On the Structure of Selmer Groups

of Λ-Adic Deformations over p-Adic Lie Extensions

Sudhanshu Shekhar and R. Sujatha

Received: September 14, 2011 Revised: August 31, 2012

Communicated by Peter Schneider

Abstract. In this paper, we consider the Λ-adic deformations of Galois representations associated to elliptic curves. We prove that the Pontryagin dual of the Selmer group of a Λ-adic deformation over certain p-adic Lie extensions of a number field, that are not necessarily commutative, has no non-zero pseudo-null submodule. We also study the structure of various arithmetic Iwasawa modules associated to such deformations.

2010 Mathematics Subject Classification: 14H52, 11F80, 11R34 Keywords and Phrases: Elliptic curve, Galois representation, deformations, Galois cohomology, p-adic Lie extensions, Selmer Groups.

Introduction

Let E be an elliptic curve defined over a number field K. Assume that p is an odd prime such that E has good ordinary reduction at the primes of K lying over p. Let K_{∞} be a (not necessarily commutative) p-adic Lie extension of K containing the cyclotomic ℤ_{p}-extension K_{cyc} of K with the additional property that the Galois group Gal(K_{∞}/K) is pro-p with no elements of order p. The dual Selmer group X(E/K_{∞}) of the elliptic curve over K_{∞} of the associated Galois representation has been extensively studied (see [OV], [OV1], [HV]). In recent works, Greenberg proved broad generalizations on the structure of the dual Selmer group for deformations, when the extension K_{∞}/K is commutative (see [Gr3] and [Gr4]). The aim of this paper is to establish analogues of these results when K_{∞}/K is a non-commutative p-adic Lie extension. More precisely, we consider R-adic deformations of the Galois representation associated to the elliptic curve; here R is a complete, local Noetherian domain which is finite flat over the Iwasawa algebra ℤ_{p}[[X]]. Such deformations give rise to a big Galois representation

ρ : G_{K} → GL_{2}(R).

In this context, there are associated Iwasawa modules that are finitely generated over the Iwasawa algebra R[[G]] where G := Gal(K_{∞}/K). We study the ranks of these Iwasawa modules and also analyse the dual Selmer group of the deformation, proving in particular that it has no non-zero pseudo-null R[[G]]-submodule. Our investigation was inspired by the article of Greenberg [Gr3] but the methods used are different as Greenberg’s methods crucially rely on the Iwasawa algebra R[[G]] being a commutative Noetherian local ring. To prove the corresponding results in this non-commutative setting, we need to use subtle methods from the theory of non-commutative Iwasawa algebras. We then adapt the techniques of Ochi and Venjakob [OV] and Jannsen [J] to our context along with specialisation arguments. We stress that our main result on the non-existence of non-zero pseudo-null submodules is for the strict Selmer group, and in section 6 we compare the strict Selmer group with the Greenberg Selmer group. As an application of our methods, we study the variation of the Iwasawa invariants of the Selmer groups in the Hida family. Analogous results for the cyclotomic ℤ_{p}-extension were proved by Emerton-Pollack-Weston [EPW], while Chandrakant [Sh] and Jha [Jh] study non-commutative p-adic Lie extensions, and our results strengthen those of [Sh] and [Jh]. Specifically, it is assumed in [Sh] that the cyclotomic μ-invariant of the dual Selmer group at a specialisation vanishes, whereas we do not make this strong assumption.

The paper consists of eight sections. Section 1 is preliminary in nature while in sections 2 and 3 we formalize the tools and techniques needed to adapt the methods of Ochi and Venjakob [OV] and Jannsen [J]. In sections 4 and 5, we prove the reflexivity of some local Iwasawa modules and establish results on the R[[G]]-ranks of certain cohomology modules. In section 6, we prove the main result on the dual Selmer group and in section 7, we discuss the surjectivity of the global to local map defining the Selmer group. In section 8, we apply the main result to study the invariance of various Iwasawa invariants at the specialisations.

Acknowledgements: The authors gratefully acknowledge the hospitality of the Tata Institute of Fundamental Research Centre for Applied Mathematics (TIFRCAM), Bangalore, while part of this paper was being written. The second author was supported by an NSERC grant. We would also like to thank the referee for a careful reading and commenting extensively, which helped improve the exposition in the paper.

1. Notation and preliminaries

Let ℚ̄ be a fixed algebraic closure of ℚ and for every prime integer l let ℚ̄_{l} be a fixed algebraic closure of ℚ_{l}. Fix an embedding ℚ ↪ ℚ̄_{l} for every prime l. Throughout, p will denote an odd prime number. Let G_{ℚ} denote the absolute Galois group of ℚ and G_{l} denote the decomposition subgroup above a prime of l given by this embedding. Suppose that E is an elliptic curve defined over ℚ such that E has ordinary reduction at p. We denote by T_{p}E the Tate module of the elliptic curve E. Let

ρ_{0} : G_{ℚ} ⟶ Aut_{ℤ_{p}}(T_{p}E)

be the associated Galois representation giving the action of G_{ℚ} on T_{p}E. Since E has ordinary reduction at p, there exists a G_{p}-submodule F^{+}T_{p}E of T_{p}E of ℤ_{p}- rank one such that the action of G_{p} on F^{−}T_{p}E := T_{p}E/F^{+}T_{p}E is unramified. In fact, F^{−}T_{p}E is canonically identified with the Tate module T_{p}Ẽ of Ẽ where Ẽ denotes the reduction of E modulo p. Let Λ be a commutative Noetherian local domain isomorphic to the power series ring ℤ_{p}[[X]]. Our aim is to study the structure of Iwasawa modules arising from the Galois cohomology of certain Λ-adic deformations of ρ_{0}. Specifically, we consider an R-adic representation

ρ : G_{ℚ} ⟶ GL_{2}(R) (1)

for a complete Noetherian local domain R which is a finite flat extension of Λ, such that ρ satisfies the following hypotheses:

Hypothesis 1.

(a) There exists a continuous ℤ_{p}-algebra homomorphism λ : R ⟶ ℤ_{p} such that λ ∘ ρ = ρ_{0}.

(b) ρ is unramified outside the primes p, the infinite prime and the primes dividing the conductor N of E.

(c) Let T denote the free R-module of rank two on which G_{ℚ} acts by ρ. Then there exists an R-submodule F^{+}T which is invariant under the action of G_{p} and such that both F^{+}T and T /F^{+}T are free R-modules of rank one. The decomposition group G_{p} acts on T /F^{+}T via an unramified character η_{p}.

An important example we have in mind is the Hida deformation. We recall this here briefly. For more details see [Gr1], [Wi], [Hi]. Let χ : G_{ℚ} ⟶ ℤ_{p}^{∗} = μ_{p−1} × (1 + pℤ_{p}) be the p-th cyclotomic character, where μ_{p−1} denotes the group of (p − 1)-th roots of unity. Let κ denote the composition of χ with the projection onto 1 + pℤ_{p}. Then κ induces an isomorphism Gal(ℚ_{cyc}/ℚ) = Γ ⟶^{∼} 1 + pℤ_{p}, where ℚ_{cyc} denotes the cyclotomic ℤ_{p}-extension of ℚ. Let Λ = ℤ_{p}[[Γ]] := \underleftarrow{lim} ℤ_{p}[Γ/p^{i}Γ] be the Iwasawa algebra of Γ. Then the ring Λ is a commutative Noetherian local domain isomorphic to the power series ring ℤ_{p}[[X]] in one variable. If E is defined over ℚ and the residual Galois representation associated to ρ_{0} is absolutely irreducible, then by Hida theory, there exists a complete Noetherian local domain R which is finite flat over Λ, and an R-adic form F that gives rise to a representation ρ̃ : G_{ℚ} ⟶ GL_{2}(R) which satisfies all the properties stated in Hypothesis 1. Further, it also satisfies the following two properties:

(i) Let ϵ : 1 + pℤ_{p} ⟶ ℚ̄_{p}^{∗} be the inclusion. Then for every k ∈ ℤ_{p} the homomorphism ϵ^{k} : 1 + pℤ_{p} ⟶ ℚ̄_{p}^{∗} induces a ℤ_{p}-algebra homomorphism ϵ^{k} : Λ ⟶ ℚ̄_{p}. Let k ≥ 2 and let ϕ ∈ Hom(R, ℚ̄_{p}) be a ℤ_{p}-algebra homomorphism such that ϕ|_{Λ} = ϵ^{k}. Then, ϕ ∘ ρ̃ : G_{ℚ} ⟶ GL_{2}(ℚ̄_{p}) is equivalent (over ℚ̄_{p}) to the ℚ̄_{p}-representation corresponding to a weight k cusp form for Γ(Np).

(ii) Let 𝔭_{k} denote the prime ideal of Λ defined by the kernel of the homomorphism ϵ^{k}. If k ∈ ℤ and k ≥ 2, then 𝔭_{k} remains unramified in R.

For property (ii) we refer the reader to [EPW, Theorem 2.1.3(3)] (or [GS, Theorem 2.6(a)]). For more details and examples, see the last part of §6. In general R is not equal to Λ. One needs to impose extra conditions to get R = Λ. In fact, R = Λ holds whenever E is a CM elliptic curve whose residual representation is absolutely irreducible. When E is not a CM elliptic curve, then there exists a set of primes of ℚ of density one for which this condition is satisfied (for instance see [NP, section 4]). Throughout this article, R will denote a commutative, complete local Noetherian domain with maximal ideal m and residue field 𝔽, which is a finite field extension of ℤ/pℤ. In Sections 4 and 8, we shall assume that R is a regular local ring to prove our results. Our main results in Section 6 and Section 7 are however valid for more general rings, namely Noetherian local domains of Krull dimension two. For a profinite group G and a commutative, Noetherian ring S, we denote by S[[G]] the completed group algebra \underleftarrow{lim}_{U} S[G/U] where U varies over open normal subgroups of G, and the inverse limit is taken with respect to the natural maps. All modules considered over the ring S[[G]] will be as left modules. For a compact (resp. discrete) S[[G]]-module M, we denote by M̂ the Pontryagin dual Hom(M, ℚ_{p}/ℤ_{p}) of M which is a discrete (resp. compact) S[[G]]-module. Let D be a discrete R-module with a continuous action of G such that the dual D̂ is a free R-module of rank r. For a finitely generated R[[G]]-module M, we define

M[D] := { \atop Hom_{R}(M, D)} = M ⊗_{R} D̂,

where G acts diagonally on the tensor product.

2. Fox Lyndon and The Big Diagram

Let K be a number field. Fix an R-adic representation

σ : Gal(K̄/K) ⟶ GL_{r}(R),

and let T be the free R-module of rank r corresponding to this representation. We denote by D be the discrete R[[Gal(K̄/K)]]-module defined by T ⊗ R̂. Throughout, we assume that the action of Gal(K̄/K) on D is unramified outside a finite set of primes of K. We denote by S(K) a finite set of primes of K containing each prime above p, every archimedean prime and every prime whose inertia group acts non-trivially on D. Let S_{f}(K) denote the set of all finite primes in S(K) and put S_{p}(K) = {v ∈ S_{f}(K) : v|p}. By K_{S}, we shall mean the maximal S(K)-ramified extension of K. Let K(D) be the extension of K defined by the kernel of σ and let K_{1} be the extension of K defined by the kernel of the residual representation of σ.

Lemma 2.1. The extension K(D) is contained in K_{S} and K(D)/K_{1} is pro-p.

Proof. The first assertion is clear. Let K_{n} be the extension defined by the kernel of the representation σ_{n} : Gal(K̄/K) ⟶ GL_{r}(R/m^{n})) and let G_{n} = Gal(K(D)/K_{n}). We have Gal(K(D)/K_{1}) = \underleftarrow{lim}_{n≥1} G_{1}/G_{n}. Let GL_{r}^{n} be the subgroup of GL_{r} consisting of matrices (a_{i,j}) such that a_{i,j} ∈ m^{n} if i ≠ j and a_{i,j} ∈ 1 + m^{n} if i = j. Then

\frac{G_{n}}{G_{n+1}} ↪ \begin{matrix}GL & n r \\ GL & {n+1 \atop r} \end{matrix} ≅ M_{r} [\frac{m^{n}}{m^{n+1}} ] ,

where M_{r}[\frac{m^{n}}{m^{n+1}} ] is the corresponding additive subgroup of the matrix ring M_{r}[\frac{R}{m^{n+1}} ]. This shows that each G_{n}/G_{n+1} is a p-group. Therefore Gal(K(D)/K_{1}) is pro-p. □

Let Ω be the maximal S-ramified p-extension of K(D) (resp. K_{1}). The extension Ω/K is Galois. Let K_{∞} be a p-adic Lie extension of K such that (i) K_{∞} ⊂ Ω, (ii) the Galois group Gal(K_{∞}/K) is a pro-p, p-adic Lie group with no elements of order p and (iii) K_{∞} contains the cyclotomic ℤ_{p}-extension K_{cyc} of K; here Gal(K_{∞}/K) is not assumed to commutative. Such an extension is called an admissible extension of K. Put

Gal(Ω/K) = G, Gal(K_{S}/K) = G_{S}, Gal(Ω/K_{∞}) = H,
Gal(K_{S}/K_{∞}) = H_{S} and Gal(K_{∞}/K) = G.

