The Main Theorems of Global Class Field Theory

18.785 Number theory I Lecture #22 22 Fall 2021 11/24/2021 The main theorems of global class field theory In this lecture we refine the correspondence between quotients of ray class groups and subfields of ray class fields given by the Artin map so that we can more precisely state the main theorems of global class field theory (for number fields) in their ideal-theoretic form. Let us first recall the notational setup. We have a number field K and a modulus m : MK → Z≥0 that we as a formal Q view m(p) product over the places of K; we may write m is a product Q= m0 m∞ , where m0 := p over primes (finite places) of K and m∞ := v|∞ v m(v) defines a subset of the real places of K (recall that for v|∞ we have m(v) ≤ 1 with m(v) = 0 if v is not real). The moduli for K are partially ordered by the divisibility relation m|n, which holds if and only if m(v) ≤ n(v) for all v ∈ MK . We then define m ⊆ I , the subgroup of fractional ideals prime to m (equivalently, m ); • IK 0 K m; • K m ⊆ K × , the subgroup of α ∈ K × for which (α) ∈ IK • K m,1 ⊆ K m , the subgroup of α ∈ K m for which vp (α − 1) ≥ vp (m0 ) for p|m0 and αv > 0 for v|m∞ (here αv ∈ R is the image of α under the real-embedding v); m m m,1 (the ray group for m); • Rm K ⊆ IK the subgroup of ideals (α) ∈ IK with α ∈ K m m • Clm K := IK /RK (the ray class group for m); • Spl(L) := Spl(L/K), the set of primes of K that split completely in an extension L; m m → Gal(L/K), Artin map of an abelian extension L/K unramified at p - m. • ψL/K : IK In the previous lecture we defined the ray class field of K for the modulus m as a finite m abelian extension L/K unramified at all p - m such that the kernel of the Artin map ψL/K is equal to the ray group Rm K . We did not prove that such fields exist, but we did prove that there is at most one of them; see Theorem 21.20. Let K(m) denote this field. Assuming the ray class field K(m) exists, it follows from the surjectivity of the Artin m m → Gal(K(m)/K) proved in Theorem 21.19 that we have a canonical map ψK(m)/K : IK isomorphism m m Clm K = IK /RK ' Gal(K(m)/K) between the ray class group and the Galois group of the ray class field. More generally, if L is any intermediate field between K and K(m), the kernel of the Artin map is a subgroup m that contains the ray group C ⊆ IK m Rm K ⊆ C ⊆ IK , and we have an isomorphism m IK /C ' Clm K /C ' Gal(L/K) m m where C denotes the image of C in Clm K = IK /RK under the quotient map. m m containing Rm (a Thus if L is a subfield of K(m) then ker ψL/K is a subgroup of IK K congruence subgroup, as defined below). To prove that a given abelian extension L/K lies in a ray class field, it is enough to show that there exists a modulus m for K such that m Rm K ⊆ ker ψL/K , since we then have Spl(K(m)) - Spl(L) and L ⊆ K(m), by Theorem 21.18. In this lecture we want to better understand the structure of congruence subgroups, and to specify a minimal modulus m for which we should expect a given finite abelian extension L/K to lie in a subfield of the ray class field K(m); this minimal modulus is known as the conductor of the extension. So far we have not addressed this question even for K = Q (but see Problem Set 10); our proof of the Kronecker-Weber theorem showed that every abelian extension lies in some cyclotomic field Q(ζm ), but we made no attempt to determine such an integer m (or more precisely, a modulus m of the form m = (m)∞ or m = (m)). 