Course Notes, Global Class Field Theory Caltech, Spring 2015/16

COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16

M. FLACH

1. Global class field theory in classical terms Recall the isomorphism Z Z × −→∼ Q Q 7→ 7→ a ( /m ) Gal( (ζm)/ ), a (ζm ζm) where ζm is a primitive m-th root of unity. This is the reciprocity isomorphism of global class field theory for the number field Q. What could a generalization of this to any number field K be? Of course one has a natural map (going the other way) ∼ × Gal(K(ζm)/K) → Aut(µm) = (Z/mZ) , σ 7→ (ζ 7→ σ(ζ)) called the cyclotomic character. It is even an isomorphism if K is linearly disjoint from Q(ζm). But that’s not class field theory, or rather it’s only a tiny part of class field theory for the field K. The proper, and much more stunning generalisation is a reciprocity isomorphism ∼ Hm −→ Gal(K(m)/K) between so called ray class groups Hm of modulus m and Galois groups of corresponding finite abelian extensions K(m)/K. In a first approximation one can think of a modulus m as an integral ideal in OK but it is slightly more general. The ray m × class group H isn’t just (OK /m) although the two groups are somewhat related. In any case Hm is explicit, or one should say easily computable as soon as one can O× O compute the unit group K and the class group Cl( K ) of the field K. Note that both are trivial for K = Q, so that ray class groups are particularly easy for K = Q. So the first part of class field theory states the existence of this family of abelian extensions K(m)/K with explicit Galois groups. Then there is a complementary statement saying that any abelian extension L/K is contained in some ray class field and one can pin down which in terms of the ramification of L/K. For K = Q this is the Kronecker Weber theorem which can be proved directly (independently of CFT) using some ramification theory (it’s a sequence of exercises in Marcus’ book). 1.1. Places. In order to rigorously define what a modulus m is, we talk about the concept of places. Recall that OK is a Dedekind ring and any integral ideal m has a unique factorization ∏ m = pnp p ≥0 into a product of prime ideals where np ∈ Z and almost all np = 0. We have to add the ”infinite primes” to define a general modulus. Definition 1.1.1. For a field K a valuation is a function |−| : K → R≥0 satisfying 1 2 M. FLACH

(i) |a| = 0 if and only if a = 0. (ii) |ab| = |a| · |b|. (iii) There is a constant C so that |a + 1| ≤ C whenever |a| ≤ 1. Remark 1.1.1. We leave it as an exercise to show that if one can take C = 2 in (iii) then | − | satisfies the triangle inequality |a + b| ≤ |a| + |b| and if one can take C = 1 then it satisfies the even stronger inequality |a + b| ≤ max{|a|, |b|}. Valuations of this last type are called non-archimedean. Example 1.1.1. The trivial valuation is the valuation given by |a| = 1 for all a ≠ 0. Example 1.1.2. If ι : K → L is an embedding of K into a local field , then the valuation of L induces one on K. In particular any prime ideal p of OK induces ∼ r1 r2 a valuation via K → Kp. If L is a direct factor of R ⊗Q K = R × C then the usual archimedean absolute value on R or C induces one on K.

Deﬁnition 1.1.2. Two valuations | − |1 and | − |2 are equivalent if there is c > 0 | | | |c ∈ with a 1 = a 2 for all a K.

Lemma 1.1. Two valuations | − |1 and | − |2 are equivalent if and only if

∀α ∈ K |α|1 < 1 ⇔ |α|2 < 1.

Proof. One direction is clear. Let y ∈ K be such that |y|1 > 1 which exists since × | |1 is nontrivial. We show that we can take c = log |y|1/ log |y|2. For any x ∈ K there is b ∈ R with | | | |b x 1 = y 1.

Approximating b by rational numbers b < mi/ni from above we have m /n | | | |b | | i i ⇒ | ni mi | x 1 = y 1 < y 1 x /y 1 < 1 which implies m /n | ni mi | ⇒ | | | | i i x /y 2 < 1 x 2 < y 2 and hence | | ≤ | |b x 2 y 2 by passing to the limit. Using a sequence approximating b from below we obtain | | ≥ | |b | | | |b x 2 y 2 and hence x 2 = y 2. So | | | |b | |c b | |b c | |c x 1 = y 1 = ( y 2) = ( y 2) = x 2.

