Positively ramified extensions of algebraic number fields

By Alexander Schmidt of Heidelberg

By a theorem of Grothendieck the structure of the étale fundamental group of a smooth, projective curve over an algebraically closed field is completely known for the part prime to the characteristic. Precisely there are 2g (g the genus) generators with the one defining relation

∏_{1}^{g}[x_{i}, y_{i}] = 1.

The proof uses the existence of a lift to characteristic 0 and eventually the topological structure of the associated manifold implies the result. For the substantial weaker statement about the structure of the maximal pro-p-factor (p a prime number different to the characteristic) there is also a purely algebraic proof, using Poincaré-duality and the classification of Demushkin groups (see [21]). The last approach should also be applicable to the number field case, where the passage to the algebraic closure of the constant field is replaced by the cyclotomic ℤ_{p}-extension. But the étale fundamental group is too small in the number field case and the cyclotomic extension itself is already ramified. Interpreting the results in the function field case in the sense ”unramified means, there is no tame ramification”, we get an analogous picture in the number field case: The maximal p-extension without tame ramification has an analogue structure like the fundamental group of a smooth,affine curve over a finite field. Both are (in the generic case) duality groups of dimension 2 and their dualizing modules have a similar description (see [20],[15]). It was the idea of K.Wingberg to compactify the affine curve Spec(O_{K}) by restricting the wild ramification using the infinite places of K. Starting from a CM-field K with maximal totally real subfield K^{+} and a prime number p, which satisfies the condition, that no prime over p splits in K/K^{+} he constructed a canonical p-extension K̃ over the cyclotomic ℤ_{p}-extension K_{∞} of K. The field K̃ is the maximal p-extension, which is unramified outside p and which is ”positively decomposed” at the places above p. The Galois group of K̃/K_{∞} has exactly the same structure as in the function field case. The idea underlying the construction of K̃ is to allow only such wild ramification, which locally at all places above p already comes from the totally real subfield K^{+}. The proof uses the duality theorem of Tate-Poitou and Iwasawa-theory, where the vanishing of the Iwasawa-μ-invariant (proved for abelian number fields) is always assumed. The aim of the present paper is to generalize Wingbergs construction to fields, that are not necessary of CM-type and also to study non-nilpotent extensions rather than p-extensions. Since the introduced extension type a posteriori only depends on the ramification behavior we prefer the name ”positive ramification”. Roughly speaking we extend Wingbergs results to fields, that are ”locally of CMtype” at all primes above p. This is the case for fields that are extensions of odd degree of an underlying CM-field and for prime numbers lying in a set P^{ns}(L) which has positive density and which will be defined in section 2. For a finite p-extension of number fields we proof a formula that is analogous to the Riemann-Hurwitz formula in algebraic geometry and which generalizes Kida’s formula of Iwasawa-theory. In section 7 we investigate an associated zeta-function, which enjoys a functional equation as soon as a primitive p^{th}-root of unity is contained in the ground field. According to the main conjecture of Iwasawa-theory (proved by Mazur and Wiles) whenever the ground field is of CM-type it splits off the 𝔭-adic zeta-function of the real subfield as its minus-part.

The author is deeply indebted to K.Wingberg for his interest and guidance.

1 The main results

Let L be a number field, which is contained in a Galois extension of odd degree of a CM-field or a function field of transcendental degree 1 over a finite constant field and let p be a prime number. In the number field case we assume that p ∈ P^{ns}(L) (see section 2), in the function field case we assume p ≠ char(L). In the number field case we denote by L̃(p) the maximal p-extension of L, that is unramified outside p and positively ramified at all places above p (see section 2). In the function field case we denote by L̃(p) the maximal unramified p-extension of L. Further let L_{∞,p} be the cyclotomic ℤ_{p}-extension of L. A primitive p^{th} root of unity is denoted by ζ_{p}. From now on we assume the Iwasawa-μ-invariant of the Λ-module Gal(L̃(p)/L_{∞,p})^{ab} to be 0 (Assumption 1, see section 9). Then the following holds:

Theorem 1 For ζ_{p} ∈ L the Galois group Gal(L̃(p)/L_{∞,p}) is as a pro-p-group generated by 2g_{p} generators x_{1}, …, x_{2g_{p}} with the one defining relation

[x_{1}, x_{2}] ⋯ [x_{2g_{p}−1}, x_{2g_{p}}] = 1.

Here g_{p} is a non-negative integer, that we call the p-genus of L. The group Gal(L̃(p)/L_{∞,p}) is a Demushkin group for g_{p} ≠ 0.

Remark: In the function field case it holds g_{p} ≤ g, where g is the genus of the smooth, projective curve which is associated to L. Equality holds if and only if the p-torsion points of the Jacobian are already rational. It would be interesting to know, whether there is a bound for the p-genera independent of p in the number field case too. By a general theorem about the behavior of Poincaré-groups in group extensions we get the Corollary 2 For ζ_{p} ∈ L the Galois group Gal(L̃(p)/L) is isomorphic to ℤ_{p} or a Poincaré-group of dimension 3 with dualizing module μ_{p^{∞}}.

For a p-extension M/L, which is positively ramified at p in the number field case we get a formula which is quite similar to the Riemann-Hurwitz-formula for smooth, projective curves: Proposition 3

2g_{p}(M) − 2 = [M_{∞,p} : L_{∞,p}](2g_{p}(L) − 2) + ∑_{𝔭∤p} (e_{𝔭}(M_{∞,p}/L_{∞,p}) − 1)

where e_{𝔭} denotes the ramification index of the prime 𝔭 of M_{∞,p} in M_{∞,p}/L_{∞,p}.

Remark: The above formula is a natural generalization of Kida’s formula (see [10]). Now we consider the analogy to the well-known structure of the fundamental group of a smooth, in finitely many points punctured Riemann surface. For a finite set of places S (S ⊇ S_{p} in the number field case) we denote by L̃_{S}(p) the maximal outside S unramified and in the number field case at p positively ramified p-extension of L. As usual we denote the maximal p-extension of a field L by L(p). Then it holds:

Theorem 4 For ζ_{p} ∈ L and S⊋S_{p} the group Gal(L̃_{S}(p)/L_{∞,p}) is a free pro-p-group of rank 2g_{p}(L) + #(S\S_{p}(L_{∞,p})) − 1 and has a presentation by generators

x_{1}, ⋯, x_{2g_{p}}, u_{𝔭}; 𝔭 ∈ S\S_{p}(L_{∞,p})

and the one relation

( ∏_{𝔭∈S\S_{p}(L_{∞,p})} u_{𝔭})[x_{1}, x_{2}] ⋯ [x_{2g_{p}−1}, x_{2g_{p}}] = 1,

where the x_{i}Gal(L̃_{S}(p)/L̃(p)) generate the group Gal(L̃(p)/L_{∞,p}) and there are suitable prolongations 𝔓|𝔭 to L(p), 𝔭 ∈ S\S_{p}(L_{∞,p}) such that

⟨u_{𝔭}⟩ = T_{𝔓}(L(p)/L_{∞,p}) ⊆ Gal(L̃_{S}(p)/L_{∞,p}), where

T_{𝔓}(L(p)/L_{∞,p}) denotes the inertia group for the prime 𝔓 in the extension L(p)/L_{∞,p}.

In order to prove theorem 4 one needs an analog to the Riemann existence theorem. Here we can omit the assumption on the finiteness of the considered sets of places and we do not assume the existence of a primitive p^{th} root of unity in the ground field. To formulate the proposition we need the notion of a free product of profinite groups over a locally profinite index set, which we will develop in section 3 as a natural generalization of the free products, defined by Neukirch ([13]).

Proposition 5 Let S ⊇ T (⊇ S_{p} in the number field case) be not necessary finite sets of primes of L_{∞,p}. Then there are the following decompositions into free pro-p-products:

⋆_{𝔭∈S\T (L̃_{T} (p))} T_{𝔭}(L(p)/L)⟶^{∼}Gal(L̃_{S}(p)/L̃_{T} (p)), if ζ_{p} ∉ L or T ≠ S_{p} or g_{p}(L) ≠ 0,

and

⋆_{𝔭≠𝔭_{0}^{𝔭∈S\Sp(L∞,p)}} T_{𝔭}(L(p)/L)⟶^{∼}Gal(L̃_{S}/L_{∞,p}), if ζ_{p} ∈ L and g_{p}(L) = 0.

In the second case 𝔭_{0} denotes an arbitrary prime in S\S_{p}(L_{∞,p}). The maps above are induced by

T_{𝔭}(L(p)/L) = T_{𝔭}(L(p)/L̃_{T} (p)) ⟶ Gal(L(p)/L̃_{T} (p)) −_{↠} Gal(L̃_{S}(p)/L̃_{T} (p)).

Analogous to the étale site over a function field (see [17]) there is a degeneration in absence of a p^{th} root of unity in the ground field for the positively ramified extensions of a number field, too.

Theorem 6 For ζ_{p} ∉ L the Galois group Gal(L̃(p)/L_{∞,p}) is a free pro-p-group of finite rank.

This rank depends on certain Iwasawa-λ-invariants.

Corollary 7 For ζ_{p} ∉ L the Galois group Gal(L̃(p)/L) is isomorphic to ℤ_{p} or a duality group of dimension 2.

Setting δ(F) = 1 for a field F, which contains the p^{th} roots of unity and δ(F) = 0 otherwise, the following holds for a finite set of places S ⊃ S_{p}:

Proposition 8 For ζ_{p} ∉ L the group Gal(L̃_{S}(p)/L_{∞,p}) is a free pro-p-group of rank

rg(Gal(L̃(p)/L_{∞,p})) + ∑_{𝔭∈S\S_{p}(L_{∞,p})} δ((L_{∞,p})_{𝔭}).

Now the question arises, whether it is possible to proof analogous results for nonnilpotent extensions, too. In the function field case this can be done by means of the GAGA-principle and topological methods, but most of these techniques applied in our situation fail. However, using group theoretic results we can proof weakened variants also for the number field case. For a field L which is contained in an odd extension of a CM-field we introduce the following notations:

L(odd) the maximal Galois extension of odd degree of L L̃ the maximal subextension of L in L(odd), which is positively ramified at all

primes of P^{ns}(L) and which has no tame ramification (which exists and is normal over L).

Our results are formulated in terms of profinite duality groups, the definition of which we briefly recall: A profinite group G is called strict Cohen-Macaulay at a prime number p if G has finite cohomological p-dimension n and if the p-dualizing module I_{p,G} = lim ; ⟶_{{U,cor^{∗} \atop m}}H^{n}(U, ℤ/p^{m}ℤ)^{∗}

has the following property:

For every finite p-torsion module M and all i the canonical trace map tr : H^{n}(G, I_{p,G}) → ℚ/ℤ induces an isomorphism

H^{n−i}(G, Hom(M, I_{p,G}))⟶^{∼}Hom(H^{i}(G, M), ℚ/ℤ).

