Arithmetical Koch Groups

by Kay Wingberg at Heidelberg

Version: November 23, 2007

Let k be a number field, p a prime number and S a finite set of primes of k. The Galois group G(k_{S}(p)|k) of the maximal p-extension of k which is unramified outside S is an important object in order to study the arithmetic of k. If all primes dividing p are in S, then a lot is known about the structure of G(k_{S}(p)|k), in particular, it is of cohomological dimension less or equal to 2 (if p = 2 one has to require that k is totally imaginary).

If S is disjoint to the set S_{p} of primes above p, the group G(k_{S}(p)|k) is very mysterious. By a famous theorem of Golod and Šafarevič, it is in general infinite, but on the other hand it is a so-called fab pro-p-group, i.e. the maximal abelian quotient of every open subgroup of G(k_{S}(p)|k) is finite. Furthermore, nothing was known on the cohomological dimension of G(k_{S}(p)|k) so far.

Recently, J. Labute [2] showed that pro-p-groups who have a presentation in terms of generators and relations of a certain type, so-called mild pro-p-groups, are of cohomological dimension equal to 2. A special case are pro-p-groups of Koch type, with certain further conditions on the relations (the linking diagram of the considered group has to be a non-singular circuit, see the definitions in the next section).

If k = ℚ, then the group G(ℚ_{S}(p)|ℚ), S ∩ S_{p} = ∅, is of Koch type, see H. Koch [1]. Labute used these results on the relation structure of G(ℚ_{S}(p)|ℚ) and ended up with a criterion on the set S for the group G(ℚ_{S}(p)|ℚ) to be of cohomological dimension 2. A. Schmidt [5] extended the result of Labute by arithmetic methods and weakened Labute’s condition on S.

There is another case when the Galois group G(k_{S}(p)|k), S ∩ S_{p} = ∅, is of Koch type: k has to be an imaginary quadratic number field not containing the p-th roots of unity and its class number is not divisible by p. Therefore, if the linking diagram of G(k_{S}(p)|k) is a non-singular circuit, then this group is of cohomological dimension equal to 2, see the paper of D. Vogel [6]. It seems that there are no other algebraic number fields k and sets S as the cases mentioned above such that the Galois group G(k_{S}(p)|k) is of Koch type.

In this paper we will consider the maximal p-extension k_{S}^{T} (p) of the number field k with restricted ramification at a finite set S containing S_{p}, which, in addition, is completely decomposed at the finite set T. The groups G(k_{S}^{T} (p)|k) are a rich source of pro-p-groups of Koch type. Under certain conditions on T and S (and conditions on k) we will show that G(k_{S}^{T} (p)|k) is a pro-p Schur group (i.e. has as many generators as relations), is of Koch type, its maximal abelian quotient is finite, and the cohomological dimension is equal to 2. Moreover, if p is odd and k is totally real, and assuming that the Leopoldt conjecture holds for totally real number fields, then G(k_{S}^{T} (p)|k) is a fab pro-p-group.

The author wants to thank J. Gärtner and A. Schmidt for helpful conversations concerning this paper.

1 Pro-p-groups of Koch type

Let p be a prime number and let G be a pro-p-group. We denote the cohomology groups H^{i}(G, ℤ/pℤ) by H^{i}(G), and put h^{i}(G) = dim_{𝔽_{p}} H^{i}(G) and

χ_{2}(G) = ∑_{i=0}^{2}(−1)^{i} h^{i}(G).

Let G_{n} be the n-th term in the lower p-central series defined recursively by G_{1} = G and G_{n+1} = (G_{n})^{p}[G_{n}, G]. We recall some definitions.

Definition 1.1 A pro-p-group G is called Schur group if h^{1}(G) = h^{2}(G).

Definition 1.2 A pro-p-group G is called fab if U^{ab} is finite for all open subgroups U of G.

For the notion of a pro-p duality group we refer to [4] III §4.

Proposition 1.3 Let G be a fab pro-p-group of cohomological dimension equal to 2. Then G is a duality group. Furthermore, the strict cohomological dimension of G is equal to 3.

Proof: In order to prove the first part of the proposition it suffices to show that the terms

D_{i}(G, ℤ/pℤ) = \underrightarrow{lim}_{U} H^{i}(U)^{∨}

are trivial for i = 0, 1; here U runs through the open subgroups of G, and the transition maps are the duals of the corestriction maps, see [4] (3.4.6). For i = 0 this is clear, since G is infinite. For i = 1 we have

D_{1}(G, ℤ/pℤ) = \underrightarrow{lim}_{U} U^{ab}/p.

Since U^{ab} is finite for all open subgroups U of G, it follows from the group theoretical form of the principal ideal theorem, see [3] VI. (7.6), that

D_{1}(G, ℤ/pℤ) = 0.

Suppose that scd_{p} G = 2, i.e. H^{2}(U, ℚ_{p}/ℤ_{p}) = 0 for all open subgroups U of G. From the exact sequence 0 → ℤ/pℤ → ℚ_{p}/ℤ_{p} →^{p} ℚ_{p}/ℤ_{p} → 0, we obtain the exact sequence

0 →(_{p}U^{ab})^{∨} → H^{2}(U) →_{p}H^{2}(U, ℚ_{p}/ℤ_{p}) → 0.

