A BRIEF SUMMARY OF THE STATEMENTS OF CLASS FIELD THEORY

BJORN POONEN

0. Profinite completions of topological groups

Let G be a topological group. The profinite completion of G is

Ĝ := \underleftarrow{lim}_{U} \frac{G}{U},

where U ranges over the finite-index open normal subgroups of G. There is a natural continuous homomorphism G → Ĝ through which every other continuous homomorphism from G to a profinite group factors uniquely. If G is profinite already, then G → Ĝ is an isomorphism.

In general, G → Ĝ need not be injective or surjective. Nevertheless, we think of G as being almost isomorphic to Ĝ: The finite-index open subgroups of G are in bijection with those of Ĝ. And finite-index open subgroups of certain Galois groups are what we are interested in…

1. Local class field theory

1.1. Notation associated to a discrete valuation ring.

O: a complete discrete valuation ring
K := Frac(O)
v: the valuation K^{×} ↠ ℤ
𝔭: the maximal ideal of O
k: the residue field O/𝔭
K^{s} : a fixed separable closure of K
K^{ab} : the maximal abelian extension of K in K^{s}
K^{unr} : the maximal unramified extension of K in K^{s}
k^{s} : the residue field of K^{unr}, so k^{s} is a separable closure of k.

Equip K and its subsets with the topology coming from the absolute value |x| := exp(−v(x)).

Date: February 17, 2012.

I thank David Zywina for comments. The writing of this article was supported by NSF grant DMS-0301280.

1.2. Local fields.

Definition 1.1. A nonarchimedean local field is a complete discrete-valued field K as in Section 1.1 such that the residue field k is finite. An archimedean local field is ℝ or ℂ.

Facts:

• A nonarchimedean local field of characteristic 0 is isomorphic to a finite extension of ℚ_{p}.

• A (nonarchimedean) local field of characteristic p > 0 is isomorphic to 𝔽_{q}((t)) for some power q of p.

1.3. The local Artin homomorphism. Let K be a local field. Local class field theory says that there is a homomorphism

θ: K^{×} → Gal(K^{ab}/K)

that is almost an isomorphism. The homomorphism θ is called the local Artin homomorphism. It cannot be literally an isomorphism, because Gal(K^{ab}/K) is a profinite group, hence compact, while K^{×} is not. What is true is that θ induces an isomorphism of topological groups K̂^{×} → Gal(K^{ab}/K).

If K is archimedean, then θ: K^{×} → Gal(K^{ab}/K) is surjective and its kernel is the connected component of the identity in K^{×}.

For the rest of Section 1.3, we assume that K is nonarchimedean. Then θ is injective: The choice of a uniformizer π ∈ O lets us write K^{×} = O^{×}π^{ℤ} ≃ O^{×} × ℤ, and O^{×} is already profinite, so K̂^{×} ≃ O^{×} × ℤ̂. Thus local class field theory says that there is an isomorphism

O^{×} × ℤ̂ → Gal(K^{ab}/K).

More canonically, without choosing π, the two horizontal exact sequences below are almost isomorphic:

(1)  [figure: 0; O^{×}; K^{×}; v; θ; ℤ; 0; 0; Gal(K^{ab}/K^{unr}); Gal(K^{ab}/K); res; Gal(K^{unr}/K); 0]

With the identification of the group at lower right

Gal(K^{unr}/K) ≃ Gal(k^{s}/k) ≃ ℤ̂

mapping the Frobenius automorphism to 1 ∈ ℤ̂, the right vertical map in (1) becomes the natural inclusion ℤ ↪ ℤ̂. In other words, θ maps K^{×} isomorphically to the set of σ ∈ Gal(K^{ab}/K) inducing an integer power of Frobenius on the residue field (as opposed to a ℤ̂-power). The bottom row of (1) is simply the profinite completion of the top row.

Also from (1), one sees that θ(O^{×}) is the inertia subgroup Gal(K^{ab}/K^{unr}) of Gal(K^{ab}/K), and that θ maps any uniformizer to a Frobenius automorphism in Gal(K^{ab}/K). Moreover, the descending chain

O^{×} ⊃ 1 + 𝔭 ⊃ 1 + 𝔭^{2} ⊃ ⋯

is mapped isomorphically by θ to the descending chain of ramification subgroups of Gal(K^{ab}/K) in the upper numbering.

