 where ⎧ ⎨⎪Q/Z p nonarchimedean ∼ 1 invp :Br(Kp) = Z/Z p real ⎩⎪ 2 0 p complex. COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 49

We ﬁrst need some local as well as semilocal preparations. Lemma 3.3. For any ﬁnite Galois extension L/K with group G and i ∈ Z one has ˆ i × ∼ ˆ i × H (G, AL ) = H (Gp,LP) p and for i>0 i A× ∼ i ¯ × H (K, K¯ ) = H (Kp, Kp ). p Proof. Since cohomology commutes with products and using Shapiro’s Lemma we get i × ∼ i × i × ˆ A ˆ p O × ˆ p H (G, L,S) = H (G , LP ) H (G ,LP). p∈/S p∈S

But if S contains all ramiﬁed primes the decomposition groups Gp for p ∈/ S are cyclic. Hence from (30) and the fact that cohomology of ﬁnite groups commutes with colimits we get ˆ i A× ˆ i A× ˆ i × ˆ i × (31) H (G, L ) = lim−→ H (G, L,S) = lim−→ H (Gp,LP)= H (Gp,LP). S S p∈S p

Now by deﬁnition A× A× A× K¯ = L = lim−→ L L L and ˆ i A× ˆ i A× ˆ i × H (G, K¯ ) =− lim→ H (G, L ) =− lim→ H (Gp,LP) L L p ˆ i × i ¯ × = −lim→ H (Gp,LP)= H (Kp, Kp ) p L p ¯ since L LP = Kp by Krasner’s Lemma. The following Lemma summarizes the statements from local class ﬁeld theory we will need.

Lemma 3.4. For any ﬁnite Galois extension LP/Kp of local ﬁelds with group Gp there is an isomorphism 1 2 × ∼ Z Z invLP/Kp : H (Gp,LP) = / [LP : Kp] which is related to the local reciprocity map × → ab ρp = ρLP/Kp : Kp Gp by the formula ∪ χ(ρp(α))=invLP/Kp (α δχ) 1 2 where δ : H (Gp, Q/Z) → H (Gp, Z) is the connecting homomorphism in the long exact cohomology sequence induced by 0 → Z → Q → Q/Z → 0. 50 M. FLACH

Proof. The existence of the invariant map follow from the determination of the Brauer group of the local field Kp, the key fact being that any central simple algebra has an unramified splitting field. So there is an isomorphism v 2 ¯ × ∼ 2 ur ¯ × p 2 ur ∼ Q Z invKp : H (Kp, Kp ) = H (Kp /Kp, Kp ) −→ H (Kp /Kp, Z) = / . For this isomorphism one proves directly that there is a commutative diagram with exact rows 2 × 2 × 2 × 0 −−−−→ H (Gp,L ) −−−−→ H (Kp, K¯p ) −−−−→ H (LP, K¯p ) ⏐ P ⏐ ⏐ ⏐ ⏐ ⏐ ∼invLP/Kp ∼invKp ∼invLP

·[LP:Kp] 0 −−−−→ 1 Z/Z −−−−→ Q/Z −−−−−−→ Q/Z. [LP:Kp] Setting −1 1 2 × γ := invK ∈ H (Gp,LP) p [LP : Kp] one deﬁnes the reciprocity isomorphism ˆ −2 Z Z ab −∪−→γ × × ˆ 0 × H (Gp, )=H1(Gp, )=Gp Kp /NLP/Kp LP = H (Gp,LP).

In other words, setting sα = ρp(α) one has by deﬁnition

γ ∪ sα =¯α and hence α¯ ∪ δχ = γ ∪ (sα ∪ δχ). Since δ commutes with cup products we get sα ∪ δχ = δ(sα ∪ χ) with

−1 ∼ 1 sα ∪ χ ∈ Hˆ (Gp, Q/Z) = Z/Z. [LP : Kp] For any ﬁnite group G, s ∈ Gab and χ ∈ Hom(G, Q/Z) one checks that s∪χ = χ(s) (Serre, Local ﬁelds, Appendix to Ch. XI, Lemma 3) and also that δ(r/n)=¯r under the boundary map −1 ∼ 1 δ ∼ 0 Hˆ (Gp, Q/Z) = ker ·n|Q/Z = Z/Z −→ Z/nZ = coker (·n|Z) = Hˆ (Gp, Z) n where n = |Gp| =[LP : Kp]. Finally then ∪ ∪ invLP/Kp (¯α δχ)=invLP/Kp (u r¯)=r/n.

Remark 3.2.1. In the next section we shall define a reciprocity map for any class formation of which the formula in Lemma 3.4 is a special case. Proof. (of Theorem 3.3) We shall write Hi(L/K, M)forHi(Gal(L/K),M). It is clear that the top sequence is exact at the left because Proposition 3.3 implies 1 1 H (K, CK¯ ) = lim−→ H (L/K, CL)=0 L and by Lemma 3.4 the bottom sequence is then also exact at the left. It is also clear that the top sequence is exact in the middle and the bottom sequence is exact at the right but unfortunately it will not be easy to construct the map invK so that the right square commutes. In order to exploit Theorem 3.2 one first looks at the analogous situation for a finite and then cyclic extension L/K. For finite L/K the COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 51

1 vanishing of H (L/K, CL) gives the analogous commutative diagram (without the dotted arrow) (32) / 2 × / 2 A× / 2 / 0 H (L/K, L ) H (L/K, L ) H (L/K, CL) 0 φ 2 × invL/K / 2 × / H (LP/Kp,LP) / 1 Z/Z / 0 H (L/K, L ) p [L:K] 0 with rows which are exact at the left and which we use to deﬁne the global invariant map

invL/K (c):= invLP/Kp (cp) p as the sum of local invariant maps. Note that invL/K is certainly not always surjective since there are global extensions for which the least common multiple of all local degrees is a proper divisor of [L : K]. We ﬁrst record the functoriality of the invariant map. Lemma 3.5. Let L/K be Galois with group G. a) For K ⊆ K ⊆ L there are commutative diagrams

invL/K 2 × / 2 × / 1 H (L/K ,L ) H (L/K , A ) Z/Z O O L [L:KO]

res cor res cor [K:K] incl invL/K H2(L/K, L×) /H2(L/K, A×) / 1 Z/Z O O L [L:KO]

inf inf incl

invK/K 2 × / 2 × / 1 Z Z H (K /K, K ) H (K /K, A ) / O O K [K :KO]