Since G is fab, we obtain

h^{1}(U) = dim_{𝔽_{p}}(U^{ab}/p) = dim_{𝔽_{p}}(_{p}U^{ab}) = h^{2}(U),

i.e. χ_{2}(U) = 1. Since cd_{p} G = 2, we have χ_{2}(U) = (G : U)χ_{2}(G). This contradiction finishes the proof of the proposition. □

Let G be a finitely represented pro-p-group and let 1 → R → F → G → 1 be a minimal presentation, where F is the free pro-p-group on the generators x_{1}, …, x_{d} and R = (w_{1}, …, w_{r}) is the normal subgroup of F generated by the elements w_{i}, i = 1, …, r.

Definition 1.4 The minimal presentation < x_{1}, …, x_{d}|w_{1}, …, w_{r} > of the prop-group G is said to be of Koch type if r ≤ d and the relations w_{i} satisfy a congruence of the form

w_{i} ≡ x_{i}^{p a_{i}} ∏_{i≠j}[x_{i}, x_{j}]^{a_{ij}} mod F_{3}

with a_{i}, a_{ij} ∈ ℤ. The group G is of Koch type if it has a presentation of Koch type.

Examples: 1. Let p be an odd prime and S a finite set of prime numbers not containing p. Let G = G(ℚ_{S}(p)|ℚ) be the Galois group of the maximal pextension of ℚ unramified outside S. We can assume that S = {q_{1}, ⋯, q_{d}} with q_{i} ≡ 1 mod p. Work of Koch [1] shows that G =< x_{1}, …, x_{d}|w_{1}, …, w_{d} > where

w_{i} ≡ x_{i}^{q_{i}−1} ∏_{i≠j}[x_{i}, x_{j}]^{b_{ij}} mod F_{3},

and q_{i} ≡ g_{j}^{b_{ij}} mod q_{j}, where g_{j} is a primitive root for the prime q_{j}.Observe that r = d.

2. Let p be an odd prime number and k an imaginary quadratic number field whose class number is not divisible by p, and which is different from ℚ(\sqrt{−3}) if p = 3. Let S be a set of primes of k whose norm is congruent to 1 mod p. If G = G(k_{S}(p)|k) is the Galois group of the maximal p-extension of k unramified outside S, then G has a presentation of Koch type with r = d, see [1] or [6].

Let G be a pro-p-group of Koch type. Following Labute, we associate to G =< x_{1}, …, x_{d}|w_{1}, …, w_{r} > and S = {x_{1}, …, x_{d}} a directed graph, denoted by Γ_{S}(p), with vertices the elements of S and a directed edge x_{i}x_{j} from x_{i} to x_{j} if

l(x_{i}, x_{j}) := a_{ij} mod p ≠ 0.

The graph Γ_{S}(p), together with the l(x_{i}, x_{j}) ∈ ℤ/pℤ, i, j ≤ d, is called the linking diagram of (G, S).

Definition 1.5 Let G =< x_{1}, …, x_{d}|w_{1}, …, w_{r} > be a pro-p-group of Koch type and let Γ_{S}(p) be the associated linking diagram of (G, S). The set S is called strictly circular with respect to p (and Γ_{S}(p) a non-singular circuit) if there exists an ordering S = {v_{1}, …, v_{d}} of the elements in S such that the following conditions are fulfilled:

(1) The vertices v_{1}, …, v_{d} of Γ_{S}(p) form a circuit v_{1}v_{2} … v_{d}v_{1}.

(2) If i, j are both odd, then v_{i}v_{j} is not an edge of Γ_{S}(p).

(3) If l_{ij} = l(v_{i}, v_{j}), then l_{12}l_{23} ⋯ l_{d−1,d}l_{d1} − l_{1d}l_{21}l_{32} ⋯ l_{d,d−1} ≠ 0.

We remark that condition (2) implies that d is even and d ≥ 4 and that condition (3) is satisfied if there exists an edge v_{i}v_{j} of the circuit v_{1}v_{2} ⋯ v_{d}v_{1} such that v_{j}v_{i} is not an edge Γ_{S}(p).

Theorem 1.6 (Labute [2], Thm. 1.6.) Let G be a pro-p-group of Koch type on the minimal set of generators S. If S is strictly circular with respect to p, then cd G = 2.

2 Galois extensions of number fields which are completely decomposed at given primes

We will use the following notation. Let S, T be sets of primes of k. Then

k_{S}(p) is the maximal p-extension of k which is unramified outside S,
k_{S}^{T} (p) is the maximal p-extension of k which is unramified outside S

and completely decomposed at T.

Furthermore, k(p) denotes the maximal p-extension of k. For a prime 𝔭 of k, let k_{𝔭} be the completion of k with respect to 𝔭, U_{𝔭} the group of units and μ(k_{𝔭}) the group of roots of unity in k_{𝔭}. We denote the decomposition group and inertia group of G(k(p)|k) with respect to 𝔭 by G_{𝔭}(k) = G_{𝔭}(k(p)|k) and T_{𝔭}(k) = T_{𝔭}(k(p)|k), respectively.