As D is fixed now, for a compact R[[G]]-module M, we will denote M[D] by M^{#}. Then the profinite group G is a pro-C-group which is topologically finitely generated such that cd_{p}(G) ≤ 2 (see [OV]); here C is a class of finite groups closed under taking subgroups, homomorphic images and group extensions. Let G be generated by d elements. Then there is the following commutative diagram with exact rows and columns

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

where F_{d} is a free pro-C-group of rank d. We fix the following notation.

X = X_{H,D} := { \atop H^{1}(H, D)}

Y = Y_{H,D} := (I_{G}^{#})_{H} J = J_{H,D} := ker(R[[G]]_{H}^{#} ⟶ (D̂)_{H}). Note that there is an isomorphism { \atop H^{1}(H_{S}, D)} ≃ { \atop H^{1}(H, D)}.

Lemma 2.2. We have a commutative diagram with exact rows and columns as below

[figure: 0; 0; J; J; 0; { \atop H^{2}(H_{S}, D)}; H_{1}(N , R)_{H}^{#}; R[[G]]^{dr}; Y; 0; 0; { \atop H^{2}(H_{S}, D)}; H_{1}(N , D̂)_{H}; H_{1}(R, D̂); { \atop H^{1}(H_{S}, D)}; 0; 0; 0.]

Also H_{1}(N , R)^{#} is a projective R[[G]]-module and H_{1}(N , R)_{H}^{#} is a projective R[[G]]-module.

□

There is a similar lemma when the number field K is replaced by a finite extension of ℚ_{l}, l a prime number. In this case, in place of K_{S}, we take an algebraic closure K̄ of K. We also have the following two lemmas.

Lemma 2.3. The R[[G]]-module R[[G]][D] is free of rank r and therefore if M is a projective R[[G]]-module, then M[D] is also a projective R[[G]]-module.

Lemma 2.4. There is a canonical exact sequence

0 ⟶ H_{1}(N , R) ⟶ R[[G]]^{d} ⟶ R[[G]] ⟶ R ⟶ 0.

The proof of the above statements are entirely analogous to the proof of [OV, Lemma 4.5, Lemma 4.2] and [NSW, Theorem 5.6.6] respectively.

3. A spectral sequence

Let S be a not necessarily commutative ring. For a finitely generated S-module M, we will denote by E_{S}^{i}(M) the (right) S-module Ext_{S}^{i}(M, S) for any integer i ≥ 0 and set E_{S}^{i}(M) = 0 for i < 0. However, we shall always consider the case when S is a completed group ring and hence E_{S}^{i}(M) is endowed with a natural left module structure (see [J, Section 1, page 175]). The subscript S from the notation will be dropped whenever there is no ambiguity. We also write M^{+} for E^{0}(M).

Definition 1. If M ≠ 0 is an S-module, then j(M) := min{i|E^{i}(M) ≠ 0} is called the grade of M.

Next, we recall the definition of the “transpose functor”. For a finitely presented S-module M, take any projective resolution P_{1} → P_{0} → M → 0 of M by finitely generated projective modules P_{0} and P_{1}. Then DM is defined as the cokernel of the induced map (P_{0})^{+} → (P_{1})^{+}. This is well defined up to homotopy. We will also need the following exact sequence (see [J]).

0 → E^{1}(DM) → M ⟶^{ϕ_{M}} M^{++} → E^{2}(DM) → 0.

A finitely generated S-module M is said to be reflexive if ϕ_{M} is an isomorphism and M is said to be torsion free (resp. torsion) if ϕ_{M} is injective (resp. ϕ_{M} is zero). If S is a Noetherian domain then the torsion elements of M form a submodule which coincides with E^{1}(DM). We say that a finitely generated S-module M is pseudo-null if grade j(M) ≥ 2. In this section, we prove various results which are generalizations of analogous results proved in [NSW, (Section 5.4, 5.6)] and [J]. Again, the proofs are not substantially different but we include them for the sake of completeness, checking that the appropriate modifications are valid in this general setting. Let G be any profinite group and M be a finitely generated R[[G]]-module. Then

M^{+} = Hom_{R[[G]]}(M, R[[G]])

≅ \underleftarrow{lim}_{U◁_{open}G} Hom_{R[G/U]}(M, R[G/U])

≅ \underleftarrow{lim}_{U} Hom_{R[G/U]}(M_{U}, R[G/U])

≅ \underleftarrow{lim}_{U} Hom_{R}(M_{U}, R) (limit being taken via the norm map)

≅ \underleftarrow{lim}_{i} \underleftarrow{lim}_{U} Hom_{R}(M_{U}, R/m^{i})

The third isomorphism is given by

Hom_{R}(M_{U}, R) ⟶ Hom_{R[G/U]}(M_{U}, R[G/U])

f ↦ (x ↦ ∑_{σ∈G/U} f(σ^{−1}x)σ).

The above calculation shows that

M̂^{+} ≅ \underrightarrow{lim}_{i} \underrightarrow{lim}_{U} Hom_{R}(M_{U}, R/m^{i})′

where N′ denotes Hom_{ℤ_{p}}(N, ℚ_{p}/ℤ_{p}) for an abelian group N. Note that Hom_{R}(M_{U}, R/m^{i}) is a discrete (in fact finite) abelian group. Therefore we have Hom(M_{U}, R/m^{i})′ ≅ { \atop Hom(M_{U}, R/m^{i})} (see also [J, section 2]). For an open normal subgroup U ⊂ G

Hom_{R}(M_{U}, R/m^{i})′
≅ Hom_{R}(R ⊗_{R[[U]]} M, R/m^{i})′
≅ Hom_{R[[U]]}(R, Hom_{R}(M, R/m^{i}))′
≅ (Hom_{R}(M, R/m^{i})^{U})′.

Here, Hom_{R}(M, R/m^{i}) is a discrete G-module because R/m^{i} is a discrete Gmodule. The proof of this fact follows from [Br, Lemma 3.4]. Therefore M̂^{+} can be written as a composition of the following two covariant functors:

M ⟶^{LC_{p}} ({ \atop Hom_{R}(M, R/m^{i})}) = (M ⊗_{R} { \atop R/m^{i}})

and

(M_{i}) ⟶^{⟶^{lim} D_{0}(ˆ−)} \underrightarrow{lim}_{i,U}((M̂_{i})^{U})′,

where D_{0}(ˆ−) := \underrightarrow{lim}_{U} H^{0}(U, ˆ−)′ is the functor defined by Tate [S, Appendix 1] whose rth derived functor is D_{r}(ˆ−) := \underrightarrow{lim}_{U} H^{r}(U, ˆ−)′. Let P be a finitely generated projective R[[G]]-module. Since Hom_{R}(P, R/m^{i}) is a direct summand of a coinduced R[[G]]-module, it is cohomologically trivial. Therefore { \atop Hom_{R}(P, R/m^{i})} is a D_{0}(ˆ−) -acyclic module for every i ≥ 1. This shows that the first functor takes projective objects to acyclic objects of the second functor. Therefore if R[[G]] is a Noetherian ring, then we have the convergence of the following spectral sequence of homological type.

Theorem 3.1. Let G be any profinite group such that R[[G]] is a Noetherian ring and let M be a finitely generated R[[G]]-module. Then,

\underrightarrow{lim}_{i} D_{r}({ \atop L^{s}C_{p}(M)}) ⇒ { \atop E^{r+s}(M)}

where L^{s}C_{p}(M) = Tor_{R}^{s}(M, { \atop R/m^{i}}) ≅ { \atop Ext_{R}^{s}(M, R/m^{i})}.

Corollary 3.2. Let D be a discrete R[[G]]-module such that D̂ is a finitely generated free R-module. Then, as U varies over the open normal subgroups of G, for every r ≥ 0 we have

\underrightarrow{lim}_{i} \underrightarrow{lim}_{U} H^{r}(U, Hom_{R}(D̂, R/m^{i}))′ ≅ { \atop E^{r}(D̂)}.

Proof. Since D̂ is free R-module Ext_{R}^{s}(D̂, R/m^{i}) = 0 for all s ≥ 1. Therefore the spectral sequence degenerates and the result follows. □

Remark 3.3. In the proof of Theorem 3.1 we have assumed that the ring R[[G]] is Noetherian. Without this assumption, the category of finitely generated R[[G]]-modules need not be an abelian category and therefore we can not use the spectral sequence arguments to prove the theorem. Nevertheless, Corollary 3.2 can be proved without the assumption that the ring R[[G]] is Noetherian. Indeed, since D̂ is a finitely generated free R-module, it follows from Lemma 2.3 that D̂ has a resolution by finitely generated projective (in fact free) R[[G]]−modules. For any finitely generated projective module P, the module Hom_{R}(P, R/m^{i}) is cohomologically trivial as explained in the proof of Theorem 3.1. Therefore if P_{•} ⟶ D̂ ⟶ 0 is a resolution of D̂ by finitely generated projective R[[G]]-modules, then 0 ⟶ Hom(D̂, R/m^{i}) ⟶ Hom(P_{•}, R/m^{i}) is a resolution of the discrete G-module Hom(D̂, R/m^{i}) by cohomologically trivial G-modules. Thus we have,

{ \atop E^{r}(D̂)} = H^{r}(P̂_{•}^{+})

= H^{r}(\underrightarrow{lim}_{i} \underrightarrow{lim}_{U}(Hom_{R}(P_{•}, R/m^{i})^{U})′)

= \underrightarrow{lim}_{i} \underrightarrow{lim}_{U} H^{r}(Hom_{R}(P_{•}, R/m^{i})^{U})′

= \underrightarrow{lim}_{i} \underrightarrow{lim}_{U} H^{r}(U, Hom_{R}(D̂, R/m^{i}))′

Let K be a finite field extension of ℚ or a finite field extension of ℚ_{l}, l a prime integer. We follow the notation of the previous section.

Lemma 3.4. The R[[G]]-module H_{1}(N , R) is finitely generated.

Proof. Let U be an open normal pro-p subgroup of G. Then it suffices to show that H_{1}(N , R) is a finitely generated R[[U]]-module. Since R is a complete Noetherian local domain with finite residue field of characteristic p, there exists an integer n ≥ 0 such that R is a finitely generated module over the power series ring ℤ_{p}[[T_{1}, ⋯, T_{n}]]. Thus we may assume that R = ℤ_{p}[[T_{1}, ⋯, T_{n}]]. Using induction on the number of variables n, and Nakayama’s lemma, it is enough to prove the lemma when R = ℤ_{p}. Now the assertion of the lemma follows from [J, Theorem 5.1(c)]. □

Proposition 3.5. Let Z_{D} = Z_{D,K_{∞}} denote the Pontryagin dual of (\underrightarrow{lim} D_{2}(Hom_{R}(D, R/m^{i})))^{H} Then,

(i) Z_{D} = DY = E_{R[[G]]}^{2}(D̂)_{H}.

(ii) Z_{D}^{+} = { \atop H^{2}(H_{S}, D)}.

Proof. From Lemma 3.4 and the proof of [NSW, Proposition 5.6.8] we have, DY = E_{R[[G]]}^{2}(D̂)_{H}. From Corollary 3.2 we have, Z_{D} = E_{R[[G]]}^{2}(D̂)_{H}. This proves the first assertion. From Lemma 2.2 we have the following exact sequence

0 ⟶ { \atop H^{2}(H_{S}, D)} ⟶ H_{1}(N , R)_{H}^{#} ⟶ R[[G]]^{dr} ⟶ Y ⟶ 0,

where the two terms in the middle are projective R[[G]]-modules. The second assertion follows from the proof of [J, Lemma 4.6] and the above exact sequence.

□

4. Grade and Reflexivity

For a compact (resp. discrete) R[[G]]-module M, let M^{∗} denote the discrete (resp. compact) R[[G]]-module Hom_{ℤ_{p}}(M, μ_{p^{∞}}) where μ_{p^{∞}} is the group of all p-power roots of unity. Throughout this section, we assume that R is a regular local domain with residue field 𝔽_{p}. Then R[[G]] is an Auslander regular ring (see [V, Theorem 3.30] and [E, Theorem 4.3]). In fact, using [Wa, Lemma 2.6] it can be shown that if R is of mixed characteristic, then R[[G]] is a complete Auslander regular local domain with residue field 𝔽_{p}. More generally, if G is a uniform pro-p group and R is a complete regular local domain of characteristic p, then R[[G]] is a complete Auslander regular local domain with finite residue field (see for instance [V] for the definition of uniform group).

Lemma 4.1. Suppose that K is a finite extension of ℚ_{l}, where l is a prime number. Then, Z_{D} is isomorphic to the Pontryagin dual of Hom_{R}(D̂, R)^{∗}^{H} and in particular, is a finitely generated R-module.

Proof. We have,

Ẑ_{D} = (\underrightarrow{lim}_{i} D_{2}(Hom(D̂, R/m^{i})))^{H} = (\underrightarrow{lim}_{i} \underrightarrow{lim}_{U} H^{2}(U, Hom(D̂, R/m^{i}))′)^{H}

= (\underrightarrow{lim}_{i} \underrightarrow{lim}_{U}(Hom(D̂, R/m^{i})^{∗})^{U})^{H} ( by local duality)

= (\underrightarrow{lim}_{i} Hom_{R}(D̂, R/m^{i})^{∗})^{H} ≅ (Hom_{R}(D̂, R)^{∗})^{H}.

