ON THE IWASAWA THEORY OF p-ADIC LIE EXTENSIONS

OTMAR VENJAKOB

Abstract. In this paper the new techniques and results concerning the structure theory of modules over non-commutative Iwasawa algebras ([32],[34]) are applied to arithmetic: we study Iwasawa modules over p-adic Lie extensions k_{∞} of number fields k “up to pseudo-isomorphism”. In particular, a close relationship is revealed between the Selmer group of abelian varieties, the Galois group of the maximal abelian unramified p-extension of k_{∞} as well as the Galois group of the maximal abelian outside S unramified p-extension where S is a finite set of certain places of k. Moreover, we determine the Galois module structure of local units and other modules arising from Galois cohomology.

1. Introduction

The starting point of the Iwasawa theory of (non-commutative) p-adic Lie groups was M. Harris’ thesis [13] in 1979. For an elliptic curve E over a number field k without complex multiplication he studied the Selmer group Sel(E, k_{∞}) over the extension k_{∞} = k(E(p)) which arises by adjoining the p-division points of E to k. Then, the Galois group G = G(k_{∞}/k) is an open subgroup of Gl_{2}(ℤ_{p}) - due to a celebrated theorem of Serre [33] - and so a (compact) p-adic Lie group. Following Iwasawa’s general idea, he studied the Pontryagin dual Sel(E, k_{∞})^{∨} of the Selmer group as module over the Iwasawa algebra

Λ(G) = ℤ_{p}[[G]],

i.e. the completed group algebra of G with coefficients in ℤ_{p}. In the late 90^{th} J. Coates and S. Howson ([5], [6], [3], [15]) as well as Y. Ochi [28] revived this Iwasawa-theoretic approach. Among other things they proved a remarkable Euler characteristic formula for the Selmer group, studied ranks respectively torsion-properties and projective dimensions of standard local and global Iwasawa modules. The contributions of this work to the Iwasawa theory of p-adic Lie groups are obtained by applying some new techniques we have developed in [34]. There we introduced the concept of pseudo-null modules over Λ = Λ(G), which is based on a general dimension theory for Auslander regular rings (for the definition see subsection 2.2 and note that Λ is a non-commutative ring in general). Therefore it is fundamental for our applications that Λ(G) is an Auslander regular ring if G is a compact p-adic Lie group without p-torsion (cf. [34, Thm. 3.26]). As a first example in which this approach proves effective we consider the following generalization of a theorem of R. Greenberg [11] and T. Nguyen-Quang-Do [26] (who considered the case G ≅ ℤ_{p}^{d}): For a finite set S of places of a number field k let k_{∞}|k be a Galois extension unramified outside S such that the Galois group G = G(k_{∞}/k) is a torsion-free p-adic Lie-group and let k_{S} be the maximal outside S unramified extension of k. Then there is a basic result on the structure of the Galois group

2000 Mathematics Subject Classification. 11R23, 11R34, 11G10, 14K15.

Key words and phrases. Iwasawa theory, p-adic analytic groups, pseudo-isomorphism, abelian varieties.

X_{S} = G(k_{S}/k_{∞})^{ab}(p)

of the maximal abelian p-extension of k_{∞} unramified outside S considered as Λ(G)-module.

Theorem (Theorem 4.5) If H^{2}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p}) = 0, then the Λ(G)-module X_{S} does not contain any non-trivial pseudo-null submodule.

Once having available the concept of pseudo-null modules one is tempted to study Iwasawa modules ”up to pseudo-isomorphism”. We will write M ∼ N if there exists a Λ-homomorphism M → N whose kernel and cokernel is pseudo-null. Since in general ∼ is not a symmetric relation we consider also the quotient category Λ-mod/PN with respect to subcategory PN of pseudo-null Λ-modules, which is a Serre subcategory, i.e. closed under subobjects, quotients and extensions. If two Λ-modules M, N become isomorphic in this category, we shall write M ≡ N modPN where this time we forced ≡ to be an equivalence relation. Now it turns out that - as in the classical ℤ_{p}-extension case - the Λ(G)-module X_{S} is closely related to the modules X_{nr} and X_{cs}^{S} which denote the Galois groups of the maximal abelian unramified pro-p-extension of k_{∞} and the maximal abelian unramified pro-p-extension of k_{∞} in which every prime above S is completely decomposed, respectively.

Theorem (Theorem 4.9) If H^{2}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p}) = 0, μ_{p^{∞}} ⊆ k_{∞}, and dim(G_{ν}) ≥ 2 for all ν ϵ S_{f}, then

X_{nr}(−1) ∼ X_{cs}^{S}(−1) ∼ E^{1}(Y_{S}) ∼ E^{1}(tor_{Λ}Y_{S}) ≅ E^{1}(tor_{Λ}X_{S}).

If, in addition, G ≅ ℤ_{p}^{r}, r ≥ 2, then even the following holds:

X_{nr}(−1) ∼ X_{cs}^{S}(−1) ∼ (tor_{Λ}X_{S})^{∘},

where ^{∘} means that G operates via the involution g ↦ g^{−1}.

Here E^{1}(M) denotes the Iwasawa adjoint Ext_{Λ}^{1}(M, Λ) of a Λ-module M.

Drawing our attention to cohomology groups associated with an abelian varieties A defined over k, we set k_{∞} = k(A(p)) and mention that H^{1}(G_{S}(k_{∞}), A(p))^{∨} has no non-zero pseudo-null submodule (Theorem 4.38). With respect to the (p-)Selmer group Sel(A, k_{∞}) of A over k_{∞} = k(A(p)) we generalize a result of P. Billot in the case of good, supersingular reduction, i.e. Ã_{k_{ν}}(p) = 0, at any place dividing p. Over a ℤ_{p}-extension an analogous statement was proved by K. Wingberg [35, cor. 2.5]. We shall write A^{∨} for the dual abelian variety of A.

Assume that G(k_{∞}/k) is a pro-p-group without any p-torsion. Then the following holds (corollary 4.37):

X_{cs} ⊗_{ℤ_{p}} (A^{∨}(p))^{∨} ∼ E^{1}(tor_{Λ}Sel(A, k_{∞})^{∨}).

We also refer the reader to our joint work with Y. Ochi [29] where we prove under certain conditions that the Pontryagin dual of the Selmer group of an elliptic curve without CM and good ordinary reduction at any place dividing p does not contain any non-zero pseudo-null Λ-submodule.

Furthermore, we proved a structure theory for the ℤ_{p}-torsion part of a Λ-module M in [34]. Up to pseudo-isomorphism any ℤ_{p}-torsion Λ-module is of the form

⨁_{i} Λ/p^{n_{i}}.

In particular, we obtained a natural definition of the μ-invariant

μ(M) := ∑_{i} n_{i}(tor_{ℤ_{p}}M)

of M, which is additive on short exact sequences of Λ-torsion modules. Hence, we can formulate and prove a generalization of theorem 11.3.7 of [24]:

Theorem (Theorem 4.17) Let k_{∞}|k be a p-adic Lie extension such that G is without p-torsion and 𝔽_{p}[[G]] is an integral ring. Then G = G(k_{S}(p)/k_{∞}) is a free pro-p-group if and only if μ(X_{S}) = 0 and the weak Leopoldt conjecture holds, i.e. H^{2}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p}) = 0.

In theorem 4.18 we describe how the weak Leopoldt conjecture and the vanishing of μ(X_{S}) - if considered simultaneously - behave under change of the base field. Furthermore, we get a formula for the μ-invariants for different S.

We briefly outline further results. In section 3 we generalize Wintenberger’s result on the Galois module structure of local units. Let k be a finite extension of ℚ_{p} and assume that k_{∞}|k is a Galois extension with Galois group G ≅ Γ ⋊_{ρ} Δ, where Γ is a pro-p Lie group of dimension 2 (e.g. Γ = ℤ_{p} ⋊ ℤ_{p}) and Δ is a profinite group of possibly infinite order prime to p, which acts on Γ via ρ : Δ → Aut(Γ). Then we determine the Λ(G)-module structure of the Galois group G_{k_{∞}}^{ab}(p) = G(k_{∞}(p)/k_{∞}), where k_{∞}(p) is the maximal abelian p-extension of k_{∞}, see theorem 3.10. Then we apply these results to the local study of elliptic curves E with CM, i.e. we determine the structure of local cohomology groups with certain division points of E as coefficients. Section 4 is devoted to the study of “global” Iwasawa modules. Besides the themes already mentioned above we study the norm-coherent S-units of k_{∞}

𝔼_{S} := lim ; ⟵_{k′}(O_{k′,S}^{×} ⊗ ℤ_{p})

by means of Jannsen’s spectral sequence for Iwasawa adjoints. Using Kummer theory, we compare 𝔼_{S} to

E_{S}(k_{∞}) := (E_{S}(k_{∞}) ⊗_{ℤ} ℚ_{p}/ℤ_{p})^{∨},

where E_{S}(k_{∞}) = lim ; ⟶_{k′}E_{S}(k′) denotes the (discrete module of) S-units of k_{∞}.

In particular, we show that E^{0}(𝔼_{S}) ≅ E^{0}E^{0}(E_{S}(k_{∞})), where E^{0}(M) denotes Hom_{Λ}(M, Λ) for any Λ-module M, and thus

rk_{Λ}𝔼_{S} = rk_{Λ}E_{S} = r_{2}(k)

under some assumptions, see corollary 4.26. If E^{0}(𝔼_{S}) is projective, its structure can be described more precisely. A criterion which tells us when this is the case is given in proposition 4.27.

Acknowledgements.

This article is based on a part of my Dissertation, Heidelberg 2000, and I would like to thank my supervisor Kay Wingberg most warmly for leading me to the nice field of “higher dimensional” Iwasawa theory. John Coates receives my special gratitude for his great interest and valuable suggestions. I wish to express my warm thanks to Yoshi Ochi for our fruitful interchange of ideas when he stayed in Heidelberg in 1999/2000. Susan Howson is heartily acknowledged for providing me with a copy of her Ph.D. thesis and for the discussions in Heidelberg as well as in Cambridge.

2. Algebraic properties of Λ-modules

2.1. Notation and Preliminaries. We follow the notation in the paper [34] and recall some basic facts which are thoroughly discussed there. For any compact padic Lie group G the completed group algebra Λ = Λ(G) is Noetherian (see [21]V 2.2.4). If, in addition, G is uniform, then Λ(G) is an integral domain, i.e. the only zero-divisor in Λ(G) is 0 ([21]). In fact, the latter property also holds for any p-adic analytic group without elements of finite order (see [25]). For instance, for p ≥ n + 2, the group Gl_{n}(ℤ_{p}) has no elements of order p, in particular, GL_{2}(ℤ_{p}) contains no elements of finite p-power order if p ≥ 5 (see [15] 4.7) .

In this case (i.e. Λ is both left and right Noetherian and without zero-divisors) we can form a skew field Q(G) of fractions of Λ (see [9]). This allows us to define the rank of a Λ-module:

Definition 2.1. The rank rk_{Λ}M is defined to be the dimension of M ⊗_{Λ} Q(G) as a left vector space over Q(G)

rk_{Λ}M = dim_{Q(G)}(M ⊗_{Λ} Q(G)).

Obviously, the rank is finite for any M in the category Λ-mod of finitely generated Λ-modules. For the rest of this section, we assume that all Λ-modules considered are assumed to be finitely generated.

By Ho(Λ) we denote the category of “Λ-modules up to homotopy”and we write M ≃ N, if M and N are homotopy equivalent, i.e. isomorphic in Ho(Λ), which holds if and only if M ⊕ P ≅ N ⊕ Q with projective Λ-modules P and Q. In particular, M ≃ 0 if and only if M is projective.

For M ϵ Λ-mod we define the Iwasawa adjoints of M to be

E^{i}(M) := Ext_{Λ}^{i}(M, Λ), i ≥ 0,

which are a priori right Λ-modules by functoriality and the right Λ-structure of the bi-module Λ but will be considered as left modules via the involution of Λ. By convention we set E^{i}(M) = 0 for i < 0. The Λ-dual E^{0}(M) will also be denoted by M^{+}.

It can be shown that for i ≥ 1 the functor E^{i} factors through Ho(Λ) defining a functor

E^{i} : Ho(Λ) → Λ-mod.

By D we denote the transpose

D : Ho(Λ) → Ho(Λ),

which is a contravariant duality functor, i.e. it satisfies D^{2} = Id. Furthermore, if pd_{Λ}M ≤ 1 then DM ≃ E^{1}(M). The next property will be of particular importance:

Proposition 2.2. For M ϵ Λ-mod there is a canonical exact sequence

\xymatrix{0 \ar[r] & E^{1}DM \ar[r] & M \ar[r]^{ϕ_{M}} & M^{++} \ar[r] & E^{2}DM \ar[r] & 0,}

where ϕ_{M} is the canonical map from M to its bi-dual. In the following we will refer to the sequence as “the” canonical sequence (of homotopy theory).

A Λ-module M is called reflexive if ϕ_{M} is an isomorphism from M to its bi-dual M ≅ M^{++}.

As Auslander and Bridger suggest the module E^{1}DM should be considered as torsion submodule of M. Indeed, if Λ is a Noetherian integral domain this submodule coincides exactly with the set ^{1} of torsion elements tor_{Λ}M. Hence, a Λ-module M is called Λ-torsion module if ϕ_{M} ≡ 0, i.e. if tor_{Λ}M := E^{1}DM = M. We say that M is Λ-torsion-free if E^{1}DM = 0. It turns out that a finitely generated Λ-module M is a Λ-torsion module if and only if M is a Λ(G′)-torsion module (in the strict sense) for some open pro-p subgroup G′ ⊆ G such that Λ(G′) is integral. Since M^{++} embeds into a free Λ-module the torsion-free Λ-modules are exactly the submodules of free modules.

Sometimes it is also convenient to have the notation of the 1^{st} syzygy or loop space functor Ω : Λ-mod → Ho(Λ) which is defined as follows (see [18, 1.5]): Choose a surjection P → M with P projective. Then ΩM is defined by the exact sequence

\xymatrix{0 \ar[r] & ΩM \ar[r] & P \ar[r] & M \ar[r] & 0.}

1A priori it is not clear whether this sets forms a submodule if Λ is not commutative.

2.2. Dimension theory for the Auslander regular ring Λ(G). Let G be any compact p-adic group without p-torsion. In [34] we proved that Λ = Λ(G) is an Auslander regular ring, i.e. Λ has finite projective dimension d := pdΛ = cd_{p}G + 1 (by a result of Brumer) and satisfies the Auslander condition: For any Λ-module M, any integer m and any submodule N of E^{m}(M), the grade of N satisfies j(N) ≥ m. Recall that the grade j(N) is the smallest number i such that E^{i}(N) ≠ 0 holds. Therefore there is a nice dimension theory for Λ-modules which we will recall briefly (for proofs and further references see [34]). A priori, any M ϵ Λ-mod comes equipped with a finite filtration

T_{0}(M) ⊆ T_{1}(M) ⊆ ⋯ ⊆ T_{d−1}(M) ⊆ T_{d}(M) = M.

If we call the number δ := min{i | T_{i}(M) = M} the dimension δ(M) then T_{i}(M) is just the maximal submodule of M with δ-dimension less or equal to i. We should mention that for abelian G the dimension δ(M) coincides with the Krull dimension of supp_{Λ}(M).

The filtration is related to the Iwasawa adjoints via a spectral sequence, in particular we have

T_{i}(M)/T_{i−1}(M) ⊆ E^{d−i}E^{d−i}(M)

and either of these two terms is zero if and only the other is. Furthermore, the equality δ(M) + j(M) = d holds for any M ≠ 0.

A Λ-module M is called pseudo-null if its codimension d − δ(M) is greater or equal to 2. As in the commutative case we say that a homomorphism φ : M → N of Λ-modules is a pseudo-isomorphism if its kernel and cokernel are pseudo-null. A module M is by definition pseudo-isomorphic to a module N, denoted

M ∼ N,

if and only if there exists a pseudo-isomorphism from M to N. In general, ∼ is not symmetric even in the ℤ_{p}-case. While in the commutative case ∼ is symmetric at least for torsion modules, we do not know whether this property still holds in the general case. If we want to reverse pseudo-isomorphisms, we have to consider the quotient category Λ-mod/PN with respect to subcategory PN of pseudo-null Λ-modules, which is a Serre subcategory, i.e. closed under subobjects, quotients and extensions. By definition, this quotient category is the localization (PI)^{−1}Λ-mod of Λ-mod with respect to the multiplicative system PI consisting of all pseudoisomorphisms. Since Λ-mod is well-powered, i.e. the family of submodules of any module M ϵ Λ-mod forms a set,these localization exists, is an abelian category and the universal functor q : Λ-mod → Λ-mod/PN is exact. Furthermore, q(M) = 0 in Λ-mod/PN if and only if M ϵ PN .

Note that for any pseudo-isomorphism f : M → N the induced homomorphism E^{1}(f) is a pseudo-isomorphism, too. If M, N are Λ-torsion modules, also the converse statement holds. By the universal property of the localisation, we obtain a functor

E^{1} : Λ-mod/PN → mod−Λ/PN ,

which is exact if it is restricted to the full subcategory Λ−mod^{≥1}/PN of Λ-mod/PN consisting of all Λ-modules of codimension greater or equal to 1. More precisely, there is a natural isomorphism of functors:

E^{1} ∘ E^{1} ≅ Id : Λ−mod^{≥1}/PN → Λ−mod^{≥1}/PN .

It is known that any torsion-free module M embeds into a reflexive module with pseudo-null cokernel while any torsion module M is pseudo-isomorphic to E^{1}E^{1}(M) (cf. [34, Prop.3.13]). Moreover, there is a canonical pseudo-isomorphism E^{1}(M) ∼ E^{1}(tor_{Λ}M) for any Λ-module M.

By Λ-mod(p) we shall write the plain subcategory of Λ-mod consisting of ℤ_{p}- torsion modules while by PN (p)“ = PN ∩ Λ-mod(p)” we denote the Serre subcategory of Λ-mod(p) the objects of which are pseudo-null Λ-modules. In other words M belongs to PN (p) if and only if it is a Λ/p^{n}-module for an appropriate n such that E_{Λ/p^{n}}^{0}(M) = 0. Recall that there is a canonical exact functor q : Λ-mod(p) → Λ-mod(p)/PN (p). Then, there is the following structure theorem on the ℤ_{p}-torsion part of a finitely generated Λ-module:

Theorem 2.3. ([34, Thm. 3.40]) Assume that G is a p-adic analytic group without p-torsion such that both Λ = Λ(G) and Λ/p are integral. Let M be in Λ-mod(p). Then there exist uniquely determined natural numbers n_{1}, …, n_{r} and an isomorphism in Λ-mod(p)/PN (p)

M ≡ ⨁_{1≤i≤r} Λ/p^{n_{i}} mod PN (p).

We define the μ-invariant of a Λ-module M as

μ(M) = ∑_{i} n_{i}(tor_{ℤ_{p}}M),

where the n_{i}s are determined uniquely by the structure theorem. This invariant is additive on short exact sequences of Λ-torsion modules and stabil under pseudoisomorphisms. Alternatively, it can be described as

μ(M) = rk_{𝔽_{p}[[G]]} ⨁_{i≥0}_{p^{i+1}}M/_{p^{i}}M = rk_{𝔽_{p}[[G]]} ⨁_{i≥0} p^{i}tor_{ℤ_{p}}M/p^{i+1}tor_{ℤ_{p}}M.

Very recently, P. Schneider [32] found a general structure theorem for Λ-torsion modules. He proved that any finitely generated Λ(G)-torsion module decomposes into the direct sum of cyclic modules up to pseudo-isomorphism, i.e. in the quotient category Λ−mod^{≥1}/PN .

Theorem (Chamarie-Schneider). Let G be an extra-powerful uniform pro-p-group. Then, for any finitely generated Λ(G)-torsion module M there exist finitely many left ideals J_{1}, …, J_{r} such that

M ≡ ⨁_{1≤i≤r} Λ/J_{i} mod PN

Recall that a pro-p-group G is called extra-powerful, if the relation [G, G] ⊆ G^{p^{2}} holds.