The main result about valuations on general ﬁelds is:

Theorem 1.1. (Weak approximation) If | |1,..., | |N are inequivalent valuations on a ﬁeld K (not necessarily global), α1, . . . , αN ∈ K, ϵ > 0 then there exists ξ ∈ K with |αn − ξ|n < ϵ for 1 ≤ n ≤ N. COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 3

Remark 1.1.2. If K is a global ﬁeld and all | |n are non-archimedean, then this is basically the Chinese remainder theorem. Take a Dedekind ring O ⊆ K so that the | |n correspond to prime ideals pn of O, multiply the αn by a common denominator in Z to make them integral (this will rescale ϵ by a positive factor) then take en −en large enough so that Npn < ϵ. Then by the Chinese remainder theorem there − ∈ O ≡ en | − | ≤ en exists ξ , ξ αn mod pn , i.e. αn ξ n Npn < ϵ.

Proof. We note ﬁrst that it will be enough to ﬁnd θn ∈ K with

|θn|n > 1, |θn|m < 1 n ≠ m. Then we have { θr 1 w.r.t. | | lim n = n →∞ r r 1 + θn 0 w.r.t. | |m, m ≠ n and ∑N θr ξ = n · α 1 + θr n n=1 n will have the required property for r suﬃciently large. By symmetry it is enough to show existence of θ = θ1 with the above property. This we do by induction on N. For N = 2 by Lemma 1.1 we ﬁnd α and β with

|α|1 < 1, |α|2 ≥ 1, |β|1 ≥ 1, |β|2 < 1 and can take θ = β/α. By the induction assumption there is ϕ ∈ K with

|ϕ|1 > 1, |ϕ|n < 1, 2 ≤ n ≤ N − 1 and ψ ∈ K with |ψ|1 > 1, |ψ|N < 1. Then set  ϕ if |ϕ|N < 1 r θ = ϕ ψ if |ϕ|N = 1  r ϕ | | 1+ϕr ψ if ϕ N > 1 where r ∈ Z is sufficiently large. Proposition 1.1. If K is a global field there are natural bijections between a) The equivalence classes of nontrivial valuations on K. b) The equivalence classes of dense embeddings ι : K → L where L is a locally compact field and two embeddings ι, ι′ are equivalent if there is a continuous ∼ ′ ′ isomorphism ϕ : L = L so that ι = ϕ ◦ ι. c) If K is a number field, 2 {p|{0} ̸= p ⊂ OK prime} ∪ {e ∈ R ⊗Q K|e = e, e indecomposable}

Proof. (sketch). By restriction one obtains a valuation on the subfield Q or Fp(T ). These are classified by Ostrowski’s theorem which we won’t prove. On Q they are the p-adic valuations | − | and the usual absolute value | − | = | − |∞. Then ∼ p Qp ⊗Q K = K1 ×· · ·×Kr and each Ki is a finite extension of Qp, hence has a unique extension | − |i of | − |p and the Ki are complete. The given valuation on K induces a function on Qp ⊗Q K by continuity. This function is nonzero (since nonzero on Qp) so must be nonzero on one of the Ki and is a valuation there, so is equivalent to | − |i. Since K is dense in Qp ⊗Q K an element (0,...,y,..., 0) corresponding to 4 M. FLACH y ∈ Ki, |y|i > 1 can be approximated arbitrarily close by an element in K which shows that | − |i is independent from the other valuations. Finally, one can classify locally compact fields (i.e. fields with a locally compact topology so that addition, multiplication and inversion are continuous). Any such field L is isomorphic to either R, C, a finite extension of Qp or a Laurent series field Fq((T )), in other words to a completion of a global field at some valuation.