One can show that then the module I_{p,G} is p-divisible. G is called of Poincaré-type at p , if I_{p,G} is isomorphic to ℚ_{p}/ℤ_{p} as an abelian group. In this case the cohomology of G with values in finite p-torsion modules is always finite ([18] Prop. 4.6.). If G is strict Cohen-Macaulay at p but not necessary of Poincaré-type, it is called a duality group at p, if the cohomology of G with values in finite p-torsion modules is always finite. Assume now, that assumption 1 is true for all finite extensions of L in L̃ and for all p ∈ P^{ns}(L). Then the following holds for the profinite group G = Gal(L̃/L) and all prime numbers p ∈ P^{ns}(L):

Theorem 9 Either the p-Sylow-subgroup of G is isomorphic to ℤ_{p} (in which case every positively ramified p-extension above L is cyclotomic) or the following holds: If ζ_{p} ∈ L̃ then G is of Poincaré-type of dimension 3 at p and the dualizing module is isomorphic to μ_{p^{∞}}. If ζ_{p} ∉ L̃ then G is a duality group at p of dimension 2.

In analogy to the function field case it would be natural to consider the situation over the field L_{∞}, i.e. over the composition of all cyclotomic ℤ_{p}-extensions. Lacking good finiteness theorems we restrict our consideration on an arbitrary chosen finite set of primes. We introduce the following notations:

C = C(L) any finite subset in P^{ns}(L) 𝔠 the category of finite groups the order of which

is divisible only by prime numbers in C

L̃(𝔠) the maximal pro-𝔠 subextension of L in L̃ L_{∞,C} the composite of the cyclotomic ℤ_{p_{i}}-extensions (p_{i} ∈ C) of L

By its definition the field L̃(𝔠) contains L_{∞,C}. We now assume assumption 1 to be true for all finite extensions of L in L̃(𝔠). We further assume the λ-invariants of the Iwasawa-modules in assumption 1 to be bounded when passing to L_{∞,C} (Assumption 2, see section 9). Then the following holds for the pro-𝔠-group G_{∞,C} := Gal(L̃(𝔠)/L_{∞,C})

Theorem 10 (i) If ζ_{p} ∈ L̃(𝔠) and p divides the order of G_{∞,C} then G_{∞,C} is of Poincaré-type of dimension 2 at p and the p-dualizing module is isomorphic to ℚ_{p}/ℤ_{p} with trivial action.

(ii) If ζ_{p} ∉ L̃(𝔠) then cd_{p}(G_{∞,C}) ≤ 1

(iii) The abelianization of every open subgroup U ⊆ G_{∞,C} is topologically finitely generated and for every prime number p ∈ C the maximal pro-p-factor group U(p) has the following structure:

(a) If ζ_{p} is not contained in the fixed field L̃(𝔠)^{U} of L̃(𝔠) under U then U(p) is a free pro-p-group of finite rank.

(b) If ζ_{p} ∈ L̃(𝔠)^{U} then U(p) is as a pro-p-group generated by 2g_{p}(U) generators x_{1}, ⋯ x_{2g_{p}(U)} with the one defining relation

[x_{1}, x_{2}] ⋯ [x_{2g_{p}(U)−1}, x_{2g_{p}(U)}] = 1.

One should remark, that the p-genera are different for different prime numbers. This is also true in the function field case, but there all p-genera are bounded by the genus of the associated smooth, projective curve and this genus is attained if sufficiently many prime numbers divide the degree. As it is well-known, the étale fundamental group of a smooth, projective curve over an algebraically closed field is topologically finitely generated. (see [4] X 2. Theorem 2.9.). Unfortunately there is still no pure algebraic proof for this fact. An analogous statement for the group G_{∞,C} = Gal(L̃(𝔠)/L_{∞,C}) would have far-reaching corollaries. For example one would have bounds for the growth of Iwasawa-invariants in positivelyramified extensions. At least the Riemann-Hurwitz-formula implies the

Proposition 11 Let K be a CM-field, ζ_{p} ∈ K and L/K an at p positively ramified p-extension. Then it holds

λ_{p}(L) ≤ 2 + [L_{∞,p} : K_{∞,p}](2λ_{p}^{−}(K) − 2) + ∑_{𝔭∤p} (e_{𝔭}(L_{∞,p}/K_{∞,p}) − 1)

where λ_{p}(L) denotes the Iwasawa-λ-invariant of L with respect to p and λ_{p}^{−}(K) is the minus part of the λ-invariant of K with respect to p.

The p-genera that have been introduced in this work are bounds for the Iwasawa-λ-invariants and they enjoy better functorial properties.

2 The definition of positive ramification

Let K be a finite number field of CM-type, i.e. K is a finite extension of ℚ, totally imaginary and of degree 2 over a totally real subfield K^{+}. As already mentioned, the idea of positive ramification in the CM-case is, to allow only that wild ramification, which locally already comes from the totally real subfield. This yields a reasonable restriction only for that primes, which are not above 2, and which do not split in the extension K/K^{+}. We define

P^{ns}(K) =^{def} {prime numbers p ≠ 2 | no prime in S_{p}(K^{+}) splits in K}.

An extension of the theory of positively ramified extensions is possible to such number fields, for which we can construct a local analog of the totally real subfield at all primes above p. We carry out such a construction for certain odd extensions of the CM-field K, which we call admissible at p.

We call a field extension odd, if it is contained in a Galois extension of odd degree and denote the maximal odd extension of a field F by F (odd). Definition: An extension L ⊂ K(odd) of K is called admissible at p ∈ P^{ns}(K) if for every prime 𝔭|p of L it holds:

L_{𝔭} ⊂ K+_{𝔭}(odd) · K_{𝔭} .

Obviously the composite of two admissible at p extensions of K is admissible at p again and we denote the maximal odd extension of K, that is admissible at p by K^{ad,p}(odd). Since a conjugate of an admissible extension is admissible too, we see that K^{ad,p}(odd) is Galois over K^{+}. As a factor group of Gal(K(odd)/K) the Galois group of K^{ad,p}/K is characterized by the following proposition. As usual we denote the commutator xyx^{−1}y^{−1} by [x, y] and the inertia group resp. decomposition group of a prime ideal 𝔭 in a field extension by T_{𝔭}(−) resp. D_{𝔭}(−). We denote the normal subgroup, generated by a subgroup U by (U).

Proposition 2.1 Let p ∈ P^{ns}(K). Then it holds:

Gal(K^{ad,p}(odd)/K) = Gal(K(odd)/K)/([T_{𝔭}(K(odd)/K), ρ_{𝔭}])_{𝔭∈S_{p}(K(odd))}

where ρ_{𝔭} is an arbitrary lift of the generator of Gal(K/K^{+}) to D_{𝔭}(K(odd)/K^{+}) and 𝔭 runs through the places over p of K(odd).

Proof: From the definition of admissible extensions it follows that

Gal(K^{ad,p}(odd)/K) = Gal(K(odd)/K)/([D_{𝔭}(K(odd)/K), ρ_{𝔭}])_{𝔭∈S_{p}(K(odd))},

since a Galois extension M/K+_{𝔭} of degree 2n, n odd, is contained in K+_{𝔭}(odd) · K_{𝔭} if and only if the sequence

1 → Gal(M/K_{𝔭}) → Gal(M/K+_{𝔭}) → Gal(K_{𝔭}/K+_{𝔭}) → 1

splits into a direct sum. We consider the exact sequence of groups with action of

Gal(K_{𝔭}/K+_{𝔭}) =< ρ_{𝔭} >≅ ℤ/2ℤ:

1 → T_{𝔭}(M/K_{𝔭}) → Gal(M/K_{𝔭}) → Gal(M̄/K̄_{𝔭}) → 1,

where M̄ denotes as usual the residue class field of M. The action on the cokernel is trivial, which is trivial in the case that K_{𝔭}/K+_{𝔭} is ramified and which follows otherwise from the fact, that the absolute Galois group of a finite field is abelian. Denoting the subgroup of ρ_{𝔭}-invariant elements of a group G by G^{ρ_{𝔭}} we get the following diagram, the columns of which are exact, because all occurring groups are of odd order, i.e. of order prime to the order of the acting group.

1 → T_{𝔭}(M/K_{𝔭}) → Gal(M/K_{𝔭}) → Gal(M̄/K̄_{𝔭}) → 1
[figure: ∪; ∪; ∥]
1 → T_{𝔭}(M/K_{𝔭})^{ρ_{𝔭}} → Gal(M/K_{𝔭})^{ρ_{𝔭}} → Gal(M̄/K̄_{𝔭})^{ρ_{𝔭}} → 1 .

Hence ρ_{𝔭} acts trivially on T_{𝔭}(M/K_{𝔭}) if and only if it acts trivially on Gal(M/K_{𝔭}). Therefore the normal subgroup in Gal(K(odd)/K) which is generated by the commutator [T_{𝔭}(K(odd)/K), ρ_{𝔭}] is the same as the normal subgroup that is generated by the commutator [D_{𝔭}(K(odd)/K), ρ_{𝔭}]. □ Remark: In particular the property of an odd extension to be admissible at p ∈ P^{ns}(K) only depends on the inertia groups at the places above p. The extensions of K that are admissible at p are just the extensions which are at all places over p ”locally of CM-type”. Actual this mean the following: Let L/K be an odd extension, admissible at p ∈ P^{ns}(K) and 𝔭 ∈ S_{p}(L) a prime. Let

L ⊂ M ⊂ K^{ad,p}(odd),

be a finite extension, which is Galois over K^{+}. We choose a prolongation 𝔓 of 𝔭 to M. Obviously M/K is also admissible at p, and hence the decomposition group D_{𝔓}(M/K^{+}) contains exactly one involution

ρ_{𝔓} : M⟶M,

that extends the complex conjugation ρ ∈ Gal(K/K^{+}). Though we do not have necessarily ρ_{𝔓}(L) ⊂ L, i.e. the field of ρ_{𝔓}-invariant elements of L has in general an index greater than 2 in L: However locally the situation is better: The Galois group

Gal(M_{𝔓}/K+_{𝔭}) = D_{𝔓}(M/K^{+})

is of the form ℤ/2ℤ ⊕ H, where H is a group of odd order. Hence L_{𝔭} corresponds to a subgroup of the form 0 ⊕ U with a subgroup U of H. Therefore

ρ_{𝔓} : M_{𝔓}⟶M_{𝔓}

leaves the subfield L_{𝔭} ⊂ M_{𝔓} invariant and the fixed field corresponds to the subgroup ℤ/2ℤ ⊕ U and has index 2 in L_{𝔭}. Denoting the fixed field Lρ_{𝔓}_{𝔭} by L+_{𝔭} we conclude that it is characterized as the unique intermediate field K+_{𝔭} ⊂ L+_{𝔭} ⊂ L_{𝔭} of index 2 in L_{𝔭}. Since M/K^{+} is a Galois extension of degree 2n, n odd, we conclude from the Sylow theorems, that all elements of order 2 in Gal(M/K^{+}) are conjugated. Hence the involution ρ_{𝔓} is the generator of the decomposition group for an infinite prime of M. In other words, the involution ρ_{𝔭} ∈ Gal(L_{𝔭}/L+_{𝔭}) is induced by the complex conjugation with respect to an imbedding of L to ℂ. An involution in Gal(M/K^{+}), which is conjugated to ρ_{𝔓} in Gal(M/K^{+}) (and all are) generates the decomposition group for a prime, which is conjugated to 𝔓 over K^{+}. Putting all together we have proved the following:

Lemma 2.2 Let L be an odd extension of K, that is admissible at p ∈ P^{ns}(K). Then the following holds:

(i) For every prime 𝔭|p of L there exists exactly one intermediate field L_{𝔭} ⊃ L+_{𝔭} ⊃ K+_{𝔭}

with [L_{𝔭} : L+_{𝔭}] = 2.