□

For an R[[G]]-module M and an ideal I of R, put M[I] = {x ∈ M : rx = 0 ∀r ∈ I}. We prove the following algebraic lemma which will be needed later in this section .

Lemma 4.2. Let M be a finitely generated R[[G]]-module and assume that λ ∈ R such that the following holds:

(i) The grade of M/λM as an (R/λ)[[G]]-module is > k.

(ii) The grade of M[λ] as an (R/λ)[[G]]-module is > k − 1. Then the grade of M as an R[[G]]-module is > k.

Proof. For any ring S and λ ∈ S a central element, we have the following spectral sequence

Ext_{S/λ}^{p}(M/λ, Ext_{s}^{q}(S/λ, S)) ⇒ Ext_{S}^{p+q}(M/λ, S)

[W, Exercise 5.6.3]. If λ is a non-zero divisor of S then we have an exact sequence

0 → S ⟶^{λ} S → S/λ → 0.

Thus we have,

Ext^{i}(S/λ, S) ≅ S/λ for i = 1
= 0 for i ≠ 1

Therefore the spectral sequence collapses to give Ext_{S/λ}^{p−1}(M/λ, S/λ) ≅ Ext_{S}^{p}(M/λ, S). Now, we take S = R[[G]]. Then by assumption we have Ext_{R[[G]]}^{p}(M/λ, R[[G]]) = 0 for all p ≤ k + 1. From the short exact sequence

0 → M/M[λ] ⟶^{λ} M → M/λM → 0,

for all i ≤ k we have an isomorphism

Ext_{R[[G]]}^{i}(M, R[[G]]) ˜⟶^{λ} Ext_{R[[G]]}^{i}(M/M[λ], R[[G]]).

We also have,

Ext_{R[[G]]}^{i+1}(M[λ], R[[G]]) = Ext_{R[[G]]}^{i+1}(M[λ]/λ, R[[G]])
= Ext_{R/λ[[G]]}^{i}(M[λ]/λ, R/λ[[G]])
= Ext_{R/λ[[G]]}^{i}(M[λ], R/λ[[G]]) = 0 for all i ≤ k − 1.

Therefore the short exact sequence

0 ⟶ M[λ] ⟶ M ⟶ M/M[λ] ⟶ 0,

gives an isomorphism

Ext_{R[[G]]}^{i}(M/M[λ], R[[G]]) ⟶^{∼} Ext_{R[[G]]}^{i}(M, R[[G]]) for all i ≤ k.

Thus, we have shown that for all i ≤ k, we have isomorphisms

Ext_{R[[G]]}^{i}(M, R[[G]]) ˜→^{λ} Ext_{R[[G]]}^{i}(M/M[λ], R[[G]]) →^{∼} Ext_{R[[G]]}^{i}(M, R[[G]]).

Therefore Ext_{R[[G]]}^{i}(M, R[[G]]) = λExt_{R[[G]]}^{i}(M, R[[G]]) for all i ≤ k. Now the lemma follows using Nakayama lemma. □

Proposition 4.3. Let K be a finite extension of ℚ_{l} and suppose that K_{∞}/K is a p-adic Lie extension of K with cd_{p}(G) = k, where G = Gal(K_{∞}/K). Then the grade of DY as an R[[G]]-module is greater than or equal to k .

Proof. Since Z_{D} is a finitely generated R-module, Ext_{R}^{s}(Z_{D}, R/m_{i}) is finite for all s ≥ 0. Being a pro-p, p-adic Lie group, G = Gal(K_{∞}/K) is a Poincaré p-group [La]. Therefore D_{r}(Ext_{R}^{s}(Z_{D}, R/m^{i})) = 0 for all r ≠ k (see [S, Appendix 1, Theorem 3]). From Theorem 3.1, we conclude that { \atop E^{n}(Z_{D})} ≅ \underrightarrow{lim}_{i} D_{k}(Ext_{R}^{n−k}(Z_{D}, R/m^{i})) for all n ≥ k and is zero for all n < k. Therefore the grade of DY = Z_{D} is ≥ k. □

Corollary 4.4. If cd_{p}G ≥ 2 then both Y and X are torsion free, and if cd_{p}G ≥ 3 then both Y and X are reflexive.

Proof. The assertion for Y follows from Proposition 4.3. Since X is a submodule of Y, it follows that X is also torsion free. Consider the right most vertical sequence in Lemma 2.2. It follows by an application of snake lemma that E^{2}(DY) is isomorphic to a submodule of J (see [OV, Lemma 5.4]). Since J is torsion free and E^{2}(DY) is pseudo-null, E^{2}(DY) must be zero. Hence X is a reflexive R[[G]]-module. □

Proposition 4.5. Let K be a finite extension of ℚ_{l} and K_{∞} be a p-adic Lie extension of K such that cd_{p}Gal(K_{∞}/K) ≥ 2. Suppose that there exists a prime ideal P of R such that R/P is normal, and a finite extension of ℤ_{p} with the property that Z_{D}/P Z_{D} is finite. Then the grade of Z_{D} ≥ 3 as an R[[G]]-module.

Proof. Since R/P is normal and finite over ℤ_{p}, the ring R/P is regular local. Therefore P is generated by a regular sequence. We shall prove the proposition by induction on the number of generators of P. We shall repeatedly use the fact (see Lemma 4.1) that Z_{D} is finitely generated as an R-module. First, suppose that P is generated by one element λ. Since Z_{D}/λ is finite, the grade of Z_{D}/λ as an R/λ[[G]]-module is ≥ 3 [J, Corollary 2.6]. Further, as Z_{D}[λ] is a finitely generated R/λ-module, it follows from [loc.cit.], that grade of Z_{D}[λ] is ≥ 2. It then follows from Lemma 4.2 that grade Z_{D} ≥ 3. It is easily seen using Lemma 4.1 that for an element λ ∈ R, we have Z_{D}/λ ≅ Z_{D[λ]}. Now choose λ to be an element which is part of a regular sequence of P. Then by induction hypothesis, the grade of Z_{D[λ]} as an R/λ[[G]]-module is ≥ 3. Since Z_{D}[λ] is a finitely generated R-module, Ext_{R}^{s}(Z_{D}[λ], R/m^{i}) is finite for each i and s. Thus by an argument similar to Proposition 4.3, and using the fact the dimension of G is ≥ 2, it can be seen that E^{n}(Z_{D}[λ]) = 0 for all n ≤ 1. Thus grade Z_{D}[λ] ≥ 2. Now the proposition follows using Lemma 4.2. □

Corollary 4.6. Under the assumptions of Proposition 4.5, both Y and X are reflexive. □

5. Coranks of Galois cohomology modules

In this section, we compute the ranks of certain Galois cohomology modules. We assume that R = Λ, where Λ is isomorphic to the power series ring ℤ_{p}[[X_{1}, ⋯, X_{n}]] in n variables with n ≥ 1. More generally, if R is a finite flat extension of Λ, and in addition, is a regular local domain, then the rank of a finitely generated R[[G]]-module M can be computed in terms of the rank of M over Λ[[G]] and the rank of R over Λ. Note that if G is a pro-p, p-adic Lie group such that G has no p-torsion, then by the remark of the previous section, R[[G]] is a complete local Noetherian domain which satisfies the Auslander regular condition. The non-zero elements of this ring form an Ore set and we denote the skew field of fractions of R[[G]] by Q. The rank of any module over Q is well defined. If M is a finitely generated compact R[[G]]- module, then we define the rank of M, denoted by rank(M), to be the rank of Q ⊗_{R[[G]]} M over Q. We prove the following theorem which generalizes [OV1, Theorem 3.2] to Λ-adic representations.

Theorem 5.1. Let K be a number field such that K(D)/K is pro-p and let K_{∞} be a pro-p, p-adic Lie extension of K. Assume that G = Gal(K_{∞}/K) has no p-torsion. Then rank({ \atop H^{1}(H_{S}, D)}) = r(K).r + rank({ \atop H^{2}(H_{S}, D)}), where r denotes the corank of D as an R-module and r(K) denotes the number of complex places of K.

Proof. From the right vertical exact sequence in Lemma 2.2 we have,

(3) rank({ \atop H^{1}(K_{S}/K_{∞}, D)}) = rank(Y) − rank(J)

(4) rank(J) = rank(R[[G]]^{#})_{H}

(5) rank(Y) = rank(R[[G]]^{dr}) − rank(H_{1}(N , R)^{#})_{H} + rank(H_{2}(H, D̂)).

Let P be prime ideal of R such that R/P ≅ ℤ_{p}. Then, since R and ℤ_{p} are regular local, P is generated by a regular sequence. Let λ ∈ P be a generator of a height one prime ideal (λ) of R. From the exact sequence

0 ⟶ R ⟶^{λ} R ⟶ R/λ ⟶ 0,

we get

0 ⟶ H_{1}(N , R) ⟶^{λ} H_{1}(N , R) ⟶ H_{1}(N , R/λ) ⟶ 0.

The above sequence is exact because N has cohomological dimension ≤ 1 and N acts on R trivially. This in turn gives the following exact sequence.

H_{1}(N , R)_{H}^{#} ⟶^{λ} H_{1}(N , R)_{H}^{#} ⟶ H_{1}(N , R/λ)_{H}^{#} ⟶ 0.

Therefore

\frac{H_{1}(N , R)_{H}^{#}}{λH_{1}(N , R)_{H}^{#}} ≅ H_{1}(N , R/λ)_{H}^{#}

Since H_{1}(N , R)_{H}^{#} is R[[G]]-projective, and R[[G]] is local, the module H_{1}(N , R)_{H}^{#} is in fact free. Therefore the rank of H_{1}(N , R)_{H}^{#} as an R[[G]]- module is the same as the rank of H_{1}(N , R/λ)_{H}^{#} as an (R/λ)[[G]]-module. Choosing a height one prime ideal P, and applying this argument successively, we get that the rank of H_{1}(N , R/P)_{H}^{#} as an R/P [[G]]-module is the same as the rank of H_{1}(N , R)_{H}^{#} as an R[[G]]-module. Now it follows from [OV1, Theorem 3.2] that the rank of H_{1}(N , R)_{H}^{#} as an R[[G]]-module is r(d − r(K) − 1). Recall that d denotes the rank of the free pro-C-group F_{d} defined in (2). Therefore from equation (5), we see that the rank of Y is r(r(K) + 1). From equation (4) and Lemma 2.3 the rank of J is r. Substituting the rank of Y and the rank of J in equation (3), we get rank(H_{1}(K_{S}/K_{∞}, D̂)) = r(K)r + rank(H_{2}(K_{S}/K_{∞}, D̂)). □

Theorem 5.2. Let K be a finite extension of ℚ_{l}, l a prime number. Suppose that K_{∞} is a pro-p, p-adic Lie extension such that Gal(K_{∞}/K) has no p torsion. Then rank({ \atop H^{1}(K_{∞}, D)}) = r[K : ℚ_{p}] if l = p and is 0 if l ≠ p.

Proof. The proof of this theorem is analogous to the global case. Notice that H^{2}(K_{∞}, D) = 0 because cd_{p}Gal(K̄/K_{∞}) ≤ 1. □

We have the following more general theorem, for which we need not assume that K(D)/K is pro-p.

Theorem 5.3. Let K be number field and let K_{∞} be a p-adic Lie extension of K. Then

rk_{R[[G]]}({ \atop H^{1}(K_{∞}, D)}) − rank_{R[[G]]}({ \atop H^{2}(K_{∞}, D)})

= (r_{1}(K) + r_{2}(K))r − ∑_{v real} dim_{𝔽_{p}}(D[m])^{+},

where (−)^{+} denotes the invariant part with respect to the complex conjugation and D[m] = {x ∈ D|ax = 0 ∀ a ∈ m}.

