Galois Groups of Local and Global Type

by Kay Wingberg at Heidelberg

In this article we are considering the structure of the Galois group

G_{S} = G(k_{S}(p)|k)

of the maximal p-extension k_{S}(p) of an algebraic number field k which is unramified outside a finite set S of primes of k. Here p is a prime number and we assume that S contains the set Σ = S_{∞} ∪ S_{p} of archimedean primes and primes above p. More precisely, we are interested in the arithmetical question about the relation between the group G_{S} and its decomposition groups

G_{𝔭} = G((k_{S}(p))_{𝔭}|k_{𝔭}) ^{⊆} G_{S}

with respect to the non-archimedean primes 𝔭 ^{∈} S. ∗^{)} If G_{𝔭} = G(k_{𝔭}(p)|k_{𝔭}) denotes the Galois group of the maximal p-extension k_{𝔭}(p) of the local field k_{𝔭}, then G_{𝔭} is obviously a quotient of G_{𝔭}. In some (rare) cases it can happen that there exists a prime 𝔭 ^{∈} S such that G_{𝔭} is almost equal to G_{S}.

Definition. G_{S} is of local type, if there exists a prime 𝔭 ^{∈} S such that G_{𝔭} = G_{S}. G_{S} is potentially of local type, if an open subgroup of G_{S} is of local type, and otherwise G_{S} is called of global type. Furthermore, we say that G_{S} is (potentially) of maximal local type, if G_{S} is (potentially) of local type with respect to some prime 𝔭 ^{∈} S and the natural map G_{𝔭} ↠ G_{𝔭} is bijective.

In the first paragraph we will characterize the cases where G_{S} is of (maximal) local type in terms of invariants of the base field k. In the following sections we deal with the question whether the group G_{S} is a free pro-p-group or a Demuškin group, i.e. a Poincaré group of dimension 2. If k contains the group μ_{p} of p-th roots of unity, we will see that this only can happen if G_{S} is potentially of local type. In contrast to this, if μ_{p} ^{⊆} / k, then there are number fields k such that G_{Σ} is a free pro-p-group of global type. But it is not known whether there are Galois groups of global type being Demuškin groups.

∗_{)}Actually one should denote this group by G_{𝔓} where 𝔓 is an extension of 𝔭 to k_{S}(p).

1 The group G_{S}

Let p be a prime number and let μ_{p} be the group of p -th roots of unity. For an algebraic number field k and a finite set S of primes of k containing Σ = S_{∞} ∪ S_{p} we introduce the following notations:

S^{f} the set of finite primes in S, S_{IR}, S_{C} the set of real and complex primes of k, r_{1}, r_{2} the number of real and complex primes of k, n_{𝔭} the local degree [k_{𝔭} : Q_{ℓ}] of a non-archimedean prime 𝔭|ℓ of k, Cl_{S}(k) the S-ideal class group of k, C_{S} the S-idèle class group C_{S}(k_{S}(p)), C_{S^{f}} the S^{f}-idèle class group of k_{S}(p), G_{S} the Galois group G(k_{S}(p)|k) of the maximal p-extension k_{S}(p) of k

which is unramified outside S, G_{𝔭} the decomposition group of G_{S} with respect to the prime 𝔭, T_{𝔭} the inertia subgroup of G_{𝔭}, G_{𝔭} the full local group G(k_{𝔭}(p)|k_{𝔭}), T_{𝔭} the inertia subgroup of G_{𝔭}, tor_{p}A the p-torsion subgroup of an abelian group A.

Furthermore we set

δ = \{ \begin{matrix}1, & μ_{p} ^{⊆} k, \\ 0, & μ_{p} ^{⊆} / k \end{matrix} and δ_{𝔭} = \{ \begin{matrix}1, & μ_{p} ^{⊆} k_{𝔭}, \\ 0, & μ_{p} ^{⊆} / k_{𝔭}, \end{matrix}

where 𝔭 is a prime of k. Assume that we are given a subset S_{0} ^{⊆} S, then we put

V_{S_{0}}^{S} = {a ^{∈} k^{×} | a ^{∈} k_{𝔭}^{×p} for 𝔭 ^{∈} S_{0}; a ^{∈} U_{𝔭}k_{𝔭}^{×p} for 𝔭 ^{∈ /} S}/k^{×p},

where U_{𝔭} is the unit group of the local field k_{𝔭} (by convention U_{𝔭} = k_{𝔭}^{×} if 𝔭 is archimedean). The dual group is denoted by � _{S_{0}}^{S} = Hom(V_{S_{0}}^{S}, ZZ/pZZ).

Observe that we have canonical inclusions

V_{S}^{S} ^{⊆} V_{S_{0}}^{S_{0}} ^{⊆} V_{S_{0}}^{S}

and that

V_{S_{0}}^{S} = ker (H^{1}(k_{S}|k, μ_{p}) ⟶ ∏_{𝔭^{∈}S_{0}} H^{1}(k_{𝔭}, μ_{p})).

where k_{S} is the maximal extension of k which is unramified outside S.

First we will see that the question whether G_{S} is of local or global type depends (i.g.) only on the decomposition groups G_{𝔭} for 𝔭|p.

Proposition 1.1. For 𝔭_{0} ^{∈} S ∖ S_{p} the canonical homomorphism

G_{𝔭_{0}} ⟶ G_{S}

is injective. The image is not open, i.e. of infinite index in G_{S}, except in the following cases: k is totally real, p is odd, ∑_{S^{f} \{𝔭_{0}}} δ_{𝔭} = 0, � _{Σ}^{Σ} = 0 and

either δ_{𝔭_{0}} = 0 (then G_{S} ≅ ZZ_{p}) or δ_{𝔭_{0}} = 1 and 𝔭_{0} does not split in the cyclotomic ZZ_{p}-extension k_{∞} of k

(then G_{𝔭_{0}} = G_{S} is a Demuškin group of rank 2).

Proof: By [1] th. 10.6.1 the map is injective and if k is not totally real, then the image is of infinite index in G_{S}. If p = 2, then G_{S}(k) is potentially of local type with respect to 𝔭_{0} if and only if G_{S}(k(i)) has this property. But this is impossible by the cited theorem. Now let p be odd and k totally real.

If δ_{𝔭_{0}} = 0, then G_{𝔭_{0}} ≅ ZZ_{p} is open in G_{S} if and only if G_{S} = G(k_{∞}|k). But the last assertion is equivalent to ∑_{S^{f}} δ_{𝔭} = 0 and � _{Σ}^{Σ} = 0.

