GLOBAL NILPOTENT CLASS FIELD THEORY

G. ABRAMOV, H. KOCH, K-F. LAI

Let p be an arbitrary, fixed prime, K an algebraic number field and S a set of places of K which ramify in the maximal p-extension of K. This means a 𝔭 ∈ S is of the form N(𝔭) ≡ 1 (mod p) or 𝔭|p or 𝔭 real if p = 2.

Let U_{S} be the subgroup of the group U_{K} of unit ideles consisting of the ideles with components 1 at the places of S. Let I_{K} be the idele group of K. Then to the maximal abelian extension of K unramified outside S corresponds in the sense of class field theory the group ℭ_{S} := I_{K}/U_{S}K^{×}. Let ℭ̂_{S} be the pro-p-completion of ℭ_{S}. Then the Artin map induces an isomorphism ϕ_{1} from ℭ̂_{S} onto the Galois group G_{S} of the maximal p-extension K_{S} of K unramified outside S.

The groups G_{S} are studied in [Ko1970]. We adopt some notation from this book. In particular B̃_{S} denotes the character group of the finite elementary p-group V_{S}/K^{×p} with

V_{S} := {α ∈ K^{×}|(α) = 𝔞^{p}, α ∈ K_{𝔭}^{p} for 𝔭 ∈ S},

where (α) is the principal ideal associated to α. If S is big enough, then B̃_{S} = {0}.

In [KoKuLa] we introduced the algebraic setting of a nilpotent clas field theory which in the global case means the following. We associate to ℭ̂_{S} the

graded Lie algebra ⨁_{n=1}^{∞} L_{n}(ℭ̂_{S}) and to G_{S} the graded Lie algebra ⨁_{n=1}^{∞} L_{n}(G_{S}) =

⨁_{n=1}^{∞} G_{S}^{(n)}/G_{S}^{(n+1)}, where {G_{S}^{(n)}|n = 1, 2, … } is the filtration of G_{S} by means of

the central decending series. The Artin map ϕ_{1} induces a map

ϕ_{n} : L_{n}(ℭ̂_{S}) → L_{n}(G_{S})

again denoted as Artin map.

In particular L_{2}(ℭ̂_{S}) is the pro-p-completion of the alternating product ℭ̂_{S} ∧ ℭ̂_{s}, which was studied in the local case, i.e. K^{×} ∧K^{×}, in [Zi1982] and [Zi1983]. In [Zi1982] it was shown as the basis of an arithmetical theory that ϕ_{2} is an isomorphism in the local case of class field theory. We want to show here, that this result is true in the global case as well if certain conditions are fulfilled:

Theorem 0.1. Let B̃_{S} = {0} and 𝔭 ∈ S for one place 𝔭 of K with 𝔭|p. Then the Artin map ϕ_{2} is an isomorphism of L_{2}(ℭ̂_{S}) onto L_{2}(G_{S}).

Part of the research of this paper was done while the second author was guest of the University of Sydney. He would like to thank this university and in particular the third author for hospitality.

1. The proof of Theorem 0.1 uses the results of [Ko1970], §11, which we remember first.

We introduce some further notations. For any field define δ(L) := 1 if the p-th roots of unity are in L and δ(L) := 0 if not. A place 𝔭 of K is ramified in the maximal p-extension \overline{K} of K only if N(𝔭) ≡ 1 (mod p) or 𝔭|p or 𝔭 real (in the case p = 2). Hence in the following we assume that a 𝔭 in S is of one of these types. For technical reasons we introduce the number e_{𝔭} for 𝔭 ∈ S:

- e_{𝔭} := p^{ν_{p}(N(𝔭)−1)} if N(𝔭) ≡ 1 (mod p),

- e_{𝔭} := p^{K} if 𝔭|p and the p^{K}-th roots of unity are in K and the p^{k+1}-th roots of unity are not in K,

- e_{𝔭} := 2 if 𝔭 is real.

Let

R → F → G_{S}  (1)

be a minimal representation of G_{S} by the free pro-p-group F. We construct it by means of a minimal system \{\overline{τ}_{i}|i ∈ I\}, τ_{i} ∈ G_{S} of generators of G_{K}/G_{K}^{(2)}G_{K}^{p} ≅ ℭ_{S}/ℭ_{S}^{p}. Let F be the free pro-p-group with generator system {t_{i}|i ∈ I}, and let θ be a continuous section of G_{K} into F with θτ_{i} = t_{i}.