Proof. We shall prove the theorem for R = ℤ_{p}[[X]]. A proof for higher dimensional power series rings can be given using a similar method. Using the Hochschild-Serre spectral sequence and following the proof of [HV, 7.4] we obtain

∑_{i≥0}(−1)^{i+1}corank_{R[[G]]}H^{i}(K_{S}/K_{∞}, D)

= ∑_{i,j≥}(−1)^{i+j+1}corank_{R}H^{j}(G, H^{i}(K_{S}/K_{∞}, D))

= ∑_{n≥0}(−1)^{n+1}corank_{R}H^{n}(K_{S}/K, D)

By [Gr3, 2.1, 3.5], we can choose a height one prime ideal λ such that R/λ ≅ ℤ_{p} and corank_{R}(H^{n}(K_{S}/K, D)) = corank_{ℤ_{p}}(H^{n}(K_{S}/K, D[λ])). Therefore we have,

∑_{i≥0}(−1)^{i+1}corank_{R[[G]]}H^{i}(K_{S}/K_{∞}, D)

= ∑_{n≥0}(−1)^{n+1}corank_{ℤ_{p}}H^{n}(K_{S}/K, D[λ])

= ∑_{n≥0}(−1)^{n+1}dim_{𝔽}H^{n}(K_{S}/K, D[m])

= (r_{1}(K) + r_{2}(K))r − ∑_{v real} dim_{𝔽_{p}}(D[m])^{+}

For the last equality see for example [NSW, 8.6.4] or [HV, 7.4]. The R[[G]]-rank of H^{0}(K_{S}/K_{∞}, D) is zero and H^{i}(K_{S}/K_{∞}, D) = 0 for all i ≥ 3. Hence the theorem follows. □

6. Selmer groups

In this section, we study the Selmer groups of R-adic representations over p-adic Lie extensions of a number field. We will mainly consider two particular types of admissible extensions, namely the False Tate extensions [HV] and certain higher dimensional p-adic Lie extensions of K. We continue with the notation of sections 1 and 2. From now on Λ will denote the power series ring ℤ_{p}[[X]] in one variable. Fix an R-adic representation

ρ : G_{ℚ} ⟶ GL_{2}(R)

as in section 1 and let T be the underlying R[[G_{ℚ}]]-module. We assume that ρ satisfies Hypothesis 1. In this section, we shall assume that R is a complete Noetherian local domain which is a finite flat extension of Λ. Let K be a number field and suppose that K_{∞} is a pro-p, p-adic Lie extension of K such that G = Gal(K_{∞}/K) does not have any p-torsion. As before, let S(K) be the set of primes of K containing the primes over p, the primes where ρ is ramified and the infinite primes. Let S_{f} denote the set of finite primes of S(K) and S_{p} denote the primes in S(K) lying over p. Consider the discrete R[[G_{ℚ}]]-module defined by

A = T ⊗_{R} R̂

where R̂ = Hom(R, ℚ_{p}/ℤ_{p}) is the Pontryagin dual of R with trivial action of G_{ℚ}. Let F^{+}A denote the submodule F^{+}T ⊗_{R} R̂ and F^{−}A denote the quotient A/F^{+}A ≅ T /F^{+}T ⊗_{R} R̂ of A. Put A = T_{p}E ⊗ℚ_{p}/ℤ_{p}. By Hypothesis 1, there exists a prime ideal P with P = ker λ such that T /P corresponds to T_{p}E. Then A ≅ A[P] [Gr3, section 1]. Recall that the Selmer group of A over K_{∞} is defined by

Sel(A/K_{∞}) = Ker(H^{1}(K_{S}/K_{∞}, A) ⟶^{γ_{∞}} ⨁_{v∈S(K)} J_{v}(A, K_{∞}))

where

J_{v}(A, K_{∞}) = \{ \begin{matrix}∏_{w|v} H^{1}(K_{∞,w}, A/F^{+}A) & if v|p \\ ∏_{w|v} H^{1}(K_{∞,w}, A) & if v ∤ p \end{matrix}

We remark that dimension theory for Auslander regular rings (see for example [V]) enables one to prove results on the existence of non-trivial pseudo-null submodules for a finitely generated module over an Auslander regular domain. But when R is not regular local, the ring R[[G]] is not necessarily Auslander regular. The following algebraic lemma is therefore useful in the study of pseudo-null submodules. This may be well known to experts, but we give a proof of this as it is important for establishing results in a more general context.

Lemma 6.1. Let M be a finitely generated R[[G]]-module. Then M is a pseudonull R[[G]]-module if and only if M is a pseudo-null Λ[[G]]-module.

Proof. We shall show that for a finitely generated R[[G]]-module M, j_{R[[G]]}(M) = j_{Λ[[G]]}(M). Let r denote the rank of R over Λ. We have

R ⊗_{Λ} Ext_{Λ[[G]]}^{i}(M, Λ[[G]]) ≅ Ext_{R[[G]]}^{i}(R ⊗_{Λ} M, R[[G]])
≅ Ext_{R[[G]]}^{i}(M^{r}, R[[G]])
≅ Ext_{R[[G]]}^{i}(M, R[[G]])^{r}.

Note that R ⊗_{Λ} Ext_{Λ[[G]]}^{i}(M, Λ[[G]]) = 0 if and only Ext_{Λ[[G]]}^{i}(M, Λ[[G]]) = 0. Therefore Ext_{Λ[[G]]}^{i}(M, Λ[[G]]) = 0 if and only if Ext_{R[[G]]}^{i}(M, R[[G]]) = 0. This proves the assertion. □

Our next result is a generalization of the results proved by Greenberg and Hachimori-Venjakob. Greenberg has considered the case when G is commutative and A is a cofinitely generated module over R [Gr3, Theorem 1]. Hachimori and Venjakob consider the case of an elliptic curve with G not necessarily commutative [OV, Theorem 4.7].

Theorem 6.2. If H^{2}(K_{S}/K_{∞}, A) = 0, then { \atop H^{1}(K_{S}/K_{∞}, A)} has no non-zero pseudo-null R[[G]]-submodule.

Proof. From Lemma 2.2 and the proof of [OV, Theorem 4.6], we have { \atop H^{1}(K_{S}/K_{∞}, A)} does not have any non-trivial pseudo-null Λ[[G]]-submodule. Note that here we have used that Λ[[G]] is an Auslander regular domain. Now the theorem follows from Lemma 6.1. □

Lemma 6.3. If a finitely generated Λ-module M is torsion, then there exist infinitely many height one prime ideals 𝔭 of Λ such that M/𝔭 is finite.

Proof. By the structure theorem for finitely generated torsion modules over Λ, any prime ideal 𝔭 of height one that does not divide the characteristic ideal of M has the desired property.

□

Let us denote A/F^{+}A by B. For a prime v of K and w be a prime of K_{∞} lying over v, put H_{w} = Gal(K̄_{v}/K_{∞,w}). Let G_{w} (resp. T_{w}) denote the decomposition group (resp. the inertia group) of w in G. To prove the non-existence of pseudonull submodules of the dual Selmer group, we need to prove the following result related to the module Z_{B,K_{∞,w}} which we considered in Section 3. Recall that the decomposition group G_{p} at p acts on T /F^{+}T via an unramified character η. Let Fr_{p} denote the Frobenius at p.

Lemma 6.4. Suppose that dim G_{w} = dim T_{w} for all primes w of K_{∞} lying above primes v ∈ S_{p}(K). If η(Fr_{p}) is not of finite order, then Z_{B,K_{∞,w}} is a finitely generated torsion Λ-module.

Proof. From Lemma 4.1, we have { \atop Z_{B,K_{∞,w}}} ≅ (Hom_{R}(B̂, R)^{∗})^{H_{w}}, where H_{w} denotes the absolute Galois group of K_{∞,w}. Since K_{∞} contains K_{cyc}, H_{w} acts trivially on μ_{p^{∞}}, thus we have an isomorphism of H_{w}-modules Hom_{R}(B̂, R)^{∗} ≅

{ \atop Hom_{R}(B̂, R)}. Since G_{w} and T_{w} have the same dimension, there exists a finite extension F_{v} of K_{v} in K_{∞,w} such that K_{∞,w} is totally ramified over F_{v}. Since B is unramified at p, we have that

{ \atop Hom_{R}(B̂, R)}^{H_{w}} = { \atop Hom_{R}(B̂, R)}^{G_{Fv}} ,

where G_{F_{v}} denotes the absolute Galois group of F_{v}. Now taking the Pontryagin dual, we get that

Z_{B,K_{∞,w}} ≅ Hom_{R}(B̂, R)_{G_{Fv}} ≅ \frac{Hom_{R}(B̂, R)}{(Fr_{v} − 1)Hom_{R}(B̂, R)},

Documenta Mathematica 17 (2012) 573–606

where Fr_{v} denotes the Frobenius at v in G_{F_{v}}. By assumption, η(Fr_{p}) is not of finite order. Therefore η(Fr_{v}) is not the identity element. Since Hom_{R}(B̂, R) is a free R-module of rank one and R is domain, it follows that Z_{B,K_{∞,w}} is a torsion R-module, and hence a torsion Λ-module. □

Lemma 6.5. Suppose that the assumptions of Lemma 6.4 hold. Then for every prime w of K_{∞} lying above a prime v ∈ S_{p}(K) such that dim G_{w} = 2, we have that J_{v}(A, K_{∞}) is a coreflexive Λ[[G]]-module.

Proof. Choose a prime w of K_{∞} such that w|v for some v ∈ S_{p}(K). Then we know that the Λ[[G]]-module J_{v}(A, K_{∞}) can also be written as Coind_{G}^{G_{w}}(H^{1}(K_{∞,w}, B)). Therefore { \atop J_{v}(A, K_{∞})} = Ind_{G}^{G_{w}}(H_{1}(K_{∞,w}, B̂)). Since Λ[[G]] is flat over Λ[[G_{w}]], it follows that

E_{Λ[[G]]}^{0}E_{Λ[[G]]}^{0}(Ind_{G}^{G_{w}}(H_{1}(K_{∞,w}, B̂)) ≅ Ind_{G}^{G_{w}}E_{Λ[[G_{w}]]}^{0}E_{Λ[[G_{w}]]}^{0}(H_{1}(K_{∞,w}, B̂).

From Lemma 6.4 and 6.3 we conclude that there exists a height one prime ideal 𝔭 of Λ such that Λ/𝔭 is a finite extension of ℤ_{p} and Z_{B,K_{∞,w}}/𝔭 is finite. Now it follows from Corollary 6.7 that H_{1}(K_{∞,w}, B̂) is a reflexive Λ[[G_{w}]]-module. Therefore

E_{Λ[[G_{w}]]}^{0}E_{Λ[[G_{w}]]}^{0}(H_{1}(K_{∞,w}, B̂) ≅ H_{1}(K_{∞,w}, B̂).

This shows that

E_{Λ[[G]]}^{0}E_{Λ[[G]]}^{0}(Ind_{G}^{G_{w}}(H_{1}(K_{∞,w}, B̂)) ≅ Ind_{G}^{G_{w}}(H_{1}(K_{∞,w}, B̂)),

thereby proving the assertion of the proposition. □

Remark 6.6. If η(Fr_{p}) is of finite order, then there exists a finite extension F′ of ℚ and a prime v of F′ lying above p such that η factors through F′_{v}. Thus if we take F′ = K, then the image of Frobenius in G_{F} acts by multiplication by one. In this case we see that Z_{B,K_{∞,w}} is free of rank one. Thus the above proposition does not hold and the local Galois cohomology groups fail to be coreflexive.

We mention that if P is a prime ideal of R such that T_{P} := T /P is a lattice of the Galois representation associated to a Hecke eigenform f_{P} of weight k ≥ 3 with ordinary reduction at p, then it is known that the action of Frobenius in G_{p} on the unramified quotient of T_{P} is given by multiplication by an element whose complex norm is not one. If T_{P} is the Tate module T_{p}E of E and E has good ordinary reduction at p, then the same assertion holds. From this, it follows that if T is the Hida deformation of E as discussed in Section 1, then η(Fr_{p}) is not of finite order, and hence the assumptions of Lemma 6.5 are satisfied.

Proposition 6.7. Suppose that for every prime w of K_{∞} lying over a prime v|p of K, dim G_{w} ≥ 3. Then J_{v}(A, K_{∞}) is a coreflexive Λ[[G]]-module.

Proof. From our assumption and Corollary 4.4, it follows that for every prime w of K_{∞} lying above a prime v|p in K, the Λ[[G_{w}]]-module H_{1}(K̄_{w}/K_{∞,w}, Â) is reflexive. It now follows from the proof of Proposition 6.5 that J_{v}(A, K_{∞}) is a coreflexive Λ[[G]]-module for all v|p. □

Proposition 6.8. Let v be a prime of K such that v ∤ p and w be a prime of K_{∞} such that w|v. If dim G_{w} ≥ 2 then we have H^{1}(K_{∞,w}, A) = 0.

Proof. From Corollary 4.4 and the assumption of the proposition, it follows that H^{1}(K_{∞,w}, A) is cotorsion free as a Λ[[G]]-module. But Theorem 5.2 implies that H^{1}(K_{∞,w}, A) is a cotorsion module. This forces H^{1}(K_{∞,w}, A) = 0. □

Lemma 6.9. Consider the following exact sequence of Λ[[G]]-modules:

0 ⟶ U ⟶ V ⟶ W ⟶ 0.

Suppose V does not have any non-trivial pseudo-null Λ[[G]]-submodule and U is a reflexive Λ[[G]]-submodule. Then W does not contain any non-trivial pseudonull submodule.

Proof. See [HO, Lemma 3.5]. □

We finally state the following important theorem for R-adic representations which generalises a similar theorem proved in [HV, Theorem 2.6(ii)] for p-adic representations.

Theorem 6.10. Suppose that the following assumptions hold:

(i) For every prime w of K_{∞} lying above a prime v ∈ S_{p}(K), dim G_{w} = dim T_{w} = 2 and η(Fr_{p}) is not of finite order.

(ii) For every prime w of K_{∞} lying above a prime v ∈ S(K) \ S_{p}(K),

dim G_{w} ≥ 2. (iii) H^{2}(K_{S}/K_{∞}, A) = 0.

If the global to local map γ_{∞} in the exact sequence defining the Selmer group is surjective, then { \atop Sel(A/K_{∞})} has no non-trivial R[[G]] pseudo-null submodule.