(ii) The involution ρ_{𝔭}, that generates the group Gal(L_{𝔭}/L+_{𝔭}) is induced by the complex conjugation with respect to an imbedding of L to ℂ.

(iii) Vice versa, to every complex embedding of L there exists a prime 𝔭 over p, such that the generator of Gal(L_{𝔭}/L+_{𝔭}) is induced by the complex conjugation.

For an odd extension L/K we define:

P^{ns}(L) := {p ∈ P^{ns}(K) | L/K is admissible at p}.

The prime numbers in P^{ns}(L) are those, for which we can extend the definition of positive ramification to the field L.

Lemma 2.3 The set P^{ns}(L) has a density of at least 1/n, where n is the absolute degree of the Galois closure of L over ℚ.

Proof: We denote the Galois closure of L over ℚ by N. The complex conjugations with respect to the archimedian primes of N define a conjugacy class in Gal(N/ℚ). Now the Chebotarev density theorem tells us, that there is a set of prime numbers of density 1/[N : ℚ] such that this conjugacy class is realized by the Frobenius automorphisms. It is easy to see that all prime numbers in this set are in P^{ns}(L). □

Remark: The local extensions L_{𝔭}/L+_{𝔭} can be unramified as well as ramified. However as one easily sees from the above proof, for a set of primes of positive density we are in the unramified situation.

Definition: Let L be admissible at p. An odd extension M/L is called positively ramified at p ∈ P^{ns}(L), if for every prime 𝔭|p of M it holds:

(i) The ramification index e_{𝔭} of 𝔭 in M/L is a p-power, i.e. at most pure-wild ramification occurs.

(ii) M_{𝔭} ⊂ L+_{𝔭}(odd) · L_{𝔭},

where the fields L+_{𝔭} in (ii) are just the fields from lemma 2.2.

Corollary 2.4 (i) Conjugates and composites of at p positively ramified extensions are positively ramified at p.

(ii) If E/L is admissible at p and M/L positively ramified at p then EM/E is positively ramified at p.

If L_{1}/L is positively ramified at p ∈ P^{ns}(L) then it holds p ∈ P^{ns}(L_{1}).

Corollary 2.5 If L_{2}/L_{1}/L is a tower of odd extensions with p ∈ P^{ns}(L) be a prime number. The extension L_{2}/L is positively ramified at p if and only if the extensions L_{1}/L and L_{2}/L_{1} are positively ramified at p.

By these corollaries the maximal at p ∈ P^{ns}(L) positively ramified p-extension L^{pos,p}(p) of L exists, is normal over L and we can describe the Galois group Gal(L^{pos,p}(p)/L) analogous to proposition 2.1.

Proposition 2.6 Let p ∈ P^{ns}(L). Then it holds:

Gal(L^{pos,p}(p)/L) = Gal(L(p)/L)/([T_{𝔭}(L(p)/L), ρ_{𝔭}])_{𝔭∈S_{p}(L(p))}

where ρ_{𝔭} is an arbitrary lift of the generator of Gal(L_{𝔭}/L+_{𝔭}) to Gal(L(p)_{𝔭}/L+_{𝔭}) , and 𝔭 runs through the places of L(p) over p.

For p ∈ P^{ns}(L) the cyclotomic ℤ_{p}-extension L_{∞,p}/L is positively ramified at p. This is trivial because of L_{∞,p} = Lℚ_{∞,p}.

3 Free products of profinite groups over locally profinite index sets

The free product in the category of pro-𝔠-groups (𝔠 a thick, full subcategory of the category of finite groups) was defined by Neukirch [13] by the universal property:

Hom( ⋆_{j∈J} H_{j}, G) = lim ; ⟵_{U} ⨁_{j∈J} Hom(H_{j}, G/U) ,

where U runs over the open normal subgroups of G. For a finite index set J this is the categorical coproduct. In general this definition is suited for its application on Galois groups of global fields and takes the fact into account, that only finitely many prime ideals ramify in a finite extension of finite global fields. Applying corresponding techniques also to infinite global fields we have to investigate further work. As usual we call a map of sets finite if the preimages of finite subsets are finite. Now let (I_{α})_{α∈A} be a directed, projective system of index sets, where all transition maps π_{α,β} (α > β) are finite. The set I := lim ; ⟵_{A}I_{α} carries a locally profinite topology.

Assume that for every α ∈ A a family (H_{i_{α}})_{i_{α}∈I_{α}} of profinite groups is given and we have continuous group homomorphisms

ϕ_{α,β} : H_{i_{α}}⟶H_{π_{α,β}(i_{α})},

satisfying the usual compatibilities. For i ∈ I we define H_{i} =^{df} lim ; ⟵_{α∈A}H_{π_{α}(i)}. We define the free product of the H_{i} by the

formula

⋆_{i∈I} H_{i} =^{df} lim ; ⟵_{α∈A} ⋆_{i_{α}∈I_{α}} H_{i_{α}}

where the free products on the right side are the free product defined by Neukirch.

Remark: In all our applications the family {H_{i}}_{i∈I} will be locally constant in the sense, that for every i ∈ I there is an open neighborhood U of i such that for all j ∈ U there is an isomorphism H_{i} ≅ H_{j}. In this case the free product that we have just defined agrees with the free product over the topologized index set I defined by Gildenhuys and Ribes (see [7] and [8] Prop. 3.1) and fulfills an universal property involving a certain étale space E which is build of the H_{i}’s by a patching procedure (loc.cit.). However the above isomorphism H_{i} ≅ H_{j} is not canonical and since we frequently use the action of Galois groups on free products we have to work with the ad hoc but canonical definition given above. Examples: (A) Let K be a not necessary finite global field and S ⊂ Spec(O_{K}) a subset, such that the restriction of the Zariski topology of Spec(O_{K}) on S is the cofinite topology. Let L/K be an arbitrary Galois extension. We define the free product of inertia groups

⋆_{𝔓∈S(L)} T_{𝔓}(L/K)

by the following data:

1. L = ⋃_{α∈A} K_{α} with K_{α}/K finite Galois , A an index set.

2. I_{α} = S(K_{α})

3. H_{i_{α}} := T_{i_{α}}(K_{α}/K). The maps are the canonical ones. We have I = S(L) and for 𝔭 ∈ I: H_{𝔭} = T_{𝔭}(L/K). (B) Assume that in addition to the data of (A) we have further given a Galois intermediate field K ⊂ M ⊂ L. We define the free product of inertia groups

⋆_{𝔭∈S(M)} T_{𝔓}(L/K)

by the following data:

1. For every 𝔭 ∈ S(M) we choose a fixed prolongation 𝔓 ∈ S(L).

2. I_{α} := { _{ 𝔭̃} ∈ S(K_{α}) — one of the 𝔓’s fixed in 1. lies over _{ 𝔭̃} }

3. H_{i_{α}} = T_{i_{α}}(K_{α}/K).

Remark:

1. We have a natural inclusion

⋆_{𝔭∈S(M)} T_{𝔓}(L/K) ⟶ ⋆_{𝔓∈S(L)} T_{𝔓}(L/K) ,

which however depends on the choice of the prolongations of the primes in S(M) to primes in S(L).

2. The set of places S can be an arbitrary subset of the spectrum of the ring of integers of a field K if K is -a finite global field or -a constant field extension of a finite function field or -a cyclotomic ℤ_{p_{1},···,p_{n}}-extension of a finite number field.

Proposition 3.1 Let H = ⋆_{i∈I} H_{i} and A be a trivial H-module. Then the following holds:

H^{i}(H, A) = lim ; ⟶_{α∈A} ⨁_{i_{α}∈I_{α}} H^{i}(H_{i_{α}}, A) ,

where the transition map for α > β is given as follows:

\begin{matrix}H^{i}(H_{i_{β}}, A) & ⟶ & ⨁_{i_{α}∈I_{α}}H^{i}(H_{i_{α}}, A) \\ x & ↦ & ∑_{i_{α}∈π_{α,β}^{−1}(i_{β})}H^{i}(ϕ_{α,β})x . \end{matrix}

(Observe that the sum is finite.) Proof: This follows from the definition together with [13] Satz 4.1. □ One easily verifies that conjugation induces a continuous action of the Galois group G = Gal(L/K) on the free product of inertia groups constructed in example (A). Hence the cohomology of this free product carries in a canonical way the structure of a discrete G-module.

Proposition 3.2 For T = ⋆_{𝔓∈S(L)} T_{𝔓}(L/K) and a trivial T -module A it holds

H^{i}(T, A) ≅ ⨁_{𝔭∈S(K)} Coind_{G_{𝔓}(L/K)}^{G}H^{i}(T_{𝔓}(L/K), A) ,

where as usual 𝔓 is a fixed extension of 𝔭 to L.

Proof: As

⋆_{𝔓∈S(L)} T_{𝔓}(L/K) = ⋆_{𝔭∈S(K)} ⋆_{𝔓∈S(L) , 𝔓|𝔭} T_{𝔓}(L/K)

we assume without loss of generality that S(K) consists of exactly one prime 𝔭_{0}. We fix a prolongation 𝔓_{0} of 𝔭_{0} to L. Now we write both sides as limits, where U runs through the open normal subgroups of G. By proposition 3.1 it holds

H^{i}( ⋆_{𝔓∈S(L)} T_{𝔓}(L/K), A) = lim ; ⟶_{U} ⨁_{𝔭∈S(L^{U})} T_{𝔭}(L^{U}/K) .

Furthermore we have by definition

Coind_{G_{_{𝔓}_{0}}}^{G}H^{i}(T_{_{𝔓}_{0}}(L/K), A) = Hom_{G_{_{𝔓}_{0}}}(G, H^{i}(T_{_{𝔓}_{0}}(L/K), A)) =

= lim ; ⟶_{U}Hom_{G_{_{𝔓}_{0}}U/U}(G/U, H^{i}(T_{𝔓_{0}∩L^{U}} (L^{U}/K), A)).