0 0 0 where the bottom portion only applies if K/K is Galois. b) The columns are exact. × 1 c) Moreover, for each α ∈ AK and χ ∈ H (G, Q/Z) we have

(33) χ(ρL/K (α))) = invL/K (α ∪ δχ) where × ab ρL/K : AK → G is the global Artin map deﬁned as the product of local ones

ρL/K ((αp)) = ρLP/Kp (αp). p

Proof. The commutativity involving the middle and right hand columns follows from the corresponding properties of the local invariant map. The commutativity involving the middle and left hand columns is just functoriality of res, cor and 52 M. FLACH inf. The exactness of the left hand column follows from the inflation-restriction sequence which extends to an exact six term sequence in view of the vanishing H1(K/K, H1(L/K,L×)) = 0 by Hilbert 90. Exactness of the right hand column is clear and exactness of the middle column follows from exactness of the right hand column (for the various local field extensions) and the fact that the invariant map is an isomorphism in the local case. Alternatively, it is a consequence of Hilbert 90 for the local fields, just like the exactness of the left hand column. To see (33) first note that the global Artin map is well defined since for any p ∈O× p prime such that αp Kp and is unramified in L/K we have ρLP/Kp (αp)=1. If χp is the restriction of χ to Gp then αp ∪ δχp is the local component of α ∪ δχ and hence ∪ ∪ invL/K (α δχ)= invLP/Kp (αp δχp) p

= χp(ρLP/Kp (αp)) = χ(ρL/K (α))). p

The next theorem verifies that the bottom row in (32) is a complex. It is a reciprocity theorem and just like Theorem 1.6 we will eventually reduce its proof to cyclotomic extensions of Q. For such extensions the proof will conclude by a direct verification of Theorem 3.1 a). Theorem 3.4. For finite L/K and any c ∈ H2(L/K, L×) one has

invL/K (c)=0.

Proof. The key is to construct a diagram of ﬁelds E = LE uu uu uu uu uu L

K u E uu uu uu uu uu Q where L/Q is Galois and E/Q is cyclic cyclotomic with certain local properties. By Lemma 3.5 a) the three maps

H2(L/K, L×) −inf−→ H2(L/K, L×) −cor−→ H2(L/Q,L×) −inf−→ H2(E/Q,E×) COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 53

2 × do not change invL/K (c). Denoting by c ∈ H (E /Q,E ) the image of c, Lemma 3.6 applied to S = {p| invp(c ) =0 } and m a common denominator of the invp(c ), gives an extension E = LE so that res(c)=0∈ H2(E/E, E×). Lemma 3.5 b) then implies that c = inf(c) for some c ∈ H2(E/Q,E×). By Lemma 3.7 we have invE/Q(c ) = 0 and another application of Lemma 3.5 a) gives invE/Q(c )=invL/K (c)=0.

Lemma 3.6. Given a number ﬁeld L, ﬁnite set of places S and an integer m then there exists a cyclic cyclotomic extension E /L so that m divides [EP : Lp] for all p ∈ S.

r r−1 Proof. Let E(q ) ⊆ Q(ζqr ) be the cyclic subextension of degree q for q odd (for q = 2 we leave it as an exercise that there is a totally complex cyclic subextension r r−2 E(2 )ofQ(ζ2r ) of degree 2 ). For any prime p (equal to q or not) the local r degree [E(q )p : Qp] tends to ∞ as r →∞since this is true for the local degrees r n nk Q r Q Q r 1 ··· [ (ζq )p : p]and[ (ζq )p : E(q )p] is bounded by q.Ifm = q1 qk the r r 1 ··· k ∈ extension E = E(q1 ) E(qk ) will have local degree divisible by m for all p S for ri >ni large enough. By possibly enlarging the ri further we get the same conclusion for E/L where E = EL and L is any number ﬁeld.

Lemma 3.7. For any cyclotomic extension E/Q Theorem 3.1 a) holds, and if E/Q 2 × is cyclic we have invE/Q(c)=0for each c ∈ H (E/Q,E ). Proof. Let × ∼ × ∼ nq × ρ = ρp : AQ → Gal(Q(ζm)/Q) = (Z/mZ) = (Z/q Z) p q be the reciprocity map. Recall that the map ρp has image in the decomposition group for all places p.Forp = ∞ we have × ∼ np × np × nq × ρp : Qp → Gal(Qp(ζm)/Qp) = (Z/p Z) ×

⊆ (Z/p Z) × (Z/q Z) q= p × ∼ × Z and in terms of this product decomposition and the decomposition Qp = Zp × p the map ρp is given by k −1 k ρp(u · p )=(u ,p ). This is the content of Dwork’s theorem (see Serre, Local ﬁelds). For p = ∞ (and m>2) the map × ∼ × ρ∞ : R → Gal(C/R) =< ±1 >⊆ (Z/mZ) . is given by ρ∞(u) = sgn(u). It suﬃces to show that ρ(−1) = 1 and ρ(l)=1 × ∼ nq × for every prime number l.Fora ∈ (Z/mZ) = q(Z/q Z) denote by (a)q its q-component. Then we have ⎧ ⎪1 p m, p = l ⎪ ⎨ −1 (l )p p | m, p = l ρp(l)= ⎪ (l)q p = l ⎩⎪ q|m,q= l 1 p = ∞ 54 M. FLACH · −1 and so ρ(l)= q|m,q= l(l)q p|m,p= l(l )p = 1. Moreover ⎧ ⎨⎪1 p m ρp(−1) = (−1)p p | m ⎩⎪ −1 p = ∞ and so ρ(−1)=(−1) p|m(−1)p =1.IfG = Gal(E/Q) is cyclic pick χ ∈ H1(G, Q/Z) so that δχ ∈ H2(G, Z) =∼ Z/|G|Z is a generator. Then

∪δχ : Hˆ 0(G, M) → Hˆ 2(G, M)

× × × is an isomorphism for both M = E and M = AE. Writing c = α∪δχ with α ∈ Q Lemma 3.5 c) and the ﬁrst part of this Lemma give

invE/Q(c)=invL/K (α ∪ δχ)=χ(ρ(α)) = 0.