It is still not known whether the ideals J_{i} can be chosen as principal ideals as in the commutative case. Anyway, if we restrict to this kind of modules, we can define a second involution

∘ : Λ−mod_{pr}^{≥1}/PN → Λ−mod_{pr}^{≥1}/PN

on the full subcategory Λ−mod_{pr}^{≥1}/PN of Λ−mod^{≥1}/PN consisting of those objects which are isomorphic (in the quotient category) to a direct sum of cyclic modules of the form Λ/Λf, f ϵ Λ. For any such f we set (Λ/Λf)^{∘} := Λ/Λf^{∘}, where ^{∘} : Λ → Λ also denotes the involution of the group algebra (induced by g ↦ g^{−1}). The following proposition implies among other things that this definition is invariant under pseudo-isomorphism and therefore it extends to the whole category Λ(G)−mod_{pr}^{≥1}/PN .

Proposition 2.4. Let G be a profinite group such that Λ = Λ(G) is a Noetherian integral ring.Then the following holds:

(i) For any f ϵ Λ there is an isomorphism E^{1}(Λ/Λf) ≅ Λ/Λf^{∘}.

(ii) Assuming that G is a p-adic analytic group without p-torsion the above two involutions coincide:

−^{∘} ≅ E^{1}(−) : Λ−mod_{pr}^{≥1}/PN → Λ−mod_{pr}^{≥1}/PN

The proof is standard, see for example the proof of proposition 2.12, where we denote the involution on Λ by ι.

We conclude this section with a technical result which will be needed in the arithmetic applications.

Proposition 2.5. Let Λ be an Auslander regular ring. For any Λ-module M such that pd_{Λ}E^{0}(M) ≤ 1 (e.g. if pd Λ = 3 or if pd Λ = 4 and E^{4}E^{1}(M) = 0) its double dual E^{0}E^{0}(M) is a 2-syzygy of E^{1}E^{0}(M), i.e. there is an exact sequence

\xymatrix{0 \ar[r] & E^{0}E^{0}(M) \ar[r] & P_{0} \ar[r] & P_{1} \ar[r] & E^{1}E^{0}(M) \ar[r] & 0}

with projective modules P_{0} and P_{1}. Furthermore, in the case of pd Λ = 3 or 4, it holds that E^{1}E^{0}(M) ≅ E^{3}E^{1}(M). If, in addition, M itself is reflexive and pd Λ = 3, then E^{3}E^{1}M ≅ E^{1}(M)^{∨}.

Proof. First observe that E^{0}(M) is a 2-syzygy of D(M) due to the definition of the latter module, i.e. pd_{Λ}E^{0}(M) ≤ pd Λ − 2 = 1, if pd Λ = 3. In the case of pd Λ = 4 it holds E^{3}E^{0}(M) = E^{4}E^{0}(M) = 0 and E^{2}E^{0}(M) ≅ E^{4}E^{1}(M) due to Björk’s spectral sequence (see [34, 3.1]). Hence, if E^{4}E^{0}(M) vanishes, it follows that pd_{Λ}E^{0}(M) ≤ 1. Now, choosing a projective resolution of E^{0}(M)

\xymatrix{0 \ar[r] & E^{0}(P_{1}) \ar[r] & E^{0}(P_{0}) \ar[r] & E^{0}(M) \ar[r] & 0,}

we derive the exact sequence

\xymatrix{0 \ar[r] & E^{0}E^{0}(M) \ar[r] & P_{0} \ar[r] & P_{1} \ar[r] & E^{1}E^{0}(M) \ar[r] & 0.}

But E^{1}E^{0}(M) ≅ E^{3}E^{1}(M) due to Björk’s spectral sequence for pd Λ ≤ 4. If M itself is reflexive and pd Λ = 3, then E^{1}E^{1}(M) = E^{2}E^{1}(M) = 0, i.e. E^{1}(M) is finite, respectively E^{3}E^{1}(M) ≅ E^{1}(M)^{∨}. □

2.3. Some representation theory. In the following lemma we shall write I(Γ) for the kernel of the canonical map ℤ_{p}[[G]] → ℤ_{p}[G/Γ], where Γ is any closed normal subgroup of the profinite group G. By Rad_{G} we denote the radical of ℤ_{p}[[G]], i.e. the intersection of all open maximal left (right) ideals of ℤ_{p}[[G]]. Finally, we write

M_{G} = M/I_{G}M

for the module of coinvariants of M and H_{•}(G, M) for the G-homology of a compact Λ-module M, which can be defined as left derived functor of −_{G} or alternatively as Tor_{•}^{Λ}(ℤ_{p}, M), where Tor denotes the left derived functor of the complete tensor product −ˆ⊗_{Λ} − .

Lemma 2.6. Let G = Γ ⋊ Δ be the semi-direct product of a uniform pro-p-group Γ of dimension t and a finite group Δ of order k prime to p. If we write U_{n} = Γ^{p^{n}} ⊴ G, then for any compact Λ = Λ(G)-module M, the following statements are equivalent:

(i) M ≅ Λ^{d}, (ii) M_{Γ} ≅ ℤ_{p}[Δ]^{d} as ℤ_{p}[Δ]-module and for all n

rk_{ℤ_{p}}M_{U_{n}} = rk_{ℤ_{p}}ℤ_{p}[G/U_{n}]^{d} = dkp^{tn},

(iii) M_{Γ}/p ≅ 𝔽_{p}[Δ]^{d} as 𝔽_{p}[Δ]-module and for all n

rk_{𝔽_{p}}M_{U_{n}}/p = rk_{𝔽_{p}}𝔽_{p}[G/U_{n}]^{d} = dkp^{tn}.

Proof. Obviously, (i) implies (ii) and (iii). For the converse let us first assume that (ii) holds and let m − 1, …, m_{d} ϵ M be lifts of a ℤ_{p}[Δ]-basis of M_{Γ}. Then the map ϕ : ⨁_{i=1}^{d} Λe_{i} → M, which sends e_{i} to m_{i}, is surjective, because I(Γ) ⊆ Rad_{G} (compare to the proof of [24]. 5.2.14 (i), d ⇒ b) and therefore we can apply Nakayama’s lemma [24], 5.2.16 (ii), (with Rad_{G} instead of 𝔐). Hence, the induced maps ϕ_{U_{n}} : ⨁_{i=1}^{d} Λ(G/U_{n})e_{i} → M_{U_{n}}, are surjective, too. But since both modules have the same ℤ_{p}-rank by assumption, these maps are isomorphisms and (i) follows. The implication (iii) ⇒ (i) is proved analogously noting that pΛ + I(Γ) ⊆ Rad_{G}. □

For a finite group G we denote by K_{0}(ℚ_{p}[G]) = K′_{0}(ℚ_{p}[G]) the Grothendieck group of finitely generated ℚ_{p}[G]-modules (which are projective by Maschke’s theorem). If G is a profinite group and U ⊴ G an open normal subgroup we define the Euler characteristic h_{U}(M) of a finitely generated Λ = Λ(G)-module M to be the class

h_{U}(M) := ∑(−1)^{i}[H_{i}(U, M) ⊗_{ℤ_{p}} ℚ_{p}] ϵ K_{0}(ℚ_{p}[G/U]).

Before stating the next result we recall some facts about the representation theory of finite groups. So let Δ be a finite group of order n prime to p. Then, there is a decomposition

ℤ_{p}[Δ] ≅ ∏ ℤ_{p}[Δ]e_{i}, e_{i} = \frac{n_{i}}{n} ∑_{g ϵ Δ} χ_{i}(g^{−1})g

of ℤ_{p}[Δ] in “simple” components (in the sense that they are simple algebras after tensoring with ℚ_{p}), which induces a decomposition of Λ = ∏ Λ^{e_{i}}, Λ^{e_{i}} = ℤ_{p}[[Γ]][Δ]e_{i} into a product of rings. Here {χ_{i}} is the set of irreducible ℚ_{p} characters (ˆ= 𝔽_{p}-characters because n is prime to p) of Δ and n_{i} are certain natural numbers associated with χ_{i} (see below). The simple algebras ℚ_{p}[Δ]e_{i} decompose into the direct sum of their simple left ideals which all belong to the same isomorphism class, say N_{i}, i.e. there is a isomorphism of ℚ_{p}[Δ]-modules

ℚ_{p}[Δ] ≅ N_{i}^{n_{i}}.

In particular, n_{i} is the length of ℚ_{p}[Δ]e_{i} and can be expressed as

n_{i} = χ(e_{i})(dim_{ℚ_{p}} N_{i})^{−1},

where χ is the character of the left regular representation of ℚ_{p}[Δ].

Now let G be again a p-adic Lie group and set Λ := Λ(G). Recall that a finitely generated Λ-module M is a Λ-torsion module if and only if M is a Λ(G′)-torsion module for some open pro-p subgroup G′ ⊆ G such that Λ(G′) is integral.

Proposition 2.7. Let G = Γ × Δ be the product of a pro-p Lie group Γ of finite cohomological dimension cd_{p}(Γ) = m and a finite group Δ of order n prime to p and let U ⊴ Γ be an open normal subgroup. Then, for any finitely generated Λ-torsion module M, it holds

h_{U}(M) = 0.

Remark 2.8. For semi-direct products this statement is false in general. For example, it is easy to see that for G = ℤ_{p} ⋊_{ω} Δ with non-trivial ω the Euler characteristic of ℤ_{p} is not zero: h_{U}(ℤ_{p}) = [ℚ_{p}] − [ℚ_{p}(ω)] ≠ 0.

Proof. (of prop. 2.7) We claim that under the assumptions of the theorem M possesses a finite free resolution. Indeed, since the Noetherian ring Λ has finite global dimension pdΛ = m + 1, there is always a resolution of the form

\xymatrix{0 \ar[r] & P \ar[r] & Λ^{d_{m}} \ar[r] & ⋯ \ar[r] & Λ^{d_{0}} \ar[r] & 0,}

with a projective module P. Since M^{e_{i}} is a Λ(Γ)-torsion module (it is even Λ(Γ′)-torsion!) and since P^{e_{i}} is a free Λ(Γ)-module, it must hold that P^{e_{i}} ≅ (Λ(Γ))^{k_{i}d_{m+1}} as Λ(Γ)-modules, where k_{i} = χ(e_{i}) denotes the ℤ_{p}-rank of ℤ_{p}[Δ]e_{i} and d_{m+1} = ∑_{i=0}^{m}(−1)^{i}d_{m−i}. Consequently, P_{Γ}^{e_{i}} ≅ ℤ_{p}^{k_{i}d_{m+1}} as ℤ_{p}-modules, respectively P_{Γ}^{e_{i}} ⊗ ℚ_{p} ≅ ℚ_{p}^{k_{i}d_{m+1}} as ℚ_{p}-modules holds. But P_{Γ}^{e_{i}} ⊗ ℚ_{p} must be isomorphic to the direct sum of m copies of N_{i} for some m due to the semisimplicity of ℚ_{p}[Δ]. Counting ℚ_{p}-dimensions, we obtain m = n_{i}d_{m+1} and hence P_{Γ}^{e_{i}} ⊗ ℚ_{p} ≅ ℚ_{p}[Δ]e_{i}^{d_{m+1}}. Since P_{Γ}^{e_{i}} is a projective ℤ_{p}[Δ]-module, this implies P_{Γ}^{e_{i}} ≅ ℤ_{p}[Δ]e_{i}^{d_{m+1}}, respectively P^{e_{i}} ≅ Λ(G)e_{i}^{d_{m+1}} (compare to the proof of lemma 2.6) and P ≅ Λ(G)^{d_{m+1}}. This proves the claim. Furthermore, we observe that ∑(−1)^{i}d_{i} = 0 and denote the resolution by F^{•} → M. Using the fact that the Euler characteristic of a bounded complex equals the Euler characteristic of its homology, we calculate

∑(−1)^{i}[H_{i}(U, M) ⊗_{ℤ_{p}} ℚ_{p}] = ∑(−1)^{i}[H_{i}(F^{•} ⊗_{Λ} ℚ_{p}[G/U])]
= ∑(−1)^{i}[F^{•} ⊗_{Λ} ℚ_{p}[G/U]]
= ∑(−1)^{i}[ℚ_{p}[G/U]^{d_{i}}]
= (∑(−1)^{i}d_{i})[ℚ_{p}[G/U]] = 0.
□

Lemma 2.9. Let G be a profinite group, H ⊆ G a closed subgroup and U ⊴ G an open normal subgroup. Then for any compact ℤ_{p}[[H]]-module M the following is true:

(i) (Ind_{G}^{H}(M))_{U} ≅ Ind_{G/U}^{HU/U}(M_{U∩H}) and
(ii) H_{i}(U, (Ind_{G}^{H}(M))_{U}) ≅ Ind_{G/U}^{HU/U}H_{i}(U ∩ H, M) for all i ≥ 0.

Proof. The dual statement of (i) is proved in [20] while (ii) follows from (i) by homological algebra. □

Lemma 2.10. Let G = Γ × Δ be the product of a pro-p-group Γ and a finite group Δ of order prime to p. Then, for any Λ = ℤ_{p}[[Γ]][Δ]-module M and for any irreducible character χ of Δ with values in ℚ_{p}, the following is true:

(i) Hom_{Λ}(M^{e_{χ}}, Λ) ≅ Hom_{Λ}(M, Λ)^{e_{χ^{−1}}},
(ii) E_{Λ}^{i}(M^{e_{χ}}) ≅ E_{Λ}^{i}(M)^{e_{χ^{−1}}} for any i ≥ 0.

Proof. While (ii) is a consequence of (i) by homological algebra the first statement can be verified at once using the decompositions M ≅ ⨁ M^{e_{χ}} and Λ ≅ ⨁ Λ^{e_{χ}} :

Hom_{Λ}(M, Λ)^{e_{χ^{−1}}} ≅ Hom_{Λ}(M, Λ^{e_{χ}})
≅ Hom_{Λ}(M^{e_{χ}}, Λ^{e_{χ}})
≅ Hom_{Λ}(M^{e_{χ}}, Λ).

□

2.4. Modules associated with group presentations. Let C be a class of finite groups closed under taking subgroups, homomorphic images and group extensions. Given an exact sequence of pro-C-groups

1 → H → G → G → 1,

where G is assumed to be finitely generated, we choose a presentation F ↠ G of G by a free pro-C-group F_{d} of rank d and we associate the following commutative diagram to it:

[figure: 1; 1; 1; H; G; G; 1; 1; R; F_{d}; G; 1; N; N; 1; 1]

Here, R and N are defined by the exactness of the corresponding sequences. In general, the p-relation module N^{ab}(p) of G with respect to the chosen free presentation (and similarly R^{ab}(p) with respect to G instead of G) fits into the following exact sequence, which is called Fox-Lyndon resolution associated with the above free representation of G :

(2.1)  \xymatrix{0 \ar[r] & N^{ab}(p) \ar[r] & Λ(G)^{d} \ar[r] & Λ(G) \ar[r] & ℤ_{p} \ar[r] & 0.}

Hence, if cd_{p}(G) ≤ 2, then N^{ab}(p) is a projective Λ(G)-module.

Furthermore, the augmentation ideal I_{F_{d}}, i.e. the kernel of Λ(F_{d}) → ℤ_{p}, is a free Λ(F_{d})-modules of rank d:

I_{F_{d}} ≅ Λ(F_{d})^{d}

(for a proof of these facts, see [24] Chap V.6).

Let A be a p-divisible p-torsion abelian group of finite ℤ_{p}-corank r with a continuous action of G.

Definition 2.11. For a finitely generated Λ = Λ(G)-module M we define the finitely generated Λ-module

M[A] := M ⊗_{ℤ_{p}} A^{∨} = Hom_{cont.,ℤ_{p}}(M, A)^{∨}

with diagonal G-action.We shall also write M(ρ) for this r-dimensional twist where ρ : G → Gl_{r}(ℤ_{p}) denotes the operation of G on A^{∨}.

Note that the functor −[A] is exact and that Λ[A] is a free Λ-module of rank r (cf. [29, lem 4.2]).

Proposition 2.12. For every i ≥ 0

E^{i}(M(ρ)) ≅ E^{i}(M)(ρ^{d}),

where ρ^{d} is the contragredient representation, i.e. ρ^{d}(g) = ρ(g^{−1})^{t} is the transpose matrix of ρ(g^{−1}).

Proof. By homological algebra (and using a free presentation of M) it suffices to prove the case i = 0 for free modules. Finally, we only have to check the commutativity of the following diagram which is associated to an arbitrary homomorphism ϕ : Λ → Λ

\xymatrix{Hom_{Λ}(Λ(ρ), Λ) \ar[r]^{ϕ(ρ)^{∗}} \ar[d] & Hom_{Λ}(Λ(ρ), Λ) \ar[d] \\ Λ^{r} \ar[d] & Λ^{r} \ar[d] \\ Hom_{Λ}(Λ, Λ)(ρ) \ar[r]^{ϕ^{∗}(ρ^{d})} & Hom_{Λ}(Λ, Λ)(ρ).}

First note that via the identification Λ^{r} ^{ψ_{ρ}} Λ(ρ) the matrix representing ϕ(ρ) is A := ∑ a_{g}gρ(g^{−1}), where we assume for simplicity that ϕ(1) =: a = ∑ a_{g}g ϵ ℤ_{p}[G]. We denote by ι both, the involution Λ → Λ, g ↦ g^{−1} (also extended to matrices with coefficients in Λ) and the isomorphism of left Λ-modules Λ → Hom_{Λ}(Λ, Λ), g ↦ (1 ↦ g^{−1}). Then its easy to see that the following two diagrams commute

\xymatrix{Hom_{Λ}(Λ(ρ), Λ) \ar[r]^{ϕ(ρ)^{∗}} \ar[d]^{(ψ_{ρ})^{∗}} & Hom_{Λ}(Λ(ρ), Λ) \ar[d]^{(ψ_{ρ})^{∗}} \\ Hom_{Λ}(Λ^{r}, Λ) \ar[d]^{i^{r}} & Hom_{Λ}(Λ^{r}, Λ) \ar[d]^{i^{r}} \\ Λ^{r} \ar[r]^{ι(A^{t})} & Λ^{r},}

\xymatrix{Λ^{r} \ar[r]^{B} \ar[d]^{ψ_{ρ^{d}}} & Λ^{r} \ar[d]^{ψ_{ρ^{d}}} \\ Λ(ρ) \ar[r]^{ι(a)(ρ^{d})} \ar[d]^{i(ρ^{d})} & Λ(ρ) \ar[d]^{i(ρ^{d})} \\ Hom_{Λ}(Λ, Λ)(ρ) \ar[r]^{ϕ^{∗}(ρ^{d})} & Hom_{Λ}(Λ, Λ)(ρ),}

where B = ∑ a_{g}g^{−1}ρ^{d}(g), because ι(a) = ∑ a_{g}g^{−1}. We are done if we can verify B = ι(A^{t}). But

ι(A^{t}) = ∑ a_{g}g^{−1}ρ(g^{−1})^{t}
= ∑ a_{g}g^{−1}ρ^{d}(g) = B.
□

With the notation

X := X_{H,A} := H^{1}(H, A)^{∨}
Y := Y_{H,A} := (I_{G}[A])_{H}
J := J_{H,A} := ker(Λ(G)[A]_{H} → (A^{∨})_{H}),

we get the following proposition:

Proposition 2.13. ([29, lem. 4.5]) We have a commutative and exact diagram

[figure: 0; 0; J; J; 0; H^{2}(H, A)^{∨}; (N^{ab}(p)[A])_{H}; Λ(G)^{dr}; Y; 0; 0; H^{2}(H, A)^{∨}; (H^{1}(N, A)^{H})^{∨}; H^{1}(R, A)^{∨}; X; 0.; 0; 0]

Furthermore, if cd_{p}(G) ≤ 2, then N^{ab}(p)[A] is a projective Λ(G)-module and (N^{ab}(p)[A])_{H} a projective Λ(G)-module.

Remark 2.14. Assume A is trivial as H-module. Then the above diagram can be easily obtained by twisting Jannsen’s original diagram (i.e. with coefficients ℚ_{p}/ℤ_{p}): diagram(A) = diagram(ℚ_{p}/ℤ_{p})[A]. Also the higher Iwasawa adjoints of the occurring modules can be calculated via proposition 2.12:

E^{i}(X_{H,A}) ≅ E^{i}(X_{H,ℚ_{p}/ℤ_{p}})(ρ^{d})
E^{i}(Y_{H,A}) ≅ E^{i}(Y_{H,ℚ_{p}/ℤ_{p}})(ρ^{d})
⋯

The following theorem is a consequence of the diagram. The restriction to p-adic Lie groups without p-torsion is necessary in order to apply the dimension theory developed in [34].