Now we define the isomorphism on the level U by the following map:

\begin{matrix}Hom_{G_{_{𝔓}_{0}}U/U}(G/U, H^{i}(T_{𝔓_{0}∩L^{U}} (L^{U}/K), A)) & ⟶ & ⨁_{𝔭∈S(L^{U})} H^{i}(T_{𝔭}(L^{U}/K), A) \\ f & ↦ & ∑_{g∈G_{_{𝔓}_{0}}U/U} H^{i}(g)f(g^{−1}) \end{matrix}

where

H^{i}(g) : H^{i}(T_{𝔓_{0}∩L^{U}} (L^{U}/K), A)⟶H^{i}(T_{g(𝔓_{0}∩L^{U})}(L^{U}/K), A)

denotes the map which is induced by the isomorphism ∗^{g} : T_{𝔓_{0}∩L^{U}} ⟶^{∼}T_{g(𝔓_{0}∩L^{U})} on the i^{th} cohomology group. □ In the following we will investigate a situation which slightly generalizes example (A): Assume that for every 𝔓 ∈ S(L) we have given a subgroup T_{𝔓} ⊂ T_{𝔓}(L/K). Analogous to the preceding we define the group

⋆_{𝔓∈S(L)} T_{𝔓}.

As a further restriction we require the following condition (∗) to be satisfied: (∗): T_{𝔓} is a normal subgroup in G_{𝔓} and the isomorphisms g : T_{𝔓}(L/K)⟶^{∼}T_{g𝔓}(L/K) which are induced by the conjugation by g ∈ G map the groups T_{𝔓} isomorphically onto T_{g𝔓} .

If the T_{𝔓} ⊂ T_{𝔓}(L/K) satisfy condition (∗), we have a natural, continuous action of G on the free product of the T_{𝔓} and the inclusion

⋆_{𝔓∈S(L)} T_{𝔓} ⟶ ⋆_{𝔓∈S(L)} T_{𝔓}(L/K)

is a homomorphism of G-operator groups.

Proposition 3.3 Assume that the T_{𝔓} ⊂ T_{𝔓}(L/K) satisfy condition (∗). Then for a trivial module A it holds:

H^{i}( ⋆_{𝔓∈S(L)} T_{𝔓}, A) ≅ ⨁_{𝔭∈S(K)} Coind_{G_{𝔓}(L/K)}^{G}H^{i}(T_{𝔓}, A) ,

where as usual 𝔓 is a fixed prolongation of 𝔭 to L.

Proof: Analogous to the proof of proposition 3.2. □ Generalizing example (B) suppose that we have given a Galois intermediate field K ⊂ M ⊂ L. Analogous to example (B) we define the free product

⋆_{𝔭∈S(M)} T_{𝔓}.

Lemma 3.4 Let T_{𝔓} ⊂ T_{𝔓}(L/M), A be a trivial module and assume that for a fixed i the decomposition group G_{𝔓}(L/M) acts trivial on H^{i}(T_{𝔓}, A). Then G(M/K) acts in a natural way on the i^{th} cohomology group of the free product and the restriction homomorphism induced by the canonical inclusion:

H^{i}( ⋆_{𝔓∈S(L)} T_{𝔓}, A)⟶^{res}H^{i}( ⋆_{𝔭∈S(M)} T_{𝔓}, A).

induces an isomorphism of G(M/K)-modules:

H^{i}( ⋆_{𝔓∈S(L)} T_{𝔓}, A)^{G(L/M)}⟶^{∼}H^{i}( ⋆_{𝔭∈S(M)} T_{𝔓}, A).

Proof: By proposition 3.1 we have

H^{i}( ⋆_{𝔓∈S(L)} T_{𝔓}, A)^{G(L/M)} = lim ; ⟶_{U}( ⨁_{_{𝔭̄}∈S(L^{U})} H^{i}(T_{𝔭̄}, A))^{G(L/M)U/U} .

where U runs through the open normal subgroups of G and T_{𝔭̄} denotes the image of T_{𝔓} in T_{𝔭̄}(L^{U}/K), which obviously does not depend of the choice of the prolongation 𝔓. Remembering that we have fixed a prolongation 𝔓 to L for every 𝔭 ∈ S(M) we define for every open normal subgroup U:

I_{U} = {_{𝔭̃} ∈ S(L^{U})| one of the fixed 𝔓 lies over _{ 𝔭̃}}.

It holds

H^{i}( ⋆_{𝔭∈S(M)} T_{𝔓}, A) = lim ; ⟶_{U} ⨁_{_{𝔭̃}∈I_{U}} H^{i}(T_{𝔭̃}, A) .

Hence it is clear that the image of H^{i}(⋆_{𝔓∈S(L)} T_{𝔓}, A)^{G(L/M)} in H^{i}(⋆_{𝔭∈S(M)} T_{𝔓}, A) is

lim ; ⟶_{U} ⨁_{_{𝔭̃}∈I_{U}} H^{i}(T_{𝔭̃}, A)^{G_{𝔭̃}(L^{U} /M∩L^{U})} .

Since G_{𝔓}(L/M) acts trivial on H^{i}(T_{𝔓}, A) by assumption the lemma is proved. □ In the following it will also be necessary to consider factor groups :

As before let T_{𝔓} ⊂ T_{𝔓}(L/K) be given, satisfying condition (∗) and let H_{𝔓} =^{def} T_{𝔓}(L/K)/T_{𝔓}. We define the free product of the factors

⋆_{𝔓∈S(L)} H_{𝔓}

by the obvious data. Then we have a canonical map

ϕ : ⋆_{𝔓∈S(L)} T_{𝔓}(L/K)⟶ ⋆_{𝔓∈S(L)} H_{𝔓}.

Proposition 3.5 (i) ϕ is surjective.

(ii) ker(ϕ) is generated by ⋆_{𝔓∈S(L)} T_{𝔓} as a normal subgroup in ⋆_{𝔓∈S(L)} T_{𝔓}(L/K).

(iii) G acts continuously on H = ⋆_{𝔓∈S(L)} H_{𝔓} by conjugation and for a trivial Hmodule A it holds

H^{i}( ⋆_{𝔓∈S(L)} H_{𝔓}, A) ≅ ⨁_{𝔭∈S(K)} Coind_{G_{𝔓}(L/K)}^{G}H^{i}(H_{𝔓}, A).

Proof: (i) and (ii) follow from the definition of the free products and from the corresponding properties for the classical free products as they are formulated in [13] Satz 1.2. The proof of (iii) is analogous to the proof of proposition 3.2 and we leave it out . □ For our number theoretical applications we need a lemma, which supplies sufficient conditions for a Galois group to be the free pro-p-product of certain inertia groups.

Lemma 3.6 Let S be a set of places of the global field K such that the Zariski topology restricted to S is the cofinite topology. Let F be a Galois p-extension of K , which is unramified outside the places of S. Suppose that for every 𝔓 ∈ S(F) a fixed normal subgroup T_{𝔓} of T_{𝔓}(K(p)/F) is given, and the T_{𝔓} satisfy the condition (∗). Let further F ⊂ E ⊂ K_{S}(p) be an intermediate field lying Galois over K. Then the maps

ϕ_{𝔓} : T_{𝔓} ⟶ T_{𝔓}(K(p)/F) −_{↠} T_{𝔓}(E/F) ⟶ Gal(E/F)

induce an isomorphism

ϕ : ⋆_{𝔓∈S(F)} T_{𝔓}⟶^{∼}Gal(E/F)

if the following holds:

(i) H^{2}(Gal(E/F), ℚ_{p}/ℤ_{p}) = 0 , H^{1}(Gal(F/K), H^{1}(Gal(E/F), ℚ_{p}/ℤ_{p}) = 0 .

(ii) For 𝔓 ∈ S the group G_{𝔓}(E/F) acts trivial on T ab_{𝔓} for every 𝔓 ∈ S.

(iii) ϕ_{𝔓} induces an isomorphism

H^{1}(Gal(E/F), ℚ_{p}/ℤ_{p})^{Gal(F/K)}⟶^{∼} ⨁_{𝔭∈S(K)} H^{1}(T_{𝔓}, ℚ_{p}/ℤ_{p})^{G_{𝔓}(E/K)}.

Remark: The homomorphism in (iii) maps into the direct sum because the restriction of the Zariski topology on S(K) is the cofinite topology by assumption, and hence in every finite extension of F only prime ideals over finitely many places of K ramify. Proof of the lemma: Let C be the cokernel of the well-defined map ϕ. By lemma 3.4, (ii) and the Hochschild-Serre sequence it follows the exactness of

0 ⟶ H^{1}(C, ℚ_{p}/ℤ_{p})^{Gal(F/K)} ⟶ H^{1}(Gal(E/F), ℚ_{p}/ℤ_{p})^{Gal(F/K)}⟶^{ϕ̄}

⟶^{ϕ̄}H^{1}( ⋆_{𝔓∈S(F)} T_{𝔓}, ℚ_{p}/ℤ_{p})^{Gal(F/K)}.

By lemma 3.4 and proposition 3.3 the last term is isomorphic to

⨁_{𝔭∈S(K)} H^{1}(T_{𝔓}, ℚ_{p}/ℤ_{p})^{G_{𝔓}(E/K)}.

By (iii) it holds H^{1}(C, ℚ_{p}/ℤ_{p})^{Gal(F/K)} = 0 , hence C = 0 since all occurring groups are p-groups. Let K be the kernel of ϕ. Because of (i) and applying the Hochschild-Serre sequence twice we get the exact sequence

0⟶H^{1}(Gal(E/F), ℚ_{p}/ℤ_{p})^{Gal(F/K)}⟶^{ϕ̄}H^{1}( ⋆_{𝔓∈S(F)} T_{𝔓}, ℚ_{p}/ℤ_{p})^{Gal(F/K)}⟶

⟶H^{1}(K, ℚ_{p}/ℤ_{p})^{Gal(E/K)}⟶0.

Again by lemma 3.4, proposition 3.3 and by condition (iii) it follows K = 0. □

4 Unit groups and Iwasawa-invariants

As before let L be an odd extension of a CM-field K and p ∈ P^{ns}(L) is a prime number that we will fix for the following. We denote the cyclotomic ℤ_{p}-extension of L by L_{∞} and the intermediate field of degree p^{n} over L by L_{n}. We define

U(L_{n}) = ⨁_{𝔭∈S_{p}(L_{n})}U^{1}(L_{n,𝔭})

E^{1,p}(L_{n}) -the global units of L_{n},

which are 1-units at all places above p

Ē(L_{n}) − the topological closure of E^{1,p}(L_{n}) in U(L_{n})

U = U_{∞} = lim ; ⟵_{n}U(L_{n})

Ē = lim ; ⟵_{n}Ē(L_{n}).