If δ_{𝔭_{0}} = 1, then G_{𝔭_{0}} is a Demuškin group of rank 2 and this group is open in G_{S} if and only if G_{S} has the same structure. Assuming this, it follows that G_{Σ} is isomorphic to ZZ_{p}, thus ∑_{S_{p}} δ_{𝔭} = 0, � _{Σ}^{Σ} = 0 and G_{Σ} = G(k_{∞}|k). From [1] th. 10.5.1 we obtain the exact sequence

1 ⟶ ∗_{𝔭^{∈}S^{f} (k_{∞})} T_{𝔭} ⟶ G_{S} ⟶ G_{Σ} ⟶ 1 .

It follows that ∑_{S^{f} \{𝔭_{0}}} δ_{𝔭} = 0 and 𝔭_{0} does not split in k_{∞}|k.

Conversely, from the conditions given in the theorem it follows that G_{Σ} = G(k_{∞}|k) and that we have the exact sequence

1 ⟶ T_{𝔭_{0}} ⟶ G_{S} ⟶ G(k_{∞}|k) ⟶ 1 ,

which shows that G_{S} is equal to the Demuškin group G_{𝔭_{0}}, since 𝔭_{0} does not split in k_{∞}|k. This completes the proof of the theorem. □

Remark: k = Q, p = 3, S = {3, 7, ∞} gives an example for the exceptional case with δ_{7} = 1. In this situation we have G_{S} = G_{7}.

Furthermore we want to mention that in the case p ≠ 2 and k totally imaginary G_{S} is potentially of local type if and only if it is of local type. More generally we have by [1] th. 10.6.2

Proposition 1.2. Let k be totally imaginary and let 𝔭 ^{∈} S. Suppose that G_{𝔭} is open in G_{S}. Then 𝔭 ^{∈} S_{p} and either

G_{𝔭} = G_{S} or p = 2 and (G_{S} : G_{𝔭}) = 2, #S^{f}(k) = 1, S^{f}(k_{S}(2)) = 2.

If the base field k contains the group μ_{p}, then we have the following assertion concerning the structure of G_{S}, see [2] or [1] ch.X §7 and ch.III §4 for the notion of a (virtual) duality group.

Theorem 1.3. If μ_{p} ^{⊆} k, then the group G_{S} is of one of the following forms.

(i) If � _{{𝔭_{0}}}^{S} = 0 for a prime 𝔭_{0} ^{∈} S^{f}, then G_{S} is of local type and it exists a finite set of primes T ^{⊇} S, such that the canonical homomorphism

∗_{𝔭^{∈}S\{𝔭_{0}}} G_{𝔭} ∗ ∗_{𝔭^{∈}T \S} G_{𝔭}/T_{𝔭} ⟶ G_{S}

is an isomorphism. (ii) Assume � _{{𝔭}}^{S} ≠ 0 for all primes 𝔭 ^{∈} S^{f}.

If p ≠ 2, then G_{S} is of global type and it is a duality group of dimension 2 with dualizing module tor_{p}C_{S^{f}}. For p = 2 the following cases occur: If #S^{f}(k_{S}(2)) > 2, then G_{S} is of global type and it is a virtual duality group of dimension 2 with dualizing module tor_{2}C_{S^{f}}. If #S^{f}(k_{S}(2)) = 2, then G_{S} is potentially of maximal local type and it is a virtual Poincaré group of dimension 2 with dualizing module μ_{2^{∞}}.

Remarks: 1. In (i) the prime 𝔭_{0} is unique but the set T is not.

2. We use the opportunity here to mention that the corollary to theorem A in [2] is only correct if we assume that μ_{2p} ^{⊆} k (see below). Furthermore, there is a mistake in the subsequent paper to [2] which appeared in J. reine u. angew. Math. 416 (1991). The question whether G_{S} is a duality group in the case that μ_{p} ^{⊆} / k remains open.

The theorem above gives us a complete characterization for G_{S} to be (potentially) of local or global type if μ_{p} ^{⊆} k. Without this assumption we at least can describe the situation when G_{S} is a free product of (full) decomposition groups and a free group (see [2] th. 6).

Theorem 1.4. For a subset S_{0} ^{⊆} S^{f} the following assertions are equivalent.

(i) It exists a finite set of primes T ^{⊇} S, such that the canonical homomorphism

∗_{𝔭^{∈}S\S_{0}} G_{𝔭} ∗ ∗_{𝔭^{∈}T \S} G_{𝔭}/T_{𝔭} ⟶ G_{S}

is an isomorphism.

(ii) � _{S_{0}}^{S} = 0 and ∑_{𝔭^{∈}S_{0}} δ_{𝔭} = δ.

If μ_{p} ^{⊆} k, then (i) and (ii) are equivalent to

(ii)’ S_{0} = {𝔭_{0}} and G_{𝔭_{0}} = G_{S}.

Furthermore, if (i) and (ii) hold, then

#(T \S) = 1 + ∑_{𝔭^{∈}S_{0}∩S_{p}}[k_{𝔭} : Q_{p}] − #(S\S_{0}).

Corollary 1.5. The group G_{S} is of maximal local type if and only if k is totally imaginary if p = 2 and there exists a prime 𝔭_{0} ^{∈} S^{f} such that

∑_{S^{f} \\{𝔭_{0}\}} δ_{𝔭} = δ , � _{S\\{𝔭_{0}\}}^{S} = 0 and r_{2} = \{ \begin{matrix}n_{𝔭_{0}}, & 𝔭_{0} | p , \\ 0, & 𝔭_{0} ∤ p . \end{matrix}

Proof: We have G_{𝔭_{0}} →^{∼} G_{S} if and only if in (i) of the theorem above S\S_{0} = {𝔭_{0}} and #(T \S) = 0. Thus S_{0} = S^{f}\{𝔭_{0}} and k has to be totally imaginary if p = 2. Furthermore, the assertion #(T \S) = 0 is now equivalent to

#(S\S_{0}) = r_{1} + r_{2} + 1 = 1 + ∑_{𝔭^{∈}S_{p}\\{𝔭_{0}\}}n_{𝔭} = 1 + r_{1} + 2r_{2} − \{ \begin{matrix}n_{𝔭_{0}}, & 𝔭_{0} | p , \\ 0, & 𝔭_{0} ∤ p . \end{matrix}

Finally, for p ≠ 2 or k totally imaginary we have � _{S^{f} \{𝔭_{0}}}^{S} = � _{S\{𝔭_{0}}}^{S}. □

Now we give a characterization for G_{S} being of local type.