22.1 Congruence subgroups Our presentation of congruence subgroups in this section follows [1, 3.3], but our notation differs slightly. Definition 22.1. Let K be a number field and let m be a modulus for K. A congruence m that contains Rm . We use C to denote subgroup for the modulus m is a subgroup C of IK K m m m the image of C in IK /RK = ClK under the quotient map. As explained above, congruence subgroups are precisely the groups we expect to arise as m m → Gal(L/K) associated to a finite abelian extension the kernel of an Artin map ψL/K : IK L/K, for a suitable choice of modulus m. The choice of m is critical; as can be seen in m Example 22.2 below, ker ψL/K need not be a congruence subgroup for the modulus m; there are constraints on the modulus m that must be satisfied beyond the basic requirement that m m must be divisible by all the primes of K that ramify in L (so that ψL/K is defined). Example 22.2. Let K = Q, and consider the cyclic cubic extension L := Q[x]/(x3 −3x−1), m which is ramified only at 3. The Artin map ψL/K is well-defined for any modulus m divisible by (3). The√ray class field for m = (3) is Q(ζ3 )+ = Q, and the ray class field for m = (3)∞ is m Q(ζ3 ) = Q( −3), neither of which contains L, so ker ψL/K does not contain Rm K for either of these moduli and is not a congruence subgroup. On the other hand, L is equal to Q(ζ9 )+ , m the ray class field for m = (9), so ker ψL/K contains (and is equal to) Rm K , and is thus a congruence subgroup for the modulus m = (9). m n If ker ψL/K is a congruence subgroup for the modulus m, then ker ψL/K is a congruence n n subgroup for each modulus n divisible by m. If m divides n then RK ⊆ Rm K and ψL/K is n m n the restriction of ψL/K to IK , which contains RK . If m and n are supported on the same m = I n and ψ m n m n primes, then IK K L/K = ψL/K , but the ray groups RK and RK may differ. To deal with these complications, we define an equivalence relation on congruence subgroups and show that each equivalence class has a canonical representative whose modulus divides the modulus of every equivalent congruence subgroup. Definition 22.3. Let K be a number field with moduli m1 and m2 . If C1 is a congruence subgroup for m1 and C2 is a congruence subgroup for m2 , then we say that (C1 , m1 ) and (C2 , m2 ) are equivalent and write (C1 , m1 ) ∼ (C2 , m2 ) whenever m1 m2 IK ∩ C 2 = IK ∩ C1 . Note that when m1 = m2 this reduces to C1 = C2 . Proposition 22.4. Let K be a number field. The relation (C1 , m1 ) ∼ (C2 , m2 ) is an equivm1 m2 alence relation. If (C1 , m1 ) ∼ (C2 , m2 ) then IK /C1 ' IK /C2 are related by a canonical isomorphism that preserves cosets of fractional ideals prime to both m1 and m2 . 18.785 Fall 2021, Lecture #22, Page 2 Proof. The relation ∼ is clearly symmetric, and reflexive. To show that it is transitive, let C1 , C2 , C3 be congruence subgroups for moduli m1 , m2 , m3 and suppose (C1 , m1 ) ∼ (C2 , m2 ) m3 m1 m2 m3 and (C2 , m2 ) ∼ (C3 , m3 ). Let I ∈ IK ∩ C1 and pick α ∈ K m1 m3 ,1 so that αI ∈ IK (this m1 m3 m1 is possible by Lemma 21.7 and Theorem 8.5). Then (α) ∈ RK ⊆ RK ⊆ C1 and I ⊆ C1 , m1 m2 m3 m2 so αI ∈ C1 , and we also have αI ∈ IK ⊆ IK , so m2 m1 αI ∈ IK ∩ C 1 = IK ∩ C2 ⊆ C2 , m1 m2 m3 m3 since C1 ∼ C2 , and αI ∈ IK ⊆ IK , so m3 m2 αI ∈ IK ∩ C 2 = IK ∩ C3 ⊆ C3 , −1 ∈ C , since C 1 m3 3 since C2 ∼ C3 . We have (α) ∈ Rm ⊆ Rm 3 3 K K , so (α) ∈ C3 and therefore (α) m1 m1 is a group. Thus α−1 αI = I ∈ C3 , and we also have I ∈ C1 ⊆ IK , so I ∈ IK ∩ C3 . Since m3 I ∈ IK ∩ C1 was chosen arbitrarily, this proves that m3 m1 IK ∩ C1 ⊆ IK ∩ C3 . The reverse inclusion follows by symmetry, so (C1 , m1 ) ∼ (C3 , m3 ) as desired. m1 For the last statement, for any fractional ideal I ∈ IK we can pick α ∈ K m1 ,1 so that m2 m2 αI ∈ IK (via Lemma 21.7 and Theorem 8.5). The image of αI in IK /C2 does not depend on m m2 0 m ,1 0 2 1 the choice of α, since for any α ∈ K with α I ∈ IK we have (αI)/(α0 I) = (α/α0 ) ∈ IK m1 m1 m2 m2 m2 0 1 and (α/α0 ) ∈ Rm K , so (α/α ) ∈ IK ∩ RK = IK ∩ RK ⊆ RK . This defines a group m1 m2 m2 m1 homomorphism ϕ : IK → IK /C2 . For I ∈ C1 , we have αI ∈ IK ∩ C1 = IK ∩ C2 ⊆ C2 , but m1 m2 