Theorem 2.15. Let cd_{p}(G) ≤ 2 and G a p-adic Lie group of dimension h without p-torsion. If the “weak Leopoldt conjecture holds for A and H”, i.e. if H^{2}(H, A) = 0, then neither Y nor X have non-zero pseudo-null submodules: T_{h−1}(X) = T_{h−1}(Y) = 0.

Proof. Apply proposition 3.10 of [34] to Y, which has pd(Y) ≤ 1 according to the above diagram, and note that T_{h−1}(X) ⊆ T_{h−1}(Y) by proposition 3.2 of (loc. cit.). □

Let Z = Z_{H,A} := (D_{2}^{(p)}(G, A)^{H})^{∨} where

D_{2}^{(p)}(G, A) = lim ; ⟶_{U⊆_{o}G,n}(H^{2}(U,_{p^{n}}A))^{∨}

and the direct limit is taken with respect to the p-power map and the dual of the corestriction. Then there is a description of the Λ(G)-module Y as follows:

Proposition 2.16. Assume that cd_{p}(G) = 2 and that N^{ab}(p) is a finitely generated Λ(G)-module. Then

Y ≃ DZ and E^{0}(Z) ≅ H^{2}(H, A)^{∨},

thus Y is determined by Z up to projective summands. Suppose, in addition, that H^{2}(H, A) = 0. Then

E^{1}(Y) ≅ Z.

For a proof of the proposition see [24] 5.6.8 and [28] thm 3.13.

3. Local Iwasawa modules

3.1. The general case. In this section we study the structure of Iwasawa modules arising from “p-adic representations” G → Aut(A), where G = G_{k} is the absolute Galois group of a finite extension k of ℚ_{ℓ} and A is a p-divisible p-torsion abelian group of finite ℤ_{p}-corank r. Having fixed a p-adic Lie extension k_{∞} of k with Galois group G, we write H = G(\overline{k}/k_{∞}) ⊆ G where \overline{k} denotes the algebraic closure of k. We are going to apply the general results of section 2.4 to the module

X_{A} := X_{H,A} = H^{1}(H, A)^{∨} = H^{1}(k_{∞}, A)^{∨},

i.e. we will determine the Λ(G)-modules occurring in the canonical exact sequence

\xymatrix{0 \ar[r] & E^{1}D(X_{A}) \ar[r] & X_{A} \ar[r] & E^{0}E^{0}(X_{A}) \ar[r] & E^{2}D(X_{A}) \ar[r] & 0.}

For the sake of completeness and for the convenience of the reader we restate some general results from [30], but see also [28]. Since we have fixed H, we shall omit it in the notation and write Y_{A}, Z_{A}, etc.

Lemma 3.1. (cf. [28])

(i) If k is a finite extension of ℚ_{ℓ} and k_{∞} is a Galois extension of k, then Z = A^{∗}(k_{∞})^{∨}, where A^{∗} = (T_{p}A)^{∨}(1) by definition,

(ii) E^{1}D(X_{A}) ≅ E^{1}(A^{∗}(k_{∞})^{∨}), (iii) E^{2}D(X_{A}) ⊆ E^{2}D(Y_{A}) ≅ E^{2}(A^{∗}(k_{∞})^{∨}), (iv) If cd_{p}(G) ≤ 1 or cd_{p}(G) = 2 and A(k_{∞})^{∨} is ℤ_{p}-torsion-free, then DX_{A} ≃

E^{1}(X_{A}).

Proof. (i) is just local Tate duality while (ii) is a consequence of (i):

E^{1}D(X_{A}) ≅ E^{1}D(Y_{A}) ≅ E^{1}(Z_{A}) ≅ E^{1}(A^{∗}(k_{∞})^{∨})

(Note that the first isomorphism holds because J_{A} is torsion-free as Λ(U)-module for a suitable open pro-p-subgroup U ⊆ G, such that Λ(U) is integral). By the same reason and using the snake lemma, one sees that E^{2}D(X_{A}) ⊆ E^{2}D(Y_{A}).

To prove (iv) just note that in these cases pdX_{A} ≤ 1 by the diagram 2.13, the defining sequence of J_{A}, corollary [34, Cor. 6.3] and [34, Cor. 4.8]. □

For a finitely generated abelian p-primary group A we denote by A_{div} the quotient of A by its maximal p-divisible subgroup. The next result generalizes a result of Greenberg [12]:

Proposition 3.2. (cf. [30, § 2]) Let n = [k : ℚ_{ℓ}], ℓ = p, be the finite degree of k over ℚ_{p} and k_{∞} a Galois extension of k with Galois group G ≅ Γ ⋊_{ω} Δ, where Γ ≅ ℤ_{p} and Δ is a finite group of order t prime to p, which acts on Γ via the character ω : Δ → ℤ_{p}^{∗}. If χ = ω^{−1} denotes the inverse of the character which determines the action on the p-dualizing module of G, the canonical sequence becomes

\xymatrix{0 \ar[r] & T_{p}A^{∗}(k_{∞})(χ) \ar[r] & X_{A} \ar[r] & P \ar[r] & M \ar[r] & 0,}

where P is a projective Λ(G)-module of rk_{Λ(Γ)}P = rnt and M is determined by the exact sequence

\xymatrix{0 \ar[r] & M \ar[r] & A^{∗}(k_{∞})_{div}(χ) \ar[r] & tor_{ℤ_{p}}(A(k_{∞})^{∨})} .

Furthermore,

(i) if A^{∗}(k_{∞}) is finite, then T_{p}A^{∗}(k_{∞})(χ) = 0. If, in addition, A(k_{∞})^{∨} is

ℤ_{p}-free, then M ≅ A^{∗}(k_{∞}).

(ii) if A^{∗}(k_{∞})^{∨} is ℤ_{p}-free, then X_{A} ≅ P ⊕ T_{p}A^{∗}(k_{∞})(χ). In particular, X_{A} is

projective, if A^{∗}(k_{∞}) = 0.

Proof. First note that according to lemma 3.1 and [34, Cor. 4.8]

E^{1}D(X_{A}) ≅ E^{1}(A^{∗}(k_{∞})^{∨})
≅ E^{1}(A^{∗}(k_{∞})^{∨}/tor_{ℤ_{p}})
≅ (A^{∗}(k_{∞})^{∨} ⊗ ℚ_{p}/ℤ_{p}(χ^{−1}))^{∨}
≅ T_{p}A^{∗}(k_{∞})(χ).

To determine E^{2}D(X_{A}) ≅ E^{2}E^{1}(X_{A}) we use the short exact sequences

\xymatrix{0 \ar[r] & X_{A} \ar[r] & Y_{A} \ar[r] & J_{A} \ar[r] & 0,}
\xymatrix{0 \ar[r] & J_{A} \ar[r] & Λ(G)^{d} \ar[r] & A(k_{∞})^{∨} \ar[r] & 0,}

i.e. E^{1}(J_{A}) ≅ E^{2}(A(k_{∞})^{∨}) ≅ A(k_{∞})_{div}(χ) by [34, Cor. 4.8] and

\xymatrix{A(k_{∞})_{div}(χ) \ar[r] & E^{1}(Y_{A}) \ar[r] & E^{1}(X_{A}) \ar[r] & 0}

is exact. Forming the long exact Ext-sequence and applying lemmas 3.1 and [34, Cor. 4.8] again, gives the desired result. □

Let us now consider the case ℓ ≠ p :

Proposition 3.3. (cf. [30, § 2]) In the situation of the last theorem but with ℓ ≠ p there is an isomorphism

X_{A} ≅ T_{p}A^{∗}(k_{∞})(χ).

Proof. In [28], prop. 3.12, it was calculated that the Λ(Γ)-rank of X_{A} equals the Λ(Γ)-corank of H^{2}(k_{∞}, A), but the latter module vanishes because the order of G is divisible by p^{∞} (cf. [24] 7.1.8). □

Proposition 3.4. (cf. [30, § 2]) Let n = [k : ℚ_{p}] be the finite degree of k over ℚ_{p} and k_{∞} a p-adic Lie extension of k such that its Galois group G has cohomological dimension cd_{p}(G) = 2. Let Γ ⊆ G be an arbitrary open uniform pro-p-subgroup, i.e. Λ(Γ) is integral, and let t be the index (G : Γ). If χ denotes the inverse of the character which determines the action of G on the p-dualizing module, then the canonical sequence becomes

\xymatrix{0 \ar[r] & X_{A} \ar[r] & R \ar[r] & E^{2}D(X_{A}) \ar[r] & 0,}

where R is a reflexive Λ(G)-module with rk_{Λ(Γ)}R = rnt. If, in addition, A(k_{∞})^{∨} is ℤ_{p}-free, then E^{2}D(X_{A}) is determined by the exact sequence

\xymatrix{0 \ar[r] & E^{2}D(X_{A}) \ar[r] & T_{p}A^{∗}(k_{∞})(χ) \ar[r] & Hom(T_{p}A(k_{∞}), ℤ_{p})} .

Proof. Using again the lemmas 3.1 and [34, Cor. 4.8], the proof is completely analogous to that in the one-dimensional case 3.2. □

Note that in the case p ≠ l and cd_{p}(G) ≥ 2 we have H = 0, i.e. X_{A} = 0, because the Galois group G_{k}(p) ≅ ℤ_{p}(1)⋊ℤ_{p} of the maximal p-extension of any local field over ℚ_{ℓ} does not have any non-trivial quotient G which satisfies these conditions.

Proposition 3.5. (cf. [30, § 2]) Let n = [k : ℚ_{p}] be the finite degree of k over ℚ_{p} and k_{∞} a p-adic Lie extension of k such that its Galois group G has cohomological dimension cd_{p}(G) ≥ 3. Let Γ ⊆ G be an arbitrary open uniform pro-p-subgroup, i.e. Λ(Γ) is integral, and let t be the index (G : U). Then

X_{A} ≅ E^{0}E^{0}X_{A}

is a reflexive Λ(G)-module with rk_{Λ(Γ)}R = rnt.

Proof. This follows from the lemmas 3.1 and [34, Cor. 4.8] as above. □

At the end of this part we want to restate the results concerning the ranks of the considered modules . The result was obtained independently by S. Howson [15, 6.1] and Y. Ochi [28, thm 3.3], see also [30, thm. 2].

Proposition 3.6. (Howson, Ochi) Let k be a finite extension of ℚ_{ℓ} and k_{∞} be a pro-p Lie extension of k with Galois group G = G(k_{∞}/k). As before r denotes the ℤ_{p}-rank of rank(A^{∨}). Assume that Λ = Λ(G) is integral. Then

rk_{Λ}H^{1}(k_{∞}, A)^{∨} = \{ \begin{matrix}r[k : ℚ_{p}] & if ℓ = p \\ 0 & otherwise \end{matrix} .

Proof. (cf. Ochi) Noting the vanishing of H^{2}(k_{∞}, A) and that N^{ab}(p) ≅ Λ(G) for d = [k : ℚ_{p}] + 2 (conferring [18] thm. 5.1 c)), the result follows from the diagram 2.13 and the above remarks with respect to the case ℓ ≠ p. □

3.2. The case A = ℚ_{p}/ℤ_{p}.

3.2.1. Local units. If we specialize to the important case A = ℚ_{p}/ℤ_{p} with trivial Galois action, we are able to determine the module structure more exactly using local class field theory: X := X_{ℚ_{p}/ℤ_{p}} ≅ H^{ab}(p)^{2} is the Galois group of the maximal abelian p-extension of k_{∞}, which is canonically isomorphic to the inverse limit

X ≅ 𝔸(k_{∞}) := lim ; ⟵_{k′}𝔸(k′)

of the p-completions 𝔸(k′) of the multiplicative groups of finite subextensions k′ of k in k_{∞} :

𝔸(k′) = lim ; ⟵_{m}(k′)^{∗}/(k′)^{∗p^{m}},

where the limit is taken via the norm maps. Since the Galois module structure of 𝔸(k′) is well known if tensored with ℚ_{p}, we get

Theorem 3.7. Let n = [k : ℚ_{ℓ}], ℓ = p, be the finite degree of k over ℚ_{p} and k_{∞} a Galois extension of k with Galois group G ≅ Γ ⋊_{ω} Δ, where Γ ≅ ℤ_{p} and Δ is a finitely generated profinite group of order prime to p, which acts on Γ via the character ω : Δ → ℤ_{p}^{∗}. We write k_{0} for the fixed field of Γ and denote by χ = ω^{−1} the inverse of the character which determines the action on the p-dualizing module of G.

(i) If μ_{p^{∞}} ⊆ k_{∞}, i.e. k_{∞} is the cyclotomic ℤ_{p}-extension of k_{0} and G = Γ×Δ,

then it holds

𝔸(k_{∞}) ≅ Λ^{n} ⊕ ℤ_{p}(1).

(ii) Let μ(k_{∞})(p) be finite. Then there is an exact sequence of Λ-modules

\xymatrix{0 \ar[r] & 𝔸(k_{∞}) ⊕ I_{G} \ar[r] & Λ^{n+1} \ar[r] & μ(k_{∞})(p)(χ) \ar[r] & 0.}

For any representation

\xymatrix{1 \ar[r] & K \ar[r] & F_{d′} \ar[r] & G \ar[r] & 1}

by a free profinite group F_{d′} on d′ ≤ n + 1 generators, there exists an exact sequence

\xymatrix{0 \ar[r] & 𝔸(k_{∞}) \ar[r] & Λ^{n−d′+1} ⊕ K^{ab}(p) \ar[r] & μ(k_{∞})(p)(χ) \ar[r] & 0.}

Remark 3.8. (i) The existence of a representations in (ii) is always guaranteed by [17] theorem 4.3. Indeed, one can choose d′ = 2. (ii) Using the Krull-Schmidt theorem and Maschke’s theorem, it is easily proved (see the proof below) that

E^{0}(I_{G})(ω) ⊕ I_{G} ≅ ℤ_{p}[[G]]^{2}
⨁_{i=1}^{m−1} I_{G}(ω^{i}) ⊕ I_{G} ≅ ℤ_{p}[[G]]^{m},

2This notation refers to the diagram of section 2.4 where we represent the absolut local Galois group G of k by a free profinite group of rank d = [k : ℚ_{ℓ}] + 2 according to [24] theorem 7.4.1.

where m denotes the order of ω. Hence, from the isomorphism K^{ab}(p) ⊕ I_{G} ≅ ℤ_{p}[[G]]^{d} according to the Lyndon sequence (2.1), we get isomorphisms (for m ≤ d)

K^{ab}(p) ≅ ℤ_{p}[[G]]^{d−2} ⊕ E^{0}(I_{G})(ω)
≅ ℤ_{p}[[G]]^{d−m} ⊕ ⨁_{i=1}^{m−1} I_{G}(ω^{i}).

In particular, if ω is an involution and d = 2, then K^{ab}(p) ≅ E^{0}(I_{G})(ω) ≅ I_{G}(ω) holds.

Proof. Let us first consider the case that Δ is a finite group, which grants that Λ(G) is Noetherian. Then the statements are consequences of theorem 3.2 once having determined the structure of P = E^{0}E^{0}X. We will apply the Krull-Schmidt theorem and we first observe that for any open normal subgroup U ⊴ Γ and Ḡ := G/U it holds: X_{U} ⊗ ℚ_{p} ≅ P_{U} ⊗ ℚ_{p} and, if k′ denotes the fixed field of U, there are exact sequences of Ḡ-modules

\xymatrix{0 \ar[r] & U^{ab}(p) \ar[r] & (I_{G})_{U} \ar[r] & ℤ_{p}[Ḡ] \ar[r] & ℤ_{p} \ar[r] & 0,}
\xymatrix{0 \ar[r] & X_{U} \ar[r] & Ḡ_{k′}^{ab}(p) \ar[r] & U^{ab}(p) \ar[r] & 0.}

Hence, by Maschke’s theorem and using Ḡ_{k′}^{ab}(p) ⊗ ℚ_{p} ≅ ℚ_{p}[Ḡ]^{n} ⊕ ℚ_{p} (cf. [24] 7.4.3), we get

P_{U} ⊗ ℚ_{p} ⊕ (I_{G})_{U} ⊗ ℚ_{p} ≅ ℚ_{p}[Ḡ]^{n+1},

i.e.

P ⊕ I_{G} ≅ Λ^{n+1}.

Now, taking U-coinvariants of the augmentation sequence

\xymatrix{0 \ar[r] & I_{G} \ar[r] & ℤ_{p}[[G]] \ar[r] & ℤ_{p} \ar[r] & 0}

and tensoring with ℚ_{p}(ω^{i}) gives

ℚ_{p}[Ḡ] ⊕ ℚ_{p}(ω^{i+1}) ≅ (I_{G}(ω^{i}))_{U} ⊗ ℚ_{p} ⊕ ℚ_{p}(ω^{i}).

For (i) just note that I_{G} is projective and ω trivial because Δ acts trivially on Γ, hence: I_{G} ≅ ℤ_{p}[[G]. The first sequence in (ii) is immediate while the second one results from the isomorphism K^{ab}(p) ⊕ I_{G} ≅ ℤ_{p}[[G]]^{d} according to the Lyndon sequence (2.1). Now let us assume that Δ is infinite. If Δ′ ⊆ Δ is an open subgroup then the functor obtained by taking Δ′-coinvariants is exact because H_{1}(Δ′, M) = 0 for any Λ-module M. Since the automorphism group is virtually pro-p, there is an open normal subgroup Δ_{0} of Δ which acts trivially on Γ, in particular any open normal subgroup Δ′ of Δ which is contained in Δ_{0} is normal with in G. Now a free presentation of G

\xymatrix{1 \ar[r] & K \ar[r] & F_{d′} \ar[r] & G \ar[r] & 1}

induces a free presentation of G′ := G/Δ′

\xymatrix{1 \ar[r] & K_{Δ′} \ar[r] & F_{d′} \ar[r] & G/Δ′ \ar[r] & 1.}

Using the Lyndon sequence, it is easy to verify that (I_{G})_{Δ′} ≅ I_{G/Δ′} and K^{ab}(p)_{Δ′} ≅ K_{Δ′}^{ab}(p). Now the strategy is as follows. Take a Λ(G)-module M and show that for any Δ′ as above its Δ′-coinvariants are isomorphic to certain finitely generated Λ(G′)-modules of the same type, e.g. 𝔸(k′) ⊕ I_{G′}, where k′ is the fixed field of k_{∞} by Δ′. Then it follows easily (using a compactness argument to grant the existence of a compatible system of isomorphisms) that M ≅ 𝔸(k_{∞})⊕I_{G}. As an example we prove the first statement in (ii): choose a surjection Λ(G)^{n+1} ↠ μ(k_{∞})(p)(χ) and define M to be the kernel of it. Taking Δ′-coinvariants and comparing it with the result for k′, i.e. for (finite) Δ/Δ′, we obtain an isomorphism M_{Δ′} ≅ 𝔸(k′)⊕I_{G′} by Schanuel’s lemma (see [18, 1.3] for a generalized version). The other statements follow by similar arguments.

The second isomorphism of the remark can be deduced by summing up (I_{G}(ω^{i}))_{U}⊗ ℚ_{p} for 0 ≤ i ≤ m. For the first one, use that due to the projectivity of I_{G}

E^{0}(I_{G})_{U} ⊗ ℚ_{p} ≅ Hom_{ℤ_{p}[[G]]}(I_{G}, ℤ_{p}[[G]])_{U} ⊗ ℚ_{p}
≅ Hom_{ℤ_{p}[Ḡ]}((I_{G})_{U}, ℤ_{p}[Ḡ]) ⊗ ℚ_{p}
≅ Hom_{ℚ_{p}}((I_{G})_{U}, ℚ_{p}) ⊗ ℚ_{p}

holds. □

Theorem 3.9. In the situation of the last theorem but with ℓ ≠ p there is an isomorphism

X ≅ \{ \begin{matrix}ℤ_{p}(1)(χ) & if μ_{p} ⊆ k_{0} \\ 0 & otherwise \end{matrix} .

The next theorem generalizes results of Wintenberger [36] who restricts himself to the case in which G is abelian. It applies for example to Γ ≅ ℤ_{p} ⋊ ℤ_{p}.