Considering the extension k_{S}(p)|k, the following primes cannot ramify in a p-extension, and are therefore redundant in S:

1. Complex primes.

2. Real primes if p ≠ 2.

3. Primes 𝔭 ∤ p with N(𝔭) ≢ 1 mod p. Removing all these redundant places from S, we obtain a subset S_{min} ⊆ S which has the property that G(k_{S}(p)|k) = G(k_{S_{min}}(p)|k). Let

S̃ = S\(S_{p} ∪ S_{∞})

the subset of finite primes of S not above p, and let

n_{S} = ∑_{𝔭∈S_{p}∩S} n_{𝔭}, δ_{S} = ∑_{𝔭∈S_{p}∩S} δ_{𝔭} − δ,

where n_{𝔭} = [k_{𝔭} : ℚ_{p}],

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}

Furthermore, θ = θ(S) is equal to 1 if μ_{p} ⊆ k and S_{min} = ∅, and zero in all other cases. Finally, � _{S}(k) denotes the dual of the Kummer group

V_{S}(k) = {a ∈ k^{×} | a ∈ k_{𝔭}^{×p} for 𝔭 ∈ S and a ∈ U_{𝔭}k_{𝔭}^{×p} for 𝔭 ∉ S}/k^{×p} .

Proposition 2.1 Let p be a prime number and assume that the number field k is totally imaginary if p = 2. Let T and S = S_{min} be finite sets of primes of k such that T ∩ S = ∅. Then

\begin{aligned}χ_{2}(G(k_{S}^{T} (p)|k)) & ≤ & θ + r_{1} + r_{2} − n_{S} + #T, \\ h^{1}(G(k_{S}^{T} (p)|k)) & ≥ & 1 + #S̃ + δ_{S} + n_{S} + dim_{𝔽_{p}} � _{S} − r_{1} − r_{2} − #T. \end{aligned}

Proof: Since T ∩ S = ∅, we have a surjection

⨁_{𝔭∈T}G_{𝔭}(k)/T_{𝔭}(k) ↠ (G(k_{S}(p)|k_{S}^{T} (p))^{ab})_{G(k_{S}^{T} (p)|k)}

(here M_{G} denotes the G-coinvariants of a G-module M). Thus we obtain

dim_{𝔽_{p}} H^{1}(G(k_{S}(p)|k_{S}^{T} (p)))^{G(k_{S}^{T} (p)|k)} ≤ #T.

Using [4] (8.7.11), the exact 5-term sequence 0 ⟶ H^{1}(G(k_{S}^{T} (p)|k)) ⟶ H^{1}(G(k_{S}(p)|k)) ⟶ H^{1}(G(k_{S}(p)|k_{S}^{T} (p)))^{G(k_{S}^{T} (p)|k)}

⟶ H^{2}(G(k_{S}^{T} (p)|k)) ⟶ H^{2}(G(k_{S}(p)|k))

gives us the inequalities

h^{2}(G(k_{S}^{T} (p)|k)) − h^{1}(G(k_{S}^{T} (p)|k))

≤ h^{2}(G(k_{S}(p)|k)) − h^{1}(G(k_{S}(p)|k)) + dim_{𝔽_{p}} H^{1}(G(k_{S}(p)|k_{S}^{T} (p)))^{G(k_{S}^{T} (p)|k)}
≤ θ − 1 + r_{1} + r_{2} − n_{S} + #T

and h^{1}(G(k_{S}^{T} (p)|k)) ≥ h^{1}(G(k_{S}(p)|k))−#T = 1+#S̃+δ_{S}+n_{S}+dim_{𝔽_{p}} � _{S}−r_{1}−r_{2}−#T.

□

Corollary 2.2 With the assumptions of proposition (2.1) let

c(S, T) = max{0, θ + r_{1} + r_{2} − n_{S} + #T }.

Assume that

#S̃ ≥ (1 + \sqrt{c(S, T)} )^{2} − (δ_{S} + dim_{𝔽_{p}} � _{S} + θ).

Then the group G(k_{S}^{T} (p)|k) is infinite.

Proof: Let G = G(k_{S}^{T} (p)|k) and suppose that this group is finite. Then, by the Golod Safarevič inequality, see [4] (3.9.7),

h^{2}(G) > \frac{h^{1}(G)^{2}}{4} .

From proposition (2.1) it follows that c(S, T) − 1 ≥ θ − 1 + r_{1} + r_{2} − n_{S} + #T ≥ h^{2}(G) − h^{1}(G) > h^{1}(G)^{2}/4 − h^{1}(G), hence

#S̃ + (δ_{S} + dim_{𝔽_{p}} � _{S} + θ) − c(S, T) + 1 ≤ h^{1}(G) < 2 + 2\sqrt{c(S, T)},

which contradicts the assumption on #S̃. □

Let K_{1}, …, K_{ρ} be independent ℤ_{p}-extensions of k such that k̃ = ⋃_{i=1}^{ρ} K_{i} is the compositum of all ℤ_{p}-extensions of k. Recall that k̃ ⊆ k_{S}(p), if S_{p} ⊆ S. We say that a finite set T of primes of k has the property (∗) if the following holds:

Property (∗): The cardinality of T is equal to ρ, and if T = {𝔭_{1}, …, 𝔭_{ρ}}, then

𝔭_{i} does not decompose in K_{i}|k, i.e. G_{𝔭_{i}}(K_{i}|k) = G(K_{i}|k), i = 1, …, ρ.