1.4. Functoriality. Let L be a finite extension of K. Let N_{L/K}: L^{×} → K^{×} be the norm map. Let θ_{L}, θ_{K} be the local Artin homomorphisms associated to L, K, respectively. Let res: Gal(L^{ab}/L) → Gal(K^{ab}/K) be the homomorphism mapping an automorphism σ of L^{ab} to its restriction σ|_{K^{ab}}. Then the square

\xymatrix{L^{×} \ar[r]^{θ_{L}} \ar[d]_{N_{L/K}} & Gal(L^{ab}/L) \ar[d]^{res} \\ K^{×} \ar[r]^{θ_{K}} & Gal(K^{ab}/K)}

commutes.

1.5. Finite abelian extensions. Because θ is almost an isomorphism, and because of Galois theory, the following sets are in bijection:

• The finite-index open subgroups of K^{×}.

• The (finite-index) open subgroups of Gal(K^{ab}/K).

• The finite abelian extensions of K contained in K^{s}. Going backwards, if L is a finite abelian extension of K in K^{s}, the corresponding subgroup of K^{×} is N_{L/K}L^{×}. (This could be viewed as a consequence of the functoriality above.)

The composition

K^{×} → Gal(K^{ab}/K) ↠^{res} Gal(L/K)

is surjective with kernel N_{L/K}L^{×}, and O^{×} maps to the inertia subgroup I_{L/K} ⊴ Gal(L/K), and any uniformizer π maps to a Frobenius element of Gal(L/K).

2. Global class field theory (via ideles)

2.1. Global fields.

Definition 2.1. A number field is a finite extension of ℚ. A global function field is a finite extension of 𝔽_{p}(t) for some prime p, or equivalently is the function field of a geometrically integral curve over a finite field 𝔽_{q} (called the constant field), where q is a power of some prime p. A global field is a number field or a global function field.

Throughout Sections 2 and 3, K is a global field. If v is a nontrivial place of K (given by an absolute value on K), then the completion K_{v} is a local field. If v is nonarchimedean, let O_{v} be the valuation subring of K_{v}; if v is archimedean, let O_{v} = K_{v}.

2.2. The adele ring. The adele ring of K is the restricted direct product

A_{K} := \begin{matrix}∏ & ′ \\ v \end{matrix}(K_{v}, O_{v}) := \{(a_{v}) ∈ ∏_{v} K_{v} : a_{v} ∈ O_{v} for all but finitely many v\} .

It is a topological ring: the topology is uniquely characterized by the condition that ∏_{v} O_{v} is open in A_{K} and has the product topology. The diagonal map K → A_{K} is like ℤ → ℝ: it embeds K as a discrete co-compact subgroup of A_{K}.

2.3. The idele group and idele class group. The idele group of K is

A_{K}^{×} = \begin{matrix}∏ & ′ \\ v \end{matrix}(K_{v}^{×}, O_{v}^{×}) := \{(a_{v}) ∈ ∏_{v} K_{v}^{×} : a_{v} ∈ O_{v}^{×} for all but finitely many v\} .

It is a topological group: the topology is uniquely characterized by the condition that ∏_{v} O_{v}^{×} is open in A_{K}^{×} and has the product topology.^{1} The diagonal map K^{×} → A_{K}^{×} is like ℤ^{×} → ℝ^{×}: it embeds K^{×} as a discrete subgroup of A_{K}^{×}, but the quotient C_{K} := A_{K}^{×}/K^{×} is not compact. The topological group C_{K} is called the idele class group.

2.4. The global Artin homomorphism. Let K^{s} be a fixed separable closure of K. Let K^{ab} be the maximal abelian extension of K contained in K^{s}.

The group C_{K} plays the role in global class field theory played by K^{×} in local class field theory. Namely, if K is a global field, there is a global Artin homomorphism

θ: C_{K} → Gal(K^{ab}/K)

that induces an isomorphism Ĉ_{K} →^{∼} Gal(K^{ab}/K).

If K is a number field, then θ is surjective and its kernel is the connected component of the identity in C_{K}.

If K is a global function field with constant field k, then θ is injective and θ(C_{K}) equals the set of σ ∈ Gal(K^{ab}/K) whose restriction in Gal(k^{s}/k) is an integer power of the Frobenius generator.