We now come back to diagram (32) and assume that L/K is cyclic. Then the following holds: Step 1. The top row is exact since

H3(G, L×) =∼ H1(G, L×)=0. Step 2. There is a map φ so that the diagram commutes since

2 × invL/K (im(H (G, L ))) = 0

by Theorem 3.4. Step 3. The map invL/K is surjective. Indeed, if L/K is cyclic of prime degree then we know |CL/NL/K CK | =[L : K], hence there must be a prime p of K with × × Gp = G since otherwise NL/K : AL → AK would be surjective. The same conclusion holds if L/K is cyclic of prime power degree by looking at the unique subextension of prime degree. So in this case invL/K is surjective and surjectivity for general cyclic L/K follows from Lemma 3.5. Step. 4 The map φ is an isomorphism since it is surjective by Step 3 and

2 0 |H (L/K, CL)| = |Hˆ (L/K, CL)| =[L : K],

again by Theorem 3.2. We conclude that for cyclic L/K both rows in (32) are exact and φ exists and is an isomorphism. As remarked above, for general ﬁnite L/K the rows in (32) need not be exact at the right. Instead we pass to the direct limit over all cyclic (or even just cyclotomic) extensions L/K and note that 2 A× 2 A× 2 A× H (K, K¯ ) = lim−→ H (L/K, L )= H (L/K, L ) L/K cyclic L/K cyclic

2 × in view of the fact that any element cp ∈ H (Kp, K¯p ) in the local Brauer group has ∈ 2 A× an unramified splitting field, hence any (necessarily finite) family (cp) H (K, K¯ ) COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 55 has a suitable global cyclic cyclotomic splitting field L/K. Passing to the limit over all finite L/K in Lemma 3.5 a) we obtain the inclusions 0 −−−−→ H2(K, K¯ ×) −−−−→ H2(K, A× ) K¯ ⏐ ⏐ ⏐ ⏐

0 −−−−→ H2(K, K¯ ×) −−−−→ H2(K, A× ) K¯ ⏐ ⏐ ⏐ ⏐

2 × 2 × 0 −−−−→ H (K /K, K ) −−−−→ H (K /K, A ) K ⏐ ⏐ ⏐ ⏐

00 we also obtain (diagram chase) 2 ¯ × 2 × 2 × H (K, K ) =− lim→ H (K /K, K )= H (K /K, K ). K /K cyclic K/K cyclic This then implies exactness of the bottom row in Theorem 3.3 by taking the direct limit over all cyclic L/K of the bottom row of (32) (the system of cyclic L/K is not filtered but at this point we only need to show right exactness which is ok for any index category since colimits commute with colimits). This concludes the computation of the Brauer group of the global field K. One also obtains Q/Z as 2 a subgroup (and hence direct summand) of H (K, CK¯ ) but at this point we still don’t know equality of the two groups. For any finite extension L/K one has the commutative diagram with exact columns 2 0 −−−−→ Q/ZL −−−−→ H (L, C ¯ ) K ⏐ ⏐ ⏐[L:K]=res ⏐res

ι 2 0 −−−−→ Q/ZK −−−−→ H (K, C ¯ ) K ⏐ ⏐ ⏐ ⏐ ι 1 L/K 2 0 −−−−→ Z/Z −−−−→ H (L/K, CL) [L:K] ⏐ ⏐ ⏐ ⏐

00 where 2 Q/ZK = H (K /K, CK ) K/K cyclic is mapped to Q/ZL under the restriction map in view of the commutative diagram GL −−−−→ Gal(LK /L) ⏐ ⏐ ⏐ ⏐

GK −−−−→ Gal(K /K) and the fact that LK/L is again cyclic. A simple diagram chase shows that 1 Z Z ⊆ 2 ι( [L:K] / ) H (L/K, CL) and hence the existence of ιL/K . To show that ι 56 M. FLACH is an isomorphism it suffices to show that ιL/K is an isomorphism for all L/K. This in turn will follow if we can prove that 2 (34) |H (L/K, CL)|≤[L : K]. We know equality for cyclic L/K. For a tower K ⊆ K ⊆ L there is an exact inflation-restriction sequence 2 2 2 0 → H (K /K, CK ) → H (L/K, CL) → H (L/K ,CL) 1 in view of the vanishing of H (L/K ,CL). Hence by an easy induction we get (34) for all solvable L/K. Finally recall that for any finite group G with Sylow subgroups Gp,anyG-module M and any i ≥ 0 i resp i H (G, M) −−→ H (Gp,M) p||G| is injective. Hence, if Kp denotes the fixed field of the Sylow p-subgroup of Gal(L/K), we get 2 i |H (L/K, CL)|≤ |H (L/Kp,CL)|≤ [L : Kp]=[L : K]. p p This finally completes the proof of Theorem 3.3. We summarize our computation of the cohomology of the idele class group in the following theorem. Theorem 3.5. Assume the statement of Theorem 3.2 holds. Then for any Galois extension L/K of number fields one has ⎧ ⎨⎪CK i =0 i H (L/K, CL)= 0 i =1 ⎩⎪ 1 Z Z [L:K] / i =2 and for CK¯ =− lim→L CL one has ⎧ ⎨⎪CK i =0 i i H (K, CK¯ ) = lim H (L/K, CL)= 0 i =1 −→ ⎩⎪ L Q/Z i =2.

4. Class formations and duality Deﬁnition 4.0.1. A class formation consists of a proﬁnite group G and a discrete G-module C together with isomorphisms

2 V ∼ 1 invU/V : H (U/V,C ) = Z/Z [U : V ] for each pair V U ⊆ G of open subgroups such that the following hold a) H1(U/V,CV )=0 b) If W U and W ⊆ V the diagram H2(U/V,CV ) −−−−inf→ H2(U/W,CW ) −−−−res→ H2(V/W,CW ) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ inv∼ inv∼ inv∼

· U V 1 Z Z −−−−⊆ → 1 Z Z −−−−[ : →] 1 Z Z [U:V ] / [U:W ] / [V :W ] / COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 57

is commutative. If (G, C) is a class formation then ⎧ G ⎨⎪C i =0 Hi(G, C)= 0 i =1 ⎩⎪ 1 Z Z 1 Z Z |G| / :=− lim→U [G:U] / i =2. So if for any integer n there is an open subgroup U ⊆ G of index divisible by n we have H2(G, C) =∼ Q/Z. This is the case if there is a surjection G → Zˆ like in examples a)- e) below. Here are some examples of class formations.