The weak Leopoldt-conjecture, which is proved for the cyclotomic ℤ_{p}-extension, claims:

Ē ≅ lim ; ⟵_{n}E^{1,p}(L_{n}) ⊗ ℤ_{p}.

We decompose for every 𝔭 ∈ S_{p}(L_{n}) the group U^{1}(L_{n,𝔭}) with respect to the action of the generating involution of Gal(L_{n,𝔭}/L_{n,𝔭}^{+}) (see lemma 2.2) in eigenspaces. So we get decompositions:

U(L_{n}) = U(L_{n})^{+} ⊕ U(L_{n})^{−},
U = U^{+} ⊕ U^{−}.

The following lemma should be seen in analogy to the fact, that the global sections of the structure sheaf of a smooth, projective curve over an algebraically closed field are just the elements of the ground field. As usual we set : δ = δ(L) equal to 1, if ζ_{p} ∈ L and 0 otherwise. Analogous δ_{𝔭} = δ(L_{𝔭}), δ+_{𝔭} = δ(L+_{𝔭}).

Lemma 4.1 Ē ∩ U^{−} ≅ ℤ_{p}(1)^{δ}.

Proof: Because of the weak Leopoldt-conjecture (see above) it suffices to prove the following : A global unit of L_{n}, which is contained locally at every place 𝔭|p in U^{1}(L_{n,𝔭})^{−}, is a root of unity. Let e be such an unit. Because of lemma 2.2 it holds ē = e^{−1} where ¯ is the complex conjugation with respect to an arbitrary chosen complex imbedding of L_{n}. Hence the unit e has the absolute value 1 at every archimedian prime of L_{n} and hence is a root of unity. □ The above groups are modules under the Iwasawa-algebra Λ = ℤ_{p}[[T]]. Under the assumption that all places above p are totally ramified in the extension L_{∞,p}/L they have the following structure:

Lemma 4.2

\begin{aligned}U^{+} & ≅ & Λ^{r_{2}} ⊕ ⨁_{𝔭∈S_{p}(L_{∞})} ℤ_{p}(1)^{δ+_{𝔭}} \\ U^{−} & ≅ & Λ^{r_{2}} ⊕ ⨁_{𝔭∈S_{p}(L_{∞})} ℤ_{p}(1)^{δ_{𝔭}−δ+_{𝔭}} \\ Ē & ≅ & Λ^{r_{2}} ⊕ ℤ_{p}(1)^{δ} \end{aligned}

where r_{2} = \frac{1}{2}[L : ℚ] is the number of complex primes of L.

Proof: The first two results follow from [22] Theorem 6.5. and the last is Proposition A1 in [23]. □ In the following we will always assume, that the places above p are totally ramified in L_{∞,p}/L and we reach this by substituting L by L_{n} with suitable n .

Corollary 4.3 U/U^{−}Ē is a Λ-torsion module.

5 The Iwasawa-module structure on the cohomology

In this section we fix the prime number p and the notations of the last section. For a set of primes S ⊃ S_{p} we denote by L̃_{S}(p) the maximal p-extension of L, which is positively ramified at p an unramified outside the places of S. Let T_{p} be the free product of inertia groups (see section 3)

T_{p} = ⋆_{𝔓∈S_{p}(L(p))} T_{𝔓}(L_{𝔓}(p)/L_{∞,𝔓})

and

T_{p}^{+} = ⋆_{𝔓∈S_{p}(L(p))} T_{𝔓}(L+_{𝔓}(p)/L_{∞,𝔓}^{+}),

where the +-fields are defined by lemma 2.2. We have a canonical surjection T_{p} −_{↠} T_{p}^{+} and we denote its kernel by T_{p}^{−}. Since the local maximal p-extension is realized by L(p) by the theorem of Grunwald-Hasse-Wang (see [12] proof of Theorem 11.3) we have an operation of G := Gal(L(p)/L_{∞}) on T_{p} and because of lemma 2.2 also on T_{p}^{+}. Considering the canonical homomorphism

ψ : T_{p}⟶G,

it is clear, that ψ(T_{p}^{−}) is a normal subgroup and by proposition 2.6 we get for a finite set of places S ⊇ S_{p} after composition with the canonical surjection G −_{↠}^{π} Gal(L_{S}(p)/L_{∞}) the equality

Gal(L̃_{S}(p)/L_{∞}) = Gal(L_{S}(p)/L_{∞})/π(ψ(T_{p}^{−})).

For a G-module M we denote the module of coinvariants under G by M_{G}.

Lemma 5.1 Local class field theory induces the commutative and exact diagram

0 → (T_{p}^{−}/[T_{p}^{−}, T_{p}])_{G} → (T_{p}^{ab})_{G} → (T_{p}^{+ ab})_{G} → 0
↑^{≀}  ↑^{≀}  ↑^{≀}
0 →  U^{−}  → U → U^{+} → 0

Proof : By Proposition 3.2 we have

H^{1}(T_{p}, ℚ_{p}/ℤ_{p})^{G} ≅ ⨁_{𝔭∈S(L_{∞})} H^{1}(T_{𝔓}, ℚ_{p}/ℤ_{p})^{G_{𝔓}}.

Hence

(T_{p}^{ab})_{G} = ⨁_{𝔭∈S_{p}(L_{∞})}(T ab_{𝔓})_{G_{𝔓}} ≅ U,

where the last isomorphism is induced by local class field theory. Analogously one gets by proposition 3.5

H^{1}(T_{p}^{+}, ℚ_{p}/ℤ_{p})^{G} ≅ ⨁_{𝔭∈S(L_{∞})} H^{1}(T_{𝔓}(L+_{𝔓}(p)/L_{∞,𝔓}^{+}), ℚ_{p}/ℤ_{p})^{G_{𝔓}},

and hence

(T_{p}^{+ ab})_{G} = ⨁_{𝔭∈S_{p}(L_{∞})} T_{𝔓}(L+_{𝔓}(p)/L_{∞,𝔓}^{+})_{G_{𝔓}}^{ab}.

The action of G_{𝔓} on T_{𝔓}(L+_{𝔓}(p)/L_{∞,𝔓}^{+}) factors over G_{𝔓}(L+_{𝔓}(p)/L_{∞,𝔓}^{+}), and hence local class field theory implies

(T_{p}^{+ ab})_{G} ≅ U^{+}.

For a 𝔓 ∈ S_{p}(L(p)) we set

T −_{𝔓} := ker(T_{𝔓}(L(p)/L_{∞}) → T_{𝔓}(L+_{𝔓}(p)/L_{∞,𝔓}^{+})),

and get in an analogous way

(T_{p}^{−}/[T_{p}^{−}, T_{p}])_{G} = ⨁_{𝔭∈S_{p}(L_{∞})}(T −_{𝔓}/[T −_{𝔓}, T_{𝔓}])_{G_{𝔓}} ≅ U^{−} .

Thus the lemma is proved. □ From the surjection T_{p}^{−} −_{↠} Gal(L_{S}(p)/L̃_{S}(p)) we get the surjection

U^{−} ≅ (T_{p}^{−}/[T_{p}^{−}, T_{p}])_{G} −_{↠} Gal(L_{S}(p)/L̃_{S}(p))_{G}^{ab},

since the action of T_{p} on the right side factors through T_{p}⟶^{can}G. Hence we get the injection

H^{1}(Gal(L_{S}(p)/L̃_{S}(p)), ℚ_{p}/ℤ_{p})^{G} ⟶ (U^{−})^{∗}.

By lemma 4.1 , lemma 5.1 and global class field theory we get for S = S_{p} the following commutative and exact diagram

0 → Ē → U → Gal(L_{S_{p}}(p)/L_{∞})^{ab} → Gal(L_{∞}^{unv}(p)/L_{∞})^{ab} → 0
[figure: ∪; ∪; ∥; (1)]
0 → ℤ_{p}(1)^{δ} → U^{−} → Gal(L_{S_{p}}(p)/L_{∞})^{ab} → Gal(L̃(p)/L_{∞})^{ab} → 0

where L_{∞}^{unv}(p) denotes the maximal (everywhere) unramified p-extension of L_{∞}. The snake lemma implies the exact sequence

0 → U/U^{−}Ē → Gal(L̃(p)/L_{∞})^{ab} → Gal(L_{∞}^{unv}(p)/L_{∞})^{ab} → 0,

and hence by corollary 4.3 the module M^{pos} := Gal(L̃(p)/L_{∞})^{ab} is a Λ-torsion module. From now on we will always assume the truth of the following conjecture. Assumption 1: The μ-invariant of the Λ-module M^{pos} is 0. This assumption is true for abelian number fields (using the theorem of Ferrero-Washington) and is in the case of CM-fields equivalent to the classical conjecture μ = 0 (see section 9). By the weak Leopoldt-conjecture it holds

H^{2}(Gal(L_{S_{p}}(p)/L_{∞}), ℚ_{p}/ℤ_{p}) = 0.

Dualizing the lower column of diagram 1 and comparing with the Hochschild-Serre sequence we get the following commutative and exact diagram

0  0
↓  ↓
H^{1}(Gal(L̃(p)/L_{∞}), ℚ_{p}/ℤ_{p}) = (Gal(L̃(p)/L_{∞})^{ab})^{∗}
↓  ↓
H^{1}(Gal(L_{S_{p}}(p)/L_{∞}), ℚ_{p}/ℤ_{p}) = (Gal(L_{S_{p}}(p)/L_{∞})^{ab})^{∗}
↓  ↓  (2)
H^{1}(Gal(L_{S_{p}}(p)/L̃(p)), ℚ_{p}/ℤ_{p})^{G} ⟶  (U^{−})^{∗}
↓  ↓
H^{2}(Gal(L̃(p)/L_{∞}), ℚ_{p}/ℤ_{p}) ⟶  ℤ_{p}(1)^{∗}^{δ}
↓  ↓
0  0  .

Denoting the kernel and cokernel of the p-multiplication map of an abelian group A by _{p}A and A_{p} the above diagram implies an injection

pH^{2}(Gal(L̃(p)/L_{∞}), ℚ_{p}/ℤ_{p}) ⟶ ℤ/pℤ(−1)^{δ}.

The exact sequence 0 → ℤ/pℤ → ℚ_{p}/ℤ_{p} →^{·p} ℚ_{p}/ℤ_{p} → 0 implies the exactness of the sequence 0 ⟶ H^{1}(Gal(L̃(p)/L_{∞}), ℚ_{p}/ℤ_{p})_{p} ⟶ H^{2}(Gal(L̃(p)/L_{∞}), ℤ/pℤ) ⟶

⟶ _{p}H^{2}(Gal(L̃(p)/L_{∞}), ℚ_{p}/ℤ_{p}) ⟶ 0.