Theorem 3.10. Let n = [k : ℚ_{p}] be the finite degree of k over ℚ_{p} and k_{∞} an Galois extension of k with Galois group G ≅ Γ⋊_{ρ} Δ, where Γ is a pro-p Lie group of dimension 2 and Δ is a profinite group of order prime to p, which acts on Γ via ρ : Δ → Aut(Γ). Let k_{0} be the fixed field of Γ and let χ = detρ^{−1} denote the inverse of the character which determines the action on the p-dualizing module of G.

(i) If μ(k_{0})(p) = 1, then X ⊕ Λ ≅ R^{ab}(p). If ρ is trivial, then X ≅ Λ^{n}. (ii) If μ_{p^{∞}} ⊆ k_{∞} and G is without p-torsion and such that its dualizing module is not isomorphic to μ_{p^{∞}}, then there is an exact sequence of Λ-modules

\xymatrix{0 \ar[r] & X ⊕ Λ \ar[r] & R^{ab}(p) \ar[r] & ℤ_{p}(1)(χ) \ar[r] & 0.}

If ρ is trivial, then

\xymatrix{0 \ar[r] & X \ar[r] & Λ^{n} \ar[r] & ℤ_{p}(1) \ar[r] & 0}

is exact.

(iii) If μ(k_{∞})(p) and Δ are finite, then X ≅ E^{0}E^{0}(X) is reflexive, i.e. there is an exact sequence

\xymatrix{0 \ar[r] & X \ar[r] & R^{ab}(p) \ar[r] & Λ \ar[r] & μ(k_{∞})(p).}

If, in addition, μ(k)(p) = 1, but μ(k_{∞})(p) ≠ 1 and χ^{−1} ≠ χ_{cycl}, then the right map is also surjective (in particular X is not free in this case).

Remark 3.11. For extensions k_{∞}|k of the type G ≅ Γ × Δ with Γ ≅ ℤ_{p}^{s}, s ≥ 3 and finite Δ, we can consider the relative situation

\xymatrix{0 \ar[r] & X(k_{∞})_{Γ′} \ar[r] & X(K_{∞}) \ar[r] & ℤ_{p} \ar[r] & 0,}

where Γ′ is direct factor of Γ isomorphic to ℤ_{p}, i.e. Γ ≅ Γ′ × ℤ_{p}^{s−1}, and K_{∞} is the fixed field of k_{∞} with respect to Γ′. By induction and applying Diekert’s theorem ([24]) one reobtains at once Wintenberger’s results (but now more generally with not necessarily abelian Δ): For any irreducible character χ ≠ 1, χ_{cycl} the component X(k_{∞})^{e_{χ}} is a free Λ(G)^{e_{χ}}-module of rank n

X(k_{∞})^{e_{χ}} ≅ (Λ(G)^{e_{χ}})^{n}.

But since we already know that pd_{Λ}X = s − 2 for s ≥ 3, X can not be projective in this case, i.e. X(k_{∞})^{e_{χ}} or X(k_{∞})^{e_{χ_{cycl}}} is definitely not of this type.

We will prove the theorem only for finite Δ because the general case follows similarly as in theorem 3.7. Just note that also in this case the automorphism group of Γ is virtually pro-p (see [8, 5.6]). But before giving the proof we need some preparation:

Lemma 3.12. Let G = Γ×Δ be the product of a pro-p Lie group Γ with cd_{p}(Γ) = 2 and a finite group Δ of order prime to p. Then

R^{ab}(p) ≅ Λ^{n+1}.

Proof. Let U_{n} := p^{n}Γ ⊴ G. By the Lyndon sequence and using proposition 2.7, we calculate the Euler characteristic h_{U_{n}}(R^{ab}(p)) = h_{U_{n}}(ℤ_{p}) + h_{U_{n}}(Λ^{n+1}) = h_{U_{n}}(Λ^{n+1}). The result follows.

□

Lemma 3.13. If in the situation of the theorem μ(k_{∞})(p) is infinite, then both E^{0}(X) and E^{0}E^{0}(X) are projective.

Proof. Since E^{0}(−) preserves projectives and E^{0}E^{0}E^{0}(X) ≅ E^{0}(X) by [34, Prop. 3.11], it is sufficient to prove the statement for E^{0}E^{0}(X). But according to proposition 2.5 the latter module is the 2-syzygy of E^{3}E^{1}(X). We claim that

Y ≃ X ⊕ Λ,

i.e. that E^{3}E^{1}(X) ≅ E^{3}E^{1}(Y) ≅ E^{3}(μ(k_{∞})(p)^{∨}) = 0, which implies the lemma. Indeed, due to Poincaré-duality

H^{2}(G, μ(k_{∞})(p))^{∨} ≅ Hom_{G}(μ(k_{∞})(p), D_{2}^{(p)}) = 0,

if D_{2}^{(p)} ≠ μ_{p^{∞}}. Hence, Y ≃ X ⊕ Λ by the second description of 4.5 b) in [18]^{3}. □

Proof (of the theorem). Let U_{m} = p^{m}Γ ⊴ G and denote the fixed field of U_{m} by k_{m}. From the exact sequence

\xymatrix{1 \ar[r] & G_{k_{∞}} \ar[r] & G_{k_{m}} \ar[r] & U_{m} \ar[r] & 1}

we obtain the associated homological Hochschild-Serre sequence

\xymatrix{0 = H_{2}(k_{m}, ℤ_{p}) \ar[r] & H_{2}(U_{m}, ℤ_{p}) \ar[r] & X_{U_{m}} \ar[r] & G_{k_{m}}^{ab}(p) \ar[r] & H_{1}(U_{m}, ℤ_{p}) \ar[r] & 0.}

After tensoring with ℚ_{p}, it follows that

X_{U_{m}} ⊗ ℚ_{p} ⊕ H_{1}(U_{m}, ℤ_{p}) ⊗ ℚ_{p} ≅ ℚ_{p}[Ḡ]^{n} ⊕ ℚ_{p} ⊕ H_{2}(U_{m}, ℤ_{p}) ⊗ ℚ_{p},

where we used Maschke’s theorem and Ḡ_{k_{m}}^{ab}(p)⊗ℚ_{p} ≅ ℚ_{p}[Ḡ]^{n}⊕ℚ_{p} (cf. [24] 7.4.3). On the other hand, the Euler characteristic of the projective module R^{ab}(p) can be calculated by means of the Lyndon sequence:

[R^{ab}(p)_{U_{m}} ⊗ ℚ_{p}] = h_{U_{m}}(R^{ab}(p))
= h_{U_{m}}(ℤ_{p}) + h_{U_{m}}(Λ^{n+1})
= [ℚ_{p}] − [H_{1}(U_{m}, ℤ_{p}) ⊗ ℚ_{p}] + [H_{2}(U_{m}, ℤ_{p}) ⊗ ℚ_{p}]
+[ℚ_{p}[Ḡ]^{n+1}]

and hence X_{U_{m}} ⊗ ℚ_{p} ⊕ ℚ_{p}[Ḡ] ≅ R^{ab}(p)_{U_{m}} ⊗ ℚ_{p}.

Assume that μ(k_{0})(p) = 1, i.e. tor_{ℤ_{p}}𝔸(k_{0}) = 1 and X_{U_{0}} is ℤ_{p}-free. Therefore, since t is prime to p, it follows that X_{U_{0}} is ℤ_{p}[Δ]-projective. If ρ is trivial, we conclude, by the calculation above under consideration of h_{U_{m}}(ℤ_{p}) = 0 (by lemma 2.7) and using the Krull-Schmidt theorem, that X_{U_{0}} ≅ ℤ_{p}[Δ]^{n}. Applying lemma 2.6, gives the desired result in this case. Anyway, these arguments show that X is projective also in the case with non-trivial ρ, i.e. we obtain X ⊕ Λ ≅ R^{ab}(p) in the general case.

In order to prove (ii), we apply theorem 3.4: Since X ⊕ Λ ≃ Y in this case (see the proof of lemma 3.13), we obtain

E^{2}D(X) ≅ E^{2}D(Y)
≅ E^{2}(ℤ_{p}(−1))
≅ ℤ_{p}(1)(χ),

where we applied the lemma 3.1 and [18, 2.6]. Note that χ^{−1}(x) = det(Adx) = det ρ(x) : G → Δ →^{det ρ} ℤ_{p}^{∗} (cf. [21] V 2.5.8.1). We still have to determine the module P = E^{0}E^{0}(X), which is projective according to lemma 3.13: it is easily seen that P_{U_{m}} ⊗ ℚ_{p} ≅ X_{U_{m}} ⊗ ℚ_{p}, i.e. P ⊕ Λ ≅ R^{ab}(p), by the above calculations.

3For Γ = ℤ_{p}^{2} this statement was proved by Jannsen ([18] 5.2 c): Though there the claimed isomorphism R^{ab}(p) ≅ Λ^{d−1} is only correct if ρ is trivial, the arguments (which we restated above) still prove X ⊕ Λ ≃ Y.

If ρ is trivial, lemma 3.12 gives the desired result.

The first statement of (iii) is just theorem 3.4 and lemma 3.1. By proposition 2.5, we obtain an exact sequence

\xymatrix{0 \ar[r] & X \ar[r] & P \ar[r] & Λ^{s} \ar[r] & μ(k_{∞})(p)}

for some s. Splitting up the sequence, taking the long exact H_{i}(U_{m}, −)-sequences and using the above calculations, one immediately sees that P_{U_{m}}⊗ℚ_{p} ≅ R^{ab}(p)_{U_{m}}⊗ ℚ_{p} ⊕ℚ_{p}[Ḡ]^{s−1}, i.e. P ≅ R^{ab}(p)⊕Λ^{s−1}. After possibly changing the basis of Λ^{d} and using the Krull-Schmidt theorem, one easily sees that we can rid off the summand Λ^{s−1}. In order to prove the last statement, we assume that χ^{−1} ≠ χ_{cycl} and consider the exact sequence

[figure: 0; E^{1}(X)^{∨}; E^{1}(Y)^{∨}; E^{1}(I)^{∨}; μ(k_{∞})(p); ℚ_{p}/ℤ_{p}(χ^{−1}).]

The decomposition of the sequence with respect to the irreducible ℚ_{p}- characters of Δ gives (E^{1}(X)^{∨})^{χ_{cycl}} = μ(k_{∞})(p)^{χ_{cycl}} = μ(k_{∞})(p). □

3.2.2. Principal units. When l = p, we are also interested in the Λ-structure of the inverse limit of the principal units

𝕌^{1}(k_{∞}) := lim ; ⟵_{k′}𝕌^{1}(k′),

where k′ runs through all finite subextensions of k_{∞}|k and the limit is taken with respect to the norm maps.

Proposition 3.14. Let k be a finite extension of ℚ_{p} and k_{∞} a Galois extension of k.

(i) If k_{∞} contains the maximal unramified p-extension of k, i.e. if p^{∞} divides the degree of the residue field extension associated with k_{∞}|k, then

𝕌^{1}(k_{∞}) ≅ 𝔸(k_{∞}).

(ii) In the other case there is the following exact sequence

\xymatrix{0 \ar[r] & 𝕌^{1}(k_{∞}) \ar[r] & 𝔸(k_{∞}) \ar[r] & ℤ_{p} \ar[r] & 0.}

Proof. For finite extensions K′|K|k of k with associated residue field extensions λ′|λ|κ consider the following commutative diagram with exact rows

\xymatrix{0 \ar[r] & 𝕌^{1}(K′)/p^{m} \ar[r] \ar[d]^{N_{K′/K}} & 𝔸(K′) \ar[r] \ar[d]^{N_{K′/K}} & ℤ_{p}/p^{m} \ar[r] \ar[d]^{[λ′:λ]} & 0 \\ 0 \ar[r] & 𝕌^{1}(K)/p^{m} \ar[r] & 𝔸(K) \ar[r] & ℤ_{p}/p^{m} \ar[r] & 0.}

While in case (i) the inverse limit lim ; ⟵_{K,m}ℤ_{p}/p^{m} vanishes, because for any m and

any K there is an extension K′ such that p^{m}|[λ′ : λ], in the second case it is isomorphic to ℤ_{p}. □

Theorem 3.15. Assume in the situation of theorem 3.10 that k_{∞} contains μ_{p^{∞}} but not the maximal unramified p-extension of k. Then there exists an exact sequence

\xymatrix{0 \ar[r] & 𝕌^{1}(k_{∞}) ⊕ Λ \ar[r] & R^{ab}(p) \ar[r] & M \ar[r] & 0.}

In particular, if ρ is trivial, there exists an exact sequence

\xymatrix{0 \ar[r] & 𝕌^{1}(k_{∞}) \ar[r] & Λ^{n} \ar[r] & M \ar[r] & 0,}

where M fits into the exact sequence

\xymatrix{0 \ar[r] & ℤ_{p} \ar[r] & M \ar[r] & ℤ_{p}(1)(χ) \ar[r] & 0.}

Proof. Evaluating the long exact E^{i}-sequence associated with the exact sequence from the proposition above and noting that pd_{Λ}𝕌^{1}(k_{∞}) ≤ 1 due to pd_{Λ}𝔸(k_{∞}) ≤ 1 and pd_{Λ}ℤ_{p} = 2, one obtains that

(i) E^{0}(𝕌^{1}(k_{∞})) ≅ E^{0}(X),
(ii) E^{1}D(𝕌^{1}(k_{∞})) = 0 and an exact sequence
(iii) \xymatrix{0 \ar[r] & ℤ_{p} \ar[r] & E^{2}D(𝕌^{1}(k_{∞})) \ar[r] & ℤ_{p}(1)(χ) \ar[r] & 0.}

Here we used that E^{2}E^{2}(ℤ_{p}) ≅ ℤ_{p}, because ℤ_{p} is a Cohen-Macaulay module of dimension 2. The result follows from the canonical sequence. □

Remark 3.16. In the situation of theorem 3.7 with trivial action of Δ the structure of the principal units is described in [24] as follows:

(i) If μ_{p^{∞}} ⊆ k_{∞}, then

𝕌^{1}(k_{∞}) ≅ Λ^{n} ⊕ ℤ_{p}(1).

(ii) If μ(k_{∞})(p) is finite, then there is an exact sequence

\xymatrix{0 \ar[r] & 𝕌^{1}(k_{∞}) \ar[r] & Λ^{n} \ar[r] & μ(k_{∞})(p).}

(iii) If k_{∞}|k is unramified, then

𝕌^{1}(k_{∞}) ≅ 𝔸(k_{∞}).

But the proof of [24] works also if ω is not trivial.

3.3. The local CM-case. As a consequence of theorem 3.10 we can also determine the structure of X_{A} = H^{1}(k_{∞}, A)^{∨} in the trivializing case, i.e. k(A) ⊆ k_{∞} :

Theorem 3.17. Let n = [k : ℚ_{p}] be the finite degree of k over ℚ_{p} and k_{∞} a Galois extension of k with Galois group G ≅ Γ⋊_{ρ} Δ, where Γ is a pro-p Lie group of dimension 2 and Δ is a finite group of order t prime to p, which acts on Γ via ρ : Δ → Aut(Γ). Let k_{0} be the fixed field of Γ and let χ = detρ^{−1} denote the inverse of the character which determines the action on the p-dualizing module of G. For any A with rk_{ℤ_{p}}A^{∨} = r such that k(A) ⊆ k_{∞} the following is true.

(i) If μ(k_{0})(p) = 1, then X_{A} ⊕ Λ^{r} ≅ R^{ab}(p)[A], in particular, if ρ is trivial: X_{A} ≅ Λ^{nr}.

(ii) If μ_{p^{∞}} ⊆ k_{∞} and G is p-torsion-free and its dualizing module is not isomorphic to μ_{p^{∞}}, then there is an exact sequence of Λ-modules

\xymatrix{0 \ar[r] & X_{A} ⊕ Λ^{r} \ar[r] & R^{ab}(p)[A] \ar[r] & A^{∨}(1)(χ) \ar[r] & 0.}

In particular, if ρ is trivial, then

\xymatrix{0 \ar[r] & X_{A} \ar[r] & Λ^{nr} \ar[r] & A^{∨}(1) \ar[r] & 0}

is exact. (iii) If μ(k_{∞})(p) is finite, then X_{A} ≅ E^{0}E^{0}(X_{A}) is reflexive, i.e. there is an

exact sequence

\xymatrix{0 \ar[r] & X_{A} \ar[r] & R^{ab}(p)[A] \ar[r] & Λ^{r} \ar[r] & μ(k_{∞})(p)[A].}

If, in addition, μ(k)(p) = 1, but μ(k_{∞})(p) ≠ 1 and χ^{−1} ≠ χ_{cycl}, then the right map is also surjective (in particular, X_{A} is not free in this case).

Proof. In this case the subgroups H, R and N act trivially on A = A(k_{∞}), i.e. X_{A} ≅ X[A]. □

This result applies to the following situation: Let K be a imaginary quadratic number field, F a finite, abelian extension of K and E an elliptic curve defined over F with complex multiplication (CM) by the ring of integers O_{K} of K such that F (E_{tor}) is an abelian extension of K. Assume that the rational prime p splits in K, i.e. pO_{K} = 𝔭𝔭̄, 𝔭 ≠ 𝔭̄, and that E has good reduction at all places lying over p. Set G = G(F (E(p))/F)_{𝔓} the decomposition group at some 𝔓|𝔭. According to [7, 1.9], the prime 𝔓 ramifies totally in F (E(𝔭))|F and decomposes only finitely (and is unramified) in F (E(𝔭̄))|F. Therefore the decomposition group G is an open subgroup of G(F (E(p))/F), i.e. of type ℤ_{p}^{2} × Δ where Δ is a finite abelian group. Thus we obtain an exact sequence

\xymatrix{0 \ar[r] & H^{1}(F (E(p))_{𝔓}, E(p))^{∨} \ar[r] & Λ(G)^{2n} \ar[r] & T_{p}E \ar[r] & 0,}

where n = [F_{𝔭} : ℚ_{p}]. By the same argument, but now using theorem 3.7(ii), there exists an exact sequence

\xymatrix{0 \ar[r] & H^{1}(F (E(𝔭)_{𝔓}, E(𝔭))^{∨} \ar[r] & Λ(G′)^{n} \ar[r] & μ(F (E(𝔭)_{𝔓})[E(𝔭)] \ar[r] & 0,}

where G′ = G(F (E(𝔭))/F)_{𝔓}, and a similar one for 𝔭̄.

4. Global Iwasawa modules

Let k_{∞} be a p-adic Lie extension of the number field k contained in k_{S} with Galois group G and let A be a p-divisible p-torsion abelian group with ℤ_{p}-corank r and on which G_{S}(k) = G(k_{S}/k) acts continuously where S is a finite set of places of k containing all places S_{p} over p and all infinite places S_{∞} (and by definition all places at which A is ramified). Here k_{S} denotes the maximal S-ramified extension of k, i.e. the maximal extension of k which is unramified outside S. In order to derive information about the Λ = Λ(G)-modules H^{i}(G(k_{S}/k_{∞}), A) we would like to apply the diagram 2.4 to the group G = G_{S} := G(k_{S}/k). On the other hand we have to guarantee that G is finitely generated as a profinite group which, unfortunately, is not known for the group G_{S}. But using a theorem of Neumann, i.e. the inflation maps are isomorphisms

H^{i}(G(Ω/k_{∞}), A) ≅ H^{i}(G_{S}(k_{∞}), A), i ≥ 0,

for any (p, S)-closed extension Ω of k (i.e. Ω is a S-ramified extension of k which does not possess any non-trivial S-ramified p-extension) and for any p-torsion G(Ω/k_{∞})-module A, we are free to replace G_{S}(k) for example by the Galois group G := G(Ω/k) where Ω is the maximal S-ramified p-extension of k′(A) and k′ is a Galois subextension of k_{∞}/k such that G(k_{∞}/k′) is an open (normal) pro-p-group. Regarding this technical detail, we assume in what follows that k_{∞} is contained in such a (p, S)-closed field Ω. Then, since G has an open pro-p Sylow group, it is finitely generated and has cd_{p}(G) ≤ 2 for odd p. Note that Y_{S,A} := Y_{G(Ω/k_{∞}),A} and X_{S,A} := X_{G(Ω/k_{∞}),A} do not depend on the choice of Ω. The next lemma shows among other things that the corresponding module Z only depends on k_{∞}, A and S. Recall that T_{p}A = Hom(ℚ_{p}/ℤ_{p}, A) denotes the “Tate module” of A. We shall write H_{cts}^{∗}(G_{S}(k), T_{p}A) ≅ lim ; ⟵_{n}H^{∗}(G_{S}(k),_{p^{n}}A) for

the continuous cochain cohomology groups (see [24, II.§3.]).