For any 𝔭 ∈ S we choose an extension 𝔓 in K_{S} and we denote the decomposition group of 𝔓 by D_{𝔭}. Let G_{𝔭} be the Galois group of the maximal p-extension \overline{K}_{𝔭} of K_{𝔭}. The inclusions

K_{S} → (K_{S})_{𝔓} → \overline{K}_{𝔭}

induce a homomorphism ψ_{𝔭} of G_{𝔭} into G_{S}.

Let

R_{𝔭} ⟶ F_{𝔭} ⟶^{φ_{𝔭}} G_{𝔭}  (2)

be a minimal presentation of G_{𝔭} by the free pro-p-group F_{𝔭}. Then we have a diagram

\begin{matrix}R_{𝔭} → & F_{𝔭} → & G_{𝔭} \\ ↓ & ↓ & ↓ \\ R → & F → & G_{S}, \end{matrix}  (3)

where the map from F_{𝔭} into F is χ_{𝔭} := ψ_{𝔭}φ_{𝔭}.

The presentations (2) of G_{𝔭} are well known (see e.g. [Ko1970] for references). For our purpose we need in the moment only the fact that R_{𝔭} is generated as closed normal subgroup of F_{𝔭} by one relation r_{𝔭} beside the case that 𝔭|p and δ(K_{𝔭}) = 0 where F_{𝔭} is free. r_{𝔭} has the form

r_{𝔭} = t_{𝔭}^{e_{𝔭}}r′_{𝔭} with r′_{𝔭} ∈ F_{𝔭}^{(2)}.

Here t_{𝔭} is an element of F_{𝔭}, which is mapped onto ζ_{𝔭} by the homomorphism

F_{𝔭} → G_{𝔭}/G_{𝔭}^{(2)} → K̂_{𝔭}^{×},

where ζ_{𝔭} denotes a primitive root of unity of order e_{𝔭} in K_{𝔭}. We put r_{𝔭} := 1 in the case 𝔭|p, δ(K_{𝔭}) = 0.

According to [Ko1970], §11, the relation module R in (3) is generated as normal subgroup of F by the local relations χ_{𝔭}r_{𝔭}, 𝔭 ∈ S if B̃_{S} = {0}. If the p-th roots of unity are contained in K, one arbitrary χ_{𝔭_{∘}}r_{𝔭_{∘}} of these relations can be omitted. We put S^{∗} := S if δ(K) = 0, S^{∗} := S − {𝔭_{∘}} if δ(K) = 1.

2. In this section we use the methods and some results of [KoKuLa]. If F is a free pro-p-group, then L(F) can be identified with the free graded Lie algebra over the pro-p-group L_{1}(F) = F/F^{(2)}. We study the map

ϕ_{2} : L_{2}(ℭ̂_{S}) ⟶ L_{2}(G_{S})

by means of the two maps

ψ_{2} : L_{2}(F) ⟶ L_{2}(G_{S})

and

T_{2} : L_{2}(F) ⟶ L_{2}(ℭ̂_{S}).

T_{2} is defined by

L_{2}(F) = L_{2}(F/F^{(2)}) ⟶ L_{2}(G_{S}/G_{S}^{(2)})

and the map

L_{2}(G_{S}/G_{S}^{(2)}) ⟶ L_{2}(ℭ̂_{S}),

induced by the inverse of the Artin map ϕ_{1}.

We begin with ψ_{2}. Let R^{(2)} := R ∩ F^{(2)}. Then

Ker ψ_{2} = R^{(2)}F^{(3)}/F^{(3)}.

Proposition 0.2. Let B_{S} = {0} and 𝔭 ∈ S for one place 𝔭|p. Then Ker ψ_{2} is generated as pro-p-group by the classes

[χ_{𝔭}t_{𝔭}^{e_{𝔭}}, x]F^{(3)}, 𝔭 ∈ S^{∗}, x ∈ F.  (4)

Proof. Since R is generated as normal subgroup of F by the relations χ_{𝔓}r_{𝔭}, 𝔭 ∈ S^{∗}, it is clear, that the classes (4) are in R^{(2)}F^{(3)}/F^{(3)}. It remains to shows that they generate R^{(2)}F^{(3)}/F^{(3)}.