Hence the existence of an isomorphism

H^{2}(Gal(L̃(p)/L_{∞}), ℤ/pℤ)⟶^{∼}_{p}H^{2}(Gal(L̃(p)/L_{∞}), ℚ_{p}/ℤ_{p})

is equivalent to the fact that Gal(L̃/L_{∞})^{ab} is torsion free.

Proposition 5.2 The Iwasawa-module Gal(L̃(p)/L_{∞})^{ab} has no nontrivial finite Λ-submodules.

Proof: First assume that ζ_{p} ∈ L. In this case we can apply the arguments from the proof of Lemma 2.3. in [23] : The assertion is invariant under twists and so it is enough to show, that

Gal(L̃(p)/L_{∞})^{ab}(−1)^{Γ_{n}}

is a free ℤ_{p}-module for large enough n. We twist the lower exact sequence of diagram 1 by (−1) and divide it into two short exact sequences:

\begin{aligned}0 & → & ℤ_{p} & → & U^{−}(−1) & → & C & → & 0 \\ 0 & → & C & → & Gal(L_{S_{p}}/L_{∞})^{ab}(−1) & → & Gal(L̃(p)/L_{∞})^{ab}(−1) & → & 0 & . \end{aligned}

From this we get the exact sequence

0 → C^{Γ_{n}} → Gal(L_{S_{p}}/L_{∞})^{ab}(−1)^{Γ_{n}} → Gal(L̃(p)/L_{∞})^{ab}(−1)^{Γ_{n}} → C_{Γ_{n}},

so that it is enough to show the following for large enough n:

(i) C_{Γ_{n}} is ℤ_{p}-free

(ii) C^{Γ_{n}} ⟶^{∼}Gal(L_{S_{p}}/L_{∞})^{ab}(−1)^{Γ_{n}}.

Denoting for a Λ-module M the Λ-torsion submodule by T_{Λ}M and defining F_{Λ}M := M/T_{Λ}M, we get the commutative diagram

F_{Λ}U^{−}(−1) ⟶^{∼} F_{Λ}C
[figure]
0 ⟶ ℤ_{p} ⟶ U^{−}(−1) ⟶ C ⟶ 0
[figure: ∥; ∪; ∪]
0 ⟶ ℤ_{p} ⟶ T_{Λ}U^{−}(−1) ⟶ T_{Λ}C ⟶ 0 .

The Λ-module F_{Λ}U^{−} is free by lemma 4.2, and hence the same is true for F_{Λ}C. So we get

F_{Λ}C^{Γ_{n}} = 0

and

F_{Λ}C_{Γ_{n}} is ℤ_{p}-free.

The imbedding

0 → ℤ_{p} → T_{Λ}U^{−}(−1) ≅ ∏_{𝔭∈S_{p}(L_{∞})} ℤ_{p}

is the diagonal imbedding and hence T_{Λ}C is a free ℤ_{p}-module and Γ_{n}-trivial for sufficiently large n. The exact sequence

0 → T_{Λ}C^{Γ_{n}} → C^{Γ_{n}} → F_{Λ}C^{Γ_{n}} → T_{Λ}C_{Γ_{n}} → C_{Γ_{n}} → F_{Λ}C_{Γ_{n}} → 0

shows that C^{Γ_{n}} and C_{Γ_{n}} are ℤ_{p}-free for sufficiently large n, and hence (i) is proved. In order to prove (ii) we can now substitute C^{Γ_{n}} by T_{Λ}C. In other words we have to show the exactness of the sequence

0 → ℤ_{p} → ∏_{𝔭∈S_{p}(L_{∞})} ℤ_{p} → Gal(L_{S_{p}}/L_{∞})^{ab}(−1)^{Γ_{n}} → 0.

By the weak Leopoldt conjecture

H^{2}(Gal(L_{S_{p}}(p)/L_{∞}), μ_{p^{∞}}) = 0

we have

\begin{aligned}Gal(L_{S_{p}}(p)/L_{∞})^{ab}(−1)^{Γ_{n}} & ≅ & (H^{1}(Gal(L_{S_{p}}(p)/L_{∞}), ℚ_{p}/ℤ_{p})(1)_{Γ_{n}})^{∗} \\ & ≅ & H^{2}(Gal(L_{S_{p}}(p)/L_{n}), μ_{p^{∞}})^{∗} \end{aligned}

For sufficiently large n we can identify the dualized sequence with a part of the long Tate-sequence (see [6] Theorem 1 Prop.22 Cor.1 ).

H^{2}(Gal(L_{S_{p}}(p)/L_{n}), μ_{p^{∞}}) → ∏_{𝔭∈S_{p}(L_{n})} H^{2}(G_{𝔭}(L̄/L_{n}), μ_{p^{∞}}) →

→ lim ; ⟶_{m}H^{0}(Gal(L_{S_{p}}(p)/L_{n}), ℤ/p^{m})^{∗} → 0

and hence (ii) and the lemma are proved if we show that

lim ; ⟶_{m}ker^{2}(L_{n}, μ_{p^{m}}) = 0.

Since the S_{p}-ideal class group of L_{n} is finite, we obtain

lim ; ⟶_{m}ker^{2}(L_{n}, μ_{p^{m}}) = (lim ; ⟵_{m}ker^{1}(L_{n}, ℤ/p^{m}))^{∗}
= (lim ; ⟵_{m}Hom(Cl_{S_{p}}(L_{n})/p^{m}, ℚ_{p}/ℤ_{p})^{∗}
= 0 .
□

Proof for the case ζ_{p} ∉ L: As we saw above we can solve the problem after twisting by (−1). This is not possible in the case ζ_{p} ∉ L. So it is a reasonable idea to adjoin the missing root of unity. But here a problem appears, which has its reason in the strong restrictions that we have to put on the field and the prime number. Under the given circumstances it is not necessary true that p ∈ P^{ns}(K(ζ_{p})). Intuitively L(ζ_{p})/K(ζ_{p}) should be admissible at p as a base change of an extension which is admissible at p, but we can not give a correct sense to this. Nevertheless we can save half of the arguments and this is enough for our purposes. We assume for simplicity that L is Galois over K^{+}. By Δ we denote the Galois group Gal(L_{∞}(ζ_{p})/L_{∞}). Let ρ be an arbitrary involution in Gal(L_{∞}/K_{∞}^{+}), which is induced by the complex conjugation associated to an imbedding of L_{∞} of ℂ. There is exactly on extension of ρ to L_{∞}(ζ_{p}), which is also induced by a complex conjugation and this extension commutes with all elements of Δ. We fix a place 𝔭 ∈ S_{p}(L_{∞}) and consider the Iwasawa-module

U_{𝔭} = ∏_{𝔓∈S_{𝔭}(L_{∞}(ζ_{p}))} U^{1}(L_{∞}(ζ_{p})_{𝔓}).

Let now ρ be that involution, which corresponds via proposition 2.2 to the place 𝔭. By the above arguments we get a decomposition of U_{𝔭} with respect to Δ and ρ:

U_{𝔭} = ∏_{χ∈Δ^{∗}} U χ,+_{𝔭} × U χ,−_{𝔭}.

Carrying out this decomposition for all 𝔭 ∈ S_{p}(L_{∞}) we get a decomposition

U(L_{∞}(ζ_{p})) = U(L_{∞}(ζ_{p}))^{+} ⊕ U(L_{∞}(ζ_{p}))^{−}

in Λ[Δ]-modules. Now we consider the following exact sequence of Λ[Δ]-modules which comes from global class field theory:

0⟶Ē(L_{∞}(ζ_{p}))⟶U(L_{∞}(ζ_{p}))⟶

⟶Gal(L(ζ_{p})_{S_{p}}(p)/L_{∞}(ζ_{p}))^{ab}⟶Gal(L_{∞}(ζ_{p})^{unv}(p)/L_{∞}(ζ_{p}))^{ab}⟶0,

and we show as in lemma 4.1:

Ē(L_{∞}(ζ_{p})) ∩ U(L_{∞}(ζ_{p}))^{−} = ℤ_{p}(1).

Hence we get an exact sequence of Λ[Δ]-modules

0 → ℤ_{p}(1) → U(L_{∞}(ζ_{p}))^{−} → Gal(L(ζ_{p})_{S_{p}}(p)/L_{∞}(ζ_{p})))^{ab} → C → 0,

where C is just defined by this sequence and it holds

C^{Δ} = Gal(L̃(p)/L_{∞})^{ab}.

Now we show as in the case ζ_{p} ∈ L, that C has no nontrivial finite Λ-submodules, and hence the same is true for for Gal(L̃(p)/L_{∞})^{ab}. □

Corollary 5.3 The group Gal(L̃(p)/L_{∞}) is a free pro-p group of finite rank or a Demushkin group with torsion free factor commutator group.

Proof: By lemma 5.2 the preceding implies that Gal(L̃(p)/L_{∞})^{ab} is a finitely generated, torsion free ℤ_{p}-module. Hence we get an injection

H^{2}(Gal(L̃(p)/L_{∞}), ℤ/pℤ) ⟶ ℤ/pℤ (−1)^{δ}.

If this injection is the zero map (this is necessary the case for δ = 0), then Gal(L̃(p)/L_{∞}) is free. Otherwise it is a pro-p-group with exactly one relation. The same is true if we substitute L by a finite extension and hence for every open subgroup U ⊂ Gal(L̃(p)/L_{∞}) we get a similar injection H^{2}(U, ℤ/pℤ) ⟶ ℤ/pℤ (−1). Hence by [3] Theorem, Gal(L̃(p)/L_{∞}) is a Demushkin-group. □ Applying the considerations following proposition 5.2 on a strictly larger set of places S⊋S_{p}, we get the exact sequence

U^{−}⟶^{ϕ}Gal(L_{S}(p)/L_{∞})^{ab}⟶Gal(L̃_{S}(p)/L_{∞})^{ab}⟶0.

Claim: ϕ is injective. Proof: Let U_{S\S_{p}} be the projective limit over the p-primary parts of the idele group of L_{n} with 1’s outside S\S_{p} and with units at all other places. Denoting as before the closure of the global units in U by Ē and the closure of the global units in U ×U_{S\S_{p}} by Ē_{(S)}, the canonical projection: pr : U ×U_{S\S_{p}}⟶U induces an isomorphism Ē_{(S)}⟶^{∼}Ē (see [21] Satz 1.5.). Denoting the kernel of ϕ by Y, we get the following commutative diagram with exact rows and columns:

0 → Ē_{(S)} → U × U_{S\S_{p}} → Gal(L_{S}(p)/L_{∞})^{ab} → Gal(L_{∞}^{unv}(p)/L_{∞})^{ab} → 0
[figure: ∪; ∪; ∥]
0 → Y → U^{−} × 1 → Gal(L_{S}(p)/L_{∞})^{ab} → Gal(L̃_{S}(p)/L_{∞})^{ab} → 0
[figure: ∩; ∩; id×0]
0 → Ē ∩ U^{−} → U^{−} → Gal(L_{S_{p}}(p)/L_{∞})^{ab} → Gal(L̃(p)/L_{∞})^{ab} → 0 .