Definition 1.1.3. A place of a global field K is an element of the set in Proposition 1.1. A place is called non-archimedean (or finite), archimedean (or infinite), real, or complex if the corresponding local field is. We denote places by p, the corresponding embeddings by ιp and, if p is finite, the corresponding discrete valuations by vp. Definition 1.1.4. Define normalized valuations | − | on a global field by  p − Np vp(x) p finite | | | | x p =  ιp(x) p real  2 |ιp(x)| p complex. They satisfy the product formula ∏ |α|p = 1. all places p 1.2. Ray class groups. Definition 1.2.1. A modulus m is a formal product over all places p ∏ np m = p = mf m∞ p ≥0 with np = 0 for almost all p, np ∈ Z for non-archimedean p, np = 0 for p complex and np ∈ {0, 1} for p real. The product over the finite (resp. infinite) places mf (resp. m∞) is called the finite (resp. infinite) part of m. We also identify m∞ with the subset of infinite places for which np = 1. Definition 1.2.2. For α ∈ K× and a modulus m we say α ≡ 1 mod m

np if for all p with np > 0 we have: α ∈ OK,p and α ≡ 1 mod p if p is finite and ιp(α) > 0 if p is infinite. Definition 1.2.3. For a modulus m we let J m be the group of fractional ideals prime to mf , isomorphic to the free abelian group ⊕ Z · p

p-mf ,p ﬁnite on the set of prime ideals not dividing mf . We let P m = {(α)|α ≡ 1 mod m} ⊆ J m be the subgroup of principal, fractional ideals generated by elements ≡ 1 mod m, and call Hm = J m/P m the ray class group modulo m. COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 5

Example 1.2.1. For m = 1, i.e. np = 0 for all places p, we get 1 1 1 H = J /P = Cl(OK ). Before computing more examples we analyze the dependence of Hm on m. For m′ | m we have a commutative diagram ′ ′ 0 −−−−→ P m −−−−→ P m −−−−→ P m /P m −−−−→ 0       y y yg

′ ′ 0 −−−−→ J m −−−−→ J m −−−−→ J m /J m −−−−→ 0 The map g is surjective since ⊕ ′ ∼ J m /J m = Z p∈S { | ′ } ∈ where S = p finite np > 0, np = 0 is finite and for all collections ep, p S, there × ′ exists α ∈ K , α ≡ 1 mod m , vp(α) = ep for p ∈ S. This follows from Theorem ′ 1.1 applied to the family αp = 1 for p | m , αp any element of valuation ep for −ep−1 −np p ∈ S and ϵ < min{1,Np ,Np } (at least if ep ≥ 0; the general element is a difference of such). So we have an exact sequence ′ 0 → ker(g) → Hm → Hm → 0 and in particular the Hm form a projective system. We determine the kernel of this ′ 1 m map for m = 1. The element (α) ∈ P /P is in the kernel of g iff vp(α) = 0 for all p | mf iff (α, mf ) = 1. There is a commutative diagram of short exact sequences 0 −−−−→ {α ∈ O× |α ≡ 1 mod m} −−−−→ {α ∈ K×|α ≡ 1 mod m} −−−−→ P m −−−−→ 0 K       y yβ yγ

−−−−→ O× −−−−→ { ∈ ×| } −−−−→ ˜m −−−−→ 0 K α K (α, mf ) = 1 P 0. So ker(g) = coker(γ) and there is an exact sequence { ∈ ×| ≡ } { ∈ ×| } → α K α 1 mod mf → → α K (α, mf ) = 1 → 0 × coker(β) × 0. {α ∈ K |α ≡ 1 mod m} {α ∈ K |α ≡ 1 mod mf } × As a consequence of Theorem 1.1 there exist elements κp ∈ K indexed by the real ′ places p such that κp ≡ 1 mod mf and ιp(κp) < 0 and ιp′ (κp) > 0 for p ≠ p. This implies that { ∈ ×| ≡ } α K α 1 mod mf ∼ ∞ ∼ ∞ = {±1}m = (Z/2Z)m {α ∈ K×|α ≡ 1 mod m} and the sequence is split by the map sending the class of α in coker(β) to the class of

(sgn(ιp(α)))p|m∞ . On the other hand we have { ∈ ×| } α K (α, mf ) = 1 ∼ O × × = ( K /mf ) {α ∈ K |α ≡ 1 mod mf } and so we obtain a short exact sequence (Z/2Z)m∞ × (O /m )× (1) 0 → K f → Hm → Cl(O ) → 0. O× K im( K ) 6 M. FLACH

In particular we see Proposition 1.2. For all moduli m the ray class group Hm is ﬁnite.