a) (G, C)=(GK ,CK¯ ) where K is a global field. × b) (G, C)=(GK , K¯ ) where K is a local field. c) (G, C)=(Zˆ, Z) d) (G, C)=(GK ,CK ) where K/k is an extension of transcendence degree one over an algebraically closed field k of characteristic zero, i.e. a function field of a smooth proper curve X over k,and Q Z CK =− lim→ HomZ(Pic(XL, / ) L/Kfinite

where XL is the smooth proper curve over k with function field L. e) There is class field theory for higher local fields. A 2-local field is a complete discretely valued field K with residue field a local field, for example K = Qp((T )), Fq((T ))((S)) or ∞ i Qp{{T }} = aiT | ai ∈ Qp, inf vp(ai) > −∞, lim vp(ai)=∞ i∈Z i→−∞ i=−∞

which has residue field Fp((T )). Then one has to extend the notion of class • formation to allow a complex of GK -modules, and it turns out (G, C )= × L × (GK , K¯ ⊗Z K¯ ) is a class formation. f) (GS,CS) where S is a set of places of the global field K containing all archimedean ones, GS = Gal(KS/K) the Galois group of the maximal extension of K unramified outside S and × × CS(L)=AL /L · US

where US is the compact subgroup O× × { } US = LP 1 . P∈/S P∈S

i Note that by (30) we have Hˆ (G, US)=0fori =0, 1 (and L/K unramiﬁed outside S) and in fact for any i ∈ Z since the decomposition groups at P ∈/ S are cyclic. By the long exact cohomology sequence induced by

0 → U¯S → CL → CS(L) → 0

i ∼ i 0 ∼ we get isomorphisms Hˆ (G, CL) = Hˆ (G, CS(L)) and also H (G, CS(L)) = CS(K). 58 M. FLACH

We shall now develop a sequence of duality theorems for finite groups which will culminate in the Tate-Nakayama duality for class formations. Throughout we set A∗ = Hom(A, Q/Z) for any abelian group A. This is an exact contravariant functor from abelian groups to abelian groups. If A is finite then A∗ coincides with the Pontryagin dual of A. Theorem 4.1. Let Γ be a finite group and A a Γ-module. Then for all i ∈ Z the pairing

i ∗ −i−1 ∪ −1 ∼ 1 Hˆ (Γ,A ) × Hˆ (Γ,A) −→ Hˆ (Γ, Q/Z) = Z/Z ⊆ Q/Z |Γ| induces an isomorphism i ∗ ∼ −i−1 ∗ Hˆ (Γ,A ) = Hˆ (Γ,A) .

Proof. We ﬁrst show the statement for i = 0. A homomorphism f : A → Q/Z is a Γ-homomorphism if and only if f(IΓA) = 0, hence we obtain an isomorphism H0(Γ,A∗) =∼ H (Γ,A)∗. 0 ∗ If f ∈ NΓA , i.e. f = σ∈ σh, then for a ∈ A with NΓa = 0 we have Γ −1 f(a)= (σh)(a)= h(σ a)=h(NΓa)=0 so we obtain a map ∗ Γ ∗ → ∗ (A ) /NΓA (NΓ A/IGA) . → Q Z If g : NΓ A / is a homomorphism that vanishes on IΓA then g can be extended to a homomorphism g : A → Q/Z which is a Γ-homomorphism since g(IΓA)=0. ∈ ∗ Γ ∈ ∗ So our map is surjective. If f (A ) vanishes on NΓ A there exists g (NΓA) → with f(a)=g(NΓa) since NΓ : A/NΓ A NΓA is an isomorphism. Again g can be extended to a homomorphism g : A → Q/Z and then f = N g since Γ −1 (NΓg)(a)= g(σ a)=g(NΓa)=f(a). σ∈Γ For arbitrary i one can use dimension shifting which gives a commutative diagram ∪ ˆ i ∗ × ˆ −i−1 / −1 Q Z H (ΓO,A ) H (Γ,A) H (Γ, / ) .

i i(i+1)/2 δ δi (−1) ∪ 0 −1 / −1 Hˆ (Γ, Hom(A, Q/Z)i) × Hˆ (Γ,A−i) H (Γ, Q/Z) ∼ Since Hom(A, Q/Z)i = Hom(A−i, Q/Z) the desired result follows. We recall that for any Γ-module A one deﬁnes A1 by the exact sequence

0 → A → A ⊗Z Z[Γ] → A1 → 0 and since the middle term is cohomologically trivial we get an isomorphism i ∼ i+1 δ : Hˆ (Γ,A1) = Hˆ (Γ,A).

For p>0 one deﬁnes Ap =(Ap−1)1 and for p<0 one uses induction and the module A−1 deﬁned by the exact sequence

0 → A−1 → HomZ(Z[Γ],A) → A → 0. COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 59

Remark 4.0.2. For an arbitrary (discrete) group Γ and (discrete) Z[Γ]-module A one always has a duality between cohomology and homology i ∗ ∼ ∗ H (Γ,A ) = Hi(Γ,A) for any i ≥ 0. This follows from the Hom-⊗-adjunction ∼ HomZ[Γ](P•, HomZ(A, Q/Z)) = HomZ(P• ⊗Z[Γ] A, Q/Z) which is a special case of the adjunction ∼ HomS(P, HomR(A, B)) = HomR(P ⊗S A, B) for a right S-module P , S-R-bimodule A and right R-module B. Theorem 4.1 extends this to all i ∈ Z in case Γ is ﬁnite. Proposition 4.1. Let Γ be a ﬁnite group and A a Z-free Γ-module. Then for all i ∈ Z the pairing i −i ∪ 0 ∼ Hˆ (Γ, Hom(A, Z)) × Hˆ (Γ,A) −→ H (Γ, Z) = Z/|Γ|Z induces an isomorphism Hˆ i(Γ, Hom(A, Z)) =∼ Hˆ −i(Γ,A)∗.