for I ∈ IK − C1 we have αI ∈ IK − C1 and therefore αI 6∈ C2 , so ker ϕ = C1 . It follows m1 m2 that ϕ induces an injective homomorphism IK /C1 → IK /C2 , and by symmetry we have an m1 m2 injective homomorphism in the opposite direction, so IK /C1 ' IK /C2 as claimed. This isomorphism is independent of the choice of α used to define it (hence canonical), and for fractional ideals I coprime to both m1 and m2 we can choose α = 1, in which case m1 m2 the coset of I in IK /C1 will be identified with the coset of I in IK /C2 . We now observe that if C is a congruence subgroup for two moduli m1 and m2 , then (C, m1 ) ∼ (C, m2 ). In particular, each subgroup of IK lies in at most one equivalence class of congruence subgroups. We can thus view the equivalence relation (C1 , m1 ) ∼ (C2 , m2 ) as an equivalence relation on the congruence subgroups of IK and write C1 ∼ C2 without ambiguity. It follows from Proposition 22.4 that each equivalence class of congruence subgroups uniquely determines a finite abelian group that is the quotient of a ray class group. Within an equivalence class of congruence subgroups there can be at most one congruence subgroup for each modulus (since C1 ∼ C2 ⇔ C1 = C2 whenever C1 and C2 are congruence subgroups for the same modulus). The following lemma gives a criterion for determining when there exists a congruence subgroup of a given modulus within a given equivalence class. Lemma 22.5. Let C1 be a congruence subgroup of modulus m1 for a number field K. There exists a congruence subgroup C2 of modulus m2 |m1 equivalent to C1 if and only if m1 2 IK ∩ Rm K ⊆ C1 , 2 in which case C2 = C1 Rm K . 18.785 Fall 2021, Lecture #22, Page 3 m1 m2 m1 m2 Proof. Note that m2 |m1 implies IK ⊆ IK , so C1 ⊆ IK ⊆ IK . m1 m2 2 Suppose C2 ∼ C1 has modulus m2 . Then IK ∩ C2 = IK ∩ C1 = C1 , and Rm K ⊆ C2 , so m1 m2 m1 m2 m2 IK ∩ RK ⊆ C1 as claimed. Now suppose IK ∩ RK ⊆ C1 , and let C2 := C1 RK . Then C2 is a congruence subgroup of modulus m2 and we have m2 m1 m1 m2 m1 2 IK ∩ C1 = C1 = C1 (IK ∩ Rm K ) = IK ∩ C1 RK = IK ∩ C2 , so C1 ∼ C2 . The equivalence class of C1 contains at most one congruence subgroup of modulus 2 m2 , so if one exists it must be C2 = C1 Rm K . Proposition 22.6. Let C1 ∼ C2 be congruence subgroups of modulus m1 and m2 , respectively. There exists a congruence subgroup C ∼ C1 ∼ C2 with modulus n := gcd(m1 , m2 ). m2 m1 Proof. Put m := lcm(m1 , m2 ) and D := IK ∩ C 1 = IK ∩ C2 ; then m1 m2 m Rm K = RK ∩ RK ⊆ D ⊆ IK , so D is a congruence subgroup of modulus m, and we have m 1 IK ∩ Rm K ⊆D and m 2 IK ∩ Rm K ⊆ D, so D ∼ C1 ∼ C2 , by Lemma 22.5. To prove the existence of an equivalent congruence m ∩ Rn ⊆ D (again by Lemma 22.5). subgroup C of modulus n it suffices to show IK K n m So let a = (α) ∈ IK ∩ RK , and choose β ∈ K m ∩ K m2 ,1 so that αβ ∈ K m1 ,1 (this is possible by Theorem 8.5 because m = lcm(m1 , m2 ) and n = gcd(m1 , m2 )). Then (β) ∈ D m ∩ Rm1 ⊆ D, so β −1 βa = a ∈ D. Thus I m ∩ Rn ⊆ D and therefore C = DRn and βa ∈ IK K K K K is a congruence subgroup of modulus n equivalent to D ∼ C1 ∼ C2 . Corollary 22.7. Let C be a congruence subgroup of modulus m for a number field K. There is a unique congruence subgroup in the equivalence class of C whose modulus c divides the modulus of every congruence subgroup equivalent to C. Definition 22.8. Let C be a congruence subgroup for a number field K. The unique modulus c := c(C) given by Corollary 22.7 is the conductor of C, and we say that C is primitive if C = CRcK (the unique congruence subgroup of modulus c equivalent to C). Proposition 22.9. Let C be a primitive congruence subgroup of modulus m for a number field K. Then m is the conductor of every congruence subgroup of modulus m contained in C; in particular, m is the conductor of Rm K. Proof. Let C0 ⊆ C be a congruence subgroup