Lemma 4.1. Let k, k_{∞} and A be as above. Then

Z_{S,A} := Z_{G(Ω/k_{∞}),A} ≅ lim ; ⟵_{k⊆k′⊆k_{∞}}H_{cts}^{2}(G_{S}(k′), T_{p}A).

A basic structure result is the following

Theorem 4.2. Let G a p-adic Lie group without p-torsion. If the “weak Leopoldt conjecture holds for A and k_{∞}”, i.e. H^{2}(G_{S}(k_{∞}), A) = 0, then neither Y_{S,A} nor X_{S,A} ≅ H^{1}(G_{S}(k_{∞}), A)^{∨} have non-zero pseudo-null submodules.

Proof. The conditions of theorem 2.15 are fulfilled. □

Remark 4.3. The weak Leopoldt conjecture for A and k_{∞} holds for example if k(A) and the cyclotomic ℤ_{p}-extension of k are contained in k_{∞}. Indeed, it is a result of Iwasawa that the weak Leopoldt conjecture (for A = ℚ_{p}/ℤ_{p}) holds for the cyclotomic ℤ_{p}-extension of any number field (see [24, 10.3.25] for a cohomological proof). The claim follows by expressing H^{2}(G_{S}(k_{∞}), A) (considered as abelian group) as direct limit lim ; ⟶_{k′}H^{2}(G_{S}(k′_{cyc}), ℚ_{p}/ℤ_{p})^{r}, where k′ runs through the finite

extensions of k in k_{∞}.

Furthermore, the Λ-rank of X_{S,A} can be determined, using the diagram 2.4:

Theorem 4.4. (Ochi [30]) Let k_{∞}|k be a p-adic pro-p extension. Assume that k(A)|k is a pro-p-extension and that Λ is an integral domain. Then

rk_{Λ}H^{1}(G_{S}(k_{∞}), A)^{∨} − rk_{Λ}H^{2}(G_{S}(k_{∞}), A)^{∨} = r_{2}(k)r

Here r_{2}(k) denotes as usual the number of complex places of k.

4.1. The multiplicative group 𝔾_{m}.

4.1.1. The maximal abelian p-extension of k_{∞} unramified outside S. We still consider p-adic Lie extensions k_{∞}|k with Galois group G = G(k_{∞}/k) such that k_{∞} is contained in the maximal S-ramified extension k_{S} of k. Here, as before, S denotes a finite set of places of k containing all places S_{p} over p and all infinite places S_{∞}. For K|k finite let S_{f}(K) be the set of finite primes in K lying above S. In this paragraph we specialize to the case A = ℚ_{p}/ℤ_{p} and we shall write X_{S} for the Λ = Λ(G)-module X_{S,ℚ_{p}/ℤ_{p}}

X_{S} := X_{S,ℚ_{p}/ℤ_{p}} = H^{1}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p})^{∨} ≅ G(k_{S}/k_{∞})^{ab}(p),

and respectively for Y_{S} and Z_{S}.

In this case theorem 4.2 is a generalization of the theorems of Greenberg [11] and Nguyen-Quang-Do [26], who considered the case G ≅ ℤ_{p}^{d}. Indeed, it confirms Greenberg’s suggestion that an analogous statement also should hold for p-adic Lie extensions.

Theorem 4.5. Let G be a p-adic Lie group without p-torsion. If the “weak Leopoldt conjecture holds for k_{∞}”, i.e. H^{2}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p}) = 0, then X_{S} ≅ G_{S}(k_{∞})^{ab}(p) has no non-zero pseudo-null Λ-submodule.

Remark 4.6. Recall that the weak Leopoldt conjecture for k_{∞} holds if the cyclotomic ℤ_{p}-extension of k is contained in k_{∞}.

We will also consider the Λ-modules

X_{nr} = G(L/k_{∞}),
X_{cs}^{S} = G(L′/k_{∞}),

where L is the maximal abelian unramified pro-p-extension of k_{∞} and L′ is the maximal subextension in which every prime above S is completely decomposed.

For an arbitrary number field K, we denote the ring of integers (resp. S-integers) by O_{K} (resp. O_{K,S}) and its units by E(K) := O_{K}^{×} (resp. E_{S}(K) := O_{K,S}^{×}). Then we define

𝔼 : = lim ; ⟵_{k′}(O_{k′}^{×} ⊗ ℤ_{p}),
𝔼_{S} : = lim ; ⟵_{k′}(O_{k′,S}^{×} ⊗ ℤ_{p}),

where the limit is taken with respect to the norm maps. This should not be confused with the discrete module of units (resp. S-units) E(k_{∞}) = lim ; ⟶_{k′}E(k′)

(resp. E_{S}(k_{∞}) = lim ; ⟶_{k′}E_{S}(k′)).

Finally, we write for the local-global modules

𝔸_{S} = ⨁_{S_{f} (k)} Ind_{G}^{G_{ν}}𝔸_{ν},

𝕌_{S} = ⨁_{S_{f} (k)} Ind_{G}^{G_{ν}}𝕌_{ν},

where 𝔸_{ν} = 𝔸(k_{∞,ν}) (resp. 𝕌_{ν} = 𝕌^{1}(k_{∞,ν})) are the local modules introduced in chapter 2. The above modules are connected via global class field theory and the Tate-Poitou sequence as follows

Proposition 4.7. (Jannsen) There are the following exact and commutative diagrams of Λ-modules:

(i)

[figure: 0; H^{2}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p})^{∨}; 𝔼; 𝕌_{S}; X_{S}; X_{nr}; 0; 0; H^{2}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p})^{∨}; 𝔼_{S}; 𝔸_{S}; X_{S}; X_{cs}^{S}; 0]

(ii)

\xymatrix{0 \ar[r] & 𝔼 \ar[r] & 𝔼_{S} \ar[r] & ⨁_{S_{un}(k)} Ind_{G}^{G_{ν}}ℤ_{p} \ar[r] & X_{nr} \ar[r] & X_{cs}^{S} \ar[r] & 0,}

where S_{un} := {ν ϵ S(k) | p^{∞} ∤ f_{ν}} and f_{ν} = f(k_{∞,ν}/k_{ν}) denotes the degree of the extension of the corresponding residue class fields.

(iii)

\xymatrix{0 \ar[r] & X_{cs}^{S} \ar[r] & Z_{S,ℚ_{p}/ℤ_{p}(1)} \ar[r] & ⨁_{S_{f} (k)} Ind_{G}^{G_{ν}}ℤ_{p} \ar[r] & ℤ_{p} \ar[r] & 0,}

and, if μ_{p^{∞}} ⊆ k_{∞},

\xymatrix{0 \ar[r] & X_{cs}(−1) \ar[r] & Z_{S} \ar[r] & ⨁_{S_{f} (k)} Ind_{G}^{G_{ν}}ℤ_{p}(−1) \ar[r] & ℤ_{p}(−1) \ar[r] & 0.}

In particular, X_{cs}^{S} = X_{cs} := X_{cs}^{S_{p}} is independent of S in this case. (iv) N^{ab}(p) is a finitely generated, projective Λ(G(k_{∞}_{S}(p)/k))-module and, if

the free presentation of G = G(k_{∞}_{S}(p)/k) (cf. section 2.4) is chosen such that d ≥ r′_{1} + r_{2} + 1, then

N^{ab}(p)_{G_{S}(k_{∞})(p)} ≅ ⨁_{S′_{∞}} Ind_{G}^{G_{ν}}ℤ_{p} ⊕ Λ(G)^{d−r_{2}−r′_{1}−1},

where S′_{∞} is the set of real places of k which ramify (i.e. become complex) in k_{∞}, r′_{1} is the cardinality of S′_{∞}, and r_{2} is the number of complex places of k.

Proof. The assertions (i) and (iii) are obtained by taking inverse limits of the Tate-Poitou sequence (see [18, Thm 5.4]) and recalling lemma 4.1 while (ii) follows from (i) by the snake lemma and prop. 3.14. □

From these diagrams and the fact that Λ is Noetherian it follows that the modules X_{nr}, X_{cs}^{S} are finitely generated. Furthermore, S. Howson [15, 7.14-7.16] and independently Y. Ochi [28, 4.10] proved that X_{nr} and X_{cs}^{S} are Λ-torsion. Actually, this result was first proved by M. Harris [13, thm 3.3] but, as S. Howson remarked, his proof is incomplete because it relies on the false “strong Nakayama” lemma (loc. cit. lem 1.9), see the discussion in [1]. However, in a recent correction M. Harris [14] has given a new proof of the result. In the case G ≅ ℤ_{p}^{d}, this result is originally proved by Greenberg [10].

Corollary 4.8. (i) If H^{2}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p}(1)) = 0 (e.g. if dim(G_{ν}) ≥ 1 for

all ν ϵ S_{f}), then X_{cs}^{S} is a Λ-torsion module.

(ii) If dim(G_{ν}) ≥ 1 for all ν ϵ S_{f}, then X_{nr} is a Λ-torsion module.

For example, the conditions of the corollary are satisfied if k_{∞} contains the cyclotomic ℤ_{p}-extension.

Proof. (cf. [30]) The first statement follows from 2.16 while the second one is a consequence of the first one and the above proposition (To calculate the (co)dimension of Ind_{G}^{G_{ν}}ℤ_{p} use [34, 4.8,4.9]. Note that the condition “dim(G_{ν}) ≥ 1 for all ν ϵ S_{f}” implies, using Tate-Poitou duality,

H^{2}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p}(1)) = � ^{2}(G_{S}(k_{∞}), μ_{p^{∞}})
= lim ; ⟶_{k′,n}� ^{1}(G_{S}(k′), ℤ/p^{n})^{∨}
= lim ; ⟶_{k′}Cl_{S}(k′) ⊗_{ℤ} ℚ_{p}/ℤ_{p}
= 0

because Cl_{S}(k′) is finite. □

Theorem 4.9. If H^{2}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p}) = 0, μ_{p^{∞}} ⊆ k_{∞}, and dim(G_{ν}) ≥ 2 for all ν ϵ S_{f}, then

X_{nr}(−1) ∼ X_{cs}^{S}(−1) ∼ E^{1}(Y_{S}) ∼ E^{1}(tor_{Λ}Y_{S}) ≅ E^{1}(tor_{Λ}X_{S}).

If, in addition, G ≅ ℤ_{p}^{r}, r ≥ 2, then even the following holds:

X_{nr}(−1) ∼ X_{cs}^{S}(−1) ∼ (tor_{Λ}X_{S})^{∘},

where ^{∘} means that G operates via the involution g ↦ g^{−1}.

Remark 4.10. In case tor_{Λ}X_{S} is isomorphic in Λ-mod/PN to a direct sum of cyclic modules of the form Λ modulo a (left) principal ideal the proposition 2.4 implies that

E^{1}(tor_{Λ}X_{S}) ≡ (tor_{Λ}X_{S})^{∘} mod PN

holds under the conditions of the theorem.

Proof. The first two pseudo-isomorphisms follow again from proposition 4.7 using [34, 4.8,4.9] and 2.16. The third one is just [34, Prop. 3.13]. Note that there is even an isomorphism tor_{Λ}Y_{S} ≅ tor_{Λ}X_{S} because the augmentation ideal I_{G} is torsion-free. □

The following consequence generalizes a result of McCallum [22, thm 8] who considered the ℤ_{p}^{r}-case:

Corollary 4.11. With the assumptions of the theorem the following holds.

(i) There is a pseudo-isomorphism

tor_{Λ}X_{S} ∼ E^{1}(X_{cs}^{S}(−1)).

(ii) If dim(G) ≥ 3, then there is an isomorphism

tor_{Λ}X_{S} ≅ E^{1}(X_{cs}^{S}(−1)).

Proof. The cokernel K := coker(X_{cs}^{S}(−1) ↪ Z_{S} ≅ E^{1}(Y_{S}) is pseudo-null, i.e. E^{1}(K) = 0. If dim(G) ≥ 3, then E^{2}(K) = 0, too, as can be calculated using [34, Prop. 2.7]. Now, the long exact E-sequence gives the result observing E^{1}E^{1}(Y_{S}) ≅ E^{1}DY_{S} ≅ tor_{Λ}Y_{S} ≅ tor_{Λ}X_{S}. □

Remark 4.12. The condition “dim(G_{ν}) ≥ 2 for all ν ϵ S_{f},” is known to hold in “most” extensions arising from geometry, e.g. for the set S_{f} = S_{bad} ∪ S_{p} if k_{∞} = k(A(p)) arises by adjoining the p-division points of an abelian variety A over k with good reduction at all places dividing p and such that G(k_{∞}/k) is a pro-p-group, see (the proof of) corollary 4.37 below. Other important cases are the following ones:

(a) Let k_{∞} be the maximal multiple ℤ_{p}-extension k̃ of k, i.e. the composite of all ℤ_{p}-extensions of k, and assume that μ_{p} ⊆ k or

(b) let k_{∞} be a multiple ℤ_{p}-extension with G ≅ ℤ_{p}^{r}, r ≥ 2, and assume that there is only one prime of k lying over p. Then, as has been observed independently by T. Nguyen-Quang-Do [27, thm 3.2] and McCallum [22, proof of thm 7], the condition holds for S = S_{p} ∪ S_{∞}. Indeed, since ℚ(μ_{p}) has only one prime dividing p, it suffices to consider the second case. But then all inertia groups T_{ν}, ν ϵ S_{p}, are conjugate, thus they are all equal and hence an open subgroup of G due to the finiteness of the ideal class group.

With respect to the composite k̃ of all ℤ_{p}-extensions of k there is the following outstanding

Conjecture (R. Greenberg). For any number field k, the Λ(G(k̃/k))-module X_{nr} is pseudo-null.

Recently, W. McCallum [22] proved this conjecture for the base field k = ℚ(μ_{p}) under some mild assumptions. For a list of other cases in which this conjecture is known to hold, see [27, rem 4.6]. Assume that μ_{p} ⊆ k and that the condition “dim(G_{ν}) ≥ 2, for all ν ϵ S_{f},” holds. Then, by the above theorem and theorem 4.5, Greenberg’s conjecture is equivalent to the statement that X_{S} is Λ-torsion-free, compare with [27, 4.4] and [22, Cor 13].

The observation of the previous proof leads also to:

Proposition 4.13. If dim(G_{ν}) ≥ 2 for all ν ϵ S_{f}, then

𝔼 ∼ 𝔼_{S}.

We are also interested in the (Pontryagin duals of the) direct limits

Cl_{S}(k_{∞})(p) = lim ; ⟶_{k′}Cl_{S}(k′)(p),
E_{S}(k_{∞}) := (E_{S}(k_{∞}) ⊗_{ℤ} ℚ_{p}/ℤ_{p})^{∨},

of the p-part of the ideal class group, resp. of the global (S-)units of finite extensions k′ of k inside k_{∞}.

Proposition 4.14. Let T be a set of places of k such that S_{∞} ⊆ T ⊆ S. Assume that dim(T_{ν}) ≥ 1 for all ν ϵ S \ T, where T_{ν} ⊆ G_{ν} denotes the inertia group of ν.

(i) There is an exact sequence of Λ-modules

\xymatrix{0 \ar[r] & Cl_{S}(k_{∞})(p)^{∨} \ar[r]^{ψ} & Cl_{T} (k_{∞})(p)^{∨} \ar[r] & E_{S}(k_{∞}) \ar[r]^{φ} & E_{T} (k_{∞}) \ar[r] & 0.}

(ii) Assume that S \ T = {ν}. Then, if dim(G_{ν}) ≥ 1 (resp. dim(G_{ν}) ≥ 2),

then coker(ψ) ≅ ker(φ) is Λ-torsion (resp. pseudo-null). (iii) If dim(G_{ν}) ≥ 2 for every ν ϵ S \ T, then there are canonical pseudo-

isomorphisms

Cl_{S}(k_{∞})(p)^{∨} ∼ Cl_{T} (k_{∞})(p)^{∨}, E_{S}(k_{∞}) ∼ E_{T} (k_{∞}).

Proof. Consider the canonical exact diagram for a finite extension k′ of k in k_{∞}

\xymatrix{E_{T} (k′) ⊗_{ℤ} ℤ_{p} \ar@{^{(}->}[r]^{i_{k′}} & E_{S}(k′) ⊗_{ℤ} ℤ_{p} \ar[r] & ⨁_{(S\T)(k′)} ℤ_{p} \ar[r] & Cl_{T} (k′)(p) \ar@{->>}[r]^{π_{k′}} & Cl_{S}(k′)(p).}

Setting C(k′) := coker(i_{k′}) (resp. D(k′) := ker(π_{k′})), C_{∞} = \underrightarrow{lim}C(k′) (resp.

D_{∞} = \underrightarrow{lim}D(k′)) and tensoring with ℚ_{p}/ℤ_{p} we get the following exact sequences

\xymatrix{0 \ar[r] & E_{T} (k′) ⊗_{ℤ} ℚ_{p}/ℤ_{p} \ar[r] & E_{S}(k′) ⊗_{ℤ} ℚ_{p}/ℤ_{p} \ar[r] & C(k′) ⊗_{ℤ} ℚ_{p}/ℤ_{p} \ar[r] & 0,}
\xymatrix{0 \ar[r] & D(k′) \ar[r] & C(k′) ⊗_{ℤ} ℚ_{p}/ℤ_{p} \ar[r] & ⨁_{(S\T)(k′)} ℚ_{p}/ℤ_{p} \ar[r] & 0,}
\xymatrix{0 \ar[r] & D(k′) \ar[r] & Cl_{T} (k′)(p) \ar[r] & Cl_{S}(k′)(p) \ar[r] & 0.}

Taking the direct limit over all finite subextensions k′, we get an isomorphism D_{∞} ≅ C_{∞} ⊗ ℚ_{p}/ℤ_{p} because the transition maps for the sum of the ℚ_{p}/ℤ_{p}’s is just the multiplication with the ramification index. The first result follows after taking the Pontryagin dual. Now assume that S \ T consists of a single prime and set Ḡ := G(k′/k) Since

then Ḡ_{ν} = G_{ν}G(k_{∞}/k′)/G(k_{∞}/k′) acts trivial on ⨁_{(S\T)(k′)} ℤ_{p} ≅ IndḠ_{ν} ; Ḡℤ_{p} and

therefore also on C(k′)⊗ℚ_{p}/ℤ_{p}, it follows that G_{ν} acts trivial on (C_{∞} ⊗ℚ_{p}/ℤ_{p})^{∨}. But then any surjection Λ^{n} ↠ (C_{∞} ⊗ ℚ_{p}/ℤ_{p})^{∨} factors through (Ind_{G}^{G_{ν}}ℤ_{p})^{n} which is torsion (resp. pseudo-null) if dim(G_{ν}) ≥ 1 (resp. dim(G_{ν}) ≥ 2). The last statement is a consequence of the second one. □

The Λ-modules Cl_{S}(k_{∞})(p)^{∨} and E_{S}(k_{∞}) are related to each other and to X_{S} via Kummer theory:

Proposition 4.15. Assume that μ_{p^{∞}} ⊆ k_{∞}. Then the following holds:

(i) There are exact sequences of Λ-modules

\xymatrix{0 \ar[r] & Cl_{S}(k_{∞})(p)^{∨} \ar[r] & X_{S}(−1) \ar[r] & E_{S}(k_{∞}) \ar[r] & 0}

and, if k_{∞} is contained in k_{Σ}, where Σ = S_{p} ∪ S_{∞},

\xymatrix{0 \ar[r] & Cl(k_{∞})(p)^{∨} \ar[r] & X_{Σ}(−1) \ar[r] & E(k_{∞}) \ar[r] & 0.}

(ii) Cl_{S}(k_{∞})(p)^{∨} is Λ-torsion. If dim(G_{ν}) ≥ 1 for every ν ϵ S_{p}, then Cl(k_{∞})(p)^{∨} is Λ-torsion, too. In particular, there are exact sequences

\xymatrix{0 \ar[r] & Cl_{S}(k_{∞})(p)^{∨} \ar[r] & tor_{Λ}X_{S}(−1) \ar[r] & tor_{Λ}E_{S}(k_{∞}) \ar[r] & 0, \\ 0 \ar[r] & Cl(k_{∞})(p)^{∨} \ar[r] & tor_{Λ}X_{Σ}(−1) \ar[r] & tor_{Λ}E(k_{∞}) \ar[r] & 0.}

Proof. The long exact H^{i}(G(k_{S}/k_{∞}), −) -sequence of

\xymatrix{0 \ar[r] & μ_{p^{n}} \ar[r] & E_{S}(k_{S}) \ar[r]^{p^{n}} & E_{S}(k_{S}) \ar[r] & 0}

induces the short exact sequence

\xymatrix{0 \ar[r] & E_{S}(k_{∞})/p^{n} \ar[r] & H^{1}(G(k_{S}/k_{∞}), μ_{p^{∞}}) \ar[r] & _{p^{n}}H^{1}(G(k_{S}/k_{∞}), E_{S}(k_{s})) \ar[r] & 0,}

i.e. after taking the direct limit with respect to n

\xymatrix{0 \ar[r] & E_{S}(k_{∞}) ⊗_{ℤ} ℚ_{p}/ℤ_{p} \ar[r] & H^{1}(G(k_{S}/k_{∞}), ℚ_{p}/ℤ_{p})(1) \ar[r] & Cl_{S}(k_{∞})(p) \ar[r] & 0.}

Taking the dual, we obtain the first statement. A canonical map Cl(k_{∞}(p)^{∨}) → X_{S}(−1) which is compatible with the inclusion Cl_{S}(k_{∞})(p)^{∨} → X_{S}(−1) from the first sequence can be defined exactly as in the ℤ_{p}-case, see [24, 11.4.2 and errata]. Then the exactness of the second sequence at the first term is obtained from the first one and prop. 4.15:

Cl(k_{∞})(p)^{∨}/Cl_{Σ}(k_{∞})(p)^{∨} ⊆ E_{Σ} ≅ X_{Σ}(−1)/Cl_{Σ}(k_{∞})(p)^{∨},

i.e. Cl(k_{∞})(p)^{∨} can be considered as submodule of X_{Σ}(1) and then its quotient is E. Comparing the ranks of X_{S} and E_{S} (see 4.26) (with respect to an arbitrary open subgroup H ⊆ G such that Λ(H) is integral), we conclude that Cl_{S}(k_{∞})(p)^{∨} is Λ-torsion while the analogous result for Cl(k_{∞})(p)^{∨} follows from prop. 4.15. Now, the last sequences can be derived from the prior ones by rank considerations or by applying the snake lemma to the canonical sequence of homotopy theory (2.2). □

In the ℤ_{p}-case there exists a remarkable duality between the inverse and direct limit of the (S-) ideal class groups in the ℤ_{p}-tower, viz the pseudo-isomorphisms

X_{nr} ∼ E^{1}(Cl(k_{∞})(p)^{∨}) ∼ (Cl(k_{∞})(p)^{∨})^{∘},
X_{cs}^{S} ∼ E^{1}(Cl_{S}(k_{∞})(p)^{∨}) ∼ (Cl_{S}(k_{∞})(p)^{∨})^{∘}.