Proof. Since A is Z-free we have an exact sequence 0 → Hom(A, Z) → Hom(A, Q) → Hom(A, Q/Z) → 0 and Hom(A, Q) is cohomologically trivial. So we get a commutative diagram

i− ∗ −i ∪ / − Hˆ 1(Γ,A ) × Hˆ (Γ,A) H 1(Γ, Q/Z)

δ id δ i −i ∪ / Hˆ (Γ, Hom(A, Z)) × Hˆ (Γ,A) H0(Γ, Z) where the vertical arrows are isomorphisms. Remark 4.0.3. One can combine Theorem 4.1 and Prop. 4.1 into an isomorphism in the derived category of abelian groups ∼ ∗ RˆΓ(Γ,RHom(A, Z)) = R Hom(RˆΓ(Γ,A), Q/Z)=RˆΓ(Γ,A) ∼ for any abelian group A.IfA is free abelian then R Hom(A, Z) = Hom(A, Z) and we recover Prop. 4.1. If A is arbitrary the exact triangle R Hom(A, Z) → R Hom(A, Q) → R Hom(A, Q/Z) → induces an isomorphism RˆΓ(Γ,RHom(A, Q/Z)) =∼ RˆΓ(Γ,RHom(A, Z))[1] which implies i ∗ ∼ i+1 Hˆ (Γ,A ) = Hˆ (Γ,RHom(A, Z)) and we recover Theorem 4.1. Theorem 4.2. (Nakayama-Tate) Let Γ=U/V be a layer in a class formation and A a Z[Γ]-module, ﬁnite free over Z. 60 M. FLACH

a) If γ ∈ H2(Γ,CV ) is the canonical generator, i.e. the unique element with 1 invU/V (γ)= [U:V ] , then

i ∪γ i+2 V Hˆ (Γ,A) −−→ Hˆ (Γ,A⊗Z C )

is an isomorphism for all i ∈ Z. b) For all i ∈ Z the cup product

i V 2−i ∪ 2 V ∼ 1 Hˆ (Γ, Hom(A, C )) × Hˆ (Γ,A) −→ H (Γ,C ) = Z/Z [U : V ] induces an isomorphism of ﬁnite abelian groups

Hˆ i(Γ, Hom(A, CV )) =∼ Hˆ 2−i(Γ,A)∗.

Proof. Let 0 → CV → C(γ) → Z[G] → Z → 0 be a Yoneda 2-extension corresponding to ∈ 2 V 2 Z V γ H (Γ,C ) = ExtZ[G]( ,C ). The composite map

0 δ1 1 δ2 2 V Z/|Γ|Z = Hˆ (Γ, Z) −→ Hˆ (Γ,IΓ) −→ Hˆ (Γ,C ) induced by the short exact sequences

(35) 0 → IΓ → Z[Γ] → Z → 0 and

V (36) 0 → C → C(γ) → IΓ → 0 also coincides with the cup product ∪γ (at least up to sign) by general homological algebra. Since C was a class formation ∪γ is an isomorphism. Since δ1 is always an isomorphism this implies that δ2 is an isomorphism. The long exact sequence

1 V 1 1 δ2 2 V 2 2 H (Γ,C ) → H (Γ,C(γ)) → H (Γ,IΓ) −→ H (Γ,C ) → H (Γ,C(γ)) → H (Γ,IΓ)

1 V 2 ∼ 1 induced by (36) together with H (Γ,C )=0andH (Γ,IΓ) = H (Γ, Z) = 0 then shows that H1(Γ,C(γ)) = H2(Γ,C(γ)) = 0. The same holds for all subgroups, so C(γ) is cohomologically trivial. Statement a) then follows from tensoring (35) and (36) with A, Lemma 4.1 below, and the description of the cup product as the composite map

i δ1 i+1 δ2 i+2 V Hˆ (Γ,A) −→ Hˆ (Γ,A⊗Z IΓ) −→ Hˆ (Γ,A⊗Z C ). Similarly, since A is Z-free applying Hom(A, −) to the exact sequences (35) and (36) yields exact sequences whose middle terms are again cohomologically trivial COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 61 by Lemma 4.1 below. This then implies that the maps δ in the diagram

i− −i ∪ / Hˆ 2(Γ, Hom(A, Z)) × Hˆ 2 (Γ,A) Hˆ 0(Γ, Z)

δ id δ ∪ i−1 × 2−i / 1 Hˆ (Γ, Hom(A, IΓ)) Hˆ (Γ,A) H (Γ,IΓ)

δ id δ i V −i ∪ / V Hˆ (Γ, Hom(A, C )) × Hˆ 2 (Γ,A) H2(Γ,C ) are isomorphisms and the asserted duality follows from Prop. 4.1. Lemma 4.1. If X is a cohomologically trivial module over a finite group Γ and A is any finitely generated Z-free Γ-module then the Z[Γ]-modules HomZ(A, X) and A ⊗Z X are cohomologically trivial. Proof. A Z[Γ]-module is cohomologically trivial if and only if it has finite projective dimension if and only if it has projective dimension one. So take a resolution

0 → P1 → P2 → X → 0 by Z[Γ]-projective modules. We get induced exact sequences

0 → HomZ(A, P1) → HomZ(A, P0) → HomZ(A, X) → 0 and 0 → A ⊗Z P1 → A ⊗Z P2 → A ⊗Z X → 0. ∼ Since A is ﬁnitely generated HomZ(A, P ) = HomZ(A, Z) ⊗Z P with diagonal action which is well known to be isomorphic to HomZ(A, Z) ⊗Z P with trivial action on dimZ A ∼ dimZ A the ﬁrst factor if P is free, hence to P . Similarly, A ⊗Z P = P if P is free. So if P is free then HomZ(A, P )andA ⊗Z P are again free, hence c.t. By passing to direct summands we deduce that HomZ(A, P )andA⊗Z P are c.t., hence so are HomZ(A, X)andA ⊗Z X. Corollary 4.1. Given any layer Γ=U/V in a class formation there is an isomorphism U V ∼ ab ρ = ρU/V : C /NΓC −→ Γ given by χ(ρ(a)) = invU/V (a ∪ δχ) δ for χ ∈ H1(Γ, Q/Z) −→ H2(Γ, Z) and a ∈ CV . The map ρ is called the reciprocity map, or norm residue homomorphism. Proof. Taking A = Z and i = 0 in Theorem 4.2 gives an isomorphism U V 0 V ∼ 2 ∗ ρ˜ : C /NΓC = Hˆ (Γ,C ) = Hˆ (Γ, Z) characterized by the property

invU/V (a ∪ δχ)=˜ρ(a)(δχ) and the isomorphism Hˆ 2(Γ, Z)∗ =∼ H1(Γ, Q/Z)∗ =∼ Γab ab sends the character ψ to the group element σψ ∈ Γ with ψ(δχ)=χ(σψ). We then deﬁne ρ(a):=σρ˜(a) and obtain the above formula for χ(ρ(a)). 62 M. FLACH

Recall from Lemma 3.4 that the reciprocity map for local fields was defined via this corollary, and this is the only way I know to define this map in the ramified case. For global fields one has the alternative definition using only the Frobenius automorphisms for the unramified places, i.e. the classical Artin map. The following corollary finally proves Theorem 3.1 a) and b). Corollary 4.2. Let K be a global field, L/K a finite abelian extension and × ρL/K = ρp : AK → Gal(L/K) p × the reciprocity map. If α ∈ K then ρL/K (α)=1. Moreover, ρL/K induces an isomorphism × × × ∼ AK /NL/K AL · K = CK /NL/K CL = Gal(L/K).