of modulus m and let c be its conductor. Then m ∩ Rc ⊆ C ⊆ C, by Lemma 22.5, and this implies that there is a congruence c|m and IK 0 K subgroup of modulus c equivalent to C, and therefore m|c, so c = m. The proposition implies that a modulus m occurs as a conductor if and only if Rm K is primitive. This does not always hold: consider K = Q and m = (2), for example; the (2) (2) (2) (1) (1) (2) (2) (1) conductor of RQ = IQ is (1), since RQ ∩ IQ = IQ ∩ IQ implies RQ ∼ IQ . Thus (2) is not the conductor of any congruence subgroup for Q. 18.785 Fall 2021, Lecture #22, Page 4 22.2 Ray class characters We now want to prove a generalization of Dirichlet’s theorem on primes in arithmetic progressions. Given a congruence subgroup C for a modulus m we would like to compute the m that lie in C. We first Dirichlet density d(C) := d({p ∈ C}) of the set of prime ideals p ∈ IK need to generalize our notion of a Dirichlet character. Definition 22.10. Let K be a number field and let χ : IK → C be a totally multiplicative function with finite image; so χ(OK ) = 1, χ(IJ) = χ(I)χ(J) for all I, J ∈ IK , and χ restricts to a homomorphism from a subgroup of IK to a finite subgroup of U(1) whose m and Rm ⊆ ker χ, kernel we denote ker χ. If m is a modulus for K such that χ−1 (U(1)) = IK K then χ is a ray class character of modulus m and its kernel is a congruence subgroup of modulus m. Equivalently, χ is the extension by zero of a character of the finite abelian m m m group Clm K = IK /RK defined by setting χ(I) = 0 for I 6∈ IK . Example 22.11. For K = Q there is a one-to-one correspondence between Dirichlet characters χ : Z → C and ray class characters χ0 : IQ → C with χ(a) = χ0 ((a)) for all a ∈ Z≥1 . Each Dirichlet character χ of modulus m corresponds to a ray class character of modulus m = (m)∞ whose conductor divides (m) if and only if χ is an even Dirichlet character, meaning that χ(−1) = 1. Definition 22.12. Let χ1 , χ2 be ray class characters of moduli m1 , m2 of a number field K, m2 with m1 |m2 . If χ2 (I) = χ1 (I) for all I ∈ IK , then χ2 is induced by χ1 . A ray class character is primitive if it is not induced by any ray class character other than itself. Definition 22.13. The conductor of a ray class character χ is the conductor c(χ) := c(ker χ) of its kernel (as a congruence subgroup). Theorem 22.14. A ray class character is primitive if and only if its kernel is primitive. Every ray class character χ is induced by a unique primitive ray class character χ e. m /(ker χ) → U(1) be the Proof. Let χ be a ray class character of modulus m, let κ : IK group character induced by χ, and let C be the primitive congruence subgroup equivalent to ker χ with modulus c = c(χ) dividing m given by Corollary 22.7. By Proposition 22.4, ∼ c /C → m /(ker χ) that we can use to define a ray we have a canonical isomorphism ϕ : IK IK c /C. class character χ e of modulus c as the extension by zero of the character κ ◦ ϕ of IK m c e(I) = χ(I) for all The isomorphism ϕ preserves cosets of fractional ideals in IK ⊆ IK , so χ m and χ is induced by χ I ∈ IK e. If χ2 is a ray class character of conductor m2 induced by a ray class character χ1 of m2 m1 conductor m1 , then ker χ1 ∩ IK = ker χ2 = ker χ2 ∩ IK and ker χ1 ∼ ker χ2 , and we also m1 m2 note that if χ1 6= χ2 then IK 6= IK and m1 6= m2 . It follows that χ e is primitive, it is the unique primitive ray class character that induces χ. Thus χ is primitive if and only if it is equal to χ e, which holds if and only if ker χ = ker χ e is primitive. Theorem 22.14 is a direct generalization of Theorem 18.13 for Dirichlet characters. For a modulus m of K we use X(m) to denote the set of primitive ray class characters of conductor dividing m, which we note is in bijection with the character group of Clm K , and thus has a m group structure given by χ e1 χ e2 = χ ] 1 χ2 . Indeed, for each character of ClK , its extension by zero is a ray class character χ of modulus m induced by a primitive ray class character χ e whose conductor divides m, and each primitive ray class character χ e of conductor dividing m induces a ray class character χ of modulus m that determines a character of Clm K ; these two maps are inverses, hence bijections. This generalizes Corollary 18.16. 