Theorem 1.6. The group G_{S} is of local type if and only if there exists a prime 𝔭_{0} ^{∈} S^{f} such that

∑_{S^{f} \\{𝔭_{0}\}} δ_{𝔭} − δ + dim_{IF_{p}} � _{S\\{𝔭_{0}\}}^{S} + r_{2} = \{ \begin{matrix}n_{𝔭_{0}}, & 𝔭_{0} | p , \\ 0, & 𝔭_{0} ∤ p . \end{matrix}

Proof: Let 𝔭_{0} be a prime of S^{f}. Consider the following commutative and exact diagram

� � � �  _{res} � � � � � �  _{res_{_{𝔭}_{0}}} � � � � � � � �  0 0  _{� {S \atop S\\{𝔭_{0}\}}} H^{1}(G_{𝔭_{0}}) H^{1}(G_{S}) ker(res_{𝔭_{0}}) 0  0_{� S S} H^{1}(k_{S}|k, μ_{p})^{∗} ∏_{S} H^{1}(G_{𝔭}) H^{1}(G_{S}) (Cl_{S}/p)^{∗} 0  ∏_{S\\{𝔭_{0}\}} H^{1}(k_{𝔭}, μ_{p})^{∗} ∏_{S\\{𝔭_{0}\}} H^{1}(G_{𝔭}) 0

where the missing coefficients of the cohomology groups are ZZ/pZZ and the horizontal sequence in the middle is obtained from the global duality theorem of Tate/Poitou, see [1] ch.VIII §7. The snake lemma implies the exact sequences

(Cl_{S}/p)^{∗} ↪ ker(res_{𝔭_{0}}) → ∏_{S\{𝔭_{0}}} H^{1}(G_{𝔭}) → coker(res ) ↠ coker(res_{𝔭_{0}}) ,

0 ⟶ coker(res ) → H^{1}(k_{S}|k, μ_{p})^{∗} → � _{S}^{S} ⟶ 0 ,

0 ⟶ coker(res_{𝔭_{0}}) ⟶ � _{S\{𝔭_{0}}}^{S} ⟶ � _{S}^{S} ⟶ 0 .

Since

dim_{IF_{p}} H^{1}(k_{S}|k, μ_{p}) = dim_{IF_{p}} O_{S}^{×}/p + dim_{IF_{p}} _{p}Cl_{S} = #S − 1 + δ + dim_{IF_{p}} Cl_{S}/p

and

dim_{IF_{p}} H^{1}(G_{𝔭}) = 1 + δ_{𝔭} + \{ \begin{matrix}n_{𝔭}, & 𝔭 | p , \\ 0, & 𝔭 ∤ p , \end{matrix}

we obtain counting dimensions

dim_{IF_{p}} ker(res_{𝔭_{0}}) = ∑_{S\\{𝔭_{0}\}} δ_{𝔭} − δ + dim_{IF_{p}} � _{S\\{𝔭_{0}\}}^{S} − \{ \begin{matrix}n_{𝔭_{0}}, & 𝔭_{0} | p , \\ 0, & 𝔭_{0} ∤ p . \end{matrix}

Now G_{S} is of local type if and only if there exists a prime 𝔭_{0} ^{∈} S^{f} such that ker(res_{𝔭_{0}}) = 0. This proves the theorem. □

If k is not totally real and G_{S} is of global type, then we will see that the decomposition groups G_{𝔭}(K), 𝔭 ^{∈} S(K), can not generate the whole group G_{S}(K) for every finite extension K|k inside k_{S}(p). Even more is true:

Theorem 1.7. Assume that k is not totally real and that G_{S} is of global type. If d ^{∈} IN is given, then there exists a finite Galois extension K of k inside k_{S}(p) such that

dim_{IF_{p}} Cl_{S}(K)/p ^{≥} d .

Proof: Let K|k be a finite Galois extension inside k_{S}(p). We put G = G_{S}(K) and G_{𝔭} = G_{𝔭}(K) for short and denote the Frattini subgroup of G by G^{∗}. From the Hochschild-Serre spectral sequence

0 ⟶ H^{1}(G/G^{∗}) ⟶^{∼} H^{1}(G) ⟶ H^{1}(G^{∗})^{G} ⟶ H^{2}(G/G^{∗}) ⟶ H^{2}(G)

with coefficients ZZ/pZZ we obtain with h^{i} = dim_{IF_{p}} H^{i}(G)

dim_{IF_{p}} H^{1}(G^{∗})^{G} ^{≥} dim_{IF_{p}} H^{2}(G/G^{∗}) − dim_{IF_{p}} H^{2}(G)
= \frac{1}{2}h^{1}(h^{1} + 1) − h^{2}
≥ \frac{1}{2}h^{1}(h^{1} − 1)
≥ \frac{1}{2}(r_{2})^{2}

where we used the Künneth formula for the p-elementary abelian group G/G^{∗}. Furthermore we have the exact sequence

0 ⟶ Hom(Cl_{S}(K′), ZZ/pZZ)^{G} ⟶ H^{1}(G^{∗})^{G} ⟶ ∏_{𝔭^{∈}S(K)} H^{1}(G^{∗} ∩ G_{𝔭})^{G_{𝔭}}

where K′ is the fixed field of G^{∗}. Obviously

dim_{IF_{p}} H^{1}(G^{∗} ∩ G_{𝔭})^{G_{𝔭}} = 1 + δ_{𝔭}

for the finite primes 𝔭 ∤ p. In order to calculate the dimensions of the local terms for 𝔭|p we consider the exact sequence

0 ⟶ H^{1}((G^{∗} ∩ G_{𝔭})/G_{𝔭}^{∗}) ⟶ H^{1}(G^{∗} ∩ G_{𝔭})^{G_{𝔭}} ⟶ H^{1}(G_{𝔭}^{∗})^{G_{𝔭}}

which gives us with h_{𝔭} = dim_{IF_{p}} H^{1}(G_{𝔭}) ^{≤} n_{𝔭} + 2

dim_{IF_{p}} H^{1}(G^{∗} ∩ G_{𝔭})^{G_{𝔭}}  ^{≤} dim_{IF_{p}} H^{1}((G^{∗} ∩ G_{𝔭})/G_{𝔭}^{∗}) + dim_{IF_{p}} H^{1}(G_{𝔭}^{∗})^{G_{𝔭}}
≤ dim_{IF_{p}} H^{1}(G_{𝔭}/G_{𝔭}^{∗}) + dim_{IF_{p}} H^{1}(G_{𝔭}^{∗})^{G_{𝔭}}
≤ dim_{IF_{p}} H^{1}(G_{𝔭}) + dim_{IF_{p}} H^{2}(G_{𝔭}/G_{𝔭}^{∗})
= h_{𝔭} + \frac{1}{2}h_{𝔭}(h_{𝔭} + 1)
≤ \frac{1}{2}(n_{𝔭} + 2)(n_{𝔭} + 5) .