2.5. Functoriality. Let L be a finite extension of K of degree n. Then A_{L} ≃ A_{K} ⊗_{K} L is

free of rank n over A_{K}, so there is a norm map N_{L/K}: A_{L} → A_{K}. We write N_{L/K} also for the induced homomorphism N_{L/K}: C_{L} → C_{K}. Then

\xymatrix{C_{L} \ar[r]^{θ_{L}} \ar[d]_{N_{L/K}} & Gal(L^{ab}/L) \ar[d]^{res} \\ C_{K} \ar[r]^{θ_{K}} & Gal(K^{ab}/K)}

commutes.

2.6. Finite abelian extensions. The following sets are in bijection:

• The finite-index open subgroups of C_{K}.

• The finite-index open subgroups of Gal(K^{ab}/K).

• The finite abelian extensions of K contained in K^{s}. Going backwards, if L is a finite abelian extension of K in K^{s}, the corresponding subgroup of C_{K} is N_{L/K}C_{L}. The composition

C_{K} → Gal(K^{ab}/K) ↠^{res} Gal(L/K)

is surjective with kernel N_{L/K}C_{L}.

1Alternatively, one can use the general recipe for getting the topology on the units of a topological ring R: not the subspace topology on R^{×} as a subset of R (this may fail to make the inverse map R^{×} → R^{×} continuous), but the subspace topology on the set of solutions to xy = 1 in R × R (this is what one gets if one expresses the multiplicative group scheme 𝔾_{m} as an affine variety).

2.7. Connection between the global and local Artin homomorphisms. Let v be a place of K. Identify K_{v}^{×} with a subgroup of A_{K}^{×} by mapping α ∈ K_{v}^{×} to the idele with α in the v-th position and 1 in every other position. The composition K_{v}^{×} ↪ A_{K}^{×} ↠ C_{K} is injective. Let θ_{v} be the local Artin homomorphism for K_{v}. Then the diagram

\xymatrix{K_{v}^{×} \ar[r]^{θ_{v}} \ar@{^{(}->}[d] & Gal(K_{v}^{ab}/K_{v}) \ar@{^{(}->}[d]^{res} \\ C_{K} \ar[r]^{θ} & Gal(K^{ab}/K)}

commutes. Thus θ determines θ_{v}.

Conversely, if one knows θ_{v} for all v, one can construct θ as follows. Let L be a finite abelian extension of K contained in K^{s}. Define

A_{K}^{×} → Gal(L/K)
(a_{v}) ↦ ∏_{v} θ_{v}(a_{v});

if v is unramified in L/K, and a_{v} ∈ O_{v}^{×}, then θ_{v}(a_{v}) = 1, so all but finitely many terms in the infinite product are 1, and the product makes sense. Take the inverse limit over all possible L to get

A_{K}^{×} → Gal(K^{ab}/K).

The idelic version of the Artin reciprocity law says that K^{×} is in the kernel, so we get a homomorphism

C_{K} → Gal(K^{ab}/K),

which is θ.

2.8. Moduli.

Definition 2.2. A modulus is a formal product 𝔪 = ∏_{v} v^{e_{v}} where e_{v} ∈ ℤ_{≥0}, all but finitely many e_{v} equal 0, and e_{v} ∈ {0, 1} for real v, and e_{v} = 0 for complex v. The support supp 𝔪 is the (finite) set of nonarchimedean places v such that e_{v} ≠ 0.

If K is a number field, then a modulus can be viewed as a pair consisting of (1) an integral ideal of the ring of integers O_{K}, and

(2) a subset of the real places. If K is the function field of a smooth projective curve X over a finite field, then a modulus is the same thing as an effective divisor on X.

2.9. Ray class groups and ray class fields. In this section we assume that K is a number field. Fix a modulus 𝔪 = ∏_{v} v^{e_{v}}. We will define a finite-index open subgroup U_{𝔪,v} ⊆ O_{v}^{×} for each v. If e_{v} = 0, define U_{𝔪,v} := O_{v}^{×}. If e_{v} > 0 and v is nonarchimedean, define U_{𝔪,v} := 1 + 𝔭_{v}^{e_{v}}, where 𝔭_{v} is the maximal ideal of O_{v}. If e_{v} > 0 and v is real, define U_{𝔪,v} as ℝ_{>0} ⊆ ℝ^{×} ≃ K_{v}^{×}. Define U_{𝔪} := ∏_{v} U_{𝔪,v} ⊆ A_{K}^{×}. The image of U_{𝔪} under A_{K}^{×} ↠ C_{K} is a finite-index open subgroup U′_{𝔪} of C_{K} (this is equivalent to finiteness of the class number of K, as we will see in Section 3.4). The corresponding finite abelian extension R_{𝔪} of K is called the ray class field of modulus 𝔪, and R_{𝔪} over K is unramified at all v with e_{v} = 0. The ray class group of modulus 𝔪 is

\frac{C_{K}}{U′_{𝔪}} = \frac{A_{K}^{×}}{U_{𝔪}K^{×}} ,

which is isomorphic to Gal(R_{𝔪}/K) via the global Artin homomorphism.