18.785 Fall 2021, Lecture #22, Page 5 Definition 22.15. A ray class character χ is principal if ker χ = χ−1 (U(1)). We use denote the unique primitive principal ray class character. 1 to Remark 22.16. For Dirichlet characters, 1 is the unique Dirichlet character of conductor 1, but for ray class characters this holds only when the class group ClK is trivial (as when K = Q). In general, the extension by zero of any character of ClK is a ray class character of conductor (1) and need not be principal (but is necessarily primitive). Like Dirichlet characters, each ray class character has an associated L-function. Definition 22.17. The Weber L-function L(s, χ) of a ray class character χ for a number field K is the complex function −1 X Y χ(a)N(a)−s , L(s, χ) := 1 − χ(p)N(p)−s = p a where the product is over prime ideals of OK and the sum is over nonzero OK -ideals; the product and sum both converge to a non-vanishing holomorphic function on Re(s) > 1 (this follows from comparison with the Dedekind zeta function ζK (s), since |χ(a)| ≤ 1). Example 22.18. For K = Q, Weber L-functions are Dirichlet L-functions. For any number field K, the Weber L-function for 1 is the Dedekind zeta function: L(s, 1) = ζK (s). More generally, we have the following theorem, which is analogous to Theorem 19.15 but avoids the need to assume the existence of a ray class field. Proposition 22.19. Let χ be a ray class character of modulus m for a number field K of degree n. Then L(s, χ) extends to a meromorphic function on Re(s) > 1 − n1 that has at most a simple pole at s = 1 and is holomorphic if χ is non-principal. Proof. Associated to each ray class γ ∈ Clm K we have a Dirichlet series X ζK,γ (s) := N(a)−s a∈γ that is holomorphic on Re(s) > 1. For the trivial modulus m, our proof of analytic class number formula (Theorem 19.12) implies that ζK,γ (s) has a meromorphic continuation to 1 − n1 with a simple pole at s = 1 and residue ρ = 2r (2π)2 RK /(ωK |DK |1/2 ), independent of γ. Recall that in our proof of Theorem 19.8 we treated each γ ∈ ClK = cl(OK ) separately and obtained the same value of ρ for each γ, leading to the residue ρK = hK ρ that appears in Theorem 19.12. The same proof works for Clm K , mutatis mutandi: replace covol(OK ) with covol(m0 ), × m := covol(π(Log(O × ∩ K m,1 ))), and replace the regulator RK = covol(π(Log(OK ))) with RK K m := #(µ ∩ K m,1 ). The exact value of ρ is not important to us replace wK = #µK with wK K here, the key point is that ζK,γ (s) has a meromorphic continuation to Re(s) > 1 − n1 with a simple pole at s = 1 whose residue ρ depends only on K and m (not γ). We then have X L(s, χ) = χ(γ)ζK,γ (s) γ∈Clm K = X γ∈Clm K   X χ(γ) ζK,γ (s) − ρ ζ(s) + χ(γ)ρ ζ(s), γ∈Clm K 18.785 Fall 2021, Lecture #22, Page 6 The first sum is a finite sum of functions holomorphic on Re(s) > 1 − n1 (since ζ(s) has a simple pole at s = 1 with residue 1), and the second sum vanishes whenever χ is nonprincipal (by Corollary 18.38). The proposition follows. We now prove a generalization of Dirichlet’s theorem on primes on arithmetic progressions for arbitrary number fields. We proved the nonvanishing of Dirichlet L-functions L(1, χ) for non-principal χ using the analytic class number formula for Q(ζm ), the ray class field Q((m)∞), by writing the Dedekind zeta function for Q(ζm ) as a product of Dirichlet L-functions (see Theorem 19.15). A similar