Proof. From Proposition 6.8 and the assumption in the theorem, we have

0 ⟶ Sel(A/K_{∞}) ⟶ H^{1}(K_{S}/K_{∞}, A) ⟶^{γ_{∞}} ⨁_{v|p} J_{v}(A, K_{∞}) ⟶ 0

Taking Pontryagin duals, we get the exact sequence

0 ⟶ ⨁_{v|p} { \atop J_{v}(A, K_{∞})} ⟶ { \atop H^{1}(K_{S}/K_{∞}, A)} ⟶ { \atop Sel(A/K_{∞})} ⟶ 0.

By Proposition 6.5 the first term in the above sequence is a reflexive Λ[[G]]- module and the middle term has no non-trivial pseudo-null submodule (Proposition 6.2). From Lemma 6.9 we have Sel(A/K_{∞}) does not contain any nontrivial pseudo-null Λ[[G]]-submodule. Thus, the theorem follows on applying Lemma 6.1. □

We mention that the assumptions (i) and (ii) of the theorem hold when K = K(μ_{p}) and K_{∞} is a False Tate extension K(μ_{p^{∞}}, m^{1/p^{∞}}) of K where m is an integer prime to p such that m is divisible by the conductor N of E, and is also p-power free.

We now study an other type of Selmer group defined by Greenberg in [Gr] for cyclotomic extensions. Let v ∈ S_{p}(K) and w be a prime of K_{∞} such that w|v. We write I_{∞,w} for the inertia subgroup of the Galois group G(\overline{K_{w,∞}}/K_{w,∞}). The Selmer group Sel^{Gr}(A/K_{∞}) of A over K_{∞} is defined as

Sel^{Gr}(A/K_{∞}) = Ker(H^{1}(K_{S}/K_{∞}, A) ⟶ ⨁_{v∈S(K)} J_{v}^{Gr}(A, K_{∞}))

where

J_{v}^{Gr}(A, K_{∞}) = \{ \begin{matrix}∏_{w|v} H^{1}(I_{∞,w}, A/F^{+}A) & if v|p \\ ∏_{w|v} H^{1}(K_{∞,w}, A) & if v ∤ p \end{matrix}

We shall show that if Hypothesis 1 and assumption (i) of Theorem 6.10 hold,

then Sel(A/K_{∞}) = Sel^{Gr}(A/K_{∞}). This will show that the { \atop Sel^{Gr}(A/K_{∞})} has no non-trivial pseudo-null submodule. Let v|p be a prime of K, and w|v be a prime of K_{∞}. Then from the Hochschild-Serre spectral sequence, we have

0 ⟶ H^{1}(ℌ_{w}, (A/F^{+}A)^{I_{∞,w}}) ⟶ H^{1}(H_{w}, A/F^{+}A) ⟶ H^{1}(I_{∞,w}, A/F^{+}A)^{H_{w}}

where, ℌ_{w} = Gal(\overline{K_{w,∞}}/K_{w,∞})/I_{∞,w} ↪ Gal(\overline{K_{w}}/K_{w})/I_{w} ≅ ℤ̂, and is hence topologically cyclic. We shall show that the first term in the above exact sequence vanishes. It follows from Hypothesis 1(c) that I_{∞,w} acts trivially on A/F^{+}A. Thus we need to show that H^{1}(ℌ_{w}, A/F^{+}A) = 0. We claim that ℌ_{w} acts non-trivially on A/F^{+}A. Suppose this is not true. This implies that H_{w} = Gal(\overline{K_{w,∞}}/K_{w,∞}) acts trivially on A/F^{+}A, and hence on (Hom_{R}(B̂, R)^{∗}, where B = A/F^{+}A. Now by an argument similar to Lemma 6.4, it follows that there exists a finite extension F_{v} of K_{v} inside K_{∞,w} such that the absolute Galois group G_{F_{v}} of F_{v} acts trivially on B. But this contradicts assumption (i) of Theorem 6.10. Therefore ℌ_{w} act non-trivially on B. Now using the fact that ℌ_{w} is procyclic we conclude that H^{1}(ℌ_{w}, B) ≅ B/(s − 1)B where, s is the generator of ℌ_{w}. Since s does not act trivially and B is cofree, we see that (s − 1)B = B. Therefore H^{1}(ℌ_{w}, B) = H^{1}(ℌ_{w}, (A/F^{+}A)^{I_{∞,w}}) = 0. We consider the following commutative diagram:

\xymatrix{0 \ar[r] & Sel(A/K_{∞}) \ar[d] \ar[r] & H^{1}(K_{S}/K_{∞}, A) \ar[d] \ar[r] & ⨁_{v∈S(K)} J_{v}(A, K_{∞})) \ar[d]^{f} \\ 0 \ar[r] & Sel^{Gr}(A/K_{∞}) \ar[r] & H^{1}(K_{S}/K_{∞}, A) \ar[r] & ⨁_{v∈S(K)} J_{v}^{Gr}(A, K_{∞}).}

The kernel of the last vertical map is product of H^{1}(ℌ_{w}, (A/F^{+}A)^{I_{∞,w}}), where w varies over set of primes of K_{∞} lying above p. But we have already shown that H^{1}(ℌ_{w}, (A/F^{+}A)^{I_{∞,w}}) = 0. Now it follows from the snake lemma that Sel^{Gr}(A/K_{∞}) ≅ Sel(A/K_{∞}). Thus we have have the following Corollary to Theorem 6.10.

Corollary 6.11. Suppose that the assumptions of Theorem 6.10 hold. Then

{ \atop Sel^{Gr}(A/K_{∞})} does not have any non-trivial pseudo-null R[[G]]-submodule.

□

Suppose that E has split multiplicative reduction at p and consider the Selmer group Sel(A/K_{∞}) associated to the Galois representation of E. Then the module H^{1}(ℌ_{w}, A/F^{+}A) is a cofinitely generated non-trivial ℤ_{p}-module. This is true because on the unramified quotient A/F^{+}A, the Frobenius acts trivially. Let M denote the product of H^{1}(ℌ_{w}, A/F^{+}A), where w varies over primes of K_{∞} lying above p. If dim G = dim G_{w} = 2 for every prime v|p, then there are finitely many primes of K_{∞} above each prime v of K as the index of decomposition groups are finite. Therefore M is cofinitely generated as a ℤ_{p}-module. Now using snake lemma we see that there exists an exact sequence

0 ⟶ Sel(A/K_{∞}) ⟶ Sel^{Gr}(A/K_{∞}) ⟶ M ⟶ 0.

Therefore the Pontryagin dual of Sel^{Gr}(A/K_{∞})-contains a non-trivial ℤ_{p}[[G]]- submodule which is also a finitely generated ℤ_{p}-module and therefore a pseudonull ℤ_{p}[[G]]-module. However, we remark that this does not rule out the existence of non-trivial pseudo-null ℤ_{p}[[G]]-submodules for { \atop Sel(A/K_{∞})} as the local Galois cohomology groups above p fail to be reflexive (cf. Lemma 6.5).

In the previous section, we considered the Selmer group of A over p-adic Lie extensions satisfying condition (i) of Theorem 6.10. Now we consider the Selmer group over certain higher dimensional p-adic Lie extensions for which this condition is not needed.

Theorem 6.12. Suppose that for every prime w of K_{∞} lying above a primes v ∈ S_{p}(K), dim G_{w} ≥ 3. Suppose further that assumptions (ii) and (iii) of Theorem 6.10 hold. If the global to local map γ_{∞} defining the Selmer group Sel(A/K_{∞}) is surjective, then { \atop Sel(A/K_{∞})} has no non-trivial R[[G]] pseudonull submodule.

Proof. We notice from Proposition 6.7 that for every prime v of K lying above a prime p, the Λ[[G]]-module J_{v}(A, K_{∞}) is coreflexive. The rest of the argument is analogous to Theorem 6.10 and the result follows. □

Let K be a number field such that K = K(E_{p}) and K_{∞} be a p-adic Lie pro-p extension of K that contains K(E_{p^{∞}}). Put G = Gal(K_{∞}/K). If E has no complex multiplication, then it is well known that, the Galois group of K_{∞}/K is a pro-p, p-adic Lie group without p-torsion and is of dimension at least 4. Further, if E has good reduction at the primes of K dividing p, then for every prime v|p, the dimension of G_{v} is greater than equal to 3 ([S1, IV-43], see also [CH, Lemma 5.1]). If v ∤ p and E does not have good reduction at v, then E has potentially multiplicative reduction at v by a result of Serre-Tate [ST, 2,Corollary 2] and thus the dimension of G_{v} is ≥ 2. Let S(K) = {v|p} ∪ {v|v ∤ p and E has bad reduction at v} ∪ {v|∞}. Suppose that K_{∞} ⊂ K_{S}. Then all the assumptions of Theorem 6.12 on the dimension of the p-adic Lie group G are satisfied. We remark that if E has split multiplicative reduction at a prime v|p then the dimension of G_{w} may not be equal to 3 for a prime w|v. But if we consider the Hida deformation of E as discussed in Section 1, then for every k ≡ 2 mod p − 1 the modular form corresponding to the specialisation of weight k will be the ordinary p-stabilization of a modular form of level prime to p and trivial nebentype. If we take K_{∞} to be a pro-p, p-adic Lie extension whose Galois group has no p-torsion and contains the trivializing extension for the Galois representation of any such modular form, then one always expect that dim G_{w} ≥ 3 for every prime w of K_{∞} lying above p. Nevertheless, under certain assumptions it is proved in [Ek] that for all but finitely many such specialisations of Hida deformation this is indeed true. Thus for all those padic Lie extensions, the same result holds.

So far we have considered the case when R is a local domain. Now suppose that R is a complete semi-local reduced Noetherian ring and there exists a representation ρ satisfying Hypothesis 1(b) and 1(c). Let T denote the associated G_{K}-module. Then in this case, we can once again define the discrete R[[G_{K}]]-module A := T ⊗_{R} R̂. One may wonder about the existence of nontrivial pseudo-null R[[G]]-submodules for the corresponding dual Selmer group. We discuss this question now. In this case, R splits into a direct product of local rings R_{m}, where m varies over the set of maximal ideals of R. Then T splits as a direct sum of T_{m}. Consider the discrete R[[G]]-module A_{m} defined by T_{m} ⊗_{R} R̂ ≅ T_{m} ⊗_{R_{m}} R_{m} ⊗_{R} R̂. Then A splits into a direct sum of

A_{m}. Note that R_{m} ⊗_{R} R̂ ≅ { \atop Hom_{R}(R_{m}, R)} ≅ { \atop Hom_{R_{m}}(R_{m}, R_{m})}. Therefore

A_{m} ≅ T_{m} ⊗_{R_{m}} R̂_{m}.

Proposition 6.13. Suppose that for every maximal ideal m of R, { \atop Sel(A_{m}/K_{∞})} does not have any non-trivial, pseudo-null R_{m}[[G]]-submodule. Then { \atop Sel(A/K_{∞})} does not have any non-trivial, pseudo-null R[[G]]-submodule.

Proof. Since A splits into a direct sum of A_{m} as Galois modules, { \atop Sel(A/K_{∞})} splits into a direct sum of { \atop Sel(A_{m}/K_{∞})} as a module over R[[G]], where each component is an R_{m}[[G]]-module and the action is component wise. Now by assumption, each component { \atop Sel(A_{m}/K_{∞})} has no non-trivial, pseudo-null R_{m}[[G]]-submodule and therefore for every maximal ideal m of R, { \atop Sel(A_{m}/K_{∞})} does not have any non-trivial, pseudo-null Λ[[G]]-submodule. This shows that { \atop Sel(A/K_{∞})} does not have any non-trivial, pseudo-null Λ[[G]]-submodule. Therefore { \atop Sel(A/K_{∞})} does not have any non-trivial, pseudo-null R[[G]]- submodule. □

We mention that a local factor R_{m} of R in the above proposition may not be a domain. But our result on the non-existence of non-trivial, pseudo-null submodules of the dual Selmer group associated to a Galois representation over the local ring R assumes that it is a domain. In fact, we have used that R is a domain only in the proof of Lemma 6.4. Now we generalize the above result to a more general context. Suppose that R is a reduced, complete local Noetherian ring and let T be the rank 2 R-module associated to the Galois representation satisfying Hypothesis 1.

Lemma 6.14. Suppose that condition one of 6.10(i) is satisfied for the Galois module T/𝔞 for every minimal prime ideal 𝔞 of R. Then Z_{B,K_{∞,w}} is a finitely generated torsion Λ-module.

Proof. First recall that B denotes the Galois module F^{−}T ⊗R̂ unramified at p, where F^{−}T ≅ T /F^{+}T . Since R is a reduced ring, the natural map from R to ∏𝔞 R/𝔞 is an injection. Let X denote the cokernel of this map. Then X𝔞 = 0 for every minimal prime ideal 𝔞 of R, where X𝔞 denote the localisation of X at 𝔞. This implies that the height of a prime ideal of R in the support of X is at least 1. Therefore the zero ideal of Λ is not in the support of X, and hence X is a torsion Λ-module. Now consider the following exact sequence

0 ⟶ Hom_{R}(B̂, R) ⟶ ⊕𝔞 Hom_{R}(B̂, R)/𝔞 ⟶ X ⊗ Hom_{R}(B̂, R) ⟶ 0.