Example 1.2.2. 1) For K = Q and m = (m)∞ where m is any integer and ∞ is the unique real place of Q we get × Z/2Z × (Z/mZ) ∼ × Hm = = (Z/mZ) {±1} 2) For K = Q and m = (m) we get (Z/mZ)× Hm = {±1} √ 3) For K = Q( 3) and m = ∞1∞2 we get Z/2Z × Z/2Z ∼ {±1} × {±1} ∼ Hm = = = Z/2Z sgn(O× ) {±1} √ K √ √ since 2+ 3 is a fundamental unit and 2+ 3 and 2− 3 are both positive. 4) K imaginary quadratic of class number one.  O × √  ( K /m) Q −  {± ± ± 2} K = ( 3)  1, ζ3, ζ3 O × Hm = ( K /m) Q  {±1,±i} K = (i)  O × √ ( K /m) Q − {±1} K = ( p), p = 7, 8, 11, 19, 43, 67, 163 1.3. The main theorem of classical class field theory. Definition 1.3.1. For any extension L/K, the norm map on ideals is the multi- plicative map defined on prime ideals by

fP NL/K (P) = p where p = P∩OK and fP = [κ(P): κ(p)] is the degree of the residue field extension. Theorem 1.2. (Takagi, with previous work by Weber, Hilbert, Furtwängler)Let K be a number field. a) For every modulus m there exists a finite Galois extension K(m)/K with K(m′) ⊆ K(m) for m′ | m. b) There exists an isomorphism m ∼ ρm : H = Gal(K(m)/K) ′ compatible with Hm Hm . c) If p | disc(K(m)/K) then p | m. d) (Artin) For all p - m one has

ρm(p) = Frobp .

Note here (Ma160a) that Frobp is a well deﬁned element, not only a conjugacy class, since Gal(K(m)/K) is abelian. e) Every abelian extension L/K is isomorphic to a subextension of K(m)/K for some m. In particular m ∼ ρ : J /QL = Gal(L/K) COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 7

m m for a unique subgroup P ⊆ QL ⊆ J where ρ is the analogue of the Artin map for L. Moreover, p | disc(L/K) ⇒ p | m and m · m (2) QL := NL/K (JL ) P is the Norm subgroup attached to the extension L/K.

Remark 1.3.1. It follows from b), the Galois correspondence and (2) that for every m ⊆ ⊆ m m · subgroup P Q J there is a unique abelian extension L/K with NL/K (JL ) P m = Q. We could have stated this remark as a corollary but it will actually be a main ingredient in the proof. In other words, the proof will show that there is nothing special about the ray class fields K(m). One rather has to work with all abelian extensions L/K and their norm subgroups (for varying base field K) at once. It is only for K = Q and K imaginary quadratic that the ray class fields have some independent significance because they arise by adjoining m-division points of algebraic groups (Gm for K = Q and CM elliptic curves otherwise). Takagi had a weaker statement than d) involving only the splitting of primes in K(m)/K. Note that p splits completely in K(m)/K if and only if p = (α) with α ≡ 1 mod m. Also recall from Ma160a that a finite extension of number fields L/K is determined up to isomorphism by the set of primes of K which split completely in L/K (we will reprove this below). That is why Takagi was content with just this statement. The Artin map is a very natural idea that immediately generalizes to higher dimensional arithmetic schemes and the etale fundamental group. It is clear that the Artin map is well defined on J m where m is divisible by all ramified primes, and the key problem becomes to characterize its kernel. It turns out to always contain a group P m for m large enough. K(1) is called the Hilbert class field of K. The extension K(1)/K is unramified everywhere and totally split at all infinite places of K, and it is the maximal abelian extension of K with these properties. We have ∼ Gal(K(1)/K) = Cl(OK ). √ Example 1.3.1. For K = Q( −5) we have √ √ √ √ K(1) = F := Q( −5, 5) = Q( −5, −1). √ ∼ We know Cl(OK ) = Z/2Z generated by (2, 1 + −5) (see Marcus p.136) and so it suffices to verify that F/K is unramified. Of course one could just compute disc(F/K) but then one would have to compute the ring of integers first. Alterna- tively, note that F ⊆ Q(ζ20) and the inertia subgroup at 2 (resp. 5) of ∼ × × Gal(Q(ζ20)/Q) = (Z/4Z) × (Z/5Z) is (Z/4Z)× × {1} (resp. {1} × (Z/5Z)×). The quadratic character cutting out K is nontrivial on both. So I(Q(ζ20)/K) = I(Q(ζ20)/K(1)) for both 2 and 5. √ Example 1.3.2. For K = Q( 3) we have K(1) = K √ since OK = Z[ 3] is a PID. On the other hand Q(ζ12)/K is unramified by a similar computation as above. But there is ”ramification” at the infinite primes. In fact we have Q(ζ12) = K(∞1∞2). 8 M. FLACH