If S is a finite set of primes of k such that S ∩ T = ∅, then it follows that the homomorphism

∗_{𝔭∈T} G_{𝔭}(k(p)|k)/T_{𝔭}(k(p)|k) ⟶ G(k_{S}(p)|k) ⟶ G(k̃ ∩ k_{S}(p)|k)

is surjective, and, in particular, G(k_{S}^{T} (p)|k)^{ab} is finite.

Proposition 2.3 Let p be a prime number and assume that the number field k is totally imaginary if p = 2. Let T and S_{p} ⊆ S = S_{min} be finite sets of primes of k such that T ∩ S = ∅.

(i) If #T = r_{2} + 1, then

χ_{2}(G(k_{S}^{T} (p)|k)) ≤ 1.

(ii) Assume that the Leopoldt conjecture holds for k and p, and that T has the property (∗). Then

h^{1}(G(k_{S}^{T} (p)|k)) = h^{2}(G(k_{S}^{T} (p)|k))

and

G(k_{S}^{T} (p)|k)^{ab} ≅ Tor G(k_{S}(p)|k)^{ab}.

In particular, G(k_{S}^{T} (p)|k)^{ab} is finite. If #S\S_{p} ≥ 4, then G(k_{S}^{T} (p)|k) is infinite.

(iii) Assume in addition to the assumptions of (ii) that

dim_{𝔽_{p}} � _{S} = 0 and ∑_{𝔭∈S_{p}} δ_{𝔭} = δ.

Then

h^{1}(G(k_{S}^{T} (p)|k)) = h^{2}(G(k_{S}^{T} (p)|k)) = #S\S_{p}.

Proof: Let G = G(k_{S}^{T} (p)|k). By proposition (2.1), we have

χ_{2}(G) ≤ 0 + r_{1} + r_{2} − [k : ℚ] + #T = 1

proving (i).

From the exact sequence 0 → ℤ/pℤ → ℚ_{p}/ℤ_{p} →^{p} ℚ_{p}/ℤ_{p} → 0, we obtain the exact sequence

0 →(_{p}G^{ab})^{∨} → H^{2}(G) →_{p}H^{2}(G, ℚ_{p}/ℤ_{p}) → 0.

By assumption, the Leopoldt conjecture holds, i.e. ρ = rank_{ℤ_{p}}G(k̃|k) = r_{2} + 1. Therefore, as T has the property (∗), G^{ab} is finite. It follows that

h^{1}(G) = dim_{𝔽_{p}} _{p}G^{ab} ≤ h^{2}(G).

Since h^{1}(G) ≥ h^{2}(G) by (i), we get equality. The commutative and exact diagram

\xymatrix{ & Tor G(k_{S}(p)|k)^{ab} \ar[dr] \ar@{^{(}->}[d] \\ ⨁_{𝔭∈T}G_{𝔭}(k(p)|k)/T_{𝔭}(k(p)|k) \ar[r] \ar@{-->}[dr]^{≅} & G(k_{S}(p)|k)^{ab} \ar[r] \ar@{->>}[d] & G(k_{S}^{T} (p)|k)^{ab} \ar[r] & 0 \\  & G(k̃|k)}

shows Tor G(k_{S}(p)|k)^{ab} →^{∼} G(k_{S}^{T} (p)|k)^{ab}. Furthermore, it follows from c(S, T) = 1 and corollary (2.2), that G(k_{S}^{T} (p)|k) is infinite, if #S\S_{p} ≥ 4. This proves (ii).

From proposition (2.1) it follows that h^{1}(G) ≥ #S̃, and using [4] (8.7.11), we have h^{2}(G) ≤ #S̃. This proves (iii). □

Theorem 2.4 Let p be a prime number and assume that the number field k is totally imaginary if p = 2. Let T and S_{p} ⊆ S = S_{min} be finite sets of primes of k such that T ∩ S = ∅. Assume that

(a) T has the property (∗).

(b) dim_{𝔽_{p}} � _{S_{p}} = 0 and ∑_{𝔭∈S_{p}} δ_{𝔭} = δ.

Then the following holds:

(i) The canonical homomorphism

∗_{𝔭∈S\S_{p}} T_{𝔭}(k(p)|k) ↠ G(k_{S}^{T} (p)|k)

is surjective.

(ii) There is an isomorphism

⨁_{𝔭∈S\S_{p}} μ(k_{𝔭})(p) ⟶^{∼} G(k_{S}^{T} (p)|k)^{ab}.

(iii) The map

H^{2}(G(k_{S}^{T} (p)|k)) ↪ ⨁_{𝔭∈S\S_{p}} H^{2}(G_{𝔭})

is injective. (iv) The pro-p-group G(k_{S}^{T} (p)|k) is of Koch type and

h^{1}(G(k_{S}^{T} (p)|k)) = h^{2}(G(k_{S}^{T} (p)|k)) = #S\S_{p}.