Every finite-index open subgroup of A_{K}^{×} contains U_{𝔪} for some 𝔪, so every finite abelian extension of K is contained in R_{𝔪} for some 𝔪.

3. Global class field theory (via ideals)

In this section we assume that K is a number field.

3.1. Classical ray class groups. Let I be the group of fractional ideals of K, or equivalently, the free abelian group on the nonarchimedean places of K. Let P be the subgroup of principal ideals. The class group is Cl O_{K} := I/P.

We now generalize to an arbitrary modulus 𝔪 = ∏_{v} v^{e_{v}}. Let I_{𝔪} be the subgroup of fractional ideals that do not involve the primes dividing 𝔪; i.e., I_{𝔪} is the free abelian group on the nonarchimedean places v satisfying e_{v} = 0. For a ∈ K^{×}, the notation a ≡ 1 (mod^{×} 𝔪) means that a ∈ U_{𝔪,v} for every v satisfying e_{v} > 0. The group P_{𝔪} ⊆ I_{𝔪} is the group of principal ideals generated by some a ∈ K^{×} with a ≡ 1 (mod^{×} 𝔪). The classical ray class group of modulus 𝔪 is Cl_{𝔪} O_{K} := I_{𝔪}/P_{𝔪}. Section 3.4 will prove that this is isomorphic to the ray class group C_{K}/U′_{𝔪} defined in Section 2.9.

3.2. The classical Artin homomorphism. Let L/K be a finite abelian extension of number fields. Let S be a finite set of finite primes of K such that S contains every prime that ramifies in L. Let I_{S} be the group of fractional ideals that do not involve the primes in S. The classical Artin homomorphism is the map

Θ: I_{S} → Gal(L/K)

sending each prime ideal 𝔭 ∉ S to the Frobenius element Frob_{𝔭} ∈ Gal(L/K).

3.3. The main theorems. The Artin reciprocity law states that there exists a modulus 𝔪 (depending on L/K) with supp 𝔪 = S such that the subgroup P_{𝔪} ⊆ I_{𝔪} = I_{S} is contained in ker Θ. The existence theorem states that given a modulus 𝔪 and group H with P_{𝔪} ⊆ H ⊆ I_{𝔪} there exists an abelian extension L of K unramified outside supp 𝔪 such that the kernel of Θ for L/K equals H.

3.4. Comparison of ideal groups and idele groups. Consider the trivial modulus 𝔪 = 1 (with e_{v} = 0 for all v). Taking the restricted direct product of the valuation maps v: K_{v}^{×} ↠ ℤ gives a surjective homomorphism

A_{K}^{×} → I

that discards the archimedean components of its input, and its kernel is U_{1} = ∏_{v} O_{v}^{×}. Thus \frac{A_{K}^{×}}{U_{1}} ≃ I. If we take the quotient by the image of K^{×} on both sides, we find that the ray

class group \frac{A_{K}^{×}}{U_{1}K^{×}} of modulus 1 is isomorphic to the class group I/P = Cl O_{K}. The ray class field R_{1} of modulus 1 is called the Hilbert class field, which can be characterized also as the maximal abelian extension of K in K^{s} that is unramified at all places of K (including the archimedean ones). We get

\frac{C_{K}}{U′_{1}} = \frac{A_{K}^{×}}{U_{1}K^{×}} ≃ \frac{I}{P} = Cl O_{K} ≃ Gal(R_{1}/K).

This can generalized to an arbitrary modulus 𝔪 = ∏ v^{e_{v}} as follows. Let A_{K}^{𝔪} ⊆ A_{K}^{×} be the subgroup consisting of (a_{v}) with (a_{v}) with a_{v} ∈ U_{𝔪,v} for every v with e_{v} > 0. Let K^{𝔪} = A_{K}^{𝔪} ∩ K^{×}. We have an isomorphism

\frac{A_{K}^{𝔪}}{U_{𝔪}} →^{∼} I_{𝔪}.