approach works for Weber L-functions, assuming the existence of ray class fields K(m): the Dedekind zeta function of K(m) is equal to the product of the Weber L-functions for χ ∈ X(m). But we will prove the non-vanishing of L(1, χ) for non-principal χ without assuming the existence of ray class fields. For a congruence subgroup C, let X(C) denote the set of primitive ray class characters whose kernels contain C. If C is a congruence subgroup of modulus m then X(C) is a subgroup m /C and we may view X(C) as the character of X(m) isomorphic to the character group of IK m group of IK /C. Theorem 22.20. Let C be a congruence subgroup of modulus m for a number field K and m : C]. The set of primes {p ∈ C} has Dirichlet density let n := [IK ( 1 if L(1, χ) 6= 0 for all χ 6= 1 in X(C), d(C) = n 0 otherwise. In fact d(C) = n1 always holds, as we will prove in Corollary 22.22 below, but it is easier to prove the theorem as stated and then use this to derive the corollary. Proof. We proceed as in the proof of Dirichlet’s theorem on primes in arithmetic progressions (see §18.4). We first construct the indicator function for the set {p ∈ C}: ( 1 if p ∈ C, 1 X χ(p) = n 0 otherwise. χ∈X(C) Note that summing over χ ∈ X(C) is equivalent to summing over the character group of P m m /C is the IK /C, so Corollary 18.38 applies: therefore P χ(p) = 0 unless the image of p in IK identity, meaning that p ∈ C, in which case χ(p) = #X(C) = n. + As s → 1 we have X log L(s, χ) ∼ χ(p)N(p)−s , p and therefore X X X log L(s, χ) ∼ χ(p)N(p)−s χ∈X(C) p χ∈X(C) ∼n X N(p)−s . p∈C By Proposition 22.19, we may write L(s, χ) = (s − 1)e(χ) g(s) 18.785 Fall 2021, Lecture #22, Page 7 for some function g(s) that is holomorphic and nonvanishing on a neighborhood of 1, where e(χ) := ords=1 L(s, χ) is −1 when χ = 1, and e(χ) ≥ 0 otherwise. We have log X X 1 1 − e(χ) log ∼n N(p)−s . s−1 s−1 χ6=1 p∈C 1 Dividing both sides by n log s−1 yields P P −s 1 − χ6=1 e(χ) p∈C N(p) ∼ 1 n log s−1 thus P d(C) = lim s→1+ p∈C log N(p)−s 1 s−1 = 1− (as s → 1+ ), P χ6=1 e(χ) n . The e(χ) are integers and the Dirichlet density is nonnegative, so either e(χ) = 0 for all χ 6= 1, in which case L(1, χ) 6= 0 for all χ 6= 1 and d(C) = n1 , or e(χ) = 1 for exactly one of the χ 6= 1 and d(C) = 0. (in fact this never happens, as noted above). Proposition 22.21. Let C be a congruence subgroup of modulus m for a number field K m : C]. For every I ∈ I m the set {p ∈ IC} has Dirichlet density and let n := [IK K ( 1 if L(1, χ) 6= 0 for all characters χ 6= 1 in X(C), d(IC) = n 0 otherwise. Proof. The proof is the same as in Theorem 22.20, except we now use the indicator function ( 1 if p ∈ IC, 1 X −1 χ(I) χ(p) = n 0 otherwise, χ∈X(C) and obtain X χ(I)−1 log L(s, χ) ∼ χ∈X(C) X X χ(I)−1 χ(p)N(p)−s ∼ n χ∈X(C) p X N(p)−s . p∈IC The rest of the proof is the same. Corollary 22.22. Let C be a congruence subgroup of modulus m for a number field K and m : C]. For every ideal I ∈ I m the set {p ∈ IC} has Dirichlet density 1/n, and for let n := [IK K every χ 6= 1 in X(C) we have L(1, χ) 6= 0. m be a complete set of coset representatives for C ⊆ I m . All Proof. Let I1 , . . . , In ∈ IK K m , hence in one of the cosets I C partitioning I m , but finitely many primes p of K lie in IK j K therefore d(I1 C) + · · · + d(In C) = 1. By Proposition 22.21, every term in this sum is either 0 or 1/n, and the equality implies they must all be equal to 1/n, which then implies L(1, χ) 6= 0 for all χ 6= 1 in X(C). Corollary 22.23. Let L/K be a finite Galois extension of number fields and let C be a congruence subgroup for a modulus m of K. If Spl(L) - {p ∈ C} then m [IK : C] ≤ [L : K]. 