(v) G(k_{S}^{T} (p)|k)^{ab} is finite. If #S\S_{p} ≥ 4, then G(k_{S}^{T} (p)|k) is infinite.

Proof: Since dim_{𝔽_{p}} � _{S_{p}} = 0 and ∑_{𝔭∈S_{p}} δ_{𝔭} = δ, the pro-p-group G(k_{S_{p}}(p)|k) is free, see [4] (8.7.10). Therefore Leopoldt’s conjecture holds for k and p. Furthermore � _{S} = 0 as � _{S_{p}} surjects onto � _{S}. From proposition (2.3) it follows that the assertion on the dimensions in (iv) and assertion (v) are true.

The cokernel of the canonical homomorphism

∗_{𝔭∈S\S_{p}} T_{𝔭}(k(p)|k) ⟶ G(k_{S}^{T} (p)|k)

is the Galois group G(k_{S_{p}}^{T}(p)|k). Since G(k_{S_{p}}(p)|k) is a free pro-p-group of rank r_{2} + 1, we have G(k_{S_{p}}(p)|k)^{ab} ≅ ℤ_{p}^{r_{2}+1}. Using the assumption (∗) for T, we get

G(k_{S_{p}}^{T}(p)|k)^{ab} = 0,

hence G(k_{S_{p}}^{T}(p)|k) = 1, i.e. we proved (i).

Since the Leopoldt’s conjecture holds for k and p, we have

(Tor G(k_{S}(p)|k)^{ab})^{∨} ≅ H^{2}(G(k_{S}(p)|k), ℤ/p^{r}ℤ)

for r ∈ ℕ big enough. The exact sequence

H^{2}(G(k_{S}(p)|k), ℤ/p^{r}ℤ) → ⨁_{𝔭∈S}H^{2}(G_{𝔭}(k), ℤ/p^{r}ℤ) → H^{0}(G(k_{S}(p)|k), μ_{p^{r}})^{∨} → 0

implies that we obtain a surjection

H^{2}(G(k_{S}(p)|k), ℤ/p^{r}ℤ) ↠ ⨁_{𝔭∈S\S_{p}} H^{2}(G_{𝔭}(k), ℤ/p^{r}ℤ) ≅ ⨁_{𝔭∈S\S_{p}} μ(k_{𝔭})(p)^{∨}.

Using proposition (2.3)(ii), it follows that we obtain an injection

⨁_{𝔭∈S\S_{p}} μ(k_{𝔭})(p) ↪ G(k_{S}^{T} (p)|k)^{ab}.

On the other hand, by (i) the map

⨁_{𝔭∈S\S_{p}} μ(k_{𝔭})(p) ≅ ⨁_{𝔭∈S\S_{p}} T_{𝔭}(k)/[T_{𝔭}(k), G_{𝔭}(k)] ↠ G(k_{S}^{T} (p)|k)^{ab}

is surjective. This proves (ii).

In order to prove (iii), we consider the exact sequence

1 ⟶ K ⟶ ∗_{𝔭∈S\S_{p}} G_{𝔭}(k) ⟶ G(k_{S}^{T} (p)|k) ⟶ 1,

where K is the kernel of the natural map ∗_{𝔭∈S\S_{p}}G_{𝔭}(k) → G(k_{S}^{T} (p)|k) which is surjective by (i). For an abelian group A we obtain (using (i) again) the commutative and exact diagram

[figure: ⨁_{𝔭∈S\S_{p}} H_{nr}^{1}(G_{𝔭}(k), A); 0; H^{1}(G(k_{S}^{T} (p)|k), A); ⨁_{𝔭∈S\S_{p}} H^{1}(G_{𝔭}(k), A); res; H^{1}(K, A)^{G(k_{S}^{T} (p)|k)}; H^{1}(G(k_{S}^{T} (p)|k), A); ⨁_{𝔭∈S\S_{p}} H^{1}(T_{𝔭}(k), A)^{G_{𝔭}(k)}.]

If A = ℚ_{p}/ℤ_{p}, then lower map is an isomorphism by (ii). Furthermore, since the Leopoldt’s conjecture holds, we have H^{2}(G(k_{S}^{T} (p)|k), ℚ_{p}/ℤ_{p}) = 0, and so the map res is surjective. If follows that

⨁_{𝔭∈S\S_{p}} H_{nr}^{1}(G_{𝔭}(k), ℚ_{p}/ℤ_{p}) ≅ H^{1}(K, ℚ_{p}/ℤ_{p})^{G(k_{S}^{T} (p)|k)},

hence

⨁_{𝔭∈S\S_{p}} H_{nr}^{1}(G_{𝔭}(k), ℤ/pℤ) ≅ H^{1}(K, ℤ/pℤ)^{G(k_{S}^{T} (p)|k)}.

Considering the diagram above with A = ℤ/pℤ, we obtain the desired injection H^{2}(G(k_{S}^{T} (p)|k)) ↪ ⨁_{𝔭∈S\S_{p}} H^{2}(G_{𝔭}(k)).