It follows that

dim_{IF_{p}} Cl_{S}(K′)/p ^{≥} dim_{IF_{p}} Hom(Cl_{S}(K′), ZZ/pZZ)^{G}
≥ \frac{1}{2}(r_{2}(K))^{2} − ∑_{𝔭^{∈}S^{f} \S_{p}(K)} (1+δ_{𝔭}) − ∑_{𝔭^{∈}S_{p}(K)} \frac{1}{2}(n_{𝔭}(K)+2)(n_{𝔭}(K)+5)
≥ \frac{1}{2}(r_{2}(K))^{2} − ∑_{𝔭^{∈}S_{p}(K)} \frac{1}{2}(n_{𝔭}(K))^{2} − c(K)

where c(K) = ∑_{S^{f} \S_{p}(K)}(1 + δ_{𝔭}) + ∑_{S_{p}(K)} \frac{7}{2}n_{𝔭}(K) + 5. If L|K is a finite extension inside k_{S}(p), then c(L) ^{≤} [L : K]c(K). Therefore, since k is not totally real, there exists a finite extension K|k inside k_{S}(p) such that for the maximal p-elementary abelian extension K′ of K inside k_{S}(p) the inequality

dim_{IF_{p}} Cl_{S}(K′)/p ^{≥} \frac{1}{4}(r_{2}(K))^{2} − ∑_{𝔭^{∈}S_{p}(K)} \frac{1}{2}(n_{𝔭}(K))^{2} .

holds (and we have this inequality for all finite extensions of K inside k_{S}(p)). Since G_{S} is of global type, the decomposition groups G_{𝔭} for 𝔭|p are of infinite index in G_{S}. Therefore, for every m ^{∈} IN and every prime 𝔭|p of K there exists an extension inside k_{S}(p) in which 𝔭 decomposes in at least m primes. The normal closure L of the compositum of all these fields has the property that every prime 𝔭|p of K has at least m extensions. It follows for the maximal p-elementary extension L′ of L inside k_{S}(p) that

dim_{IF_{p}} Cl_{S}(L′)/p ^{≥} \frac{1}{4}(r_{2}(L))^{2} − ∑_{𝔭^{∈}S_{p}(L)} \frac{1}{2}(n_{𝔭}(L))^{2}
= \frac{[L:K]^{2}}{4} (r_{2}(K))^{2} − \frac{1}{2} ∑_{𝔭^{∈}S_{p}(K)} ∑_{𝔓|𝔭}[L_{𝔓} : K_{𝔭}]^{2}(n_{𝔭}(K))^{2}
≥ \frac{[L:K]^{2}}{4} (r_{2}(K))^{2} − \frac{1}{2} ∑_{𝔭^{∈}S_{p}(K)} \frac{[L:K]^{2}}{m} (n_{𝔭}(K))^{2}
= \frac{[L:K]^{2}}{m^{2}} · (\frac{m^{2}}{4} (r_{2}(K))^{2} − \frac{m}{2} ∑_{𝔭^{∈}S_{p}(K)}(n_{𝔭}(K))^{2}) .

Since [L : K] ^{≥} m and m can be chosen as large as one wants, the p-rank of the S-ideal class group of L′ = L′(m) exceeds every given bound. □

Together with proposition 1.2 the theorem above implies the

Corollary 1.8. Assume that p ≠ 2 and k is totally imaginary. Then G_{S} is of global type if and only if there exists a finite Galois extension K of k inside k_{S}(p) such that Cl_{S}(K)(p) ≠ 0.

We do not know whether this is also true for number fields having real primes.

2 Free pro-p-groups

In this section we consider the case where G_{S} is a free pro-p-group. The following proposition is well known, see [1] th. 8.8.10:

Proposition 2.1. G_{S} is free if and only if k is totally imaginary if p = 2 and

∑_{𝔭^{∈}S^{f}} δ_{𝔭} = δ and � _{S}^{S} = 0 .

Therefore we assume in the following that k is totally imaginary if p = 2. Two cases occur

Corollary 2.2.

(i) Assume there exists a prime 𝔭 ^{∈} S^{f} such that μ_{p} ^{⊆} k_{𝔭}. Then G_{S} is free if and only if μ_{p} ^{⊆} k, S^{f} = S_{p} = {𝔭} and B_{S}^{S} = 0 (hence Cl_{S}(p) = 0). In particular, G_{S} is of local type in this case.

(ii) Assume that μ_{p} ^{⊆} / k_{𝔭} for all 𝔭 ^{∈} S^{f}. Then G_{S} is free if and only if B_{S}^{S} = 0.

In the following we will show that in case (ii) of the corollary above the group G_{S} can be of local or of global type. Let k = Q and p ≠ 2. Then G_{p} surjects onto G_{Σ} ≅ ZZ_{p}. Hence this group is of local type. It is more difficult to find an example where G_{Σ} is free and of global type. We need the following

Lemma 2.3. Let p be an odd prime number, k a number field and let S = {𝔩}∪Σ, where 𝔩 is a non-archimedean prime of k not above p. Assume that

μ_{p} ^{⊆} / k_{𝔭} for all 𝔭|p, μ_{p} ^{⊆} k_{𝔩} and � _{Σ}^{S} = 0 .

Then there is an isomorphism

G_{𝔩}(k) ∗ F_{r_{2}} ⟶^{∼} G_{S}(k) ,

where F_{r_{2}} is a free pro-p-group of rank r_{2} = r_{2}(k). If K|k is a Galois extension inside k_{S}(p) of degree p which is ramified at 𝔩, then

G_{𝔩}(K) ∗ F_{p r_{2}} ⟶^{∼} G_{S}(K)

and

G_{Σ}(K) ⟶^{∼} F_{p r_{2}+1} .