Therefore it seems natural (though maybe very optimistic) to pose the following

Question 4.16. Is it true that for any p-adic Lie extension (at least under the assumption “dim(G_{ν}) ≥ 2, for all ν ϵ S_{f},”) there exist pseudo-isomorphisms

X_{nr} ∼ E^{1}(Cl(k_{∞})(p)^{∨}) ≡ (Cl(k_{∞})(p)^{∨})^{∘} mod PN ,
X_{cs}^{S} ∼ E^{1}(Cl_{S}(k_{∞})(p)^{∨}) ≡ (Cl_{S}(k_{∞})(p)^{∨})^{∘} mod PN ?

Observe, that X_{nr} ∼ X_{cs}^{S} and Cl_{S}(k_{∞})(p)^{∨} ∼ Cl(k_{∞})(p)^{∨} by 4.7, 4.14. Hence, it would suffice to prove the existence of one of the pseudo-isomorphisms. By prop. 4.15(ii) and theorem 4.9 the question would be true if one could show that the Λ-torsion of E_{S}(k_{∞}) is pseudo-null. But it seems difficult to prove the latter statement directly. In fact, in the case of a multiple ℤ_{p}-extension k_{∞}|k where μ_{p^{∞}} ⊆ k_{∞} and k has only one prime above p , W. McCallum [22, thm 7] answers the above question positively and then derives tor_{Λ}E_{S}(k_{∞}) = 0 just from the desired pseudo-isomorphism. This is the only case to the knowledge of the author where a positive answer to this question is known. Also J. Nekovar (unpublished) announced a result in the direction of the question.

The next result generalizes theorem 11.3.7 of [24].

Theorem 4.17. Let k_{∞}|k be a p-adic Lie extension such that G is without ptorsion and 𝔽_{p}[[G]] is an integral domain. Then G = G(k_{S}(p)/k_{∞}) is a free pro-p-group if and only if μ(X_{S}) = 0 and the weak Leopoldt conjecture holds: H^{2}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p}) = 0.

Proof. Since G is pro-p it is free if and only if H^{2}(G, ℤ/p) = 0, i.e. if and only if p(X_{S}) and H^{2}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p}) vanish. But, by remark 3.33 of [34] and since X_{S} does not contain any pseudo-null submodule, these two conditions are equivalent to the vanishing of μ(X_{S}) and the validity of the weak Leopoldt conjecture. □

The next theorem, which generalizes theorem 11.3.8 in [24], shows that the validity of the weak Leopoldt conjecture and the vanishing of the μ-invariant are properties which should be considered simultaneously if one studies the behavior of X_{S} under change of the base field.

Theorem 4.18. Let K|k be a finite Galois p-extension inside k_{S}, k_{∞}|k a p-adic pro-p Lie extension such that

(∗) G = G(k_{∞}/k) is without p-torsion and 𝔽_{p}[[G]] is integral.

Set K_{∞} = Kk_{∞} and G′ = G(K_{∞}/K). Then G′ satisfies the condition (∗), too, and the following holds

\{ \begin{matrix}μ(X_{S}(k_{∞}/k)) = 0 and \\ H^{2}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p}) = 0 \end{matrix} \} ⇔ \{ \begin{matrix}μ(X_{S}(K_{∞}/K)) = 0 and \\ H^{2}(G_{S}(K_{∞}), ℚ_{p}/ℤ_{p}) = 0 \end{matrix} \} .

In particular, if k_{∞} contains the cyclotomic ℤ_{p}-extension, then

μ(X_{S}(k_{∞}/k)) = 0 ⇔ μ(X_{S}(K_{∞}/K)) = 0.

Proof. Let H′ := H ∩ G(k_{S}(p)/K). Then, by theorem 4.17, the statements to be compared are equivalent to the freeness of H, resp. H′, thus equivalent to cd_{p}(H) = 1, resp. cd_{p}(H′) = 1. But, since H′ is open in H and cd_{p}(H) < ∞, we have cd_{p}(H′) = cd_{p}(H) by [24] 3.3.5,(ii). □

The same arguments prove the following

Theorem 4.19. Let K_{∞}|k_{∞}|k be p-adic pro-p Lie extensions (inside k_{S}) such that for both G(K_{∞}/K) and G(k_{∞}/k) the condition (∗) of the previous theorem holds. Then

\{ \begin{matrix}μ(X_{S}(k_{∞}/k)) = 0 and \\ H^{2}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p}) = 0 \end{matrix} \} ⇒ \{ \begin{matrix}μ(X_{S}(K_{∞}/k)) = 0 and \\ H^{2}(G_{S}(K_{∞}), ℚ_{p}/ℤ_{p}) = 0 \end{matrix} \} .

The next theorem, which generalizes theorem 11.3.5 in [24], describes the “difference” if we vary the finite set of places S defining the module X_{S}. By T (K/k) ⊆ G(K/k) we shall denote the inertia subgroup for a Galois extension K|k of local fields and, for an arbitrary set of places S of k and a p-adic analytic extension k_{∞}|k, we write S^{cd}(k) for the subset of finite places which decompose completely in k_{∞}|k.

Theorem 4.20. Let S ⊇ T ⊇ S_{p} ∪S_{∞} be finite sets of places of k and let k_{∞}|k be a p-adic pro-p Lie extension inside k_{T} with Galois group G. Assume that G does not contain any p-torsion element and that the weak Leopoldt conjecture holds for k_{∞}|k. Then there exists a canonical exact sequence of Λ-modules

\xymatrix{0 \ar[r] & ⨁_{(S\T)(k)} Ind_{G}^{G_{ν}}T (k_{ν}(p)/k_{ν})_{G_{k∞,ν}} \ar[r] & X_{S} \ar[r] & X_{T} \ar[r] & 0}

and the direct sum on the left is isomorphic to

⨁_{p^{∞}|f_{ν}, μ_{p}⊆k_{ν}^{(S\T)(k)}} Ind_{G}^{G_{ν}}ℤ_{p}(1) ⊕ ⨁_{(S\T)^{cd}(k)} Λ/p^{t_{ν}},

where p^{t_{ν}} = #μ(k_{ν})(p) and, as before, f_{ν} = f(k_{∞,ν}/k_{ν}) denotes the degree of the extension of the corresponding residue class fields. In particular, there is an exact sequence of Λ-torsion modules

\xymatrix{0 \ar[r] & ⨁_{(S\T)(k)} Ind_{G}^{G_{ν}}T (k_{ν}(p)/k_{ν})_{G_{k∞,ν}} \ar[r] & tor_{Λ}X_{S} \ar[r] & tor_{Λ}X_{T} \ar[r] & 0.}

Proof. Since H^{2}(G_{T} (k_{∞})(p), ℚ_{p}/ℤ_{p}) = 0, we have an exact sequence

\xymatrix{0 \ar[r] & G(k_{S}(p)/k_{T} (p))_{G_{T} (k_{∞})}^{ab} \ar[r] & X_{S} \ar[r] & X_{T} \ar[r] & 0.}

Setting G = G_{T} (k)(p) and using [24, 10.5.4,10.6.1] as well as lemma 2.9, we obtain

G(k_{S}(p)/k_{T} (p))_{G_{T} (k_{∞})}^{ab} ≅ ( ⨁_{(S\T)(k)} Ind_{G}^{G_{ν}}T (k_{T} (p)_{ν}(p)/k_{T} (p)_{ν}) )_{G_{T} (k_{∞})}
≅ ⨁_{(S\T)(k)} Ind_{G}^{G_{ν}}T (k_{ν}(p)/k_{ν})_{G_{k∞,ν}} .

Observe that, for ν ϵ S \ T,

T (k_{ν}(p)/k_{ν}) ≅ \{ \begin{matrix}ℤ_{p}(1) & if μ_{p} ⊆ k_{ν}, \\ 0 & otherwise. \end{matrix}

Since G is without p-torsion and ν ϵ S \ T is unramified in k_{∞}|k, there are only two possibilities for G_{ν} :

G_{ν} = \{ \begin{matrix}0 & if ν is completely decomposed in k_{∞}|k, \\ ℤ_{p} & if p^{∞}|f_{ν}, \end{matrix}

respectively,

G_{k_{∞},ν}(p) ≅ \{ \begin{matrix}ℤ_{p}(1) ⋊ ℤ_{p}, & if ν is completely decomposed in k_{∞}|k, \\ ℤ_{p}(1), & if p^{∞}|f_{ν}. \end{matrix}

It follows that

G(k_{S}(p)/k_{T} (p))_{G_{T} (k_{∞})}^{ab} ≅ ⨁_{p^{∞}|f_{ν}, μ_{p}⊆k_{ν}^{(S\T)(k)}} Ind_{G}^{G_{ν}}ℤ_{p}(1) ⊕ ⨁_{(S\T)^{cd}(k)} Λ/p^{t_{ν}}.

In particular, this module is Λ-torsion and therefore the second statement follows from the first. □

Recalling that μ is additive on short exact sequences of Λ-torsion modules we obtain the following

Corollary 4.21. Under the assumptions of the theorem,

μ(X_{S}) = μ(X_{T}) + ∑_{(S\T)^{cd}(k)} t_{ν},

where p^{t_{ν}} = #μ(k_{ν})(p).

4.1.2. Global units. We still consider p-adic Lie extensions k_{∞}|k with Galois group G = G(k_{∞}/k).

Recall that we denote the norm compatible S-units of k_{∞} by 𝔼_{S} := lim ; ⟵_{k′}(O_{k′,S}^{×} ⊗

ℤ_{p}). Noting that 𝔼_{S} ≅ lim ; ⟵_{k′}H^{1}(G_{S}(k′), ℤ_{p}(1)) by Kummer theory and the finiteness

of the S-ideal class group, we are going to derive some relations between 𝔼_{S} and H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨} by interpreting Jannsen’s spectral sequence ([19], see also [34, Thm. 4.5]) or for Iwasawa adjoints with respect to A = ℚ_{p}/ℤ_{p}(1) = μ_{p^{∞}}(k_{S}). We assume that G does not have any p-torsion, i.e. G is a Poincaré group at p, and we denote the character which gives the operation of G on the dualizing module by χ^{−1}.

Proposition 4.22. (i) If μ_{p^{∞}} ⊆ k_{∞}, then (a) if cd_{p}(G) = 1 :

𝔼_{S} ≅ ℤ_{p}(1)(χ) ⊕ E^{0}(X_{S}(−1))
lim ; ⟵_{k′}H^{2}(G_{S}(k′), ℤ_{p}(1)) ≅ E^{1}(X_{S}(−1)),
E^{i}(X_{S}(−1)) = 0 for i ≥ 2.

(b) if cd_{p}(G) = 2 : there is an exact sequence

[figure: 0; 𝔼_{S}; E^{0}(X_{S}(−1)); ℤ_{p}(1)(χ)]
\xymatrix{lim ; ⟵_{k′} H^{2}(G_{S}(k′), ℤ_{p}(1)) \ar[r] & E^{1}(X_{S}(−1)) \ar[r] & 0,}

and

E^{i}(X_{S}(−1)) = 0 for i ≥ 2.

(c) if cd_{p}(G) = 3 : there is an exact sequence

\xymatrix{0 \ar[r] & lim ; ⟵_{k′} H^{2}(G_{S}(k′), ℤ_{p}(1)) \ar[r] & E^{1}(X_{S}(−1)) \ar[r] & ℤ_{p}(1)(χ) \ar[r] & 0,}

and

𝔼_{S} ≅ E^{0}(X_{S}(−1)),
E^{i}(X_{S}(−1)) = 0 for i ≥ 2.

(d) if cd_{p}(G) ≥ 4 :

𝔼_{S} ≅ E^{0}(X_{S}(−1)),

lim ; ⟵_{k′}H^{2}(G_{S}(k′), ℤ_{p}(1)) ≅ E^{1}(X_{S}(−1)),

E^{i}(X_{S}(−1)) = \{ \begin{matrix}ℤ_{p}(1)(χ) & if i = cd_{p}(G) − 2, \\ 0 & otherwise, \end{matrix} for i ≥ 2.

Similar results hold for arbitrary A with k(A) ⊆ k_{∞} if 𝔼_{S} is replaced by lim ; ⟵_{k′}H^{1}(G_{S}(k′), T_{p}A), X_{S}(−1) by X_{S}[A], ...

(ii) If μ(k_{∞})(p) is finite, then (a) if cd_{p}(G) = 1 : then there is an exact sequence

\xymatrix{0 \ar[r] & 𝔼_{S} \ar[r] & E^{0}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}) \ar[r] & μ(k_{∞})(p)^{∨}(χ) \ar[r] & lim ; ⟵_{k′} H^{2}(G_{S}(k′), ℤ_{p}(1)).}

(a_{1}) If in addition H^{2}(G_{S}(k_{∞}), μ_{p^{∞}}) = 0, then the cokernel of the sequence is E^{1}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}) and E^{i}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}) = 0 for i ≥ 2.

(a_{2}) If in addition H^{2}(G_{S}(k_{∞}), ℚ_{p}/ℤ_{p}) = 0, then there is a short exact sequence

\xymatrix{0 \ar[r] & 𝔼_{S} \ar[r] & E^{0}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}) \ar[r] & μ(k_{∞})(p)^{∨}(χ) \ar[r] & 0.}
(b) if cd_{p}(G) = 2, then 𝔼_{S} ≅ E^{0}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}).

If in addition H^{2}(G_{S}(k_{∞}), μ_{p^{∞}}) = 0, then there is an exact sequence

\xymatrix{0 \ar[r] & lim ; ⟵_{k′} H^{2}(G_{S}(k′), ℤ_{p}(1)) \ar[r] & E^{1}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}) \ar[r] & μ(k_{∞})(p)^{∨}(χ) \ar[r] & 0}

and

E^{i}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}) = 0 for i ≥ 2.
(c) if cd_{p}(G) ≥ 3, then 𝔼_{S} ≅ E^{0}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}).

If in addition H^{2}(G_{S}(k_{∞}), μ_{p^{∞}}) = 0, then

𝔼_{S} ≅ E^{0}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}),
lim ; ⟵_{k′}H^{2}(G_{S}(k′), ℤ_{p}(1)) ≅ E^{1}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}),

E^{i}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}) = \{ \begin{matrix}μ(k_{∞})(p)(χ) & if i = cd_{p}(G) − 1, \\ 0 & otherwise, \end{matrix} for i ≥ 2.

(iii) If μ(k_{∞})(p) = 0, then there is in addition to the results for finite μ(k_{∞})(p) the following exact sequence:

[figure: 0; E^{1}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}); lim ; ⟵_{k′} H^{2}(G_{S}(k′), ℤ_{p}(1))]
\xymatrix{E^{0}(H^{2}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}) \ar[r] & E^{2}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}) \ar[r] & 0,}

and

E^{i}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}) ≅ E^{i−2}(H^{2}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨})).

For the proof apply Jannsen’s theorem (see [34, Thm. 4.5,Cor. 4.6]) and note the following facts: H^{1}(G_{S}(k_{∞}), A)^{∨} ≅ X_{S}[A] if k(A) ⊆ k_{∞}, H^{2}(G_{S}(k_{∞}), A) = 0 if μ_{p^{∞}} ⊆ k_{∞} because the weak Leopoldt conjecture is true for the cyclotomic extension of any number field. Furthermore, we applied several times [18, 2.6]. Also observe, that the reflexive module E^{0}(X_{S}(−1)) is projective in the case cd_{p}(G) = 1 regarding the defining sequence of the transpose functor D and using that pd(Λ) = cd_{p}(G)+1 = 2. The last statement of (ii)(a) is proved in [24] 11.3.9.

These results bear a lot of information about the structure of H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨} and 𝔼_{S}, e.g. one can derive the projective dimension of the latter module (using corollary [34, Cor. 6.3]) and some information about the dimensions of the modules occurring above, in particular whether a module is torsion(free). Furthermore, we see that 𝔼_{S} is reflexive for almost all cases with cd_{p}(G) ≥ 2 by proposition 3.11 of [34]. In order to relate 𝔼_{S} to the finitely generated Λ-module

E_{S}(k_{∞}) = (E_{S}(k_{∞}) ⊗_{ℤ} ℚ_{p}/ℤ_{p})^{∨}

we need some technical lemmas.

Lemma 4.23. (i) Let G = G(k_{∞}/k) ≅ ℤ_{p}^{d}, d ≥ 1, and G_{n} := p^{n}G.

(a) If μ_{p^{∞}} ⊆ k_{∞}, then with Γ = G(k(μ_{p^{∞}}) and Γ_{n} = p^{n}Γ the following

holds

H^{i}(G_{n}, μ_{p^{∞}}) = μ(k_{n})(p)^{\binom{d−1}{i}},

where k_{n} = k(μ_{p^{∞}})^{Γ_{n}}.

(b) If μ(k_{∞})(p) is finite, then for any n such that μ(k_{∞})(p)^{G_{n}} = μ(k_{∞})(p)

it holds

H^{i}(G_{n}, μ_{p^{∞}}) = μ(k_{∞})(p)^{\binom{d}{i}}.