Let S̃ = {𝔮_{1}, …, 𝔮_{d}} and let τ_{i} be a generator of the cyclic group T_{𝔮_{i}}(k), i = 1, …, d. Then by (i) and (iv) the set {τ_{1}, …, τ_{d}} is a minimal set of generators of the group G(k_{S}^{T} (p)|k). Let F be the free pro-p-group on the generators x_{1}, …, x_{d}, and let

1 ⟶ R ⟶ F ⟶^{π} G(k_{S}^{T} (p)|k) ⟶ 1

be a minimal presentation of the group G(k_{S}^{T} (p)|k), where π maps x_{i} to τ_{i}, i = 1, …, d. From (iv) and [4] (7.5.2) it follows that a set of defining relations is given by

w_{i} = x_{i}^{N(𝔮_{i})−1}[x_{i}, y_{i}], i = 1, …, d,

where y_{i} ∈ F denotes a pre-image of the Frobenius automorphism σ_{i} with respect to 𝔮_{i}, see [1],§11.4. Let

y_{i} ≡ ∏_{i≠j} x_{j}^{l_{ij}} mod F_{2}

with l_{ij} ∈ ℤ/pℤ. Then we obtain

w_{i} = x_{i}^{N(q_{i})−1}[x_{i}, y_{i}] ≡ x_{i}^{N(q_{i})−1}[x_{i}, ∏_{i≠j} x_{j}^{l_{ij}}] ≡ x_{i}^{N(q_{i})−1} ∏_{i≠j}[x_{i}, x_{j}]^{l_{ij}} mod F_{3}.

Thus G(k_{S}^{T} (p)|k) is a pro-p-group of Koch type. □

From theorem (2.4) and Labute’s theorem (1.6) we obtain

Theorem 2.5 Let p be a prime number and assume that the number field k is totally imaginary if p = 2. Let T and S_{p} ⊆ S = S_{min} be finite sets of primes of k such that T ∩ S = ∅. Assume that

(a) T has the property (∗),

(b) dim_{𝔽_{p}} � _{S_{p}} = 0 and ∑_{𝔭∈S_{p}} δ_{𝔭} = δ,

(c) Γ_{S\S_{p}}(p) is a non-singular circuit.

Then G(k_{S}^{T} (p)|k) is a pro-p Schur group, G(k_{S}^{T} (p)|k)^{ab} is finite and

cd_{p} G(k_{S}^{T} (p)|k) = 2.

Corollary 2.6 With the notation and assumptions of theorem (2.5) assume in addition, that p is odd and k is totally real. Assume further that the Leopoldt conjecture holds for totally real number fields.

Then G(k_{S}^{T} (p)|k) is a fab pro-p-group, and a duality group of dimension 2 and strict cohomological dimension equal to 3.

Proof: If K|k is a finite Galois extension inside k_{S}^{T} (p), then K is also totally real as p ≠ 2. Since the Leopoldt conjecture holds for K and p, there is only one ℤ_{p}-extension of K, the cyclotomic one. The prolongations of the only prime 𝔮 ∈ T (k) are inert in this extension. Therefore G(k_{S}^{T} (p)|K)^{ab} is finite. The second assertion follows from (1.3). □

It seems that among the conditions of (2.6) the assumption that k is totally real in order to show that G(k_{S}^{T} (p)|k) is fab is not necessary but we can not prove it. The next results show that theorem (2.5) is not empty. The idea of the proof is inspired by [2] prop. 6.1.

Proposition 2.7 Let k be a number field and let p be a prime number such that μ_{p} ⊆ / k. Let T and S = S_{min} be finite disjoint sets of primes of k with S_{p} ⊆ S. Assume that conditions (a) and (b) of (2.5) hold, and let

∗_{𝔭∈S̃}T_{𝔭}(k(p)|k) ↠ G(k_{S}^{T} (p)|k)

be a minimal presentation of the pro-p-group G(k_{S}^{T} (p)|k) of Koch type, where S̃ = S\S_{p} = {𝔮_{1}, …, 𝔮_{m}}. Let q_{i} = 𝔮_{i} ∩ ℚ, i = 1, …, m, be the underlying prime numbers, and assume that for all i

(i) q_{i} ≡ 1 mod p and q_{i} ≠ q_{j} if i ≠ j, (ii) the prime number q_{i} is unramified in k|ℚ, (iii) the image of 𝔮_{i} in the p-primary part Cl_{k}(p) of the ideal class group of k is

trivial. Then a prime 𝔮_{m+1} can be found satisfying (i)-(iii) such that the additional edges of the linking diagram Γ_{S̃∪{𝔮}}(p) of (G(k_{S∪{𝔮}}^{T}(p)|k), S̃ ∪ {𝔮}) are arbitrarily prescribed.

Remark: Often we identify the sets {𝔮_{1}, …, 𝔮_{m}} and {τ_{𝔮_{1}}, …, τ_{𝔮_{m}}} of primes of k and generators of G(k_{S}^{T} (p)|k), respectively, and denote them by the same letter.