If in addition K is totally imaginary and Cl_{Σ}(K)(p) ≠ 0, then G_{Σ}(K) is free and of global type.

Proof: Using theorem 1.4 the assumptions imply the statement for G_{S}(k). The subgroup theorem for free pro-p-product, see [1] th. 4.2.1, gives us the second assertion and the third follows by definition of the group G_{Σ}(K). Finally, if Cl_{Σ}(K)(p) ≠ 0, then the decomposition groups of G_{Σ}(K) with respect to the primes above p have an index greater than 2. Hence G_{Σ}(K) is of global type by proposition 1.2. □

Now we give an example of a number field for which we can apply lemma 2.3. Let k_{0} = Q(i) and let p and ℓ be odd prime numbers such that p | ℓ − 1. Let k be the unique subfield of k_{0}(ζ_{ℓ}) of degree p over k_{0}. By k_{0}^{(1)} and k^{(1)} we denote the first layers of the cyclotomic ZZ_{p}-extensions of k_{0} and k, respectively. We have the following diagram of fields:

� � � �_{p} � � �  _{p} � k_{0}  k  k_{0}(ζ_{ℓ}) K k_{0}^{(1)}  k^{(1)} K_{Σ}(p)

where K is some extension of k_{0} inside k^{(1)} of degree p and different to k_{0}^{(1)} and k. Then K is totally imaginary and has the property that μ_{p} ^{⊆} / K_{𝔭} for 𝔭|p and μ_{p} ^{⊆} K_{ℓ}. Now we assume in addition that

p^{2} ∤ ℓ − 1, ℓ is inert in k_{0}, i.e. ℓ ≡ 3 mod 4, and p^{\frac{ℓ−1}{p}} ≡ 1 mod ℓ .

The prime numbers p = 5 and ℓ = 31 give an example for this situation. We will show that

Cl_{Σ}(K)(p) ≠ 0 and � _{Σ}^{S}(K) = 0 .

In order to show the first assertion we first observe that the extension k^{(1)}|K is at most ramified at the primes above p and above ℓ. Since ℓ is ramified in k|k_{0} and unramified in k_{0}^{(1)}|k_{0}, the field k_{0}^{(1)} is the inertia field for the extension k^{(1)}|k_{0} with respect to ℓ. Hence k^{(1)}|K is unramified at ℓ. With the same argumentation one sees that this extension is unramified at S_{p}, too: The extension k^{(1)}|Q is abelian and k is the inertia field of the extension k^{(1)}|k_{0} with respect to the primes above p. From the assumption that p^{\frac{ℓ−1}{p}} ≡ 1 mod ℓ follows that p decomposes in in the extension Q(ζ_{ℓ})|Q in d primes with p|d. Hence the primes above p decompose in the extension k|k_{0} and therefore in the extension k^{(1)}|K, too. Thus k^{(1)}|K is unramified and split at S_{p}. From class field theory follows that Cl_{Σ}(K)(p) ≠ 0. In order to prove the second assertion we first show that V_{Σ}^{S}(k_{0}) = 0. Since Cl(k_{0}) = 0 and the 4-th roots of unity form the group of units of k_{0}, an element of V_{Σ}^{S}(k_{0}) is of the form x = ℓ^{a} k_{0}^{×p}, a ^{∈} ZZ. Since p^{2} ∤ ℓ − 1, the element ℓ is not a p-power in (k_{0})_{𝔭} for 𝔭|p. Hence p|a and therefore V_{Σ}^{S}(k_{0}) = 0. Finally, with G = G(K|k_{0}) the commutative and exact diagram

� � � � � � � � �  ( ∏_{𝔭^{∈}Σ(K)} H^{1}(G_{𝔭}(K), μ_{p}))^{G} H^{1}(G_{S}(K), μ_{p})^{G} (V_{Σ}^{S}(K))^{G} 0  ∏_{𝔭^{∈}Σ(k_{0})} H^{1}(G_{𝔭}(k_{0}), μ_{p}) H^{1}(G_{S}(k_{0}), μ_{p}) V_{Σ}^{S}(k_{0}) 0

shows that (V_{Σ}^{S}(K))^{G} ≅ V_{Σ}^{S}(k_{0}) = 0, hence V_{Σ}^{S}(K) = 0 and therefore � _{Σ}^{S}(K) = 0.

3 Demuškin groups

In this paragraph we want to classify number fields k and sets of primes S (containing Σ) such that G_{S} is a Demuškin group.

Proposition 3.1. If G_{S} is a Demuškin group, then either

� _{S}^{S} = 0 and ∑_{𝔭^{∈}S\S_{C}} δ_{𝔭} − δ = 1 ,

or

dim_{IF_{p}} � _{S}^{S} = 1 and ∑_{𝔭^{∈}S\S_{C}} δ_{𝔭} − δ = 0 .

Proof: This is clear, since a Demuškin group is a one relator group, i.e. dim_{IF_{p}} H^{2}(G_{S}, ZZ/pZZ) = 1, see [1] cor. 8.8.9. □

By the proposition above k must be totally imaginary if p = 2 and G_{S} is a Demuškin group, since every open subgroup of G_{S} is again a Demuškin group. We will assume this in the following.

Theorem 3.2. Let p ≠ 2 and assume that μ_{p} ^{⊆} k. Then the following assertions are equivalent:

(i) G_{S} is a Demuškin group. (ii) G_{S} is a Demuškin group of maximal local type. (iii) S = Σ = S_{∞} ∪ {𝔭_{1}, 𝔭_{2}}, � _{{𝔭_{1}}}^{S} = 0 and r_{2} = n_{𝔭_{1}}.

(iv) #S^{f}(k_{S}(p)) = 2.

Let p = 2 and assume that k is totally imaginary. Then the following assertions are equivalent:

(i) G_{S} is a Demuškin group.

(ii) G_{S} is a Demuškin group potentially of maximal local type. (iii) S = Σ = S_{∞} ∪ {𝔭_{1}, 𝔭_{2}}, � _{{𝔭_{1}}}^{S} = 0 and r_{2} = n_{𝔭_{1}} or

S = Σ = S_{∞} ∪ {𝔭} and there exists a quadratic extension k′|k in which 𝔭 splits, say 𝔭 = 𝔭_{1}𝔭_{2} such that � _{{𝔭_{1}}}^{S}(k′) = 0.