Since B is a cofree R-module we have Hom_{R}(B̂, R)/𝔞 ≅ Hom_{R/𝔞}(B̂/𝔞, R/𝔞) for every minimal prime ideal 𝔞 of R. The last term in the above exact sequence is a torsion Λ-module. We conclude that Hom_{R}(B̂, R)_{G_{Fv}} is torsion Λ-module if and only if Hom_{R/𝔞}(B̂/𝔞, R/𝔞)_{G_{Fv}} is a torsion Λ-module for every minimal ideal 𝔞 of R. Here F_{v} is the finite extension of ℚ_{p} considered in the proof of Lemma 6.4. Under the hypothesis of the lemma, it follows from the proof of Lemma 6.4 that Hom_{R/𝔞}(B̂/𝔞, R/𝔞)_{G_{Fv}} is a torsion Λ-module for every minimal prime ideal 𝔞 of R. This shows that Z_{B,K_{∞,w}} ≅ Hom_{R}(B̂, R)_{G_{Fv}} is a torsion Λ-module. □

In the remaining part of this section, we shall discuss the vanishing of the second cohomology group H^{2}(K_{S}/K_{∞}, A). This is the analogue of the classical Weak Leopoldt conjecture. Recall that this vanishing was a necessary assumption for Theorems 6.10 and 6.12. The proof of this vanishing will depend on certain ramification behaviour of prime ideals of Λ in R. We start by fixing some notation. Let 𝔖 denote a set of height one prime ideals of Λ which remain unramified in R and such that for every 𝔭 ∈ 𝔖, Λ/𝔭 is a finite extension of ℤ_{p}.

Lemma 6.15. Let 𝔭 ∈ 𝔖 be a height one prime ideal of Λ. Then the kernel and cokernel of the map

R/𝔭 ⟶ ⊕_{P |𝔭}R/P

are finite.

Proof. If 𝔭 is unramified in R, then for every minimal prime ideal P of R/𝔭 we have that R_{P} /𝔭_{P} ≅ R_{P} /P_{P}. This shows that the kernel and cokernel of the above map has support consisting of only maximal ideals m of R. Therefore it must be finite. □

Proposition 6.16. Let 𝔭 ∈ 𝔖 be a height one prime ideal of Λ. Suppose that for every height one prime ideal P of R lying above 𝔭 we have that H^{2}(K_{S}/K_{∞}, A[P]) = 0. Then H^{2}(K_{S}/K_{∞}, A) = 0.

Proof. Let 𝔭 be a height one prime ideal of Λ such that it remains unramified in R. Now from Lemma 6.15, the kernel and cokernel of the map from

Â/𝔭 ⟶ ⊕_{P |𝔭}Â ⊗ R/P

are finite. Therefore the kernel and cokernel of the map

⊕_{P |𝔭}A[P] ⟶ A[𝔭].

are finite. This shows that the kernel and cokernel of the map

⊕_{P |𝔭}H^{2}(K_{S}/K_{∞}, A[P]) ⟶ H^{2}(K_{S}/K_{∞}, A[𝔭])

are ℤ_{p}-torsion. Since we have assumed that H^{2}(K_{S}/K_{∞}, A[P]) = 0, we get that H^{2}(K_{S}/K_{∞}, A[𝔭]) is torsion as a ℤ_{p}[[G]]-module. But Proposition 3.5 says that H^{2}(K_{S}/K_{∞}, A[𝔭]) is a coreflexive ℤ_{p}[[G]]-module. Therefore it must be zero. Since 𝔭 is a principal ideal and R is flat over Λ and hence torsion free, we have a surjective map from H^{2}(K_{S}/K_{∞}, A[𝔭]) to H^{2}(K_{S}/K_{∞}, A)[𝔭]. Therefore H^{2}(K_{S}/K_{∞}, A)[𝔭] = 0, and by Nakayama’s lemma it follows that H^{2}(K_{S}/K_{∞}, A) = 0. This proves the proposition. □

Corollary 6.17. Suppose that K is an abelian extension of ℚ and Gal(K_{∞}/K_{cyc}) is a p-adic Lie extension of dimension ≤ 1. Let 𝔭 ∈ 𝔖 be a prime ideal of Λ. If for every height one prime ideal P of R lying above 𝔭, the Galois module T /P T is a lattice corresponding to the Galois representation of a Hecke eigenform of tame level N, then H^{2}(K_{S}/K_{∞}, A) = 0.

Proof. Let P be a height one prime ideal of R lying above 𝔭. If K is an abelian extension of ℚ, then it is a deep result of Kato that the group H^{2}(K_{S}/K_{cyc}, A[P]) = 0. It then follows from an argument entirely analogous to [HV, Remark 2.2] that H^{2}(K_{S}/K_{∞}, A[P]) = 0. Now the Corollary follows from Proposition 6.16. □

We mention that if K_{∞} is a False Tate extension of K, then Gal(K_{∞}/K_{cyc}) ≃ ℤ_{p} and hence has p-cohomological dimension 1. Thus the assumption of the above corollary holds. The next result proves a similar assertion for a p-adic Lie extension of larger dimension. For every height one prime ideal P of R, let K(A[P]) denote the trivialising extension of A[P].

Corollary 6.18. Let 𝔭 ∈ 𝔖 and K = K(E[p]). Assume that for every height one prime ideal P of R lying above 𝔭, the Galois module T /P T is a lattice corresponding to the Galois representation of a Hecke eigenform of tame level N. Further assume that K_{∞} is a Galois extension of K such that the group Gal(K_{∞}/K) is a pro-p, p-adic Lie extension and without p-torsion. If K(A[P]) ⊂ K_{∞} for every prime P lying above 𝔭, then H^{2}(K_{S}/K_{∞}, A) = 0.

Proof. Let P be a prime of R such that P lies above 𝔭. Since R/P is a finite ℤ_{p}-algebra, T /P is isomorphic to a finitely generated free ℤ_{p}-module with the trivial action of Gal(K_{S}/K_{∞}). Thus A[P] is a finite direct sum of ℚ_{p}/ℤ_{p} as a module over Gal(K_{S}/K_{∞}). Now it follows from the proof of [OV, Corollary 4.8] that H^{2}(K_{S}/K_{∞}, A[P]) = 0, and the assertion follows from Proposition 6.16. □

We remark that if we consider the extension K_{∞} = K(A[P]) for one particular prime P of R lying above 𝔭 in the previous corollary, then we cannot conclude using Proposition 6.16 that H^{2}(K_{S}/K_{∞}, A) = 0, as we do not know at present that, H^{2}(K_{S}/K_{∞}, A[Q]) = 0 for a prime Q ≠ P of R lying above 𝔭. But if we assume that P is principal prime ideal, then H^{2}(K_{S}/K_{∞}, A[P]) surjects onto H^{2}(K_{S}/K_{∞}, A)[P]. In that case, using Nakayama’s lemma, we may conclude that H^{2}(K_{S}/K_{∞}, A) = 0.

We end this section by recalling various rings considered in [EPW] over which there exists a Galois representation satisfying Hypothesis 1. Let T_{N} denote the universal ordinary Hecke algebra of tame level N and T_{N}^{new} denote the new quotient of T_{N}. Then it is known that T_{N}^{new} is a finite, reduced, torsion free Λ-algebra. The algebra T_{N}^{new} splits as a product of local rings (T_{N}^{new})_{m} where m varies over the maximal ideals of T_{N}^{new} and the subscript m denotes the localization at m. Then it is known that each such local factor is a finite flat extension of Λ([Hi, Section 9]). If the residual representation associated to E at p is irreducible, then there exists a maximal ideal m of T_{N}^{new} and a rank two free module T_{m} over (T_{N}^{new})_{m} satisfying Hypothesis 1(see [EPW, Proposition 2.2.7]). We mention that the ring (T_{N}^{new})_{m} need not be domain. But one can replace this ring by (T_{N}^{new})_{m}/𝔞 ≅ T_{N}^{new}/𝔞 for a minimal prime ideal 𝔞 ⊂ m of (T_{N}^{new})_{m} depending on E. For every integer k ≥ 2 we had considered a height one prime ideal 𝔭_{k} of Λ in section 1. Then it is known that the rings (T_{N}^{new})_{m} and T_{N}^{new}/𝔞 are unramified over every such prime ideal 𝔭_{k}. This follows from the fact that ((T_{N}^{new})_{m})_{𝔭_{k}} ≅ (T_{N}^{new}/𝔞)_{𝔭_{k}} ≅ (T_{N}^{new})_{𝔭_{k}} and 𝔭_{k} is unramified in T_{N}^{new} for each k ≥ 2 ([EPW, Theorem 2.1.3]). Here the subscript 𝔭_{k} denotes the localization of the corresponding ring at 𝔭_{k}. The Galois module T_{m}/𝔞 satisfies all the properties of Hypothesis 1, except that if T_{N}^{new}/𝔞 is not integrally closed, then it is not known whether this local ring is a flat extension of Λ (we thank the referee for kindly pointing this out to us). To remedy this problem, one can again replace this ring by various larger rings in its field of fractions. Let T̂ denote the integral closure of (T_{N}^{new})_{m}/𝔞 in its field of fractions. This new ring is a finite flat extension of Λ (see [EPW, Proposition 2.2.4]). For every integer k ≥ 2, let T̂_{𝔭_{k}} denote the localization of T̂ at 𝔭_{k}. Then the natural map from (T_{N}^{new}/𝔞)_{𝔭_{k}} to T̂_{𝔭_{k}} is an isomorphism and 𝔭_{k} remains unramified in T̂ (see the proof of [EPW, Proposition 2.2.4]). Now we have a rank two free module T̂ over T̂ defined as T_{m}/𝔞 ⊗ T̂. This Galois module satisfies all the assumptions of Hypothesis 1(see also [EPW, Remark 2.2.3] and [Gr2, Section 3.4]). In fact condition (b) and (c) of Hypothesis 1 are satisfied by construction. Since the natural map from (T_{N}^{new}/𝔞)_{𝔭_{k}} to T̂_{𝔭_{k}} is an isomorphism for every height one prime ideal P_{k} of T_{N}^{new}/𝔞 lying above 𝔭_{k}, we have T_{m}/𝔞 ⊗ T̂_{P_{k}} ≅ (T_{m})_{P_{k}}. Thus T̂_{P_{k}}/P_{k} is equivalent to the Galois representation of a Hecke eigenform of weight k of tame level N. In particular T̂_{P_{2}}/P_{2} ≅ T_{p}E ⊗ℚ_{p} for a height one prime ideal P_{2} lying over 𝔭_{2}. Therefore T_{p}E is a ℤ_{p}-lattice of the Galois representation T̂_{P_{2}}/P_{2}. Now using the assumption that the residual representation associated to E at p is irreducible, it can be shown that T̂/P_{2} is equivalent to T_{p}E. Thus all the conditions of Hypothesis 1 are satisfied. We mention that there is yet another ring, namely the reflexive hull of (T_{N}^{new}/𝔞) in its field of fractions over which there exists a Galois representation satisfying Hypothesis 1. If T̃ denotes the reflexive hull of T_{N}^{new}/𝔞, then we have T̃_{𝔭_{k}} ≅ T̂_{𝔭_{k}} for every k ≥ 2 ([Hi, Corollary 1.4]). Therefore T̃ is unramified at 𝔭_{k}.

7. Surjectivity of global to local map

In this section, we discuss the surjectivity of the global to local map defining the Selmer group. We continue to assume that K_{∞} is a pro-p, p-adic Lie extension of a number field K such that Gal(K_{∞}/K) has no p-torsion. Let F be a finite extension of K such that K ⊂ F ⊂ K_{∞} and let S(F) denote the set of primes of F that lie above the set of primes in S(K). Then for every i ≥ 0, the module � ^{i}(F, A) [Gr3] is defined by

� ^{i}(F, A) = Ker(H^{i}(F_{S}/F, A) ⟶ ⨁_{v∈S(F)} H^{i}(F_{v}, A)).

The compact version of the above cohomology groups with continuous cocycles taking values in T^{∗} =: Hom_{ℤ_{p}}(A, μ_{p^{∞}}) are defined by

� ^{i}(F, T^{∗}) = Ker(H^{i}(F_{S}/F, T^{∗}) ⟶ ⨁_{v∈S(F)} H^{i}(F_{v}, T^{∗})).

Let w be a prime of F such that w|v for some v ∈ S_{p}(K). Then put

L_{w,F} = Image(H^{1}(F_{w}, F^{+}A) ⟶ H^{1}(F_{w}, A))
Q_{w,F} = H^{1}(F_{w}, A)/L_{w,F}

If w is a prime of F such that w ∤ p then we take L_{w,F} = 0 and Q_{w,F} = H^{1}(F_{w}, A). Let L_{w,F}^{∗} denote the orthogonal complement of L_{w,F} in H^{1}(F_{w}, T^{∗}) under local duality. The compact version of the Selmer group Sel(T^{∗}/F) with respect to these local conditions is defined as

Sel(T^{∗}/F) = Ker(H^{1}(F_{S}/F, T^{∗}) ⟶ ⨁_{v∈S(K)} J_{v}(T^{∗}, F))

where J_{v}(T^{∗}, F) = ∏_{w|v} H^{1}(F_{w}, T^{∗})/L_{w,F}^{∗} .