We continue with some remarks about the proof of the theorem. Clearly one has to construct abelian extensions somehow and the key method√ will be Kummer × × m m theory. Recall that if ζm ∈ K and a ∈ K /(K ) then K( a)/K is abelian with Galois group a subgroup of µm, and every cyclic extension of degree dividing m arises this way. However, this does not immediately apply to most class ﬁelds. Over K = Q only the quadratic extensions arise this way.

The main feature of classical class field theory is a mix of analytic and algebraic arguments which I actually find quite appealing. Subsequent authors (Chevalley, Hochschild-Nakayama, Tate) have found ”purely algebraic” proofs centering around the concept of class formation. I do not think either formulation is more illuminating than the other and the jury is still out on a better understanding of class field theory.

1.4. Analytic theory of ray class L-functions. Part of the analytic theory should have been done in Ma160a. The techniques in this section are an extension of Dirichlet’s proof of inﬁnitude of primes in arithmetic progressions. There are two main results in this section, the ”First inequality” and the ”Frobenius density theorem”. In the proofs of global CFT it will be important to go back and forth between between extensions L/K and subgroups of J m. A key result relating the two is the following: Theorem 1.3. (First inequality) For any extension L/K and modulus m one has m hL := [J : QL] ≤ [L : K].

For the proof one must have some analytic properties of Zeta functions. Recall the Dedekind Zeta function of the number ﬁeld K ( )− ∑ 1 ∏ 1 1 (3) ζ (s) = = 1 − K Nas Nps a p or its variant with respect to a modulus m ( )− ∑ 1 ∏ 1 1 ∑ ζm(s) = = 1 − = ζ (s, c) K N s N s K a p ∈ m (a,mf )=1 p-mf c H where ∑ 1 ζ (s, c) = K Nas a∈cP m m is a partial Zeta function associated to the coset cP . The behavior of ζK (s, c) around s = 1 can be analyzed using ”geometry of numbers” techniques in exactly the same way as was hopefully done in Ma160a for ζK (s). Proposition 1.3. The limit 2r1 (2π)r2 Rm lim(s − 1)ζK (s, c) = √ s→1 m w |dK |Nm exists and only depends on K and m. Here Rm is the covolume of O×,m { ∈ O× | ≡ } K := α K α 1 mod m COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 9

× m |O ,m ∩ | · |m∞| under the regulator map, w = K µK and Nm = Nmf 2 . In fact ζ(s, c) ℜ − 1 can be analytically continued to (s) > 1 [K:Q] except for a simple pole at s = 1.

The following definition generalizes Dirichlet L-functions. Definition 1.4.1. Let χ : Hm → C× be a complex character of the finite abelian group Hm. Define the (imprimitive) L-function of χ as ( )− ∑ χ(a) ∏ χ(p) 1 ∑ L(s, χ) = = 1 − = χ(c)ζ (s, c). N s N s K a p ∈ m (a,mf )=1 p-mf c H Here is an immediate corollary of Proposition 1.3 and the orthogonality relations for characters. Corollary 1.1. One has { 0 χ ≠ 1 lim(s − 1)L(s, χ) = 2r1 (2π)r2 Rmhm s→1 √ χ = 1 m w |dK |Nm where hm := |Hm| is the ray class number modulo m. L(s, χ) can be analytically ℜ − 1 continued to (s) > 1 [K:Q] except that L(s, 1) has a simple pole at s = 1. Note that ( ) ∏ 1 L(s, 1) = ζm(s) = 1 − ζ (s). K Nps K p|mf As a reality check we verify that the residue formula in Cor. 1.1 is compatible with the analytic class number formula. So we need to show ∏ ( ) 2r1 (2π)r2 Rmhm 1 2r1 (2π)r2 Rh √ = 1 − · √ m w |dK |Nm Np w |dK | p|mf or equivalently m m ∏ ∏ R h w Np − 1 | | − · · = Nm = 2 m∞ · (Np − 1)Npnp 1. R h wm Np p|mf p|mf