(iv) #S^{f}(k_{S}(2)) = 2.

Proof: The assertion (iv) obviously implies that

S = Σ = S_{∞} ∪ {𝔭_{1}, 𝔭_{2}} and � _{{𝔭_{1}}}^{S} = 0 = � _{{𝔭_{2}}}^{S} or (if p = 2)

S = Σ = S_{∞} ∪ {𝔭} and there exists a quadratic extension k′|k in which 𝔭 splits, say 𝔭 = 𝔭_{1}𝔭_{2} such that � _{{𝔭_{1}}}^{S}(k′)=0=� _{{𝔭_{2}}}^{S}(k′) ,

since μ_{p} ^{⊆} k and by Kummer theory � _{{𝔭}}^{S} is non-zero if and only if there exists a subextension of k in k_{S}(p), in which 𝔭 splits. Using theorem 1.4 these assertions imply for G_{S}(k) (resp. for G_{S}(k′) in the second case) free product decompositions

G_{𝔭_{1}} = G_{S} ≅ G_{𝔭_{2}} ∗ (other terms) and G_{𝔭_{2}} = G_{S} ≅ G_{𝔭_{1}} ∗ (other terms).

But

rk G_{𝔭_{i}} ^{≥} rk G_{𝔭_{i}}, i = 1, 2.

Hence the other terms are not present and G_{S} = G_{𝔭_{1}} = G_{𝔭_{2}} and r_{2} = n_{𝔭_{1}}. Thus (iv) implies (iii). Similar the assertion (iii) implies (ii): Using theorem 1.4 in the first case and the argument above in the second (if p = 2) we see that G_{S} is a Demuškin group of maximal local type in the first case and a virtual Demuškin group which is potentially of maximal local type in the second. But, since k is totally imaginary, G_{S} is torsion free and therefore it is itself a Demuškin group. Obviously (ii) implies (i). Now we assume that G_{S} is a Demuškin group. The fact that tor_{p}C_{S^{f}} is the dualizing module of G_{S}, see [1] lemma 10.7.10, which has to be isomorphic to Q_{p}/ZZ_{p} as abelian group, and the exact sequence

0 ⟶ μ_{p^{∞}} ⟶ ∏_{𝔭^{∈}S^{f}} Ind_{G_{𝔭}}^{G_{S}}μ_{p^{∞}} ⟶ tor_{p} C_{S^{f}} ⟶ 0 ,

loc.cit. prop. 10.2.1, imply that #S^{f}(k_{S}(p)) = 2. This finishes the proof of the theorem. □

The situation becomes not really more difficult if we drop the assumption that μ_{p} ^{⊆} k but assume that there exists a prime 𝔭_{0} ^{∈} S such that μ_{p} ^{⊆} k_{𝔭_{0}}.

Theorem 3.3. Assume that μ_{p} ^{⊆} / k and that there exists a prime 𝔭_{0} ^{∈} S such that μ_{p} ^{⊆} k_{𝔭_{0}}. Then the following assertions are equivalent:

(i) G_{S} is a Demuškin group. (ii) G_{S} is a Demuškin group of local type. (iii) μ_{p} ^{⊆} / k_{𝔭} for all primes 𝔭 in S_{0} := S^{f}\{𝔭_{0}},

� _{S}^{S} = 0 and r_{2} + dim_{IF_{p}} � _{S_{0}}^{S} = \{ \begin{matrix}n_{𝔭_{0}}, & 𝔭_{0} | p , \\ 0, & 𝔭_{0} ∤ p . \end{matrix}

Proof: Let G_{S} be a Demuškin group. Since μ_{p^{∞}}(k_{S}(p)) = 0, we have an injection

�  tor_{p} C_{S^{f}} Ind_{G_{𝔭_{0}}}^{G_{S}}μ_{p^{∞}}

where tor_{p} C_{S^{f}} is isomorphic to Q_{p}/ZZ_{p} as abelian group, we obtain G_{S} = G_{𝔭_{0}}, hence G_{S} is of local type. From (ii) follows, since G_{S} is a Demuškin group,

1 = dim_{IF_{p}} H^{2}(G_{S}, ZZ/pZZ) = dim_{IF_{p}} � _{S}^{S} + ∑_{𝔭^{∈}S_{0}} δ_{𝔭} + 1

hence � _{S}^{S} = 0 and μ_{p} ^{⊆} / k_{𝔭} for all primes 𝔭 ^{∈} S_{0}. Since G_{S} is of local type we conclude by theorem 1.6 the last condition of (iii). Now we assume (iii). Then G_{S} is a one relator group, i.e. dim_{IF_{p}} H^{2}(G_{S}, ZZ/pZZ) = 1. Furthermore, using again theorem 1.6, we obtain that G_{S} is of local type, i.e. G_{S} = G_{𝔭_{0}}. Thus 𝔭_{0} does not split in k_{S}(p)|k and therefore

dim_{IF_{p}} H^{2}(G_{S}(K), ZZ/pZZ) = dim_{IF_{p}} � _{S}^{S}(K) + 1

for every finite Galois extension K|k inside k_{S}(p). Since � _{S}^{S}(K)^{G(K|k)} = � _{S}^{S}(k) as μ_{p} ^{⊆} / k, it follows that dim_{IF_{p}} H^{2}(G_{S}(K), ZZ/pZZ) = 1. Now we obtain (i) from from the following lemma. □

Lemma 3.4. Let G be a finitely generated pro-p-group of cohomological dimension equal to 2 such that

dim_{IF_{p}} H^{2}(N, ZZ/pZZ) = 1

for every normal open subgroup N of G. Then G is a Demuškin group.

For a proof see [1] cor. 3.7.3. As an application of the last two theorems we now consider totally real fields and CM-fields k|k^{+}, i.e. k is totally imaginary and a quadratic extension of its maximal totally real subfield k^{+}.

Theorem 3.5. Assume that k is totally real (and p ≠ 2). Then G_{S} is a Demuškin group if and only if one of the following assertions hold:

(i) There exists a exactly one prime 𝔭_{0} ^{∈} S^{f} such that μ_{p} ^{⊆} k_{𝔭_{0}}. If 𝔭_{0} ∤ p, then � _{S\{𝔭_{0}}}^{S} = 0. If 𝔭_{0}|p, then � _{Σ}^{Σ} = 0 and dim_{IF_{p}} � _{Σ\{𝔭_{0}}}^{Σ} = 1.