Notice that L_{w,F}^{∗} and H^{1}(F_{w}, T^{∗})/L_{w,F}^{∗} are compact R-modules isomorphic to the Pontryagin dual of Q_{w,F} and L_{w,F} respectively. The map H^{1}(F, T^{∗}) ⟶ { \atop Q_{w,F}} is induced by the restriction map G_{F_{w}} ⟶ G(F_{S}/F) (see [Gr2, section 2]). The compact R[[G]]-module R(T^{∗}, K_{∞}) is defined as \underleftarrow{lim}_{F} Sel(T^{∗}/F), where F varies over finite Galois extensions of K contained in K_{∞}.

Lemma 7.1. Suppose that H^{2}(K_{S}/K_{∞}, A) = 0 and that assumption (ii) of Theorem 6.10 hold. Then the compact Λ[[G]]-module R(T^{∗}, K_{∞}) is torsion free.

Proof. From the global duality theorem, for every finite extension F of K, we have { \atop � ^{2}(F, A)} ≅ � ^{i}(F, T^{∗}) [NSW, 8.6.8]. Therefore � ^{2}(K_{∞}, A) ≅ \underleftarrow{lim}_{F} � ^{1}(F, T^{∗}), where � ^{2}(K_{∞}, A) = \underrightarrow{lim}_{F} � ^{2}(F, A). Since � ^{2}(K_{∞}, A) ⊂ H^{2}(K_{S}/K_{∞}, A) and H^{2}(K_{S}/K_{∞}, A) = 0 by assumption of the lemma, we get \underrightarrow{lim}_{F} � ^{2}(F, A) = 0. This shows that

\underleftarrow{lim}_{F} H^{1}(F_{S}/F, T^{∗}) ⟶ \underleftarrow{lim}_{F} ⨁_{v∈S(F)} H^{1}(F_{v}, T^{∗})

is an injective map. On the other hand, from local duality and Corollary 4.4, it follows that \underleftarrow{lim}_{F} ⨁_{v∈S(F)} H^{1}(F_{v}, T^{∗}) is a torsion free R[[G]]-module. Therefore \underleftarrow{lim}_{F} H^{1}(F_{S}/F, T^{∗}) is a torsion free R[[G]]-module. As R(T^{∗}, K_{∞}) injects into \underleftarrow{lim}_{F} H^{1}(F_{S}/F, T^{∗}), the assertion is true. □

Theorem 7.2. Suppose that Hypothesis 1 and assumption (ii) of Theorem 6.10 hold. If H^{2}(K_{S}/K_{∞}, A) = 0 then the sequence

0 ⟶ Sel(A/K_{∞}) ⟶ H^{1}(K_{S}/K_{∞}, A) ⟶ ⨁_{v∈S(K)} J_{v}(A, K_{∞}) ⟶ 0

is exact if and only if { \atop Sel(A/K_{∞})} is a torsion R[[G]]-module.

Proof. The Poitou-Tate exact sequence gives an exact sequence

0 ⟶ Sel(A/K_{∞}) ⟶ H^{1}(K_{S}/K_{∞}, A) ⟶ ⨁_{v∈S(K)} J_{v}(A, K_{∞})

⟶ { \atop R(T^{∗}, K_{∞})} ⟶ \underrightarrow{lim}_{F} { \atop � ^{1}(F, T^{∗})} ⟶ 0.

As shown in the proof of Lemma 7.1, the last term in the above exact sequence is zero. Let s denote the rank of R as a Λ-module. We denote by m_{Λ} and m_{R} the maximal ideal of Λ and R respectively. From Theorem 5.3 we have

corank_{Λ[[G]]}(H^{1}(K_{S}/K_{∞}, A)) = 2s(r_{1}(K) + r_{2}(K)) + ⊕_{v}dim_{Λ/m_{Λ}}(D[m_{Λ}])^{+}

where r_{1}(K) (resp. r_{2}(K)) denotes the number of real (resp. complex) embeddings of K, v varies over the set of all real primes of K and superscript (−)^{+} denotes the invariant part with respect to complex conjugation. Note that Hypothesis 1(a) implies that R/m_{R} ≅ Λ/m_{Λ}. From this we conclude that dim_{Λ/m_{Λ}}(D[m_{Λ}])^{+} = s × dim_{R/m_{R}}(D[m_{R}])^{+}. Again by Hypothesis 1, we have D[m_{R}] ≅ E[p]. Now it is well known that the action of complex conjugation on E[p] has two eigenvalues namely +1 and −1. Therefore dim_{R/m_{R}}(D[m_{R}])^{+} = 1. Thus we have

corank_{Λ[[G]]}(H^{1}(K_{S}/K_{∞}, A)) = 2s(r_{1}(K) + r_{2}(K)) + sr_{1}(K) = s[K : ℚ].

Since the module A/F^{+}A is a cofree rank one R-module, it follows from Theorem 5.2 that the Λ-corank of the third term in the above exact sequence is equal to s[K : ℚ]. Thus the R[[G]]-coranks of the middle term and the last term are equal. This shows that { \atop Sel(A/K_{∞})} is torsion if and only if R(T^{∗}, K_{∞}) is torsion. But the previous lemma asserts that R(T^{∗}, K_{∞}) is torsion free. Hence { \atop Sel(A/K_{∞})} is torsion if and only if R(T^{∗}, K_{∞}) is zero. This completes the proof of the theorem. □

We remark that the dual Selmer group of the p-adic representation associated to an ordinary modular form is conjectured to be torsion as ℤ_{p}[[G]]-module, and it is hence not unreasonable to expect that the same holds in this general situation. Thus it is natural to ask whether { \atop Sel(A/K_{∞})} is R[[G]]-torsion whenever a specialisation { \atop Sel(A[P]/K_{∞})} is torsion as an R/P [[G]]-module. We show that this is indeed true in the following special case.

Proposition 7.3. Suppose that R is a finite flat extension of ℤ_{p}[[X]] and let 𝔭 be a height one prime ideal of Λ such that R/𝔭 is finite extension of ℤ_{p}. If the ideal 𝔭 remains unramified in R and { \atop Sel(A[P]/K_{∞})} is a torsion ℤ_{p}[[G]]-module for every height one prime ideal P of R lying above 𝔭, then { \atop Sel(A/K_{∞})} is a torsion Λ[[G]]-module.

Proof. Using an argument similar to Proposition 6.16 we see that the kernel and cokernel of the map

⊕_{P |𝔭}H^{1}(K_{S}/K_{∞}, A[P]) ⟶ H^{2}(K_{S}/K_{∞}, A[𝔭])

are ℤ_{p}-cotorsion. A similar assertion holds for local Galois cohomology modules. Now using the snake lemma, we conclude that the kernel and cokernel of the map

⊕_{P |𝔭}Sel(A[P]/K_{∞}) ⟶ Sel(A[𝔭]/K_{∞})

are ℤ_{p}-cotorsion modules. Therefore Sel(A[P]/K_{∞}) is a ℤ_{p}[[G]]-cotorsion module for every P if and only if Sel(A[𝔭]/K_{∞}) is ℤ_{p}[[G]]-cotorsion. Thus by assumption we get that Sel(A[𝔭]/K_{∞}) is ℤ_{p}[[G]]-cotorsion. Consider the following commutative diagram:

\xymatrix{0 \ar[r] & Sel(A[𝔭]/K_{∞}) \ar[d]^{α} \ar[r] & H^{1}(K_{S}/K_{∞}, A[𝔭]) \ar[d]^{β} \ar[r]^{f} & ⨁_{v∈S(K)} J_{v}(A[𝔭], K_{∞})) \ar[d]^{γ} \\ 0 \ar[r] & Sel(A/K_{∞})[𝔭] \ar[r] & H^{1}(K_{S}/K_{∞}, A)[𝔭] \ar[r]^{g} & ⨁_{v∈S(K)} J_{v}(A, K_{∞})[𝔭].}

Using an argument similar to [CS, Theorem 4.2], it can be shown that the kernel of γ is a cofinitely generated ℤ_{p}[[H]]-module, where H := Gal(K_{∞}/K_{cyc}), and hence ℤ_{p}[[G]]-torsion. Therefore from the snake lemma we have that kernel and cokernel of α are also cofinitely generated ℤ_{p}[[G]]-module. This shows that kernel and cokernel of α are cotorsion ℤ_{p}[[G]]-modules. Therefore Sel(A/K_{∞})[𝔭] must be a cotorsion ℤ_{p}[[G]]-module. This implies that the grade of { \atop Sel(A/K_{∞})[𝔭]} as a ℤ_{p}[[G]]-module is greater than equal to 1. Now R/𝔭[[G]] is a free ℤ_{p}[[G]]-module. Therefore the grade of { \atop Sel(A/K_{∞})[𝔭]} is also greater than or equal to 1 (see Lemma 6.1). Now from Lemma 4.2 it follows that the grade of { \atop Sel(A/K_{∞})} is also greater then equal to 1. Therefore { \atop Sel(A/K_{∞})} must be a torsion Λ[[G]]-module.

□

We state another result which can be useful in proving the surjectivity of the global to local map defining the Selmer group for certain p-adic Lie extensions of dimension 2.

Theorem 7.4. Suppose that K_{∞}/K_{cyc} is a pro-p, p-adic Lie extension of dimension 1. Assume further that H^{2}(K_{S}/K_{∞}, A) = 0. If the sequence

0 ⟶ Sel(A/K_{cyc}) ⟶ H^{1}(K_{S}/K_{cyc}, A) ⟶^{γ_{cyc}} ⨁_{v∈S(K)} J_{v}(A, K_{cyc}) ⟶ 0

is exact, then

0 ⟶ Sel(A/K_{∞}) ⟶ H^{1}(K_{S}/K_{∞}, A) ⟶^{γ_{∞}} ⨁_{v∈S(K)} J_{v}(A, K_{∞}) ⟶ 0

is also exact.

Proof. The argument is as in [CSS]. Let H = Gal(K_{∞}/K_{cyc}). One first uses the Hochschild-Serre spectral sequence and the vanishing of H^{2}(K_{S}/K_{∞}, A) to conclude that H^{1}(H, H^{1}(K_{S}/K_{∞}, A)) = 0 (see [CSS, Lemma 2.4]). Simlarly, arguing as in [CSS, Lemma 2.5], it is easily seen that H^{1}(H, Sel(A/K_{∞})) = 0. We then obtain the exact sequence (see [CSS, Lemma 2.3])

0 ⟶ Sel(A/K_{∞})^{H} ⟶ H^{1}(K_{S}/K_{∞}, A)^{H} ⟶^{γ_{∞}} ⨁_{v∈S(K)} J_{v}(A, K_{∞})^{H} ⟶ 0,

which implies that coker(γ_{∞})^{H} = 0. Thus from Nakayama’s Lemma coker(γ_{∞}) = 0, and hence γ_{∞} is surjective. □

8. Iwasawa Invariants

In this section, we discuss the vanishing of the R-torsion submodule of the dual Selmer group { \atop Sel(A/K_{∞})}. We also discuss the variation of the Iwasawa μ-invariant attached to the dual Selmer groups of the specialisations of the Hida deformations considered in section 1. For the definition of the μ-invariant of a finitely generated module M over an Iwasawa algebra, the reader is referred to [H, section 1]. Thus R is a Noetherian local domain which is finite, flat over Λ ≃ ℤ_{p}[[X]]. Recall that in the case of a Hida deformation, for every integer k ≥ 2, we have the specialisation maps which we denote by ϵ^{k}, and the prime ideal 𝔭_{k} of Λ corresponding to the kernel of the specialisation ϵ^{k}. Let P_{k} be a height one prime ideal of R lying above 𝔭_{k} and π_{k} denote a generator of the maximal ideal of R/P_{k}. (We mention that there can be more then one such prime ideal but for ease of notation, we ignore the ambiguity, and denote any such prime ideal by the same notation P_{k}, and choose one of them). Let T_{k} be the finitely generated R/P_{k}-module T /P_{k} and A_{k} denote the cofinitely generated discrete R/P_{k}-module T_{k} ⊗ ℚ_{p}/ℤ_{p} for every k ≥ 2. The definition of the Selmer groups Sel(A_{k}/K_{∞}) for A_{k}, attached to the specialisations ϵ^{k} is analogous to the definition of the Selmer group Sel(A/K_{∞}) (see [CS] for details). For simplicity, we let X(T /K_{∞}) (resp. X(T_{k}/K_{∞})) denote the module { \atop Sel(A/K_{∞})} (resp. { \atop Sel(A_{k}/K_{∞})}). We denote by μ_{k} the μ-invariant attached to X(T_{k}/K_{∞}). Let X(T /K_{∞})(R) (resp. X(T_{k}/K_{∞})(p)) denote the R-torsion submodule of X(T /K_{∞}), (resp. the p-primary part of X(T /K_{∞})) considered as an R[[G]]-submodule. Put Y (T /K_{∞}) = X(T /K_{∞})/X(T /K_{∞})(p), Y (T_{k}/K_{∞}) = X(T_{k}/K_{∞})/X(T_{k}/K_{∞})(p), and H = Gal(K_{∞}/K_{cyc}). First we prove the following lemma which we shall need in this section.