(ii) Let G be a finitely generated pro-p Lie group without p-torsion which fits into a exact sequence

\xymatrix{1 \ar[r] & U \ar[r] & G \ar[r]^{π} & Γ \ar[r] & 1,}

with Γ ≅ ℤ_{p} and let G_{n} be an open subgroup. Assume that Γ_{n} := π(G_{n}) acts via a splitting trivially on U_{n} = G_{n} ∩ U. Then H^{2}(G_{n}, μ(k_{∞})(p)) is finite and the following holds

(a) If μ_{p^{∞}} ⊆ k_{∞} and Γ = G(k(μ_{p^{∞}}), then

H^{1}(G_{n}, μ(k_{∞})(p)) ≅ μ(k_{n})(p)^{s} ⊕ ⨁_{i} μ_{p^{ν_{i}}}(k_{n}),

where U_{n}^{ab} ≅ ℤ_{p}^{s} ⊕ ⨁_{i} ℤ_{p}/p^{ν_{i}} with U_{n} = U ∩ G_{n}.

(b) If μ(k_{∞})(p) is finite, then for any n such that μ(k_{∞})(p)^{G_{n}} = μ(k_{∞})(p) there is an exact sequence

\xymatrix{0 \ar[r] & μ(k_{∞})(p) \ar[r] & H^{1}(G_{n}, μ(k_{∞})(p)) \ar[r] & μ(k_{∞})(p)^{s} ⊕ ⨁_{i} μ_{p^{ν_{i}}}(k_{∞}) \ar[r] & 0.}
(c) If cd_{p}(G) = 2, then

H^{2}(G_{n}, μ(k_{∞})(p)) ≅ \{ \begin{matrix}0 & if μ_{p^{∞}} ⊆ k_{∞}, \\ μ(k_{∞})(p) & otherwise. \end{matrix}

Proof. Consider the exact sequence

\xymatrix{1 \ar[r] & U \ar[r] & G \ar[r]^{π} & Γ \ar[r] & 1,}

and let U_{n} = G_{n} ∩ U and Γ_{n} = π(G_{n}). The Hochschild-Serre spectral sequence gives

\xymatrix{H^{1}(Γ_{n}, H^{i}(U_{n}, μ(k_{∞})(p))) \ar@{^{(}->}[r] & H^{i+1}(G_{n}, μ(k_{∞})(p)) \ar@{->>}[r] & H^{i+1}(U_{n}, μ(k_{∞})(p))^{Γ_{n}}}

for i ≥ 0. Let us first assume that μ_{p^{∞}} ⊆ k_{∞} : Since U_{n} acts trivially on μ_{p^{∞}}, we get

H^{i}(U_{n}, ℚ_{p}/ℤ_{p}(1)) = H^{i}(U_{n}, ℚ_{p}/ℤ_{p})(1) = (ℚ_{p}/ℤ_{p})^{\binom{d−1}{i}}.

in the abelian case by the Künneth formula. As ℚ_{p}/ℤ_{p}(1)_{Γ_{n}} = 0 it follows that H^{i}(G_{n}, μ_{p^{∞}}) = H^{i}(U_{n}, μ_{p^{∞}})^{Γ_{n}} = μ(k_{n})^{\binom{d−1}{i}}. In the non-abelian case we calculate

H^{1}(G_{n}, μ_{p^{∞}}) = H^{1}(U_{n}, ℚ_{p}/ℤ_{p})(1)^{Γ_{n}}
= (U_{n}^{ab})^{∨}(1)^{Γ_{n}}
= μ(k_{n})(p)^{s} ⊕ ⨁ μ_{p^{ν_{i}}}(k_{n}).

Hence H^{1}(Γ_{n}, H^{1}(U_{n}, μ(k_{∞})(p))) is finite and the finiteness of H^{2}(G_{n}, μ_{p^{∞}}) follows because H^{2}(U_{n}, μ_{p^{∞}})^{Γ_{n}} ≅ H^{2}(U_{n}, ℚ_{p}/ℤ_{p})(1)^{Γ_{n}} is also finite (H^{2}(U_{n}, ℚ_{p}/ℤ_{p}) is a cofinitely generated abelian group). Now we consider the case of finite μ(k_{∞})(p) : Here H^{1}(Γ_{n}, μ(k_{∞})(p)) = μ(k_{∞})(p) and the abelian case follows again using the Künneth formula. In the non-abelian case the finiteness of H^{2}(G_{n}, μ_{p^{∞}}) is trivial while H^{1}(U_{n}, μ(k_{∞})(p))^{Γ_{n}} can be calculated similarly as above. For the last assertion just note that U_{n} ≅ ℤ_{p}. □

Lemma 4.24. (i) In the situation of the previous lemma (ii) it holds

(a) lim ; ⟵_{m,n}_{p^{m}}H^{1}(G_{n}, E_{S}(k_{∞})/μ(k_{∞})) ≅ lim ; ⟵_{m,n}_{p^{m}}H^{1}(G_{n}, E_{S}(k_{∞})) = 0,

(b) lim ; ⟵_{n}H^{1}(G_{n}, E_{S}(k_{∞})) ⊆ X_{cs}^{S},

(c) \begin{aligned}E^{0}(E_{S}(k_{∞})) & ≅ & lim ; ⟵_{m,,n}_{p^{m}}(E_{S}(k_{∞}) ⊗ ℚ_{p}/ℤ_{p})^{G_{n}} \\ & ≅ & lim ; ⟵_{m,n}(E_{S}(k_{∞})/μ(k_{∞}))^{G_{n}}/p^{m}, \end{aligned}

(d) T_{0}(lim ; ⟵_{n}H^{1}(G_{n}, E_{S}(k_{∞})/μ(k_{∞}))) = T_{0}(E^{1}((E_{S}(k_{∞}) ⊗_{ℤ} ℚ_{p}/ℤ_{p})^{∨})),

(e) that the following sequence is exact:

\xymatrix{0 \ar[r] & lim ; ⟵_{n} H^{1}(G_{n}, E_{S}(k_{∞})/μ(k_{∞})) \ar[r] & E^{1}(E_{S}(k_{∞})) \ar[r] & lim ; ⟵_{m,n}^{p^{m}}H^{2}(G^{n}, E^{S}(k^{∞})/μ(k^{∞})) \ar[r] & 0.}

(ii) If, in addition, cd_{p}(G) ≤ 2, then with κ = \{ \begin{matrix}1 & if μ(k_{∞})(p) is finite, \\ 0 & otherwise,e \end{matrix}

there are the following exact sequences

(a) if cd_{p}(G) = 2 :

\xymatrix{0 \ar[r] & lim ; ⟵_{n} H^{1}(G_{n}, E_{S}(k_{∞})) \ar[r] & lim ; ⟵_{m,n} H^{1}(G_{n}, E_{S}(k_{∞})/μ(k_{∞}))/p^{m} \ar[r] & μ(k_{∞})(p)^{κ} \ar[r] & D \ar[r] & 0,}

[figure: 0; lim ; ⟵_{m,n}^{p^{m}}H^{2}(G^{n}, E^{S}(k^{∞})); lim ; ⟵_{m,n}^{p^{m}}H^{2}(G^{n}, E^{S}(k^{∞})/μ(k^{∞})); D]

\xymatrix{lim ; ⟵_{m,n} H^{2}(G^{n}, E^{S}(k^{∞}))/p^{m} \ar[r] & lim ; ⟵_{m,n} H^{2}(G^{n}, E^{S}(k^{∞})/μ(k^{∞}))/p^{m} \ar[r] & 0,}

where D is some finite module.

(b) if cd_{p}(G) = 1 :

\xymatrix{0 \ar[r] & E^{2}E^{1}(𝔼_{S}) \ar[r] & μ(k_{∞})(p)^{κ} \ar[r] & lim ; ⟵_{n} H^{1}(G_{n}, E_{S}(k_{∞})) \ar[r] & lim ; ⟵_{m,n} H^{1}(G_{n}, E_{S}(k_{∞})/μ(k_{∞}))/p^{m} \ar[r] & 0}

and

lim ; ⟵_{m,n}_{p^{m}}H^{2}(G_{n}, E_{S}(k_{∞})) ≅ lim ; ⟵_{m,n}_{p^{m}}H^{2}(G_{n}, E_{S}(k_{∞})/μ(k_{∞})).

Proof. If we split the long exact cohomology sequence induced by

\xymatrix{0 \ar[r] & μ(k_{∞}) \ar[r] & E_{S}(k_{∞}) \ar[r] & E_{S}(k_{∞})/μ(k_{∞}) \ar[r] & 0,}

we get the following short exact sequences

\xymatrix{0 \ar[r] & F_{n} \ar[r] & H^{1}(G_{n}, μ(k_{∞})) \ar[r] & A_{n} \ar[r] & 0,}
\xymatrix{0 \ar[r] & A_{n} \ar[r] & H^{1}(G_{n}, E_{S}(k_{∞})) \ar[r] & B_{n} \ar[r] & 0,}
\xymatrix{0 \ar[r] & B_{n} \ar[r] & H^{1}(G_{n}, E_{S}(k_{∞})/μ(k_{∞})) \ar[r] & C_{n} \ar[r] & 0}

and furthermore a map \xymatrix{C_{n} \ar@{^{(}->}[r] & H^{2}(G_{n}, μ(k_{∞})(p)).} Evaluating the associated long exact sequences of p^{m}-torsion (snake lemma) and noting the finiteness of A_{n} and C_{n} according to the previous lemma, we get

lim ; ⟵_{m}_{p^{m}}B_{n} ≅ lim ; ⟵_{m}_{p^{m}}H^{1}(G_{n}, E_{S}(k_{∞})/μ(k_{∞})),

\xymatrix{0 \ar[r] & lim ; ⟵_{m} _{p^{m}}H^{1}(G_{n}, E_{S}(k_{∞})) \ar[r] & lim ; ⟵_{m} _{p^{m}}B_{n} \ar[r] & A_{n},}

and therefore

\xymatrix{0 \ar[r] & lim ; ⟵_{m,n} ^{p^{m}}H^{1}(G^{n}, E^{S}(k^{∞})) \ar[r] & lim ; ⟵_{m,n} ^{p^{m}}H^{1}(G^{n}, E^{S}(k^{∞})/μ(k^{∞})) \ar[r] & lim ; ⟵_{n}A_{n}}

is exact.

But lim ; ⟵_{n}A_{n} is a quotient of

lim ; ⟵_{n}H^{1}(G_{n}, μ(k_{∞})(p)) = \{ \begin{matrix}μ(k_{∞})(p) & if d = 1 and μ(k_{∞})(p) is finite, \\ 0 & otherwise. \end{matrix}

(See the previous lemma and note that the transition maps are partially norm maps besides the non-trivial case where they are the natural projections, i.e. identities for n sufficiently big.). Since the middle term is ℤ_{p}-torsion free, we get the desired isomorphism, because, by the Hochschild-Serre spectral sequence, it can be seen in any case that the first group is contained in lim ; ⟵_{m,n}_{p^{m}}Cl_{S}(k_{n}) = 0.

This proves (i)(a) while (b) is again the cited spectral sequence.

The first equality of (i)(c) is just theorem 4.7 (iii) of [34] because E_{S}(k_{∞}) has no ℤ_{p}-torsion while the second one follows by the exact sequence

\xymatrix{(E_{S}(k_{∞})/μ(k_{∞}))^{G_{n}}/p^{m} \ar@{^{(}->}[r] & _{p^{m}}(E_{S}(k_{∞}) ⊗_{ℤ} ℚ_{p}/ℤ_{p})^{G_{n}} \ar@{->>}[r] & _{p^{m}}H^{1}(G_{n}, E_{S}(k_{∞})/μ(k_{∞}))}

and (a). Similar arguments apply for (i)(e), i.e.

E^{1}(E_{S}(k_{∞})) ≅ lim ; ⟵_{m,n}H^{1}(G_{n},_{p^{m}}(E_{S}(k_{∞}) ⊗_{ℤ} ℚ_{p}/ℤ_{p})).

The assertion (d) is a direct consequence of (e), because lim ; ⟵_{m,n}_{p^{m}}H^{2}(G_{n}, E_{S}(k_{∞})/μ(k_{∞})) is ℤ_{p}-torsion-free.

Now let us assume that cd_{p}(G) ≤ 2. With the notation as above and recalling that A_{n}, B_{n} and C_{n} are finite, we get exact sequences

\xymatrix{0 \ar[r] & A_{n} \ar[r] & H^{1}(G_{n}, E_{S}(k_{∞})) \ar[r] & B_{n} \ar[r] & 0,}

\xymatrix{0 \ar[r] & B_{n} \ar[r] & lim ; ⟵_{m}H^{1}(G_{n}, E_{S}(k_{∞})/μ(k_{∞}))/p^{m} \ar[r] & C_{n} \ar[r] & 0}

and

\xymatrix{0 \ar[r] & C_{n} \ar[r] & H^{2}(G_{n}, μ(k_{∞})) \ar[r] & D_{n} \ar[r] & 0.}

Passing to the limit gives the first exact sequence in (ii)(a) (Note that the transition maps of the system {C_{n}} are the canonical projections, i.e. identities for n sufficiently large). The second one is proved similarly using

\xymatrix{D_{n} \ar@{^{(}->}[r] & H^{2}(G_{n}, E_{S}(k_{∞})) \ar[r] & H^{2}(G_{n}, E_{S}(k_{∞})/μ(k_{∞})) \ar[r] & H^{3}(G_{n}, μ(k_{∞})(p)) = 0}

and \xymatrix{H^{2}(G_{n}, μ(k_{∞})(p)) \ar@{->>}[r] & D_{n}} . The proof of (ii)(b) is completely analogous, just note that lim ; ⟵_{n}F_{n} ≅ E^{2}E^{1}(𝔼_{S}) because the latter module is the cokernel of

𝔼_{S} → E^{0}E^{0}(𝔼_{S}) ≅ E^{0}(E_{S}(k_{∞})). □

Proposition 4.25. There is an exact sequence

\xymatrix{0 \ar[r] & ℤ_{p}(1)^{δ} \ar[r] & 𝔼_{S} \ar[r] & E^{0}(E_{S}(k_{∞})) \ar[r] & C}

with

C = \{ \begin{matrix}μ(k_{∞})(p) & if d = 1 and μ(k_{∞})(p) finite \\ ℤ_{p}(1) & if d = 2 and μ_{p^{∞}} ⊆ k_{∞} \\ f.g. ℤ_{p}-module & d ≥ 3 and G non-abelian \\ 0 & otherwise \end{matrix}

and

δ = \{ \begin{matrix}1 & if d = 1, μ_{p^{∞}} ⊆ k_{∞}, \\ 0 & otherwise. \end{matrix}

If in addition the weak Leopoldt conjecture holds, the right map is onto in the case d = 1 and μ(k_{∞})(p) finite.

Proof. Taking G_{n}-invariants of the exact sequence

\xymatrix{0 \ar[r] & μ(k_{∞})(p) \ar[r] & E_{S}(k_{∞}) ⊗_{ℤ} ℤ_{p} \ar[r] & (E_{S}(k_{∞})/μ(k_{∞})) ⊗_{ℤ} ℤ_{p} \ar[r] & 0}

and passing to the inverse limit, we get

\xymatrix{0 \ar[r] & lim ; ⟵_{n} μ(k_{n})(p) \ar[r] & 𝔼_{S} \ar[r] & lim ; ⟵_{m,n} (E_{S}(k_{∞})/μ(k_{∞}))^{G_{n}}/p^{m} \ar[r] & lim ; ⟵_{n} H^{1}(G_{n}, μ(k_{∞})(p))}

The result follows except the fact that E^{0} maps onto the finite group of roots of unity in the case when d = 1. But this is proved in [24] 11.3.9 under the assumption that the weak Leopoldt conjecture holds. □

Corollary 4.26. Let k_{∞}|k be a p-adic Lie extension such that G does not have any p-torsion. Then

E^{0}(𝔼_{S}) ≅ E^{0}E^{0}(E_{S}(k_{∞})) ≅ E^{0}(H^{1}(G_{S}(k_{∞}), μ_{p^{∞}})^{∨}).

In particular, if G is in addition pro-p and H^{2}(G_{S}(k_{∞}), μ_{p^{∞}}) = 0 (e.g. if μ_{p^{∞}} ⊆ k_{∞}), then

rk_{Λ}𝔼_{S} = rk_{Λ}E_{S} = r_{2}(k).

Now the question arises whether the module E^{0}(𝔼_{S}) is not only reflexive but also projective. While in the case cd_{p}(G) = 1 this is always true, in higher dimensions one needs additional conditions. We will only get a satisfying answer in the two dimensional case:

Proposition 4.27. Let k_{∞}|k be a p-adic Lie extension such that cd_{p}(G) = 2 and assume that the weak Leopoldt conjecture holds for k_{∞}. Then the following is equivalent:

(i) E^{0}(𝔼_{S}) is projective,
(ii) T_{0}E^{1}(E_{S}(k_{∞})) = 0.

Remark 4.28. These equivalent statements hold for example, if either μ_{p^{∞}} ⊆ k_{∞} or μ(k_{∞})(p) = 0, and T_{0}(X_{cs}^{S}) = 0, i.e. if X_{cs}^{S} does not have any non-zero finite submodule, because then T_{0}E^{1}(E_{S}(k_{∞})) = 0 by lemma 4.24.

Proof. Since we already know that pd(E^{0}(𝔼_{S})) ≤ 1, because E^{0}(𝔼_{S}) is the second syzygy of D𝔼_{S}, the projectivity is equivalent to the vanishing of E^{1}E^{0}(𝔼_{S}). Now the equivalence stated above follows from the next lemma. □

Lemma 4.29. In the situation of the proposition it holds

T_{0}E^{1}(E_{S}(k_{∞})) ≅ E^{1}E^{0}(𝔼_{S}) ≅ E^{3}E^{1}(𝔼_{S})

Proof. Set M := E_{S}(k_{∞}) and consider the exact sequence

\xymatrix{0 \ar[r] & M/T_{1}(M) \ar[r] & E^{0}(𝔼_{S}) \ar[r] & E^{2}D(M) \ar[r] & 0.}

The long exact sequence for E^{i} gives

\xymatrix{0 = E^{1}E^{2}D(M) \ar[r] & E^{1}E^{0}(M) \ar[r] & E^{1}(M/T_{1}(M)) \ar[r] & E^{2}E^{2}D(M).}

On the other hand there is the exact sequence

\xymatrix{0 = E^{0}(T_{1}(M)) \ar[r] & E^{1}(M/T_{1}(M)) \ar[r] & E^{1}(M) \ar[r] & E^{1}E^{1}D(M).}

Since E^{i}E^{i}D(M) is pure of codimension i, the isomorphism follows. But E^{1}E^{0}(𝔼_{S}) ≅ E^{3}E^{1}(𝔼_{S}) by the spectral sequence due to Björk, see 2.5. □

The proposition above should be compared with the following result which has already been observed by Kay Wingberg (unpublished):

Proposition 4.30. If cd_{p}(G) = 1, then for sufficiently large n there is a canonical exact sequence

\xymatrix{0 \ar[r] & E_{S}(k_{∞})^{G_{n}} \ar[r] & E_{S}(k_{∞}) \ar[r] & E^{0}(𝔼_{S}) \ar[r] & C \ar[r] & 0}

where C = E^{2}D(E_{S}(k_{∞})) is connected with E^{2}D(𝔼_{S}) by the exact sequence

\xymatrix{0 \ar[r] & E^{2}D(𝔼_{S}) \ar[r] & μ(k_{∞})(p)^{κ} \ar[r] & T_{0}X_{cs}^{S} \ar[r] & C^{∨} \ar[r] & 0.}

Proof. The first sequence is just the canonical sequence 2.2 for the module E_{S}(k_{∞}) while the second one already occurred in lemma 4.24 (ii)(b) as we show now: The fact that T_{0}(X_{cs}^{S}) ≅ lim ; ⟵_{n}H^{1}(G_{n}, E_{S}(k_{∞})) is well known (see for example [24,

XI.§3.]). Recall that E^{2}E^{1}(E_{S}(k_{∞})) ≅ T_{0}(E^{1}(E_{S}(k_{∞})))^{∨} and apply lemma 4.24 (i)(d) to recover C. Using 4.24, (i)(e) and (ii)(b) we see that E^{1}E^{1}(E_{S}(k_{∞})) ≅ E^{1}(lim ; ⟵_{m,n}_{p^{m}}H^{2}(G_{n}, E_{S}(k_{∞}))), which we will determine by means of [34, Thm. 4.7

(iii)]:

M := lim ; ⟵_{m,n}_{p^{m}}H^{2}(G_{n}, E_{S}(k_{∞}))^{∨} ≅ lim ; ⟶_{m,n}E_{S}(k_{∞})^{G_{n}}/p^{m} = lim ; ⟶_{m}E_{S}(k_{∞})^{G_{n}}/p^{m}

for n sufficiently large, because E_{S}(k_{∞}) is a finitely generated Λ-module. Hence

E^{1}(M) ≅ lim ; ⟵_{m,n}(_{p^{m}}M)_{G_{n}} = E_{S}(k_{∞})^{G_{n}}.

for n large enough. □

Proposition 4.31. Let k_{∞}/k be a p-adic Lie extension such that G ≅ Γ × Δ where Γ is a pro-p-Lie group of cd_{p}(Γ) = 2, Δ is a finite group of order prime to p. Assume that the weak Leopoldt conjecture holds for k_{∞}. Then the following is true:

(i) There is an exact sequence

[figure: 0; E^{0}E^{0}(𝔼_{S}); Λ^{r_{2}+r_{1}−r′_{1}−s} ⊕ ⨁_{S^{cd}∪S′_{∞}} Ind_{G}^{G_{ν}}(ℤ_{p})]
\xymatrix{Λ^{s} \ar[r] & T_{0}E^{1}(E_{S}(k_{∞})) \ar[r] & 0.}

(ii) If E^{0}E^{0}(𝔼_{S}) is projective, then

E^{0}E^{0}(𝔼_{S}) ≅ Λ^{r_{2}+r_{1}−r′_{1}} ⊕ ⨁_{S^{cd}∪S′_{∞}} Ind_{G}^{G_{ν}}(ℤ_{p}).