Dividing by the image of K^{𝔪} on both sides gives

(2)  \frac{A_{K}^{𝔪}}{U_{𝔪}K^{𝔪}} →^{∼} \frac{I_{𝔪}}{P_{𝔪}}.

On the other hand, A_{K}^{×} = A_{K}^{𝔪}K^{×}, so there is an isomorphism

\frac{A_{K}^{𝔪}}{K^{𝔪}} →^{∼} \frac{A_{K}^{×}}{K^{×}} = C_{K}.

Dividing by the image of U_{𝔪} on both sides, and combining with (2), we get isomorphisms

\frac{C_{K}}{U′_{𝔪}} = \frac{A_{K}^{×}}{U_{𝔪}K^{×}} ≃ \frac{I_{𝔪}}{P_{𝔪}} = Cl_{𝔪} O_{K} ≃ Gal(R_{𝔪}/K).

4. An introduction to an introduction to the Langlands program

Let K be a local or global field. Every 1-dimensional character (continuous homomorphism)

Gal(K^{s}/K) → ℂ^{×}

factors through Gal(K^{ab}/K) and has finite image. These characters form a discrete abelian group, the Pontryagin dual of the profinite group Gal(K^{ab}/K). It follows that the problem of classifying finite abelian extensions of K is more or less the same as the problem of describing all these characters.

The Langlands program is an attempt to understand Gal(K^{s}/K) more completely by describing its higher-dimensional representations: the group ℂ^{×} = GL_{1}(ℂ) is replaced by GL_{n}(ℂ), or even G(ℂ) for other linear algebraic groups G. The continuous homomorphisms

Gal(K^{s}/K) → G(ℂ)

are conjectured to correspond to certain “automorphic” objects defined intrinsically in terms of K, just as class field theory gives a description of the group Gal(K^{ab}/K) (which is defined in terms of extrinsic objects such as finite abelian extensions, which are initially mysterious) in terms of intrinsic objects (K^{×} or C_{K}) obtained directly from K.

Ultimately, the program would give information about nonabelian algebraic extensions of K.

5. Further reading

For basics on profinite groups, see [Ser02, I.§1] and [Gru86]. The latter discusses infinite Galois theory as well.

For local class field theory, see [Ser86]. For the approach to global class field theory via cohomology of ideles, see [Tat86]. For a treatment of global class field theory via ideals, see [Jan96]. All these topics are covered also in [Neu99].

For an introduction to the Langlands program, see [BG03].

References

[BG03] D. Bump, J. W. Cogdell, E. de Shalit, D. Gaitsgory, E. Kowalski, and S. S. Kudla, An introduction to the Langlands program, Birkhäuser Boston Inc., Boston, MA, 2003. Lectures presented at the Hebrew University of Jerusalem, Jerusalem, March 12–16, 2001; Edited by Joseph Bernstein and Stephen Gelbart. MR1990371 (2004g:11037) ↑5

[Gru86] K. Gruenberg, Profinite groups, Algebraic number theory (J. W. S. Cassels and A. Fröhlich, eds.), Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London. Reprint of the 1967 original, 1986, pp. 116–127. MR0225922 (37 #1512) ↑5

[Jan96] Gerald J. Janusz, Algebraic number fields, 2nd ed., Graduate Studies in Mathematics, vol. 7, American Mathematical Society, Providence, RI, 1996. MR1362545 (96j:11137) ↑5

[Neu99] Jürgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR1697859 (2000m:11104) ↑5

[Ser86] J.-P. Serre, Local class field theory, Algebraic number theory (J. W. S. Cassels and A. Fröhlich, eds.), Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London. Reprint of the 1967 original, 1986, pp. 128–161. MR0220701 (36 #3753) ↑5

[Ser02] Jean-Pierre Serre, Galois cohomology, Corrected reprint of the 1997 English edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. Translated from the French by Patrick Ion and revised by the author. MR1867431 (2002i:12004) ↑5

[Tat86] J. T. Tate, Global class field theory, Algebraic number theory (J. W. S. Cassels and A. Fröhlich, eds.), Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London. Reprint of the 1967 original, 1986, pp. 162–203. MR0220697 (36 #3749) ↑5

Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

E-mail address: poonen@math.berkeley.edu
URL: http://math.berkeley.edu/~poonen