Lemma 8.1. Suppose that M is a finitely generated R[[G]]-module such that it does not contain any non-trivial pseudo-null submodule. Then, for all but finitely many height one prime ideals P of R, we have M[P] = 0.

Proof. Since M does not contain any non-trivial pseudo-null R[[G]]-submodule, from Lemma 6.1, it follows that M does not contain any non-trivial pseudo-null Λ[[G]]-submodule. Now from the structure theory for finitely generated Λ[[G]]- modules M, it follows that M injects into a finitely generated Λ[[G]]-module N such that for all but finitely many height one prime ideals 𝔭 of Λ, we have N[𝔭] = 0 (see [H1, Theorem 2.5]). Now, if we take a height one prime ideal P of R such that its restriction to Λ does not annihilate any element of M, then M[P] = 0. □

Proposition 8.2. Assume that there exists an integer k ≥ 2 such that (i) X(T /K_{∞}) has no P_{k} torsion, (ii) μ_{k} = 0. If the assumptions of Theorem 6.10 ( resp. 6.12) hold, then X(T /K_{∞})(R) = 0.

Proof. The assertion follows from Theorem 6.10 (resp. 6.12) and [CS, Theorem 6.3]. □

In the following proposition, we prove a result on the variation of the μ-invariant with the specialisations. Results of this kind have been proven for K_{∞} = ℚ_{cyc} in [EPW]. We prove this for more general p-adic Lie extensions of K. For this proposition, we only need to assume Hypothesis 1 and that G is a uniform pro-p, p-adic Lie group without p-torsion. In the next proposition, we need the Selmer group associated to the residual representation ρ̄ defined by the action of Galois group of the R/m-module T /m. This is defined as follows:

Sel(A[m]/K_{∞}) = Ker(H^{1}(K_{S}/K_{∞}, A[m]) ⟶ ⨁_{v∈S(K)} J_{v}(A[m], K_{∞})).

The filtration on T induces a filtration on T /m. The local terms J_{v}(A[m], K_{∞})) are defined using this induced filtration on T /m. Note that the set of primes at which A[m] is ramified can be smaller than the set S(K) that we are considering. But we shall use this larger set to define this Selmer group as it is more suitable for comparing the μ-invariant of specialisations. First we state the following well known result (cf. [V, Remark 3.33]).

Lemma 8.3. Let O be the ring of integers of a finite field extension of ℚ_{p}. Let M be a finitely generated torsion O[[G]]-module. Then the μ-invariant of M vanishes if and only if the 𝔽[[G]]-rank of M/π is zero. Here π is a generator of the maximal ideal of O and 𝔽 denotes the residue field of O. □

Theorem 8.4. Assume that X(T_{k}/K_{∞}) is a torsion R/P_{k}[[G]]-module for all k ≥ 2. If there exists a k ≥ 2 such that μ_{k} = 0, then μ_{k′} = 0 for every k′ ≥ 2.

Proof. For every k ≥ 2, we have a homomorphism

X(T_{k}/K_{∞})/π_{k} ⟶^{f_{k}} X((T_{k}/π_{k})/K_{∞})

where X((T_{k}/π_{k})/K_{∞}) is the Pontryagin dual of the Selmer group defined for the the residual representation T_{k}/π_{k}. Now, it can be shown using a method similar to ([CS, Theorem 4.2]) that the kernel and cokernel of f_{k} is a finitely generated 𝔽[[H]]-module. Therefore the kernel and cokernel of f_{k} is a torsion 𝔽[[G]]-module and the 𝔽[[G]]-rank of X(T_{k}/K_{∞})/π_{k} and X((T_{k}/π_{k})/K_{∞}) are the same. Let k ≥ 2 be an integer such that μ_{k} = 0. Then from Lemma 8.3, it follows that the 𝔽[[G]]-rank of X((T_{k}/π_{k})/K_{∞}) is zero. Let k′ ≥ 2 be any integer. Since X((T_{k}/π_{k})/K_{∞}) ≅ X((T_{k}/π_{k′})/K_{∞}) we see that the 𝔽[[G]]-rank of X(T_{k′}/K_{∞})/π_{k′} is also zero. Again using Lemma 8.3 we conclude that μ_{k′} vanishes. □

From now on we shall assume in this section that R is a regular local ring.

Corollary 8.5. Suppose that the hypotheses of Theorem 6.10 (resp. 6.12) hold. If μ_{k} = 0 for some k ≥ 2, then X(T /K_{∞})(R) = 0. In particular X(T /K_{∞})(p) = 0.

Proof. Since X(T /K_{∞}) does not contain any non-trivial pseudo-null R[[G]]- submodule, we have X(T /K_{∞})[P_{k}] = 0 for all but finitely many height one prime ideals P_{k}, by Lemma 8.1. From Theorem 8.4, we have μ_{k′} = 0 for all k′ ≥ 2. Therefore we can choose a height one prime ideal P_{k′} such that μ_{k′} = 0 and X(T /K_{∞})[P_{k′}] = 0. Now the corollary follows from Proposition 8.2. □

Let O be the ring of integers of a finite field extension of ℚ_{p} and π be a generator of the maximal ideal of O. Then a torsion O[[G]]-module M is said to be in the category 𝔐_{H}(G) if M/M(π) is a finitely generated O[[H]]-module.

Proposition 8.6. Assume that Y (T /K_{∞}) is a finitely generated R[[H]]- module. Then X(T_{k}/K_{∞}) is in 𝔐_{H}(G) for all k ≥ 2. Conversely, if there exists a k ≥ 2 such that μ_{k} vanishes and X(T_{k}/K_{∞}) does not have any nontrivial pseudo-null R/P_{k}[[G]]-submodule, then X(T_{k}/K_{∞}) belongs to 𝔐_{H}(G) if and only if X(T /K_{∞}) is a finitely generated R[[H]]-module.

Proof. Consider the following exact sequence,

\xymatrix{X(T /K_{∞})/P_{k} \ar[d] \ar[r] & Y (T /K_{∞})/P_{k} \ar[d] \ar[r] & 0 \\ X(T_{k}/K_{∞}) \ar[r] & Y (T_{k}/K_{∞}) \ar[r] & 0.}

Since Y (T /K_{∞}) is a finitely generated R[[H]]-module, it follows from the above diagram that Y (T_{k}/K_{∞}) is a finitely generated R/P_{k}[[H]]-module if and only if the cokernel of the second vertical map is a finitely generated R/P_{k}[[H]]- module. But the cokernel of the first vertical map is a finitely generated R/P_{k} -module [CS, Theorem 4.2]. Therefore the cokernel of the second vertical map is finitely generated as a R/P_{k}-module, and hence as an R/P_{k}[[H]]-module. This proves the first assertion. For the proof of the converse assertion, we notice that the hypothesis implies that X(T_{k}/K_{∞}) is a finitely generated R/P_{k}[[H]]- module. Now the assertion follows from [CS, Proposition 5.4]. □

We remark that in the above proposition, the assumption that the dual Selmer group associated to some specialisation does not contain any non-trivial pseudonull submodule holds for the specialisation satisfying Hypothesis 1. It is also expected in general that the Pontryagin dual of the corresponding dual Selmer group is in 𝔐_{H}(G). Thus it seems probable that Y (T /K_{∞}) is a finitely generated R[[H]]-module. Now consider the following commutative diagram.

[figure: 0; Sel(A_{k}/K_{∞}); α_{k}; H^{1}(K_{S}/K_{∞}, A_{k}); β_{k}; ⨁_{v∈S(K)} J_{v}(A_{k}, K_{∞}); ⊕_{v}γ_{k,v}; 0; Sel(A/K_{∞})[P_{k}]; H^{1}(K_{S}/K_{∞}, A)[P_{k}]; ⨁_{v∈S(K)} J_{v}(A, K_{∞})[P_{k}].]

In [CS] it is shown that the cokernel of α_{k} is a cofinitely generated R/P_{k}[[H]]- module. But in the following lemma we shall show that under the hypothesis of Theorem 6.10 (resp. 6.12), it is infact R/P_{k}[[H]]-cotorsion.

Lemma 8.7. If the assumptions of Theorem 6.10( resp. 6.12) hold, then the cokernel of α_{k} is a cotorsion R/P_{k}[[H]]-module.

Proof. By the snake lemma, it is enough to show that the kernel of γ_{k,v} is a cotorsion R/P_{k}[[H]]-module for each v. If the hypothesis of Theorem 6.10 (resp. 6.12) holds, then for each prime v of K_{∞}, the dimension of the decomposition group G_{v} is ≥ 2. Fix a finite prime v of K_{∞}. Let H_{v} denote the decomposition subgroup of H corresponding to v. Since the dimension of G_{v}/H_{v} is equal to one for each v, the dimension of H_{v} must be greater than or equal to one for each v. Let A(K_{v,∞}) denote the group H^{0}(K_{v,∞}, A). We shall first show that Coind_{H}^{H_{v}}(A(K_{v,∞})/P_{k}) is a cotorsion R/P_{k}[[H]]-module. The Pontryagin dual of Coind_{H}^{H_{v}}(A(K_{v,∞})/P_{k}) is isomorphic to Ind_{H}^{H_{v}}({ \atop A(K_{v,∞})}[P_{k}]). Since { \atop A(K_{v,∞})}[P_{k}] is a finitely generated R/P_{k}-module and dimension of H_{v} is greater than or equal to 1, { \atop A(K_{v,∞})/P_{k}} must be a torsion R/P_{k}[[H_{v}]]-module. Now it follows from [V, Proposition 2.7(i)] that induced module of a torsion module is torsion. Therefore Coind_{H}^{H_{v}}(A(K_{v,∞})/P_{k}) is a cotorsion R/P_{k}[[H]]- module. Note that the kernel of γ_{k,v} is equal to ⨁_{w} Coind_{H}^{H_{w}}(A(K_{w,∞})/P_{k}), where w varies over the finite primes of K_{cyc} such that w|v. Since each Coind_{H}^{H_{w}}(A(K_{w,∞})) is a cotorsion R/P_{k}[[H]]-module and there are finitely many primes of K_{cyc} over each prime v of K, the kernel of γ_{k,v} is a cotorsion R/P_{k}[[H]]-module for each v. □

Theorem 8.8. Suppose that Y (T /K_{∞}) is a finitely generated R[[H]]-module. If the assumptions of Theorem 6.10 (resp. 6.12) hold, then the R/P_{k}[[H]]-rank of Y (T_{k}/K_{∞}) is same as the R[[H]]-rank of Y (T /K_{∞}) for all k ≥ 2.

Proof. Since X(T /K_{∞})(p) is a finitely generated R[[G]]-module, there exists a positive integer n such that p^{n}X(T /K_{∞})(p) = 0. Thus we can view Y (T /K_{∞}) as the submodule p^{n}X(T /K_{∞}) of X(T /K_{∞}) for such a large n. Therefore Y (T /K_{∞}) does not contain any non-trivial pseudo-null R[[G]]-submodule. We shall show that Y (T /K_{∞}) is R[[H]]-torsion free. Indeed, since R is assumed to be regular local, the ring R[[G]] is Auslander regular, and standard dimension theory in this context yields that a finitely generated R[[H]]-module is pseudonull as an R[[G]]-module, if and only if it is R[[H]]-torsion. Therefore by our assumption, Y (T /K_{∞}) is R[[H]]-torsion free. From this, we conclude that the R[[H]]-rank of Y (T /K_{∞}) is same as the R/P_{k}-rank of Y (T /K_{∞})/P_{k} for all k ≥ 2 (cf [Jh, Lemma 5]). Now consider the following commutative diagram.

\xymatrix{ & X(T /K_{∞})(p)/P_{k} \ar[r] \ar[d] & X(T /K_{∞})/P_{k} \ar[d] \ar[r] & Y (T /K_{∞})/P_{k} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & X(T_{k}/K_{∞})(p) \ar[r] & X(T_{k}/K_{∞}) \ar[r] & Y (T_{k}/K_{∞}) \ar[r] & 0.}

The cokernel of the middle vertical map in the above diagram is a finitely generated R/P_{k}-module [CS, Proposition 5.4] and therefore it is R/P_{k}[[H]]- torsion . Since X(T_{k}/K_{∞})(p) is ℤ_{p}-torsion and by Lemma 8.7, the kernel of the middle map is R/P_{k}[[H]]-torsion, it follows from the snake lemma that the kernel of the last vertical map is R/P_{k}[[H]]-torsion. Therefore, the R/P_{k}[[H]]- rank of Y (T_{k}/K_{∞}) and Y (T /K_{∞})/P_{k} are the same. Thus it follows that the R/P_{k}[[H]]-rank of Y (T_{k}/K_{∞}) is same as the R[[H]]-rank of Y (T /K_{∞}). □

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Sudhanshu Shekhar

School of Mathematics Tata Institute of Fundamental Research Homi Bhabha road Colaba, Mumbai-400005 India sudhansu@math.tifr.res.in, R. Sujatha Math Department University of British Columbia Vancouver, B.C. Canada V6T 1Z2 sujatha@math.ubc.ca