Any class of R^{(2)}F^{(3)}/F^{(3)} is of the form

∏_{𝔭∈S^{∗}}(χ_{𝔭}r_{𝔭})^{α_{𝔭}}rF^{(3)}, α_{𝔭} ∈ ℤ_{p},

with

∏_{𝔭∈S^{∗}}(χ_{𝔭}r_{𝔭})^{α_{𝔭}} ∈ F^{(2)},  (5)

where rF^{(3)} is in the pro-p-group generated by the elements (4).

For the proof of Proposition 0.2 it is sufficient to show α_{𝔭} = 0 for 𝔭 ∈ S^{∗}.

(5) is equivalent to

∏_{𝔭∈S^{∗}}(χ_{𝔭}t_{𝔭})^{e_{𝔭}α_{𝔭}} ∈ F^{(2)}.  (6)

3. Now we have our problem reduced to a question of class field theory. We have to study G_{S}/G_{S}^{(2)} ≅ ℭ̂_{S} in connection with F/F^{(2)}.

First we construct a generator system of G_{S}/G_{S}^{(2)} according to [Ko1970], §11.4. We begin by completing our local considerations in 1.

If N(𝔭) ≡ 1 (mod p) or 𝔭 real, the inertia group of G_{𝔭} is cyclic. We denote the element corresponding to ζ_{𝔭} in ℭ̂_{S} again by ζ_{𝔭} and the corresponding element in G_{S}/G_{S}^{(2)} by τ′_{𝔭}.

For 𝔭|p let ζ_{𝔭}, α_{1𝔭}, …, α_{n_{𝔭}𝔭} with n_{𝔭} := [K_{𝔭} : ℚ_{𝔭}] be a basis of the principal unit group of K_{𝔭}. We denote the corresponding elements in ℭ̂_{K} again by ζ_{𝔭}, α_{1𝔭}, …, α_{n_{𝔭}𝔭} and the corresponding elements in G_{S}/G_{S}^{(2)} by τ′_{𝔭}, τ′_{1𝔭}, …, τ′_{n_{𝔭}𝔭}.

Furthermore, let Cl(K)(p) be the p-component of the ideal class group of K and let 𝔞_{1}, …, 𝔞_{h} be ideles in I_{K} such that the corresponding ideals 𝔞′_{1}, …, 𝔞′_{h} form a basis of Cl(K)(p). Let ω_{1}, …, ω_{h} be the corresponding elements in G_{S}/G_{S}^{(2)}. Then by class field theory

{ω_{1}, …, ω_{h}} ∪ {τ′_{𝔭}|𝔭 ∈ S} ∪ {τ′_{1𝔭}, …, τ′_{n_{𝔭}𝔭}/𝔭 ∈ S, 𝔭|p}

is a generator system of G_{S}/G_{S}^{(2)}.

We present G_{S}/G_{S}^{(2)} as the quotient of the free abelian pro-p-group A with generator system

{w_{1}, …, w_{h}} ∪ {t′_{𝔭}|𝔭 ∈ S} ∪ {t′_{1𝔭}, …, t′_{n_{𝔭}𝔭}|𝔭 ∈ S, 𝔭|p}.

We put t′_{𝔭} = 1 if 𝔭|p, δ(K_{𝔭}) = 1. With the map α : A → G_{S}/G_{S}^{(2)} given by

α(w_{i}) = ω_{i}, i = 1, …, h, α(t′_{𝔭}) = τ′_{𝔭}, 𝔭 ∈ S,
α(t′_{i𝔭}) = τ′_{i𝔭}, i = 1, …, n_{𝔭}, 𝔭|p.

Then the kernel of α is generated by the local relations

t′_{𝔭}^{e_{𝔭}}, 𝔭 ∈ S  (7)

and the relations which correspond to the finiteness of Cl(K)(p) and the group E of units of K:

Let ε ∈ E. Then we have the following equation in Î_{K}:

(ε) = ∏_{𝔭∈S} ζ_{𝔭}^{a_{𝔭ε}} ∏_{{𝔭|p \atop 𝔭∈S}} ∏_{i=1}^{n_{𝔭}} α_{i𝔭}^{b_{i𝔭ε}}, a_{𝔭ε} ∈ ℤ, b_{i𝔭} ∈ ℤ_{p}.