Proof: First we observe that, if 𝔮 ∉ T ∪S is a prime of k with N_{k|ℚ} 𝔮 ≡ 1 mod p, then by theorem (2.4) the group G(k_{S∪{𝔮}}^{T}(p)|k) is also of Koch type and

∗_{𝔭∈S̃∪{𝔮}}T_{𝔭}(k(p)|k) ↠ G(k_{S∪{𝔮}}^{T}(p)|k)

is a minimal presentation of G(k_{S∪{𝔮}}^{T}(p)|k).

If k̄ is the maximal abelian p-extension of k(μ_{p}) and 𝔮 a non-archimedean prime of k(μ_{p}) not lying above p, then

G_{𝔮}(k̄|k(μ_{p})) =< σ_{𝔮}, τ_{𝔮} >⊆ G(k̄|k(μ_{p})),

where < τ_{𝔮} > is the inertia subgroup of the decomposition group G_{𝔮}(k̄|k(μ_{p})) of G(k̄|k(μ_{p})) with respect to 𝔮, and σ_{𝔮} is a Frobenius lift.

By (i) and (ii) there is a unique extension E_{i} of k(μ_{p}) contained in k(μ_{pq_{i}}) of degree p, i.e. G(E_{i}|k(μ_{p})) ≅ ℤ/pℤ. By (iii), there exists a natural number h_{i} prime to p such that 𝔮_{i}^{h_{i}} = (π_{𝔮_{i}}) is a principal ideal of k. Let

F_{i} = k(μ_{p}, \sqrt[p]{π_{𝔮_{i}}})

with Galois group G(F_{i}|k(μ_{p})) ≅ ℤ/pℤ, and let H_{k} be the p-elementary Hilbert field of k, i.e. the maximal p-elementary abelian unramified extension of k, with Galois group G(H_{k}(μ_{p})|k(μ_{p})) ≅ ℤ/pℤ^{ϵ} for some ϵ ≥ 0. The fields

E_{1}, …, E_{m}, F_{1}, …, F_{m}, H_{k}(μ_{p})

are linearly disjoint over k(μ_{p}), and let K be the composite of these fields. The field K is Galois over k and the subgroup H = G(K|k(μ_{p})) of G(K|k) is the direct product of the Galois groups of these fields over k(μ_{p}).

If σ_{𝔔} ∈ G(K|k) is the Frobenius automorphism at the unramified prime 𝔔 of K and 𝔔 lies above the prime 𝔮 of k, then σ_{𝔔} ∈ G(K|k(μ_{p})) if and only if N_{k|ℚ} 𝔮 ≡ 1 mod p. Furthermore, the restriction of σ_{𝔔} to H_{k}(μ_{p}) is the identity if and only if the image of 𝔮 in Cl_{k}(p) is trivial.

Assume a prime 𝔮_{m+1} of k is given such that the underlying prime number q_{m+1} is unramified in K, q_{m+1} ≡ 1 mod p (and so N_{k|ℚ} 𝔮_{m+1} ≡ 1 mod p), q_{m+1} ≠ q_{j} for j = 1, …, m, and the image of 𝔮_{m+1} in Cl_{k}(p) is trivial. Then we choose a prolongation of 𝔮_{m+1} to k(μ_{p}), which we also denote by 𝔮_{m+1}. Let 𝔔|𝔮_{m+1} be a prime of K; we denote σ_{𝔔} by σ_{𝔮_{m+1}} = σ_{𝔮_{m+1}}|_{K} as H is abelian. Let h ∈ ℕ be prime to p such that (𝔮_{m+1})^{h} = (π_{𝔮_{m+1}}) is a principal ideal of k. Since

G(E_{i}|k(μ_{p})) =< τ_{𝔮_{i}}G(k̄|E_{i}) >≅ ℤ/pℤ,

we get

σ_{𝔮_{m+1}}|_{E_{i}} ≡ (τ_{𝔮_{i}}|_{E_{i}})^{l_{m+1,i}} mod G(k̄|E_{i}),

where l_{m+1,i} ∈ ℤ/pℤ. Therefore the restriction of σ_{𝔮_{m+1}} to E_{i} is the identity if and only if the restriction of (σ_{𝔮_{m+1}})^{h} to E_{i} is the identity (recall that h is prime to p), and this is the case if and only if π_{𝔮_{m+1}} is a p-th power mod 𝔮_{i}. If F_{i} = k(μ_{p}, \sqrt[p]{π_{𝔮_{i}}}), then

σ_{𝔮_{m+1}}|_{F_{i}}(\sqrt[p]{π_{𝔮_{i}}}) = σ_{𝔮_{m+1}}|_{(F_{i})_{𝔮_{m+1}}}(\sqrt[p]{π_{𝔮_{i}}}) = (\frac{π_{𝔮_{i}}}{𝔮_{m+1}} )\sqrt[p]{π_{𝔮_{i}}} ,

where (\frac{π_{𝔮_{i}}}{𝔮_{m+1}}) ∈ μ_{p} ⊆ (F_{i})_{𝔮_{m+1}} is the Hilbert symbol, see [3] §8. We have