Hence the isomorphism Ē_{(S)}⟶^{∼}Ē, which is induced by the canonical projection, induces an injection

Y = Ē_{(S)} ∩ (U^{−} × 1) ⟶ Ē ∩ U^{−} = ℤ_{p}(1)^{δ}.

But the intersection of ℤ_{p}(1) with U^{−} × 1 in U × U_{S\S_{p}} is obviously trivial and hence ϕ is injective. By dualizing and comparing with the Hochschild-Serre sequence the following diagram commutes

0  0
↓  ↓
H^{1}(Gal(L̃_{S}(p)/L_{∞}), ℚ_{p}/ℤ_{p}) = (Gal(L̃_{S}(p)/L_{∞})^{ab})^{∗}
↓  ↓
H^{1}(Gal(L_{S}(p)/L_{∞}), ℚ_{p}/ℤ_{p}) = (Gal(L_{S}(p)/L_{∞})^{ab})^{∗}
↓  ↓  (3)
H^{1}(Gal(L_{S}(p)/L̃_{S}(p), ℚ_{p}/ℤ_{p})^{G} ⟶  (U^{−})^{∗}
↓  ↓
H^{2}(Gal(L̃_{S}(p)/L_{∞}), ℚ_{p}/ℤ_{p}) ⟶  0
↓  ↓
0  0  .

Hence it holds

H^{2}(Gal(L̃_{S}(p)/L_{∞}), ℚ_{p}/ℤ_{p}) = 0.

Putting together our information about S and S_{p} we get the following diagram with exact rows and columns:

0  0  0
↓  ↓  ↓
0 → ℤ_{p}(1)^{δ} →  U_{S\S_{p}}  →  Y  → 0
↓  ↓  ↓
0 → U^{−} → Gal(L_{S}(p)/L_{∞})^{ab} → Gal(L̃_{S}(p)/L_{∞})^{ab} → 0 (4)
↓  ↓  ↓
0 → U^{−}/ℤ_{p}(1)^{δ} → Gal(L_{S_{p}}(p)/L_{∞})^{ab} → Gal(L̃(p)/L_{∞})^{ab} → 0
↓  ↓  ↓
0  0  0  .

For the exactness of the middle vertical sequence see [21] Satz 1.5. Now it holds

U_{S\S_{p}} ≅ ⨁_{𝔭∈S\S_{p}(L_{∞})} ℤ_{p}(1)^{δ(L_{∞,𝔭})}

and the imbedding of ℤ_{p}(1) (for δ = 1) is diagonal so that we get the following proposition.

Proposition 5.4 For S⊋S_{p} Gal(L̃_{S}(p)/L_{∞}) is a free pro-p-group of finite rank and the following formula holds:

rg(Gal(L̃_{S}(p)/L_{∞})) = rg(Gal(L̃(p)/L_{∞})) + ∑_{𝔭∈S\S_{p}(L_{∞})} δ(L_{∞,𝔭}) − δ.

Proof: Because H^{2}(Gal(L̃_{S}(p)/L_{∞}), ℚ_{p}/ℤ_{p}) = 0 it is obviously sufficient to show that Gal(L̃_{S}(p)/L_{∞})^{ab} is torsion free. This follows from the above diagram and from the fact, that Gal(L̃(p)/L_{∞})^{ab} is ℤ_{p}-free. The statement about the ranks follows immediately. □

6 Proofs of the main theorems

Proof of theorem 1: We show that, if ζ_{p} ∈ L and the group Gal(L̃(p)/L_{∞}) is nontrivial, then it is a Demushkin group. Since we already know, that the factor commutator group is torsion free, then the result follows by the classification of Demushkin groups (see [1] or [11]). Assume that L_{∞} ⊂ L′ ⊂ L̃(p) is a non trivial extension. We choose a finite strictly larger set of places S⊋S_{p} and consider the rank equality of proposition 5.4. Observing that every place not lying above p is totally decomposed in the extension L′/L_{∞} a simple calculation shows:

rg(Gal(L̃(p)/L′)) − 2
= rg(Gal(L̃_{S}(p)/L′)) − #(S\S_{p}(L′)) − 1
= [L′ : L_{∞}](rg(Gal(L̃_{S}(p)/L_{∞})) − 1) − [L′ : L_{∞}]#(S\S_{p}(L_{∞}))
= [L′ : L_{∞}](rg(Gal(L̃(p)/L_{∞})) − 2).

The result now follows from the main theorem of [3]. □ Proof of proposition 3: The Riemann-Hurwitz-formula follows in the same way as theorem 1. We choose a sufficiently large finite set of places S ⊃ S_{p}, such that M ⊂ L̃_{S}(p). Then we carry out a similar computation like in the last proof:

2g_{p}(M) − 2 = rg(Gal(M̃_{S_{p}}(p)/M_{∞})) − 2 = rg(Gal(M̃_{S}(p)/M_{∞})) − #(S\S_{p}(M_{∞})) − 1 = [M_{∞} : L_{∞}](rg(Gal(L̃_{S}(p)/L_{∞})) − 1) − [M_{∞} : L_{∞}]#(S\S_{p}(L_{∞}))

+ ∑_{𝔭∈S\S_{p}(M_{∞})}(e_{𝔭}(M_{∞}/L_{∞}) − 1)

= [M_{∞} : L_{∞}](rg(Gal(L̃(p)/L_{∞})) − 2) + ∑_{𝔭∈S\S_{p}(M_{∞})}(e_{𝔭}(M_{∞}/L_{∞}) − 1) = [M_{∞} : L_{∞}](2g_{p}(L) − 2) + ∑_{𝔭∈S\S_{p}(M_{∞})}(e_{𝔭}(M_{∞}/L_{∞}) − 1).

Thus proposition 3 is proved. □ Proposition 3 directly implies proposition 11 : To see this we take into consideration that

λ_{p}(L) ≤ 2g_{p}(L)

by diagram 1 in section 5, and we use the equality g_{p}(K) = λ_{p}^{−}(K) for the CM-field K (see corollary 9.2 of section 9). □ Proof of theorem 4: The first part of the statement is already contained in proposition 5.4, so it remains to show the statement about the presentation by in terms of generators and relations. Using diagram 4 in section 5 we get the exact sequence

0 → ℤ_{p}(1) → U_{S\S_{p}} → Gal(L̃_{S}(p)/L_{∞})^{ab} → Gal(L̃(p)/L_{∞})^{ab} → 0.

In the case g_{p}(L) = 0 the statement becomes obvious. Otherwise we set: H := Gal(L̃_{S}(p)/L̃(p)), G = Gal(L̃_{S}(p)/L_{∞}), D = Gal(L̃(p)/L_{∞}) and get

H/[H, G] = U_{S\S_{p}}.

We choose a basis ũ_{j} of H/[H, G] by taking the images of a fixed generator of U_{𝔭} ≅ ℤ_{p}(1), 𝔭 ∈ S\S_{p}. By x̄_{i} we denote arbitrary lifts of a basis of D according to theorem 1. Now the statement follows from the group theoretic result Lemma 2.4. in [21]. □ Proof of proposition 5: We have to verify the conditions of lemma 3.6 for K = L_{∞}, F = L̃_{T} (p), E = L̃_{S}(p), S = S\T, and T_{𝔓} = T_{𝔓}(L(p)/L_{∞}) for 𝔓 ∈ S\T. It is clear, that we can restrict to the case, where S and T are finite sets of places and then apply the results of section 5. It would also have been possible to skip the finiteness assumption for the sets of places already in section 5 . We assume S ≠ S_{p}, otherwise the statement of the proposition is empty. Condition (ii) of lemma 3.6 is automatically satisfied, since we work over the cyclotomic ℤ_{p}-extension which locally contains the maximal unramified p-extension at every prime not above p. By proposition 5.4 the pro-p-group Gal(L̃_{S}(p)/L_{∞}) is free, thus

H^{2}(Gal(L̃_{S}(p)/L̃_{T} (p)), ℚ_{p}/ℤ_{p}) = 0.

We consider the Hochschildt-Serre spectral sequence

E_{2}^{ij} = H^{i}(Gal(L̃_{T} (p)/L_{∞}), H^{j}(Gal(L̃_{S}(p)/L̃_{T} (p)), ℚ_{p}/ℤ_{p}))

⟹ H^{i+j}(Gal(L̃_{S}(p)/L_{∞}), ℚ_{p}/ℤ_{p}).

Corollary 5.3 and proposition 5.4 show, that the spectral term E_{2}^{ij} is non-zero at most for i = 0, 1, 2 and j = 0, 1. Because of H^{2}(Gal(L̃_{S}(p)/L_{∞}), ℚ_{p}/ℤ_{p}) = 0 we conclude the following equality

E_{2}^{1,1} = E_{∞}^{1,1} = 0.

But this is just the second condition (i) of lemma 3.6. To verify (iii) we first notice, that the following equality holds by local class field theory:

H^{1}(T_{𝔓}, ℚ_{p}/ℤ_{p})^{G_{𝔓}(L̃_{S}(p)/L_{∞})} ≅ U^{1}(L_{∞,𝔓}),

and that every 𝔓 ∈ S\T (L_{∞}) is totally decomposed in L̃_{T} (p)/L_{∞}. We first deal with the case T = S_{p}: Comparing diagram 4 of section 5 with the dual of the above spectral sequence we get the following diagram with exact rows:

0  0
↓  ↓
ℤ_{p}(1)^{δ}  ⟶ H^{2}(Gal(L̃(p)/L_{∞}), ℚ_{p}/ℤ_{p})^{∗}
↓  ↓
U_{S\S_{p}}  ⟶ Gal(L̃_{S}(p)/L̃(p))_{Gal(L̃(p)/L_{∞})}^{ab}
↓  ↓
Gal(L̃_{S}(p)/L_{∞})^{ab} =  Gal(L̃_{S}(p)/L_{∞})^{ab}
↓  ↓
Gal(L̃(p)/L_{∞})^{ab} =  Gal(L̃(p)/L_{∞})^{ab}
↓  ↓
0  0  .

We observe that condition (iii) is satisfied for δ = 0 or δ = 1 and g_{p} ≠ 0 (i.e. the upper horizontal arrow is an isomorphism), and that in the case δ = 1 and g_{p} = 0 condition (iii) is realized by omitting one of the prime ideals in S\S_{p}(L_{∞}). Assume now T ⊋S_{p}. Again we use diagram 4 of section 5 for S and for T. Since Gal(L̃_{T} (p)/L_{∞}) is free we get the following commutative diagram with exact rows and columns:

0  0
↓  ↓
ℤ_{p}(1)^{δ}  =  ℤ_{p}(1)^{δ}
↓  ↓
0 ⟶  U_{T \S_{p}}  ⟶  U_{S\S_{p}}  −_{↠}  U_{S\T}
↓  ↓  ↓^{≀}
0 ⟶ Gal(L̃_{T} (p)/L_{∞})^{ab} ⟶ Gal(L̃_{S}(p)/L_{∞})^{ab} −_{↠} Gal(L̃_{S}(p)/L̃_{T} (p))_{Gal(L̃_{T} (p)/L_{∞})}^{ab}
↓  ↓
Gal(L̃(p)/L_{∞})^{ab} = Gal(L̃(p)/L_{∞})^{ab}
↓  ↓
0  0  .