Let q_{j} be the order of 𝔞′_{j} and β_{j} ∈ K^{×} such that 𝔞_{j}^{′q_{j}} = (β_{j}), j = 1, …, h. Then 𝔞_{j}^{q_{j}}(β_{j}^{−1}) is a unit idele whose image in Î_{K} has the form

∏_{𝔭∈S} ζ_{𝔭}^{c_{𝔭}j} ∏_{{𝔭|p \atop 𝔭∈S}} ∏_{i=1}^{n_{𝔭}} α_{i𝔭}^{d_{i𝔭j}}, c_{𝔭j} ∈ ℤ, d_{i𝔭j} ∈ ℤ_{p}.

j = 1, …, h. The correspondent relations in A are

∏_{𝔭∈S} t′_{𝔭}^{a_{𝔭}ε} ∏_{{𝔭|p \atop 𝔭∈S}} ∏_{i=1}^{n_{𝔭}} t′_{i𝔭}^{bi𝔭ε}  (8)

and

w_{j}^{−q_{j}} ∏_{𝔭∈S} t′^{c_{𝔭j}} ∏_{{𝔭|p \atop 𝔭∈S}} ∏_{i=1}^{n_{𝔭}} t′^{b_{i𝔭ε}}.  (9)

The relations (7) - (9) generate Ker α by construction.

Now we have to relate A and α to F/F^{(2)}.

F/F^{(2)} is to be generated by a minimal system of generators. We can do this by choosing among the generators of A a minimal subsystem {t_{i}|i ∈ I}. This defines a homomorphism β of A onto F/F^{(2)} with kernel generated by the relations (8) and (9). This corresponds to the procedure in [Ko1970], §11.4.

4. Now we come back to the proof of Proposition 0.2. The condition (6) is equivalent to

∏_{𝔭∈S^{∗}} t′_{𝔭}^{e_{𝔭}α_{𝔭}} ∈ Kerβ.  (10)

Since the generators ω_{1}, …, ω_{h} do not appear in (10), the relation has to be of the form

∏_{𝔭∈S} t′_{𝔭}^{a_{𝔭}ε} ∏_{{𝔭|p \atop 𝔭∈S}} ∏_{i=1}^{n_{𝔭}} t′_{i𝔭}^{b_{i𝔭ε}}  (11)

for an ε ∈ E. By assumption 𝔭_{∘} ∈ S, 𝔭_{∘}|p, hence b_{i𝔭_{∘}ε} = 0 for i = 1, …, n_{𝔭_{∘}}. This implies that ε is a root of unity. Since only powers of p contribute to (11), we can assume that ε is a root of unity of p-th power order. In the case δ(K) = 0 this implies a_{𝔭ε} = 0 and therefore α_{𝔭} = 0. In the case δ(K) = 1 we let the product in (10) run through S^{∗} = S − {𝔭_{∘}}. This means a_{𝔭_{∘}ε} = 0 and ε = 1, hence α_{𝔭} = 0. This proves Proposition 0.2.

5. Now we go over to the consideration of Ker T_{2}.

Proposition 0.3. Ker T_{2} = Ker ψ_{2}.

Proof. Since ℭ̂_{S} ≅ G_{S}/G_{S}^{(2)}, the assertion is equivalent to Ker T′_{2} = Ker ψ_{2}, where T′_{2} denotes the natural map

T′_{2} : L_{2}(F) ⟶ L_{2}(G_{S}/G_{S}^{(2)}).

It follows immediately from Proposition 1 that Ker ψ_{2} is contained in Ker T′_{2}. On the other hand ψ_{2} factors through T′_{2} and therefore Ker T′_{2} ⊆ Ker ψ_{2}.

6. Now it is easy to prove Theorem 1. We have a triangle of surjective maps

L_{2}(F)

↙ T_{2} ↘ ψ_{2}

L_{2}(ℭ̂_{S}) ⟶^{ϕ_{2}} L_{2}(G_{S}).

Since Ker T_{2} = Ker ψ_{2} the map ϕ_{2} is a isomorphism. □

7. Now we study the arithmetic properties of ϕ_{2}. Let 𝔭 be a place of K and 𝔓 an extension of 𝔭 to the maximal p-extension. We denote by D_{𝔓} the decomposition group of 𝔓 in G_{S}.