m∞ × This last quantity is the cardinality of the numerator (Z/2Z) × (OK /mf ) in (1.2), and therefore (1.2) implies ∏ m | | − h × 2 m∞ · (Np − 1)Npnp 1 = · | im(O )| h K p|mf and we are reduced to showing m R w × · = | im(O× )|−1 = [O× : O ,m]−1. R wm K K K But this follows by looking at cokernels in the map of short exact sequences with injective vertical maps ×,m ×,m − 0 −−−−→ O ∩ µ −−−−→ O −−−−→ Zr1+r2 1 −−−−→ 0 K  K K     y y y

× − −−−−→ −−−−→ O −−−−→ Zr1+r2 1 −−−−→ 0 µK K 0. 10 M. FLACH

Proof. We apply Proposition 1.3 and Cor. 1.1 as follows. One has ∞ ∑ ∑ 1 ∑ 1 log ζ (s) = ∼ K n Npns Nps p n=1 p where we write f(s) ∼ g(s) if lims→1 f(s) − g(s) exists. This is because ∞ ∞ ∑ ∑ 1 ∑ ∑ 1 ≤ [K : Q] n Npns n pns p n=2 p n=2 converges at s = 1. A similar argument gives for any K′ ⊆ K. ∑ 1 ∑ 1 (4) log ζ (s) ∼ ∼ . K Nps Nps p NK/K′ (p) is prime ∑ { | ′ } { | } 1 since p NK/K (p) is prime contains p Np prime and Np not prime Nps converges at s = 1. In other words primes of degree one, i.e. where Np is a prime number, dominate the sum for any number ﬁeld. On the other hand, the fact that ζK (s) has a simple pole at s = 1 implies

(5) log ζK (s) ∼ − log(s − 1). Similarly ∑ χ(p) (6) log L(s, χ) ∼ Nps p and (7) log L(s, χ) ∼ m(χ) log(s − 1) where m(χ) = ords=1 L(s, χ). By Cor. 1.1 m(1) = −1 and m(χ) ≥ 0 for χ ≠ 1. Set ∑ δ := m(χ) ≥ 0. χ=1̸ Using the (other) orthogonality relation for characters of the ﬁnite abelian group m J /QL and applying (6) and (7) we get ∑ h ∑ ∑ χ(p) L = ∼ (1 − δ)(− log(s − 1)). Nps Nps p∈QL χ p Applying (5) and (4) for the ﬁeld L and restricting to real s > 1 we get ∑ [L : K] ∑ 1 (8) ≥ ∼ log ζ (s) ∼ − log(s − 1). Nps NPs L ∃P,NL/K P = p NL/K P prime

By deﬁnition a prime p of K that is a norm from L lies in QL. Hence, as s → 1 from above ( ) ∑ ∑ − ≤ 1 − 1 ∼ 1 δ − 1 − − 0 s s ( log(s 1)) Np Np hL [L : K] p∈QL ∃P,NL/K P = p and since (− log(s − 1)) tends to +∞ the term in parentheses cannot be negative, ≥0 i.e. hL ≤ [L : K] and δ = 0 since δ ∈ Z . COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 11

Remark 1.4.1. The proof of Theorem 1.3 also showed nonvanishing L(1, χ) ≠ 0 for a nontrivial character χ which is trivial on the norm subgroup QL of the field L. This nonvanishing is a key ingredient in the analog of Dirichlet’s theorem for the number field K but for that we have to know it for all characters of Hm. Class field theory shows that indeed P m is the norm subgroup of the ray class field which gives nonvanishing for all χ ≠ 1. 1.4.1. Dirichlet density. We are now interested in the consequences of Prop. 1.1 for densities of certain sets of primes. Definition 1.4.2. If S is a set of prime ideals in a number field K, its Dirichlet density is the limit ∑ ∑ 1 1 ∑p∈S Nps p∈S Nps δ(S) := lim 1 = lim s→1+ s→1+ − − p Nps log(s 1) if it exists. Remark 1.4.2. One can show that if the natural density {p ∈ S|Np ≤ x} d(S) := lim x→∞ {p|Np ≤ x} exists, then δ(S) exists and δ(S) = d(S). Lemma 1.2. 1) One always has 0 ≤ δ(S) ≤ 1, 2) For S′ ⊆ S one has δ(S′) ≤ δ(S). In fact, δ(S) = δ(S′)+δ(S \S′) whenever two of the densities (and hence the third) exist. 3) δ(S) = δ(S1) where S1 ⊆ S is the subset of primes of degree one, i.e. such that Np is a prime number. Proof. This is clear from the definition