(\frac{π_{𝔮_{i}}}{𝔮_{m+1}} ) = 1 if and only if π_{𝔮_{i}} ≡ α^{p} mod 𝔮_{m+1}

for some α ∈ k(μ_{p}), i.e. the restriction of σ_{𝔮_{m+1}} to F_{i} is the identity if and only if π_{𝔮_{i}} is a p-th power mod 𝔮_{m+1}. Let G = G(k_{S∪{𝔮_{m+1}}}^{T}(p)|k), then

σ_{𝔮_{m+1}} ≡ ∏_{1≤j≤m}(τ_{𝔮_{j}})^{l_{m+1,j}} mod G_{2}

and

σ_{𝔮_{i}} ≡ ∏_{{1≤j≤m+1 \atop j≠i}}(τ_{𝔮_{j}})^{l_{ij}} mod G_{2}

with l_{ij} ∈ ℤ/pℤ. By the considerations above, l_{m+1,j} = 0 if and only if π_{𝔮_{m+1}} is a p-th power modulo 𝔮_{j} and this is the case if and only if the restriction of σ_{𝔮_{m+1}} to E_{j} is the identity, and l_{i,m+1} = 0 if and only if π_{𝔮_{i}} is a p-th power modulo 𝔮_{m+1} and this is the case if and only if the restriction of σ_{𝔮_{m+1}} to F_{i} is the identity.

By the Čebotarev density theorem, for every g ∈ H there exist infinitely many primes 𝔮 of k of degree equal to 1 such that σ_{𝔮} = g. Thus we may assume that 𝔮 = 𝔮_{m+1} is not in T, that the underlying prime number q_{m+1} is different to q_{i}, i = 1, …, m, and that q_{m+1} is unramified in K|ℚ. Since σ_{𝔮_{m+1}} ∈ H, it follows that q_{m+1} = N_{k|ℚ} 𝔮_{m+1} ≡ 1 mod p. Thus 𝔮_{m+1} satisfies (i) and (ii). Furthermore, choosing the element g ∈ H suitable, we can extend the directed graph ΓS̃(p) by a single prime 𝔮_{m+1} ∉ T ∪S satisfying (i), (ii) and, in addition, (iii) with prescribed edges joining the primes of S̃ to 𝔮_{m+1} and 𝔮_{m+1} to the primes of S̃. □

Corollary 2.8 With the notation and assumptions of (2.7) let #S̃ ≥ 2. Then S̃ can be extended to a set S̃′ with #S̃′ = 2#S̃ such that the linking diagram ΓS̃′(p) of (G(k_{S̃′∪S_{p}}^{T}(p)|k), S̃′) is a non-singular circuit.

Proof: Let S̃ = {𝔮_{1}, …, 𝔮_{m}}. We extend S̃ by a single prime 𝔯_{1} so that 𝔮_{1}𝔯_{1}, 𝔯_{1}𝔮_{2} are edges with 𝔯_{1}𝔮_{1} not an edge. Now extend the new graph Γ_{S̃∪{𝔯_{1}}}(p) by another prime 𝔯_{2} so that 𝔮_{2}𝔯_{2} and 𝔯_{2}𝔮_{2} are the only new edges. Continuing in this way, we see that we can extend ΓS̃(p) to a non-singular circuit ΓS̃′(p) having 2m vertices. If 1 ≤ i ≤ m, let v_{2i−1} = 𝔯_{i} and v_{2i} = 𝔮_{i}. Then v_{1} ⋯ v_{2m}v_{1} is the required non-singular circuit. □

Example: Let k = ℚ(ζ_{p} + ζ_{p}^{−1}), where p is an odd regular prime number and ζ_{p} a primitive p-root of unity. Then k has property (b) of theorem (2.5). Let T = {𝔭_{0}} where 𝔭_{0} is a prime of k which is inert in first step of the cyclotomic ℤ_{p}-extension of k. Then T has the property (∗). Let S̃ = {𝔮_{1}, …, 𝔮_{m}}, m ≥ 2, be a set of primes of k lying over pairwise different prime numbers q_{1}, …, q_{m} such that q_{i} ≡ 1 mod p, and 𝔭_{0} ∉ S̃. By (2.8), we can extend S̃ to a set S̃′ such that the linking diagram ΓS̃′(p) of (G(k_{S̃′∪S_{p}}^{T}(p)|k), S̃′) is a non-singular circuit.

References

[1] Koch, H. Galoissche Theorie der p-Erweiterungen. Deutscher Verlag der Wissenschaften (1970), English translation: Springer 2002

[2] Labute, J. Mild Pro-p-Groups and Galois Groups of p-Extensions of ℚ . J. Reine u. Angew. Math. 596 (2006), 155-182

[3] Neukirch, J. Algebraische Zahlentheorie. Springer 1992, English translation: Algebraic Number Theory. Springer 1999

[4] Neukirch, J., Schmidt, A., Wingberg, K. Cohomology of Number Fields. Springer 2000

[5] Schmidt, A. Circular sets of prime numbers and p-extensions of the rationals. J. Reine u. Angew. Math. 596 (2006), 115-130

[6] Vogel, D. Circular sets of primes of imaginary quadratic number fields. Preprints der Forschergruppe Algebraische Zykel und L-Funktionen Regensburg/Leipzig Nr. 5, 2006.

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