18.785 Fall 2021, Lecture #22, Page 8 Proof. We know from Theorem 21.15 that Spl(L) has polar density 1/[L : K], and this is also m : C], its Dirichlet density, by Proposition 21.12. The set {p ∈ C} has Dirichlet density 1/[IK by Theorem 22.22, and Spl(L) - {p ∈ C} (by assumption), so 1 1 = d(Spl(L)) ≤ d(C) = m . [L : K] [IK : C] 22.3 The conductor of an abelian extension We now introduce another notion of conductor, one attached to an abelian extension of number fields, which is defined as a product of local conductors attached to corresponding abelian extensions of the local field Kv for each place v ∈ MK . Definition 22.24. Let L/K be a finite abelian extension of local fields. The conductor c(L/K) is defined as follows.1 If K is archimedean then c(L/K) = 1 when K ' R and L ' C and c(L/K) = 0 otherwise. If K is nonarchimedean and p is the maximal ideal of its valuation ring OK , then c(L/K) := min{n : 1 + pn ⊆ NL/K (L× )} × × ). If L/K is a finite abelian extension , with 1 + p0 := OK (here 1 + pn is a subgroup of OK of global fields then its conductor is the modulus c(L/K) : MK → Z v 7→ c(Lw /Kv ) where Kv is the completion of K at v and Lw is the completion of L at a place w|v. (the fact that L/K is Galois ensures that c(Lw /Kv ) is the same for every w|v). As with any modulus, we may view the finite part of c(L/K) as an OK -ideal and the infinite part as a subset of ramified infinite places. It is not hard to show that conductor is supported on ramified places (in particular, it has finite support, as required for a modulus). More generally, we have the following. Proposition 22.25. Let L/K be a finite abelian extension of local or global fields. For each prime p of K we have   if and only if p is unramified, 0 vp (c(L/K)) = 1 if and only if p is ramified tamely,   ≥ 2 if and only if p is ramified wildly. Proof. See Problem Set 11. The finite part of the conductor of an abelian extension divides the discriminant ideal and is divisible by the same set of primes, but the valuation of the conductor at these primes is typically smaller than that of the discriminant. For example, the discriminant of the extension Q(ζp )/Q is (p)p−2 , but its conductor is (p)∞. Lemma 22.26. Let L1 /K and L2 /K be two finite abelian extensions of a local or global field K. If L1 ⊆ L2 then c(L1 /K) divides c(L2 /K). Proof. If K ' R, C the result is clear, and for nonarchimedean local K we may apply × × NL2 /K (L× 2 ) = NL1 /K (NL2 /L1 (L2 )) ⊆ NL1 /K (L1 ). The global case follows. 1 Many authors use f(L/K) rather than c(L/K), we use c to avoid confusion with the residue field degree. 18.785 Fall 2021, Lecture #22, Page 9 22.4 Norm groups m m → Gal(L/K). We can now identify a candidate for the kernel of the Artin map ψL/K : IK Recall from Lecture 6 that the norm map NL/K : IL → IK can be defined by Y n Y nf qi i 7→ pi i i , i i where pi := qi ∩ OK and fi := [Fqi : Fpi ] is the residue field degree. Definition 22.27. Let L/K be a finite abelian extension of number fields and let m be a modulus for K divisible by the conductor of L/K. The norm group (or Takagi group) associated to m is the congruence subgroup m m := Rm TL/K K NL/K (IL ), where ILm denotes the subgroup of fractional ideals in IL that are coprime to mOL . Proposition 22.28. Let L/K be a finite abelian extension of number fields and let m be a m . modulus for K divisible by the conductor of L/K. Then Spl(L/K) - TL/K Proof. Let p ∈ Spl(L/K) be coprime to m. Then ep = fp = 1. Pick a prime q of L above p. m . Then q - mOL and NL/K (q) = pfp = p (by Theorem 6.10), so p ∈ NL/K (ILm ) ⊆ TL/K We now record the following theorem, which is one of the two fundamental inequalities of class field theory. It was historically the “first" fundamental inequality of class field theory, proved by Weber, even though in modern terminology it is often called the second. It is also sometimes called the “universal norm index inequality”. Theorem 22.29 (Norm Index Inequality). Let L/K be a finite