(ii) For all primes 𝔭 ^{∈} S^{f} hold μ_{p} ^{⊆} / k_{𝔭} and dim_{IF_{p}} � _{Σ}^{Σ}(k_{n}) = 1 for all n ^{∈} IN, where k_{n} is the n-th layer of the cyclotomic ZZ_{p}-extension k_{∞} of k.

Remarks: 1. In both cases G_{S} is not of global type, if we assume in the second case that the Greenberg conjecture holds, i.e. the maximal unramified abelian p-extension of the the cyclotomic ZZ_{p}-extension k_{∞} of k is finite.

2. In (ii) it is sufficient that dim_{IF_{p}} � _{Σ}^{Σ}(k_{n}) = 1 for one n which is large enough.

3. Equivalent to the statements in the theorem are the assertions that the Iwasawa λ-invariant of the ZZ_{p}[[G(k_{∞}|k)]]-module G(k_{S}|k_{∞})^{ab} is equal to 1 and its μ-invariant is zero.

Proof: If there exists a prime 𝔭_{0} ^{∈} S^{f} such that μ_{p} ^{⊆} k_{𝔭_{0}}, then it follows from theorem 3.3 that G_{S} is a Demuškin group (of local type) if and only if the conditions of (i) are fulfilled (see also proposition 1.1 for an equivalent description). Furthermore, in this case the dualizing module of G_{S} is equal to μ_{p^{∞}}, hence G(k_{∞}|k) acts non-trivially on G(k_{S}(p)|k_{∞}) ≅ ZZ_{p}(1) which shows the remark 3. If μ_{p} ^{⊆} / k_{𝔭} for all primes 𝔭 ^{∈} S^{f}, then G_{S} = G_{Σ} and

dim_{IF_{p}} H^{2}(G_{Σ}(k_{n}), ZZ/pZZ) = dim_{IF_{p}} � _{Σ}^{Σ}(k_{n}) .

Thus, if G_{S} is a Demuškin group, then condition (ii) holds. Conversely, from (ii) follows that dim_{IF_{p}} H^{1}(G_{Σ}(k_{n}), ZZ/pZZ) = 2 for all n ^{≥} 1, hence the group (G(k_{Σ}(p)|k_{∞})^{ab}/p)_{Γ_{n}} is generated by one element, where Γ_{n} = G(k_{∞}|k_{n}). Since G(k_{Σ}(p)|k_{∞})^{ab} is a ZZ_{p}[[Γ]]-torsion module, there exists an n such that Γ_{n} acts trivially on G(k_{Σ}(p)|k_{∞})^{ab}/p. Therefore this group is generated by one element and it follows that the same is true for the group G(k_{Σ}(p)|k_{∞}). (For the Iwasawa theory we used here, see [1] ch.XI §3.) Thus G_{Σ} is a Demuškin group, loc.cit. th. 3.7.4. Also in this case G_{Σ} is (potentially) of local type, if we assume that the Greenberg conjecture holds, because then G_{𝔭} is of finite index in G_{Σ}. □

Examples:

1. Case (i), 𝔭_{0} ∤ p: p = 3, k = Q, S = {3, 7, ∞}.

Then G_{S}(Q) = G_{7}.

𝔭_{0} | p: p = 3, k = Q(\sqrt{6}), S = \{3, ∞\}.

Then G_{S}(k) is a Demuškin group of rank 2.

2. Case (ii): p = 37, k = Q(ζ_{37} + ζ_{37}^{−1}), S = {3, ∞}.

Then G_{S}(k) is a Demuškin group of rank 2.

In the second example we have μ_{3} ^{⊆} k_{3}. The Iwasawa μ-invariant of G(k_{S}(3)|k_{∞})^{ab} is zero by the theorem of Ferrero and Washington and its λ-invariant is equal to the λ-invariant of the maximal abelian unramified 3-extension of k′_{∞}, where k′ = Q(\sqrt{−2}). But this invariant is equal to 1. Also in the third example the Iwasawa λ-invariant of G(k_{S}(3)|k_{∞})^{ab} is equal to 1 and its μ-invariant is zero.

Since G_{S} = G_{S′}, where S′ = {𝔭 ^{∈} S^{f}\S_{p} | μ_{p} ^{⊆} k_{𝔭}} ∪ Σ ^{⊆} S, we assume for the following theorem that S = S′.

Theorem 3.6. Let p ≠ 2 and let k|k^{+} be a CM-field. Assume that there exists a prime 𝔭_{0} ^{∈} S^{f} such that μ_{p} ^{⊆} k_{𝔭_{0}}. Let 𝔭̃_{0} be the underlying prime of 𝔭_{0} of k^{+}.

Then G_{S}(k) is a Demuškin group if and only if the following conditions are fulfilled:

1. #S^{f}(k) = 1 + δ and S^{f}(k^{+}) = {𝔭̃_{0}},

2. G_{S}(k^{+}) is a Demuškin group and μ_{p} ^{⊆} k+ ; 𝔭̃_{0} ,

3. Cl_{S}(k)^{−} = 0. In this case G_{S} is of local type (with respect to the prime 𝔭_{0} which lies over p) and of maximal local type, if μ_{p} ^{⊆} k.

Proof: Let G_{S}(k) be a Demuškin group. Then by the theorems 3.2 and 3.3 G_{S}(k) is of local type with respect to 𝔭_{0}, which has to lie over p. Furthermore � _{S}^{S}(k) = 0 and #S^{f}(k) = 1+δ (by our convention that S^{f} contains only relevant primes). Hence � _{S}^{S}(k^{+}) = 0. The action of G(k|k^{+}) on the 1-dimensional IF_{p}-vector space H^{2}(G_{S}(k), ZZ/pZZ) must be trivial. Indeed, otherwise H^{2}(G_{S}(k^{+}), ZZ/pZZ) = 0 and therefore G_{S}(k^{+}) would be isomorphic to ZZ_{p}, since k^{+} is totally real. Furthermore the subspaces H^{1}(G_{S}(k), ZZ/pZZ)^{±} would be totally isotropic with respect to the nondegenerated pairing

H^{1}(G_{S}(k), ZZ/pZZ) × H^{1}(G_{S}(k), ZZ/pZZ) ⟶^{∪} H^{2}(G_{S}(k), ZZ/pZZ) .