× 1 Proof. By Lemma 3.5 c) for each α ∈ AK and χ ∈ H (G, Q/Z)wehave

χ(ρL/K (α))) = invL/K (α ∪ δχ) which means that ρL/K coincides with the map ρ of Corollary 4.1. Corollary 4.3. For a Galois extension L/K with subﬁeld K ⊆ K ⊆ L there are commutative diagrams ρ−1 o ab COK Gal(L/KO )

cor res cor res ρ−1 o ab CK Gal(L/K ) .

Proof. Since ρ−1 is given by ∪γ and res(γ)=γ it suﬃces to show res(x ∪ y)= res(x)∪res(y) and the projection formula cor(x∪res(y)) = cor(x)∪y. By dimension shifting this reduces to the degree 0 case where we have cor(x) ⊗ y = s(x) ⊗ y = s(x ⊗ y) = cor(x ⊗ y). s s

Note that in negative degrees, in particular for i = −2, we have homology groups, so the natural map is the corestriction map whereas the restriction is the ”opposite” or ”Umkehr” map. For i = −2 it is called the transfer.

4.1. Direct proof of Theorem 3.2. In this section we give a proof of Theorem 3.2 that does not use results from Part 1 except for some Lemmas which had a self-contained proof. Other Lemmas from Part 1 we will have to reprove but we will not use any results of section 1.4, the analytic theory of ray class L-functions. In the above short derivation of Theorem 3.2 we already showed directly, only using Corollary 1.6 and Lemma 1.7, that 0 |Hˆ (G, CL)| (37) q(CL):= = |G| =: n |Hˆ 1(G, CL)|

0 for a cyclic extension L/K with group G. Hence it suffices to show that |Hˆ (G, CL)| 1 divides n, or that Hˆ (G, CL) = 0 in order to prove Theorem 3.2. By induction on COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 63 the number of prime factors of n it furthermore suffices to show this for prime degree n in view of the long exact inflation restriction sequence 1 1 1 0 → H (Gal(L /K),CL ) → H (G, CL) → H (Gal(L/L ),CL) and the fact that H1 = Hˆ 1. We can further assume that K contains a primitive n-th root of unity by the following argument. The field K := K(ζn) has degree [K : K] dividing n − 1, hence prime to n, the field L := LK is cyclic of degree n over K and G := Gal(L/K) is isomorphic to G by restriction. We have maps of G-modules NL/L CL → CL −−−−→ CL whose composite is multiplication by [L : L]=[K : K]. Hence the induced map

N 1 1 K /K 1 H (G ,CL) → H (G ,CL ) −−−−→ H (G ,CL) 1 is multiplication by [K : K] but is also the zero map if we know that H (G ,CL )= 1 1 1 0. Since H (G ,CL)isann-torsion group this implies H (G ,CL)=H (G, CL)= 0. √ n Since ζn ∈ K, our field L = K( a) is a Kummer extension. We place ourselves in the situation of Proposition 1.6, making sure that LS,n contains L.SoletS be a finite set of places of K such that • p |∞⇒p ∈ S • p | n ⇒ p ∈ S •{p1,...,pk}⊆S where p1,...,pk =Cl(OK ). • vp(a) =0 ⇒ p ∈ S and define n × LS,n := K OK,S . × Clearly, L ⊆ LS,n since a ∈OL,S. By the proof of Prop. 1.6 we also know that s [LS,n : K]=n where s = |S|. What we don’t know at this point is whether L or LS,n are class fields. In fact we don’t even know surjectivity of the Artin map since this was deduced from the analytic theory. So we reprove here Corollary 1.3. Lemma 4.2. For any abelian extension L/K and modulus m divisible by all ramified primes the Artin map is surjective. Proof. We again look at the fixed field L of the image of the Artin map and conclude that all p m split completely in L/K. Let E ⊆ L be a subextension so that E/K is cyclic. Then again all p m split completely in E/K and in particular × × {p | m∞} Kp = NEP/Kp (EP) .SoifS = we have × × Kp × 1 ⊆ NE/KAE p∈/S p∈S and by the weak approximation theorem K× surjects onto × × Kp /NEP/Kp (EP) , p∈S i.e. × × × 1 × Kp ⊆ K · NE/KAE. p∈/S p∈S 64 M. FLACH

So we have ˆ 0 × × × × × × H (E/K,CE)=AK /(K · NE/KAE)= Kp /(K · NE/KAE)=1 p which together with (37) for the cyclic extension E/K implies E = K. Hence L = K, i.e. the Artin map is surjective. Using similar ideas we reprove Lemma 1.13, actually a slightly strengthened version of it. Lemma 4.3. Let S ⊆ S be a subset also satisfying the above four conditions and put × × × n OK,S,n := OK,S ∩ (Kp ) . p∈S × × n Then one has OK,S,n =(OK,S) and therefore by (23) × × × × n s OK,S/OK,S,n = OK,S/(OK,S) = n .

O× ⊇ O× n ∈O× Proof. The inclusion K,S,n ( K,S√ ) is clear. Now take α K,S,n and consider the cyclic extension E = K( n α). Then all p ∈ S are split completely in E/K and all p ∈/ S are unramiﬁed in E/K.So × × × × A O × ⊆ A K,S := Kp Kp NE/K E. p∈/S p∈S × By the fact that S contains generators of the ideal class group we have AK = × × K · AK,S . Therefore ˆ 0 × × × × × × × H (E/K,CE)=AK /(K · NE/KAE)=K · AK,S /(K · NE/KAE)=1 which together with (37) for the cyclic extension E/K implies E = K. This means × × n × n α ∈OK,S ∩ (K ) =(OK,S) .