Theorem 1.4. (Frobenius density theorem) Let L/K be a Galois extension and let C be a conjugacy class of cyclic subgroups of order m of G = Gal(L/K). Then the set of primes p of K so that < Frobp >∈ C has Dirichlet density ϕ(m)|C|/|G|. Proof. We only indicate the case of the trivial subgroup where m = |C| = 1 which is readily implied by (8). This case is all we need for the applications below. Corollary 1.2. Let L and L′ be Galois extensions of K and S (resp. S′) the set of primes of K which split completely in L/K (resp. L′/K). Then L ⊇ L′ ⇔ ∃F ⊂ S with δ(F ) = 0 S \ F ⊆ S′.

Proof. The direction ⇒ is obvious since primes splitting completely in L/K also split completely in L′/K. The primes splitting completely in E := LL′/K are those splitting completely in L/K and in L′/K, i.e. the primes in S ∩ S′. Hence if S \ F ⊆ S′ then 1 1 = δ(S ∩ S′) ≥ δ(S \ F ) = δ(S) = [E : K] [L : K] i.e. [L : K] ≥ [E : K], hence E = L, hence L ⊇ L′. 12 M. FLACH

Definition 1.4.3. If L/K is an abelian extension and m a modulus divisible by all prime divisors of disc(L/K) then the Artin map m m → 7→ ρ = ρL : J Gal(L/K), a (a, L/K) is defined by ρ(p) = Frobp, i.e. ∏ ∏ np np ρ(a) = (a, L/K) = Frobp , a = p . p p Corollary 1.3. For any abelian extension L/K and allowed modulus m the Artin map is surjective. Proof. This is immediate from the Frobenius density theorem but one can also prove it just from its special case dealing with the split primes. Namely, let F ⊆ L be the fixed field of the image of ρ. Then for any p - mf the Frobenius Frobp acts trivially on F , i.e. p splits completely in F/K. But this implies F = K by Corollary 1.2. 1.5. Reduction steps. After these analytic preliminaries we continue with the proof of the main theorem. Here is the key definition for the entire theory. Definition 1.5.1. An abelian extension L/K is called a class field if P m lies in the kernel of the Artin map ρ : J m → Gal(L/K) for some m (divisible by all primes p ramified in L/K). A subgroup Q with P m ⊆ Q ⊆ J m will be called a congruence subgroup (of modulus m). An extension L/K is called m · m a class field for the subgroup Q, if L/K is a class field and Q = NL/K (JL ) P . m Remark 1.5.1. It is immediate from the definition that NL/K (JL ) lies in the kernel of the Artin map, hence so does m · m QL = NL/K (JL ) P if L/K is a class field. Since ρ is surjective, the first inequality becomes an equality m hL = [J : QL] = [L : K] and QL = ker(ρ) for a class field L/K. One can equivalently define a class field to be an extension for which QL = ker(ρ). ′ Remark 1.5.2. If P m lies in the kernel of the Artin map, so does P m for any multiple m′ of m. This just follows from commutativity of ′ ′ ′ ρ P m −−−−→ J m −−−−→ Gal(L/K)     (9) y yι ∥

ρ P m −−−−→ J m −−−−→ Gal(L/K). The conclusions of the previous remark will then also hold for m′. Remark 1.5.3. One can define a weaker notion of class field by just requiring the m numerical equality [J : QL] = [L : K] for some m with no further condition on the Artin map or even the extension L/K. It is clear that the weak and strong notion are equivalent after one knows the full class field theory by uniqueness of the class field. What one can easily show directly is that for a non-normal extension COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 13