Proof. We calculate the Euler characteristic with respect to an arbitrary open normal subgroup U ⊴ Γ using lemma 2.7, proposition 2.13,[18] 5.4 b),

h_{U}(E^{0}E^{0}(𝔼_{S})) = h_{U}(𝔼_{S})
= h_{U}(𝔸_{S}) − h_{U}(X_{S}) + h_{U}(X_{cs}^{S})
= h_{U}(𝔸_{S}) − h_{U}(Y_{S}) + h_{U}(I_{G})
= h_{U}(𝔸_{S}) − h_{U}(Λ^{d}) + h_{U}(N_{H}^{ab}(p)) + h_{U}(Λ) − h_{U}(ℤ_{p})
= h_{U}(𝔸_{S}) − h_{U}(Λ^{r_{2}+r′_{1}}) + h_{U}(⨁_{S′_{∞}} Ind_{G}^{G_{ν}}(ℤ_{p}))
= ∑_{S} Ind_{G}^{G_{ν}}h_{U∩G_{ν}}(𝔸_{ν}) − h_{U}(Λ^{r_{2}+r′_{1}}) + h_{U}(⨁_{S′_{∞}} Ind_{G}^{G_{ν}}(ℤ_{p}))
= ∑_{S^{cd}} Ind_{G}^{G_{ν}}h_{U}(ℤ_{p}) + h_{U}(Λ^{r_{2}+r_{1}−r′_{1}}) + h_{U}(⨁_{S′_{∞}} Ind_{G}^{G_{ν}}(ℤ_{p})).

Therefore, if E^{0}E^{0}(𝔼_{S}) is projective, it follows that

E^{0}E^{0}(𝔼_{S}) ≅ Λ^{r_{2}+r_{1}−r′_{1}} ⊕ ⨁_{S^{cd}∪S′_{∞}} Ind_{G}^{G_{ν}}(ℤ_{p}).

This proves (ii) while (i) follows easily applying proposition 2.5. □

4.2. Selmer groups of abelian varieties. In this section let k be a number field, A a g-dimensional abelian variety defined over k and p a fixed rational odd prime number. For a non-empty, finite set S of places of k containing the places S_{bad} of bad reduction of A , the places S_{p} lying over p and the places S_{∞} at infinity we write H^{i}(G_{S}(k), A), respectively H^{i}(k_{ν}, A), for the cohomology groups H^{i}(G_{S}(k), A(k_{S})), respectively H^{i}(G_{ν}, A(k̄_{ν})), where G_{S}(k) denotes the Galois group of the maximal outside S unramified extension of k, k̄_{ν} the algebraic closure of the completion of k at ν and G_{ν} the corresponding decomposition group. The (p^{m})-)Selmer group Sel(A, k, p^{m}) and the Tate-Shafarevich group � (A, k, p^{m}) fit by definition into the following commutative exact diagram

[figure: 0; 0; 0; A(k)/p^{m}; Sel(A, k, p^{m}); � (A, k, p^{m}); 0; 0; A(k)/p^{m}; H^{1}(G_{S}(k),_{p^{m}}A); _{p^{m}}H^{1}(G_{S}(k), A); 0; ⨁_{S(k)} H^{1}(k_{ν}, A)(p); ⨁_{S(k)} H^{1}(k_{ν}, A)(p).]

If k_{∞} is an infinite Galois extension of k with Galois group G = G(k_{∞}/k,) we get the following commutative exact diagram by passing to the direct limit with respect to m and finite subextensions k′ of k_{∞}/k:

[figure: 0; 0; 0; A(k_{∞}) ⊗ ℚ_{p}/ℤ_{p}; Sel(A, k_{∞}, p^{∞}); � (A, k_{∞}, p^{∞}); 0; 0; A(k_{∞}) ⊗ ℚ_{p}/ℤ_{p}; H^{1}(G_{S}(k_{∞}), A(p)); H^{1}(G_{S}(k_{∞}), A)(p); 0; ⨁_{S(k)} Coind_{G}^{G_{ν}} H^{1}(k_{∞,ν}, A)(p); ⨁_{S(k)} Coind_{G}^{G_{ν}} H^{1}(k_{∞,ν}, A)(p).]

Note that lim ; ⟶_{k′} ⨁_{S(k′)} H^{1}(k′_{ν}, A)(p) ≅ ⨁_{S(k)} Coind_{G}^{G_{ν}}H^{1}(k_{∞,ν}, A)(p). Alternatively,

we can pass to the inverse limits and we will get the following commutative exact diagram

[figure: 0; 0; 0; Â_{k_{∞}}; Ŝel(k_{∞}, A); lim ; ⟵_{k′,m} � (A, k′, p^{m}); 0; 0; Â_{k_{∞}}; lim ; ⟵_{k′} H^{1}(G_{S}(k′), T_{p}A); lim ; ⟵_{k′} T_{p}H^{1}(G_{S}(k′), A); 0; lim ; ⟵_{k′} ⨁_{S(k′)} T_{p}H^{1}(k′_{ν}, A); lim ; ⟵_{k′} ⨁_{S(k′)} T_{p}H^{1}(k′_{ν}, A).]

where Â_{k_{∞}} := lim ; ⟵_{k′,m}A(k′)/p^{m} and Ŝel(k_{∞}, A) := lim ; ⟵_{k′,m}Sel(k′, A, p^{m}) (The limits are

taken with respect to corestriction maps and multiplication by p).

Henceforth we will drop the p from the notation of the Selmer group: Sel(A, k_{∞}) := Sel(A, k_{∞}, p^{∞}). Furthermore, we shall use the following notation for the localglobal modules

𝕌_{S,A} := ⨁_{S_{f} (k)} Ind_{G}^{G_{ν}}H^{1}(k_{∞,ν}, A)(p)^{∨},

𝔸_{S,A} := ⨁_{S_{f} (k)} Ind_{G}^{G_{ν}}H^{1}(k_{∞,ν}, A(p))^{∨},

𝕋_{S,A} := ⨁_{S_{f} (k)} Ind_{G}^{G_{ν}}(A(k_{∞,ν}) ⊗ ℚ_{p}/ℤ_{p})^{∨}.

As a consequence of the long exact sequence of the Tate-Poitou duality theorem we have the following (compact) analogue of proposition 4.7, where we shall write A^{∨} for the dual abelian variety of A and � _{S}^{1}(k_{∞}, A(p)) for the kernel of the localization map

H^{1}(G_{S}(k_{∞}), A(p)) → ⨁_{S(k)} Coind_{G}^{G_{ν}}H^{1}(k_{∞,ν}, A(p)).

Proposition 4.32. Let k_{∞}|k be a p-adic Lie extension with Galois group G. Then, there are the following exact commutative diagrams of Λ = Λ(G)-modules

(i)

[figure: 0; 0; Sel(A, k_{∞})^{∨}; � _{S}^{1}(k_{∞}, A(p))^{∨}; H^{1}(G_{S}(k_{∞}), A(p))^{∨}; H^{1}(G_{S}(k_{∞}), A(p))^{∨}; 𝕌_{S,A}; 𝔸_{S,A}; 𝕋_{S,A}; Ŝel(k_{∞}, A^{∨}); lim_{\overleftarrow{k′}} H^{1}(G_{S}(k′), T_{p}(A^{∨})); 𝕋_{S,A},; H^{2}(G_{S}(k_{∞}), A(p))^{∨}; H^{2}(G_{S}(k_{∞}), A(p))^{∨}; 0; 0]

(ii)

[figure: 0; Ŝel(k_{∞}, A^{∨}); lim_{\overleftarrow{k′}} H^{1}(G_{S}(k′), T_{p}A^{∨}); 𝕋_{S,A}]
\xymatrix{Sel(A, k_{∞})^{∨} \ar[r] & � _{S}^{1}(k_{∞}, A(p))^{∨} \ar[r] & 0,}

(iii)

\xymatrix{0 \ar[r] & � _{S}^{1}(k_{∞}, A(p))^{∨} \ar[r] & Z_{S,A^{∨}(p)} \ar[r] & ⨁_{S_{f} (k)} Ind_{G}^{G_{ν}}(A(k_{∞,ν})(p))^{∨} \ar[r] & A(k_{∞})(p)^{∨} \ar[r] & 0.}

For the proof, just note that by virtue of local Tate duality ([23, Cor.3.4]), the Weil-pairing and 4.1,

(i) H^{1}(k_{∞,ν}, A)(p)^{∨} ≅ { \atop (A^{∨})}_{∞,ν} := lim ; ⟵_{k′,m}A^{∨}(k′_{ν})/p^{m},

(ii) Z_{S,A^{∨}(p)} ≅ lim_{\overleftarrow{k′}}H^{2}(G_{S}(k′), T_{p}(A^{∨})),

(iii) (A(k_{∞,ν}) ⊗ ℚ_{p}/ℤ_{p})^{∨} ≅ lim ; ⟵_{k′}T_{p}H^{1}(k′_{ν}, A^{∨}) and

(iv) H^{1}(k_{∞,ν}, A(p))^{∨} ≅ lim_{\overleftarrow{k′}}H^{1}(k′_{ν}, T_{p}(A^{∨}))

hold.

By a well-known theorem of Mattuck, we have an isomorphism

A(k′_{ν}) ≅ ℤ_{l}^{g[k′_{ν}:ℚ_{l}]} × (a finite group),

for any finite extension k′_{ν} of ℚ_{l}. Recall that g denotes the dimension of the abelian variety A. Clearly

A(k′_{ν}) ⊗_{ℤ} ℚ_{p}/ℤ_{p} = 0

for all l ≠ p and ν | l, i.e.

H^{1}(k′_{ν}, A)(p) ≅ H^{1}(k′_{ν}, A(p)),

respectively

H^{1}(k′_{∞,ν}, A)(p) ≅ H^{1}(k′_{∞,ν}, A(p)),

in this case. On the other hand, Coates and Greenberg proved that for primes ν | p with good reduction

H^{1}(k_{∞,ν}, A)(p) ≅ H^{1}(k_{∞,ν}, Ã(p))

holds, if k_{∞} is a deeply ramified, where Ã denotes the reduction of A (see [4, Prop. 4.8]). We recall that an algebraic extension k of ℚ_{p} is called deeply ramified if H^{1}(k, \overline{𝔪}) vanishes, where \overline{𝔪} is the maximal ideal of the ring of integers of an algebraic closure \overline{ℚ_{p}} of ℚ_{p}; see [4, p. 143] for equivalent conditions and for the following statement (loc. cit. thm. 2.13): A field k_{∞} which is a p-adic Lie extension of a finite extension k of ℚ_{p} is deeply ramified if the inertial subgroup of G(k_{∞}/k) is infinite. For arbitrary reduction at ν | p, the same result as above holds, if one replaces Ã_{p^{∞}} by the quotient A(p)/F_{A}(\overline{𝔪})(p), where F_{A} denotes the formal group associated with the Neron model of A over a possibly finite extension of k_{ν}, such that the Neron model has semi-stable reduction. Taking these facts into account, we get the following description for 𝕌_{S,A}, where T (k_{∞,ν}/k_{ν}) denotes the inertia subgroup of G_{ν}.

Proposition 4.33. (cf. [29, lemma 5.4]) Assume that dim(T (k_{∞,ν}/k_{ν}) ≥ 1 for all ν ϵ S_{p}. Then there is an isomorphism of Λ-modules

𝕌_{S,A} ≅ ⨁_{S_{p}(k)} Ind_{G}^{G_{ν}}H^{1}(k_{∞,ν}, Ã(p))^{∨} ⊕ ⨁_{S_{f} \S_{p}(k)} Ind_{G}^{G_{ν}}H^{1}(k_{∞,ν}, A(p))^{∨}.

In particular, if dim(G_{ν}) ≥ 2 for all ν ϵ S_{f}, then

𝕌_{S,A} ≅ ⨁_{S_{p}(k)} Ind_{G}^{G_{ν}}H^{1}(k_{∞,ν}, Ã(p))^{∨}

and 𝕌_{S,A} is Λ-torsion-free.

Proof. The first assertion has been explained above while the second statement follows from the local calculations 3.4, 3.5 with respect to the p-adic representations A = Ã(p) respectively A = A(p) and the comment before 3.5. □

Before going on we would like to recall some well-known facts about abelian varieties:

Remark 4.34. (i) rk_{ℤ_{p}}(A(p)^{∨}) = 2g, where g denotes the dimension of A. (ii) There exists always an isogeny from A to its dual A^{∨}, by which the Weilpairing induces a non-degenerate skew-symmetric pairing on the Tate-module T_{p}A of A. If A = E is an elliptic curve this isogeny can be chosen as a canonical isomorphism between E and E^{∨}. Again for an arbitrary abelian variety it follows that k(μ_{p^{∞}}) ⊆ k(A(p)) = k(A^{∨}(p)) (see [31, §0 lem. 7]).

Theorem 4.35. Assume that H^{2}(G_{S}(k_{∞}), (A^{∨})(p)) = 0. If dim(G_{ν}) ≥ 2 for all ν ϵ S_{f}, then

� _{S}^{1}(k_{∞}, A(p))^{∨} ∼ E^{1}(Y_{S,A^{∨}(p)}) ∼ E^{1}(tor_{Λ}Y_{S,A^{∨}(p)}) ≅ E^{1}(tor_{Λ}X_{S,A^{∨}(p)}).

If, in addition, G ≅ ℤ_{p}^{r}, r ≥ 2, then the following holds:

� _{S}^{1}(k_{∞}, A(p))^{∨} ∼ (tor_{Λ}X_{S,A^{∨}(p)})^{∘},

where ^{∘} means that the G acts via the involution g ↦ g^{−1}.

Remark 4.36. In case tor_{Λ}X_{S,A^{∨}(p)} is isomorphic in Λ-mod/PN to a direct sum of cyclic modules of the form Λ modulo a (left) principal ideal proposition 2.4 implies that

� _{S}^{1}(k_{∞}, A(p))^{∨} ≡ (tor_{Λ}X_{S,A^{∨}(p)})^{∘} mod PN

holds under the conditions of the theorem.

Proof. The first condition implies Z_{S,A^{∨}(p)} ≅ E^{1}(Y_{S,A^{∨}(p)}) while the other condition grants that ⨁_{S_{f} (k)} Ind_{G}^{G_{ν}}(A(k_{∞,ν})(p))^{∨} is pseudo-null because A(k_{∞,ν})(p)^{∨} is a finitely generated (free) ℤ_{p}-module. Now everything follows as in 4.9 using here prop. 4.32. □

Corollary 4.37. Let A be an abelian variety over k with good supersingular reduction, i.e. Ã_{k_{ν}}(p) = 0, at all places ν dividing p. Set k_{∞} = k(A(p)) and assume that G(k_{∞}/k) is a pro-p-group without any p-torsion. Then, for Σ_{bad} := S_{bad} ∪ S_{p} ∪ S_{∞} the following holds:

X_{cs}[A^{∨}(p)] ≅ � _{Σ_{bad}}^{1}(k_{∞}, A^{∨}(p))^{∨} ∼ E^{1}(tor_{Λ}Sel(A, k_{∞})^{∨}).

In particular, if A has CM, then there is even a pseudo-isomorphism

X_{cs}[A^{∨}(p)] ∼ (tor_{Λ}Sel(A, k_{∞})^{∨})^{∘}.

Therewith, in the case of an elliptic curve with CM, we reobtain a theorem of P. Billot [2, 3.23]. Over a ℤ_{p}-extension an analogous statement was proved by K. Wingberg [35, cor. 2.5]. Of course, remark 4.36 applies literally to tor_{Λ}Sel(A, k_{∞})^{∨}, i.e. under the conditions mentioned there it holds

X_{cs}[A^{∨}(p)] ≡ (tor_{Λ}Sel(A, k_{∞})^{∨})^{∘} mod PN .

Proof. First note that by the Néron-Ogg-Shafarevich criterion the sets of bad reduction of A and its dual A^{∨} coincide. Therefore, it suffices to prove that dim(G_{ν}) ≥ 2 for all ν ϵ S_{bad} ∪ S_{p} because then the theorem applies to A^{∨} and proposition 4.33 shows that 𝕌_{S,A} = 0, i.e. X_{S,A(p)} ≅ Sel(A, k_{∞})^{∨}. So, let ν be either in S_{p} or in S_{bad}. Since k_{ν}(A(p)) contains k_{ν}(μ_{p^{∞}}), we only have to show that G(k_{ν}(A(p))/k_{ν}(μ_{p^{∞}})) is not trivial because then it automatically has to be infinite as G_{ν} ⊆ G has no finite subgroup by assumption. If ν|p, by a theorem of Imai^{4} [16]A(k_{ν}(μ_{p^{∞}})) is finite and thus k_{ν}(A(p)) ≠ k_{ν}(μ_{p^{∞}}). If ν ϵ S_{bad}, then the Néron-Ogg-Shafarevich criterion implies that G(k_{ν}(A(p))/k_{ν}(μ_{p^{∞}})) = T (k_{ν}(A(p))/k_{ν}) is non-trivial. □

By remark 4.3 and 4.34 the conditions of theorem 4.2 are fulfilled for the p-torsion points A(p) and its trivializing extension of k, i.e. the extension which is obtained by adjoining the p-torsion points of A :

Theorem 4.38. Let k_{∞} = k(A(p)) and assume that G does not have any ptorsion. Then H^{1}(G_{S}(k_{∞}), A(p))^{∨} has no non-zero pseudo-null submodule.

Recall that G does not have any p-torsion if p ≥ 2 dim(A) + 2. Otherwise one only has to replace k by a finite extension inside k_{∞}.

We should mention that the rank of the global module H^{1}(G_{S}(k_{∞}), A(p))^{∨} is g[k : ℚ], which was determined by Y. Ochi who also calculated the ranks and torsion-submodules of the local, respectively local-global modules (i.e. those global modules which are induced from local ones) that occur in proposition 4.32 (cf. [28, 5.7,5.11,5.12]). See also the results in S. Howson’s PhD-thesis [15, 5.30,6.1,6.5-6.9,6.13-6.14,7.3].

Furthermore, in the case of elliptic curves S. Howson proved the following result.

4I owe to John Coates the idea to use Imai’s theorem here.

Proposition 4.39. (Howson [15, 6.14-15]) Let E be an elliptic curve over k without complex multiplication and with good ordinary reduction at all places over p. Assume that G = G(k(E(p)/K) is pro-p without any p-torsion. Then

𝕋_{S,E} ≅ 𝔸_{S,Ẽ} ≅ ⨁_{S_{f} (k)} Ind_{G}^{G_{ν}} lim_{\overleftarrow{k′}}H^{1}(k′_{ν}, T_{p}(Ẽ))

and these modules are Λ(G)-torsion-free. Furthermore, there is an isomorphism

𝕌_{S,E} ≅ E^{0}(𝕋_{S,E}).

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Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany.

E-mail address: otmar@mathi.uni-heidelberg.de
URL: http://www.mathi.uni-heidelberg.de/~otmar/