In the situation of Lemma 4.3 we now assume that the primes {p1,...,pt} = \ { } Z ⊆ S S are chosen so that Frobp1 ,...,Frobpt is a /n-basis of Gal(LS ,n/L) Gal(LS,n/K). This we can do by Lemma 4.2. In particular, the primes pi split completely in L/K and we have t = s − 1ands = s + t =2s − 1. We therefore have × n × × × O× ⊆ A× U := (Kp ) Kp Kp NL/K ( L ) p∈S p∈S\S p∈/S and ˆ 0 × × × × × × × H (G, CL)=AK /(K · NL/K (AL )) = K · AK,S/(K · NL/K (AL )) has order dividing × × × n [A : U)] [Kp :(Kp ) ] × · A× × · K,S p∈S [K K,S : K U)] = × × × = × × [K ∩ AK,S : K ∩ U] [OK,S : OK,S,n] n2s n2s = = = n ns n2s−1 where we have also used Lemma 1.14 which had a self-contained proof. This ﬁnishes our second proof of Theorem 3.2. COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 65

5. More examples and complements 5.1. The quadratic reciprocity law for number fields. Recall the quadratic reciprocity law first proven by Gauss. For an integer a and prime number p 2a define the Legendre symbol a 1ifa is a square modulo p = p −1 otherwise.

More generally, for an integer b = p1 ···pr relatively prime to 2a one defines the Jacobi symbol a a a = ··· . b p pr 1 a Clearly, b only depends on a modulo b. Theorem 5.1. (Gauss) For positive, odd, relatively prime integers a, b one has a b a−1 b−1 =(−1) 2 2 . b a Moreover, for odd b there are the supplementary statements −1 b−1 2 b2−1 =(−1) 2 ; =(−1) 8 . b b We shall reprove the quadratic reciprocity law from the Artin reciprocity law and at the same time generalize it to number fields. Fix a number field K.Forα ∈OK and a prime ideal p 2α one can define an obvious analogue of the Legendre symbol α 1ifα is a square modulo p = p −1 otherwise and an obvious analogue of the Jacobi symbol for β ∈OK relatively prime to 2α α α α = ··· β p1 pr where (β)=p1 ···pr is the prime factorization of β. One clearly has √ α 1ifp splits in K( α)/K = √ p −1ifp is inert in K( α)/K √ with the understanding that any prime splits in K( α)/K if α is a square. This condition has a straightforward reformulation in terms of the Frobenius automor- phism √ Frobp ∈ Gal(K( α)/K) ⊆{±1} which in turn is the image of a local uniformizer πp under the local reciprocity map √ √ × ρp Kp −→ Gal(K( α)P/Kp) ⊆ Gal(K( α)/K) ⊆{±1}. p ∞ So for 2α we have α = ρp(πp). p 66 M. FLACH

In order to prove a reciprocity√ law one needs to look at ρp for all places p of K. Note that the extension Kp( α)/Kp can be ramified at places p | 2α∞ (or can be trivial if α is locally a square). If p | α but p 2∞ then √ Kp( α)/Kp unramified ⇔ vp(α) ≡ 0mod2 and if vp(α) ≡ 1 mod 2 this extension is tamely ramified. By local class field theory the map ρp then induces an isomorphism √ O× O× 2 ∼ O p × O p × 2 ∼ ∼ {± } ρp : Kp /( Kp ) = ( K / ) /(( K / ) ) = Gal(K( α)P/Kp) = 1 ∈O× and so we have for α Kp vp(α) α ρp(α )= p for the entirely different reason that the Legendre symbol is the only nontrivial × quadratic character of (OK /p) .Ifp is real then ρp is also easy to analyze. We have √ sgn σp(α)−1 Kp( α) = Kp ⇔ σp(α) < 0 ⇔ (−1) 2 = −1 sgn σp(α )−1 and in that case we have ρp(α )=(−1) 2 . So altogether sgn σp(α )−1 sgn σp(α)−1 ρp(α )=(−1) 2 2 . The hardest primes p to analyze are those dividing 2 since there are several ramified ∼ quadratic extensions of Kp. Let’s do this when Kp = Q2.Wehave Q× Q× 2 ∼ Z/2Z × − Z/2Z × Z/2Z 2 /( 2 ) = [2] [ 1] [5] and this group both parametrizes the quadratic extensions by Kummer theory and is the source of the Artin map for all quadratic extension. So we can define a bilinear symbol on this Z/2Z-vector space, the Hilbert symbol α ,α := ρp(α ) ∈{±1}. p √ α ,α Then p = 1 if and only if α is a norm of the quadratic extension Q2( α)/Q2 if and only if x2 − αy2 = α ∈ Q2 has a solution (x, y) 2 which (for nonsquare α) is true if and only if x2 − αy2 − αz2 =0 ∈ Z3 has a nonzero solution (x, y, z) 2. But this condition is symmetric in α, α which means that our bilinear form is symmetric. It is given by the following table √ 2 −1 5 Q − 2(√ 2) 1 1 1 Q − − 2( √ 1) 1 1 1 Q2( 5) −1 1 1 which is justified by the following computations √ 2 −1 5 Q 2 − · 2 2 − · 2 − − 2(√ 2) 2 2 1 =2 1 2 1 = 1 1 Q − 2 2 ≡− 2 2 2( √ 1) 1 a + b 1mod4 2 +1 =5 2 2 Q2( 5) −1 1 5 − 5 · 2 =5. COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 67 √ The remaining −1 simply results from the fact that Q2( 5) is the unramified quadratic extension of Q2, so 2 is sent to the Frobenius, i.e. to −1. Of course it also follows from a computation modulo 8. One verifies directly that 2 √ α −1 ρ (α )=(−1) 8 Q2( 2)/Q2 ∈ Z× for α 2 and α−1 α −1 √ − 2 2 ρQ2( α)/Q2 (α )=( 1) ∈ Z× for α, α 2 by checking that both sides are multiplicative and agree on the generators listed in the above table. These formulas are just a neat way to con- dense the table, for otherwise one would have to make many case distinctions. For × × 2 an extension Kp/Q2 of degree d the space Kp /(Kp ) has dimension d +2 over Z/2Z and the Hilbert symbol also gives a symmetric bilinear form which is fairly straightforward to compute in any given example. The quadratic reciprocity law for K would depend on this table as is clear from the following theorem.