m L/K the first inequality is always strict [J : QL] < [L : K] and that if one has equality for L/K one also has equality for L′/K for all K ⊆ L′ ⊆ L. Hence it follows that a weak class field is Galois with G = Gal(L/K) having the property that all subgroups are normal. If G is indeed abelian then the Artin map would be defined but one would still have to show that P m lies in its kernel. Moreover, the quaternion group Q8 provides an example of a non-abelian group with this property. What is apparently true and very striking is that products of quaternion and abelian groups are the only finite groups all of whose subgroups are normal. It will turn out that every abelian extension L/K is a class field and that every congruence subgroup has a class field but neither direction is obvious. In order to organize the relation between different class fields and congruence subgroups we make the following definition. ′ ′ Definition 1.5.2. Given two groups P m ⊆ Q ⊆ J m and P m ⊆ Q′ ⊆ J m one ′ defines Q ⊆ Q′ to mean that ι−1(Q) ⊆ (ι′)−1(Q′) where ι : J mm → J m and ′ ′ ι′ : J mm → J m are the natural inclusions. On the set of all Q (with varying m) this relation is reflexive and transitive. Then one has Lemma 1.3. If S is a set with a reflexive and transitive relation ≤, then a ∼ b ⇔ a ≤ b and b ≤ a is an equivalence relation on S compatible with ≤ (i.e. a ≤ b, a ∼ a′, b ∼ b′ implies a′ ≤ b′) and the set of equivalence classes [S] becomes an ordered set. Proof. Exercise Following this Lemma we view the set of equivalence classes of congruence subgroups Q as an ordered set. It will turn out to be in order reversing bijection with isomorphism classes of abelian extensions of K. For now we can already prove this for class fields. Proposition 1.4. If L and L′ are class fields, one has L ⊇ L′ if and only if QL ⊆ QL′ . ′ Proof. The implication L ⊇ L ⇒ QL ⊆ QL′ holds by transitivity of the norm for ′ any extension L/K. Now suppose L is a class field for QL′ . Then primes in QL′ ′ split completely in L /K since QL′ is the kernel of ρm and p splits completely if and only if Frobp = 1. If p splits completely in L/K then p is a norm from L/K, ′ ′ hence lies in QL hence in QL, hence splits completely in L /K. We then conclude from Corollary 1.2 that L ⊇ L′. Lemma 1.4. Let L/K be a class field. Then any subextension K ⊆ L′ ⊆ L is a m class field and for any group QL ⊆ Q ⊆ J there exist a (unique) class field. Proof. The Artin map for L′ is a quotient of the Artin map for L, so P m lies in its kernel. Given Q there exists L′ by the Galois correspondence, and since P m lies in ′ the kernel of the Artin map for L we have Q = QL′ . The proof of the main theorem is organized (or perhaps disorganized) into certain substatements which build on each other. The first statement, the norm index theorem, is used in the proofs of both the reciprocity theorem and the existence theorem. The reciprocity theorem is used in the proof of the existence theorem. 14 M. FLACH

Theorem 1.5. (Norm index theorem) Every cyclic extension L/K satifies m [J : QL] = [L : K], i.e. is a weak class field in the sense of Remark 1.5.3, for some modulus m only divisible by the primes ramified in L/K. Theorem 1.6. (Reciprocity theorem) Every abelian extension L/K is a class field. More precisely, there is a modulus m only divisible by the primes ramified in L/K so that P m ⊆ ker(ρ). Theorem 1.7. (Existence theorem) There exists a class field for each P m ⊆ Q ⊆ J m. Corollary 1.4. (Classification theorem) There is an inclusion reversing bijection between equivalence classes of congruence subgroups P m ⊆ Q ⊆ J m and isomorphism classes of abelian extensions L/K. Proof. The map is of course given by sending an abelian extension L/K to its norm m m m ⊆ ′ group P NL/K (JL ) for a modulus m with P ker(ρ). By (9) any multiple m of m will also have this property and since ι is injective ′ ′ ′ ′ m m ∩ m ∩ m ′ m m P NL/K (JL ) J = ker(ρ) J = ker(ρ ) = P NL/K (JL ), ′ ′ m m m m i.e. the two congruence subgroups P NL/K (JL ) and P NL/K (JL ) are equivalent. Hence we have a well defined map to equivalence classes of congruence subgroups. It is surjective by Theorem 1.7 and injective (and order reversing ) by Prop. 1.4. Corollary 1.5. (Main theorem) The statements of Theorem 1.2 all hold. Proof. This is just applying the existence theorem to Q = P m.