It would follow that

r_{2}(k) + 2 = dim_{IF_{p}} H^{1}(G_{S}(k), ZZ/pZZ) = 2 · dim_{IF_{p}} H^{1}(G_{S}(k^{+}), ZZ/pZZ) = 2

which is absurd. Thus dim_{IF_{p}} H^{2}(G_{S}(k^{+}), ZZ/pZZ) = 1. Since this argument is true for all finite CM-extensions K|K^{+} of k|k^{+} inside k_{S}(p), we obtain from 3.4 that G_{S}(k^{+}) is a Demuškin group. Furthermore the equality 1 = dim_{IF_{p}} H^{2}(G_{S}(k^{+}), ZZ/pZZ) = δ𝔭̃_{0} shows that μ_{p} ^{⊆} k𝔭̃_{0}. Thus we obtain condition 2. Finally, condition 3 holds, because G_{S} is of local type.

Conversely, let us assume that the conditions 1 to 3 are fulfilled. First we consider the case that μ_{p} ^{⊆} k. Let 𝔭̃_{0} = 𝔭_{1}𝔭_{2} in k by our assumption. Hence r_{2} = n_{𝔭_{1}} = n_{𝔭_{2}} and ∑_{𝔭^{∈}S^{f} \{𝔭_{1}}} δ_{𝔭} = δ. We will show that � _{S\{𝔭_{1}}}^{S} = 0. Suppose the contrary. Then by Kummer theory there exists a Galois extension K|k of degree p inside k_{S}(p) in which 𝔭_{2} decomposes. If this extension is Galois over k^{+}, then 𝔭_{1} decomposes, too. Since Cl_{S}(k)^{−} = 0 by condition 3 and Cl_{S}(k)^{+} = 0 by condition 2 and the remark 1 following theorem 3.5, we obtain a contradiction. Thus K is not Galois over k^{+} and we denote by K̃ the normal closure of K over k^{+}, hence G(K̃|k) ≅ (ZZ/pZZ)^{2}. Let K̃^{+} be the fixed field of G(K̃|k)^{−} ≅ ZZ/pZZ. Since Cl_{S}(k)^{+} = 0, both primes, 𝔭_{1} and 𝔭_{2}, do not split in K̃^{+}, but each of them decomposes in p primes in K̃. Thus the extension K̃|K̃^{+} is unramified and all primes of S_{p} split, hence G(K̃|K̃^{+}) ≅ Cl_{S}(k)^{−}. Again we obtain a contradiction. This proves that � _{S\{𝔭_{1}}}^{S} = 0. It follows from theorem 1.5 that G_{S}(k) is of maximal local type, i.e. G_{S}(k) = G_{𝔭_{1}}(k) = G_{𝔭_{2}}(k) and therefore G_{S}(k) is a Demuškin group.

Now let μ_{p} ^{⊆} / k. Since 𝔭̃_{0} does not split in the extension k|k^{+}, the group G(k|k^{+}) acts on G_{𝔭_{0}}(k)^{ab}. Using condition 3 we obtain the commutative diagram

� � � � � � �  G_{S}(k^{+})^{ab} G_{𝔭_{0}}(k^{+})^{ab}  G_{S}(k)^{ab} G_{𝔭_{0}}(k)^{ab}  G_{S}(k)^{ab−} G_{𝔭_{0}}(k)^{ab−}

where the map at the bottom is surjective, since G_{S}(k^{+}) is of local type by condition 2 and the remark 1 to theorem 3.5. Thus G_{S}(k) is of local type. Since H^{2}(G_{S}(k), Q_{p}/ZZ_{p})^{−} = 0 and G_{S}(k)^{ab,−} is ZZ_{p}-torsion free (because μ_{p} ^{⊆} k+ ; 𝔭̃_{0} and G_{𝔭_{0}}(k)^{ab−} ≅ G_{S}(k)^{ab,−}), we obtain H^{2}(G_{S}(k), ZZ/pZZ)^{−} = 0 and therefore dim_{IF_{p}} H^{2}(G_{S}(k), ZZ/pZZ) = dim_{IF_{p}} H^{2}(G_{S}(k^{+}), ZZ/pZZ) = 1 and � _{S}^{S}(k) = 0. Since μ_{p} ^{⊆} / k, it follows that � _{S}^{S}(K) = 0 for all finite Galois extension K|k inside k_{S}(p), and since G_{S}(k) is of local type, we have #S^{f}(K) = 1. Thus dim_{IF_{p}} H^{2}(G_{S}(K), ZZ/pZZ) = 1 for all these extensions K|k. Now lemma 3.4 gives the desired result. □

Examples: Case 1. μ_{p} ^{⊆} k: p = 3, k = Q(\sqrt{6}, \sqrt{−3}), S = \{3, ∞\}.

Then G_{S}(k) is a Demuškin group of rank 4.

Case 2. μ_{p} ^{⊆} / k: p = 3, k = Q(\sqrt{6}, \sqrt{−1}), S = \{3, ∞\}.

Then G_{S}(k) is a Demuškin group of rank 4.

We know already that G_{S}(k^{+}), where k^{+} = Q(\sqrt{6}), is a Demuškin group of rank equal to 2, see the examples for totally real fields. Furthermore we have Cl_{S}(k)^{−} ≅ Cl_{S}(Q(\sqrt{−6})) ⊕ Cl_{S}(Q(\sqrt{d})), where d = −3 or d = −1 according to we are in case 1 or 2. Thus Cl_{S}(k)^{−} = 0.

It remains to classify number fields k and sets of primes S such that G_{S} is a Demuškin group, k is not totally real and μ_{p} ^{⊆} / k_{𝔭} for all primes 𝔭 ^{∈} S^{f}. We do not know any example of this case. Moreover, does there exists a situation where G_{S} is a Demuškin group of global type? By the considerations above this can only happen in the just mentioned case.

References

[1] Neukirch, J., Schmidt, A., Wingberg, K. Cohomology of Number Fields. Springer 1999

[2] Wingberg, K. On Galois groups of p-closed algebraic number fields with restricted ramification. J. reine u. angew. Math. 400 (1989) 185-202

Mathematisches Institut der Universität Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg Germany

e-mail: wingberg@mathi.uni-heidelberg.de