Theorem 5.2. Let K be a number ﬁeld. For α, β ∈OK relatively prime to each other and relatively prime to 2 we have α β sgn σp(α)−1 sgn σp(β)−1 = ρp(β) (−1) 2 2 β α p|2 p real √ where ρp is the reciprocity map for the quadratic extension Kp( α)/Kp.If2 splits completely in K/Q this law simpliﬁes to α β σp(α)−1 σp(β)−1 sgn σp(α)−1 sgn σp(β)−1 (38) = (−1) 2 2 (−1) 2 2 β α p|2 p real × and we have the following two supplementary laws. If α ∈OK and β is prime to 2 then α σp(α)−1 σp(β)−1 sgn σp(α)−1 sgn σp(β)−1 (39) = (−1) 2 2 (−1) 2 2 . β p|2 p real If α is 2-primary and β is prime to 2 then −v (α) σp(β)2−1 2 p σp(α)−1 σp(β)−1 α vp(α) (40) = (−1) 8 (−1) 2 2 β p|2

sgn σp(α)−1 sgn σp(β)−1 × (−1) 2 2 . p real

Proof. We have α vp(β) = ρp(πp) = ρp(β) β p2α∞ p2α∞ and v α β β p( ) = = ρp(β) α p p|α p|α and sgn σp(α)−1 sgn σp(β)−1 (−1) 2 2 = ρp(β). p real p real 68 M. FLACH

Theorem 5.2 now follows from Theorem 3.1 a) (41) ρ(β)= ρp(β)=1. p β The supplementary law (39) follows from (38) since α =1if(α)=OK is the unit ideal. The supplementary law (40) is again a direct consequence of (41) but there vp(α) are no odd primes dividing α to consider. At each p | 2 we write σp(α)=2 up ∈ Z× with up 2 and note that ρp(β) is multiplicative in σp(α). Remark 5.1.1. Note that (38), (39) and (40) include a generalization of the classical reciprocity and supplementary laws to negative b. For example if K = Q and β = −b<0 we get − −b− −b b− − 1 1 −1·−1 +1 1 1 =(−1) 2 (−1) =(−1) 2 =(−1) 2 = −b b a which is in accordance with the fact that b only depends on the ideal generated by b. √ Q − Example√ 5.1.1. In K = ( 7) decide if 5+2ω is a square modulo 73 where 1+ −7 ω = 2 . Note we have 2 2 NK/Q(a + bω)=a + ab +2b and in particular N(5 + 2ω)=43. The prime 73 ≡ 3mod7is inert in K/Q since 3 is a nonsquare modulo 7.SinceK has no real places and 73 ≡ 1mod4we have 5+2ω 73 30 2 3 5 = = = . 73 5+2ω 5+2ω 5+2ω 5+2ω 5+2ω Since 5 ≡ 1mod4we have with p =(ω) 2 σ ω − 5 5+2ω ω ω¯ σp(¯ω)−1 5−1 52−1 p¯ ( ) 1 5−1 = = =(−1) 2 2 (−1) 8 (−1) 2 2 = −1. 5+2ω 5 5 Since 3 ≡−1mod4and (5 + 2ω − 1)/2=2+ω is even at p =(ω) and odd at ¯p we get 3 2+2ω = − 5+2ω 3 2 (1+σ (ω))−1 2 (σ (ω))−3 3 −1 p 3−1 3· 3 −1 p¯ 3−1 = − (−1) 8 (−1) 2 2 · (−1) 8 (−1) 8 2

= − [1 · 1] = −1

(1+σp(ω))−1 since 2 =1/σp(¯ω) is odd and so is (σp(ω)+1)/4 − 1 (σp(ω)) − 3 ¯ = ¯ 2 8 since 32 − 3+2=8≡ 0mod8but not mod 16. Finally (5 + 2ω)2 − 1 = (24 + 20ω +4ω2)/8=(24+20ω +4(ω − 2))/8=2+3ω 8 is even at p =(ω) and odd at p¯ =(¯ω) and hence 2 2 2 (5+2σp(ω)) −1 (5+2σp¯ (ω)) −1 =(−1) 8 (−1) 8 =1· (−1) = −1. 5+2ω COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 69

So we ﬁnd 5+2ω =(−1) · (−1) · (−1) = −1. 73 Actually one could have saved quite a bit of work in this example by noting that a a ∈ Z 5+2ω = 43 for a prime to 43. We just use this remark to double check our results. 5 3 = = −1 43 5 3 1 = − = −1 43 3 2 432−1 =(−1) 8 = −1 43 √ −1 Remark 5.1.2. Note that also for K = Q( −7) we can verify that β only depends on the ideal generated by β but this time the compensating sign changes occur at the two primes dividing 2, not at 2 and ∞ as was the case for K = Q. One has −1 −σp(β)−1 −σp¯ (β)−1 σp(β)−1 σp¯ (β)−1 −1 =(−1) 2 (−1) 2 =(−1) 2 (−1) 2 = . −β β

We continue with some remarks on more general reciprocity laws. If a local field Kp contains a primitive n-th root of unity, the combination of Kummer theory and local class field theory leads to the definition of the local Hilbert symbol. For ∈ α, α Kp let √ × n ρp : K → Gal(Kp( α)/Kp) → μn p √ n be the reciprocity map for the extension Kp( α)/Kp followed by the canonical homomorphism √ (42) σ → ( n α)σ−1 of Kummer theory, and define the Hilbert symbol α ,α := ρp(α ) ∈ μn. p This symbol is a bilinear map × × n × × n Kp /(Kp ) × Kp /(Kp ) → μn and one can show that it actually coincides with the cup product

1 1 ∪ 2 ⊗2 ∼ 2 inv ⊗μn H (Kp,μn) × H (Kp,μn) −→ H (Kp,μn ) = H (Kp,μn) ⊗ μn −−−−−→ μn.

Apart from bilinearity it has the following properties √ α ,α n a) = 1 if and only if α is a norm from Kp( α)/Kp. p −1 b) α ,α = α,α . p p c) α,1−α = 1 and α,−α =1. p p α,α × × n d) p = 1 for all α ∈ Kp if and only if α ∈ (Kp ) . 70 M. FLACH

Theorem 5.3. Let K be a number ﬁeld containing a primitive n-th root of unity. × Then for α, β ∈ K α, β =1. p p

Proof. This is just a rewriting of the Artin reciprocity law ρ(β)= ρp(β)=1 p √ × n where ρ : AK → Gal(K( α)/K) is the global reciprocity map. ForanumberﬁeldK containing a primitive n-th root of unity one can deﬁne the n-th power residue symbol. Given α ∈OK and p nα let α ∈ μn p n Np−1 α n be the unique root of unity congruent to α modulo p.Then p =1if n n and only if there is β ∈OK with β ≡ α mod p. From (42) and the fact that ρp(πp)=Frobp where πp is a uniformizer of Kp on deduces that α πp,α = . p n p This is the starting point for deducing a reciprocity law for the n-th power residue symbol from Theorem 5.3.