abelian extension of number fields and let m be a modulus for K divisible by the conductor of L/K. Then m m [IK : TL/K ] ≤ [L : K]. m . Proof. Apply Proposition 22.28 and Corollary 22.23 to the congruence subgroup TL/K 22.5 The main theorems of class field theory (ideal-theoretic version) We can give a more precise statement of the main theorems of class field theory. Let m be a modulus for a number field K. The three main theorems of class field theory state that: • Existence: The ray class field K(m) exists. • Completeness: If L/K is finite abelian then L ⊆ K(m) if and only c(L/K) | m. In particular, every finite abelian L/K lies in a ray class field. m m • Artin reciprocity: For each subextension L/K of K(m) we have ker ψL/K = TL/K m /T m with conductor c(L/K)|m and a canonical isomorphism IK L/K ' Gal(L/K). Artin reciprocity gives us a commutative diagram of canonical bijections: m L7→TL/K m} → {congruence subgroups C ⊆ IK ← ← ← {abelian L/K with c(L/K) | m} m /C C7→IK → → L7→Gal(L/K) → {quotients of Gal(K(m)/K)} m ψL/K ← { quotients of Clm K} 18.785 Fall 2021, Lecture #22, Page 10 22.6 The Hilbert class field Definition 22.30. Let K be global field. The Hilbert class field of K is the maximal unramified abelian extension of K (the compositum of all finite unramified abelian extensions of K inside a fixed separable closure of K). While it is not obvious from the definition, it follows from the completeness theorem of class field theory that the Hilbert class field must be the ray class field for the trivial modulus, and in particular, that it is a finite extension of K. This is a remarkable result (which we will prove in a later lecture), since infinite unramified extensions of number fields do exist (they are necessarily nonabelian). Indeed, one way to construct such an extension is by considering a tower of Hilbert class fields. Starting with a number field K0 := K, for each integer n ≥ 0 define Kn+1 to be the be the Hilbert class field of Kn . This yields an infinite tower of finite abelian extensions K0 ⊆ K1 ⊆ K2 ⊆ · · · , S and we may then consider the field L := n Kn . There are two possibilities: either we eventually reach a field Kn with class number 1, in which case Km = Kn for all m ≥ n and L/K is a finite unramified extension of K, or this does not happen and L/K is an infinite unramified extension of K (which is necessarily nonabelian). It was a longstanding open question as to whether the latter could occur, but in 1964 Golod and Shafarevich proved that indeed it can; in particular, the field √ √ K0 = Q( −30030) = Q( −2 · 3 · 5 · 7 · 11 · 13) is the base of an infinite tower of Hilbert class field extensions. One might ask whether one can use an imaginary quadratic field of smaller discriminant than this. It is known that no imaginary quadratic field of discriminant |D| ≤ 420 has an infinite Hilbert class field tower [3]; they all stabilize at either K2 or K3 . Extensions arising from Hilbert class field towers are necessarily solvable, since they are towers of finite abelian extensions. One might ask whether infinite nonsolvable unramified extensions exist. As shown by Maire [2], they do, and this can happen even when the base field has class number one and the Hilbert class field tower is trivial. Indeed, the biquadratic extension √ √ Q( 17601097, 17380678572169893) has class number one and its maximal unramified extension is an infinite extension. References [1] Henri Cohen, Advanced topics in computational number theory, Springer, 2000. [2] Christian Maire, On infinite unramfieid extensions, Pacific J. Math. 192 (2000), 135– 142. [3] Ken Yamamura, Maximal unramified extensions of imaginary quadratic number fields of small conductors, Journal de Théorie des Nombres de Bordeaux 9 (1997) 405–448. 18.785 Fall 2021, Lecture #22, Page 11