Proposition 0.4. If the natural map K̂_{𝔭}^{×} → ℭ̂_{s} is an injection, then

D_{𝔓} ∩ G_{S}^{(2)} = D_{𝔓}^{(2)}.  (12)

Proof. Let G_{𝔭} be the Galois group of the maximal p-extension of K_{𝔭}. Then by our assumption the map

G_{𝔭}/G_{𝔭}^{(2)} ⟶ G_{S}/G_{S}^{(2)}

is injective. Since the image of the restriction map ψ_{𝔭} = G_{𝔭} → G_{S} is D_{𝔓}, (12) follows immediately.

Remark 0.5. The assumption that K̂_{𝔭}^{×} → ℭ̂_{S} is injective is fulfilled if \overline{K} = K_{S} is the maximal p-extension.

The decomposition group of 𝔓 in G_{S}^{(n)}/G_{S}^{(n+1)} is (D_{𝔓} ∩ G_{S}^{(n)})G_{S}^{(n+1)}/G_{S}^{(n+1)}.

Proposition 0.6. The decomposition group of 𝔓 in G_{S}^{(n)}/G_{S}^{(n+1)} is independent of the choice of the extension 𝔓 of 𝔭.

Proof. Let 𝔓′ be any extension of 𝔭 to K_{S}. Then there is an ξ ∈ G_{S} such that 𝔓′ = ξ𝔓 and D_{𝔓′} = ξD_{𝔓}ξ^{−1}.

If η ∈ D_{𝔓} ∩ G_{S}^{(n)}, then ξηξ^{−1} ∈ D′_{𝔓} ∩ G_{S}^{(n)} and

ξηξ^{−1} ≡ η(mod G_{S}^{(n+1)}).

We denote the decomposition group of 𝔓 in L_{n}(G_{S}) by L_{n𝔭}(G_{S}).

To the map K_{𝔭}^{×} → ℭ̂_{S} there corresponds a map Λ_{𝔭} : L_{2}(K_{𝔭}^{×}) → L_{2}(ℭ̂_{S}).

Theorem 0.7. If the natural map K̂_{𝔭}^{×} → ℭ̂_{S} is an injection, then

ϕ_{2}Λ_{𝔭}L_{2}(K̂_{𝔭}^{×}) = L_{2𝔭}(G_{S}).

Proof. It is clear that the map ϕ_{2}Λ_{𝔭} maps L_{2}(K_{𝔭}^{×}) into L_{2𝔭}(G_{S}). By Proposition 3 we have

L_{2𝔭}(G_{S}) = D_{𝔓}^{(2)}G_{S}^{(3)}/G_{S}^{(3)}.

On the other hand we see from the commutative diagram

L_{2}(K_{𝔭}^{×}) ⟶^{Λ_{𝔭}} L_{2}(ℭ̂_{S})
↓  ↓
L_{2}(G_{𝔭}) ⟶ L_{2}(G_{S})

that

ϕ_{2}Λ_{𝔭}L_{2}(K_{𝔭}^{×}) = D_{𝔓}^{(2)}G_{S}^{(3)}/G_{S}^{(3)}.

References

[Ko1970] H. Koch, Galoissche Theorie der p-Erweiterungen, VEB Deutscher Verlag der Wissenschaften, Berlin

[KoKuLa] H. Koch, S. Kukkuk, J. Labute, Local nilpotent class field theory, to appear in Acta Arithmetica

[Zi1982] E.-W. Zink, Lokale projektive Klassenkörpertheorie, Grundbegriffe und erste Resultate, AdW der DDR, R-Math-01/82, Berlin

[Zi1983] E.-W. Zink, Lokale projektive Klassenkörpertheorie, II, Math. Nachr. 114, 123-150

Max-Planck-Institut für Mathematik, Bonn, Gottfried-Claren-Str. 26, D-53225 Bonn; Humboldt-Universität zu Berlin, Institut für Mathematik, Lehrstuhl Zahlentheorie, Jägerstr. 10-11, D-10 117 Berlin, Germany; Department of Mathematics, University of Sydney, NSW 206, Australia

  E-mail address: abramov@mathematik.hu-berlin.de; koch@mathematik.hu-berlin.de;
lai-k@maths.su.oz.au