This proves the required isomorphism. By lemma 3.6 proposition 5 is proved. □ Theorem 6 has already been proved within the proof of corollary 5.3. The statement of proposition 8 is contained in proposition 5.4. The corollaries 2 and 7 eventually follow from the behavior of duality groups in group extensions (see [14] Th. 3.9 and [19] Th.1.4.). Theorem 10 will be proved in section 8.

7 The zeta-function

After illustrating the analogy of the positively ramified site to the étale site of a function field, we consider the associated zeta-function.

Let L be as before an odd extension of a CM-field K and p ∈ P^{ns}(L) a prime number. For the following we fix a generator γ of the Galois group Γ of the cyclotomic ℤ_{p}- extension of L. We set

H_{pos}^{i}(L_{∞}, ℤ/p^{n}ℤ) = H^{i}(Gal(L̃(p)/L_{∞}), ℤ/p^{n}ℤ)

and define the positive p-adic cohomology H_{pos}^{i}(L_{∞}, ℚ_{p}) as the p-adic cohomology of the Galois group Gal(L̃(p)/L_{∞}):

H_{pos}^{i}(L_{∞}, ℚ_{p}) = (lim ; ⟵_{n}H_{pos}^{i}(L_{∞}, ℤ/p^{n}ℤ)) ⊗_{ℤ_{p}} ℚ_{p} .

This cohomology admits a natural action of Γ = Gal(L_{∞}/L). We define the zetafunction L and γ by

Z_{γ}(L, t) = exp(∑_{i}(−1)^{i} ∑_{r=1}^{∞} T r(γ^{−r}, H_{pos}^{i}(L_{∞}, ℚ_{p}))).

By a standard procedure (see e.g. [9] Lemma 4.1.App.C) we can transform this in a multiplicative form:

Z_{γ}(L, t) = ∏_{i} det(1 − γ^{−1}t, H_{pos}^{i}(L_{∞}, ℚ_{p}))^{(−1)^{i+1}},

whence the zeta-function is rational: Z_{γ}(L, t) ∈ ℚ_{p}(t). Assume now ζ_{p} ∈ L. By q ∈ ℤ_{p}^{×} we denote the image of γ under the cyclotomic character

κ : Γ⟶ℤ_{p}^{×},

that is given by the action of Γ on the roots of unity. (In the function field case q is the number of elements in the constant field, when γ is the Frobenius automorphism.) Then the following is true.

Proposition 7.1 For ζ_{p} ∈ L it holds

Z_{γ}(L, t) = \frac{det(1 − γt, H_{pos}^{1}(L_{∞}, ℚ_{p}))}{(1 − t) · (1 − qt)}

and Z_{γ}(L, t) satisfies the functional equation

Z_{γ}(L, \frac{1}{qt}) = (t^{2}q)^{1−g_{p}(L)}Z_{γ}(L, t).

Proof: By theorem 1 Corollary 2 for all n the cup product induces a perfect pairing

H_{pos}^{1}(L_{∞}, ℤ/p^{n}ℤ) × H_{pos}^{1}(L_{∞}, ℤ/p^{n}ℤ)⟶^{∪}H_{pos}^{2}(L_{∞}, ℤ/p^{n}ℤ) ≅ ℤ/p^{n}ℤ(−1).

Passing to the limit and tensoring by ℚ_{p} we get a perfect Γ-equivariant pairing of finite dimensional ℚ_{p}-vector spaces:

H_{pos}^{1}(L_{∞}, ℚ_{p}) × H_{pos}^{1}(L_{∞}, ℚ_{p})⟶^{∪}H_{pos}^{2}(L_{∞}, ℚ_{p}) ≅ ℚ_{p}(−1).

The statement of the proposition now follows by linear algebra (see [9] Lemma 4.3. App. C). □

8 Non-nilpotent extensions

In order to investigate the group G = Gal(L̃/L) we need the following proposition where p is a prime number in P^{ns}(L) and G(p) denotes the maximal pro-p-factor group of the pro-finite group G.

Proposition 8.1 Assume that G(p) ≅ ℤ̸_{p}. Then the inflation map

H^{i}(G(p), M)⟶H^{i}(G, M)

induces isomorphisms for all i and every finite p-primary G(p)-torsion module M.

Proof: Using the Hochschildt-Serre spectral sequence it suffices to show for I := ker(G⟶G(p)):

cd_{p}(I) ≤ 1.

For this it is enough to show that cd_{p}(J(p)) ≤ 1 for every open subgroup J of I. Such a J corresponds to a field M with: L_{∞,p} ⊂ L̃(p) ⊂ M ⊂ L̃. Since J has finite index in I we find a finite extension N of L in L̃ such that M = L̃(p)N. Hence J(p) = Gal(M̃(p)/M) = Gal(Ñ(p)/M) is a closed subgroup of Gal(Ñ(p)/N_{∞,p}). If ζ_{p} ∉ N the statement about the cohomological dimension follows from theorem 6. If ζ_{p} ∈ N we know by theorem 1, that Gal(Ñ(p)/N_{∞,p}) is a Demushkin group. Since G(p) ≅ ℤ̸_{p} we know that the p-genus of L is not zero and hence J(p) is a closed subgroup of infinite index in a Demushkin group and therefore free by a standard argument. This finishes the proof. □ Now we are ready to proof theorem 9. Using the corollaries 2 and 7 we know the structure of U(p) for every open subgroup U ⊂ G. In particular we get that either the p-Sylow subgroup of G is equal to ℤ_{p} or cd_{p}(G) = 2 if ζ_{p} ∉ L̃ and cd_{p}(G) = 3 otherwise. By [18] Prop. 4.3. it remains to show the vanishing of certain terms D_{i}, the definition of which we recall:

D_{i}(G, ℤ/pℤ) =^{df} lim ; ⟶_{U,cor^{∗}}H^{i}(U, ℤ/pℤ)^{∗} .

Here U runs through the open subgroups of G. We have to show D_{0} = D_{1} = 0 for ζ_{p} ∉ L̃ and D_{0} = D_{1} = D_{2} = 0 for ζ_{p} ∈ L̃. However by proposition 8.1 is suffices to show the corresponding statements for the terms D_{i}(U(p), ℤ/pℤ) for sufficiently small open subgroups U of G. This again follows from the corollaries 2 and 7. Finally we have to analyze the structure of the dualizing module in the case ζ_{p} ∈ L̃. Again by proposition 8.1 we obtain for a sufficiently small open subgroup U ⊂ G:

H^{3}(U, μ_{p^{n}}) ≅ ℤ/p^{n}ℤ

for all n. From this, the statement about the dualizing module follows by a standard argument (see the proof of Theorem 1 in [16] II 5.2.). □ Now let C be any finite subset in P^{ns}(L) and G_{∞,C} = Gal(L̃(𝔠)/L_{∞,C}). Thus we have an exact sequence

1⟶G_{∞,C}⟶G⟶Gal(L_{∞,C}/L)⟶1.

By the results on the behaviour of duality groups in group extensions (see [14] [19]) and theorem 9 we get that part (i) of theorem 10 is equivalent to the fact, that the cohomology of G_{∞,p} with values in finite discrete modules is finite. Further it holds for every p ∈ P^{ns}(L):

G(p) = lim ; ⟵_{L⊂N⊂L_{∞,C}}Gal(Ñ(p)/N_{∞,p}).

Using proposition 8.1, theorem 1 and theorem 6 we see, that theorem 10 follows from the following Iwasawa-theoretic assumption: Assumption 2: The λ-invariant of the Iwasawa-module Gal(L̃(p)/L_{∞,p})^{ab} is bounded when passing to L_{∞,C\{p}}.

This assumption is at least true for abelian number fields (see section 9).

9 The Iwasawa-theoretic assumptions

We need two Iwasawa-theoretic assumptions in this paper, which we investigate in this section. Assumption 1: μ(M^{pos}) = 0 .

Proposition 9.1 (i) For CM-fields assumption 1 is equivalent to the classical assertion μ = 0, i.e. to the statement, that the μ-invariant of the maximal unramified abelian p-extension over the cyclotomic ℤ_{p}-extension vanishes.

(ii) Assumption 1 is true for abelian number fields.

(iii) Assumption 1 is stable under positively ramified p-extensions.

Proof: We use the notions of section 5. If L is of CM-type then there is a global action of the complex conjugation on diagram 1 in section 5. We get a decomposition in (+) and (−)-eigenspaces:

M^{pos} = Gal(L̃(p)/L_{∞})^{ab} = Gal(L_{S_{p}}^{+}(p)/L_{∞}^{+})^{ab} ⊕ Gal(L_{∞}^{unv}(p)/L_{∞})^{−},

which proves (i), since the μ-invariants of the two Iwasawa-modules on the right side agree (see [22] Cor.7.10). (ii) follows from (i) and from the theorem of Ferrero and Washington ([2]). (iii): Assume μ(M^{pos}(L)) = 0 and let L′/L be a p-extension, positively ramified at p. We choose a sufficiently large finite set of places S such that L′ ⊂ L̃_{S}(p). By proposition 4 the group Gal(L̃′(p)/L′_{∞,p}) is a homomorphic image of Gal(L̃_{S}(p)/L′_{∞,p}). The latter is finitely generated as an open subgroup of the finitely generated pro-p-group Gal(L̃_{S}(p)/L_{∞,p}). Hence also Gal(L̃′(p)/L′_{∞,p}) is finitely generated showing μ(M^{pos}(L′)) = 0. □

Corollary 9.2 For a CM-field K with ζ_{p} ∈ K it holds:

g_{p}(K) = λ_{p}^{−}(K)

where λ_{p}^{−}(K) is the minus-part of the Iwasawa-λ-invariant of K.

Let now C = {p_{1}, …, p_{n}} be any finite subset in P^{ns}(L) . We need the Assumption 2: λ(M_{p_{1}}^{pos}) keeps bounded when passing to the cyclotomic ℤ_{p_{2},...,p_{n}}- extension.

Proposition 9.3 (i) Assumption 2 is true for abelian number fields.

(ii) Assumption 2 is stable under positively ramified p-extensions.

Proof: Part (i) is follows from Corollary 9.2 and from the result of [5] on class groups in ℤ_{p_{1}} × ⋯ × ℤ_{p_{s}}-extensions. Part (ii) follows from proposition 3. □

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