ON THE RANKS OF IWASAWA MODULES OVER p-ADIC LIE EXTENSIONS

YOSHIHIRO OCHI AND OTMAR VENJAKOB

Abstract. In this paper we study Iwasawa modules arising from Galois cohomology over general p-adic Lie extensions both in the local and global case. In particular we calculate their Λ-ranks. Then we apply the results to abelian varieties.

1. Introduction

The Iwasawa theory for an elliptic curve over a ℤ_{p}-extension of a number field was begun by B. Mazur ([Ma], 1972). Some years later, M. Harris began the study of the Iwasawa theory of an elliptic curve over the GL_{2}(ℤ_{p})-extension of a number field ([Ha1], 1979). Let E be an elliptic curve over a number field K without complex multiplication over K̄ and consider K_{∞} = K(E[p^{∞}]), the field obtained by adjoining all p-power division points E[p^{∞}] of E(K̄). By Serre’s theorem [Se2, IV 2.2], after fixing a ℤ_{p}-basis of the Tate module T_{p}E, the Galois group G = G(K_{∞}/K) is identified with an open subgroup of GL_{2}(ℤ_{p}), so G is a p-adic analytic group of dimension 4. The aim of this paper is to study Iwasawa modules, i.e. modules over the completed group algebra of G

(1)  Λ(G) := \underleftarrow{lim} ℤ_{p}[G/U],

where U runs over all normal open subgroups of G. While at the moment we restrict to GL_{2}-extensions, we should point out that most of our results are obtained for general p-adic analytic extensions K_{∞} over K.

One of the important Iwasawa modules is the Selmer group defined as follows:

(2)  Sel(E/K_{∞}){p} := ker(H^{1}(K_{∞}, E[p^{∞}]) → ∏ H^{1}(K_{∞,v}, E)(p)).

Since the cohomology groups have a natural continuous G-action as well as an action of ℤ_{p}, the Selmer group is a discrete Λ(G)-module. Let S = S(K) be a finite set of places of K such that S contains all places above p and ∞ and all places where E has bad reduction. Denote by K_{S} the maximal S-unramified extension of K and write G_{S}(K_{∞}) = Gal(K_{S}/K_{∞}). Then the Selmer group is also given by the following exact sequence

(3) 0 → Sel(E/K_{∞})\{p\} → H^{1}(G_{S}(K_{∞}), E[p^{∞}]) → \underrightarrow{lim}_{n} ⨁_{w∈S(K_{n})} H^{1}(K_{n,w}, E)(p).

The vanishing of H^{2}(G_{S}(K_{∞}), E[p^{∞}]), which immediately follows from the theorem of Iwasawa on the weak Leopoldt conjecture ([NSW, 10.3.25]), will allow us to obtain an unconditional result on the Λ(G)-(co)rank of the middle term. Our main concern is with the (co)rank of the Selmer group over K_{∞} and there is a basic conjecture on this rank (Conjecture 7.4, also [CH, Conjecture 2.4]), but at this moment we are unable to prove this conjecture but have only some inequalities (Propositions 5.1 and 7.3).

2000 Mathematics Subject Classification. Primary 11G05, 14K15; Secondary 16E65, 16S34.

The theory has recently made progress ([CH], [Ho3], [OV], [Su], [Ve1]), in particular structure theorems of torsion modules recently obtained by O. Venjakob (the p-torsion part, [Ve1]), S. Howson (the central torsion part, [Ho3]) and by J. Coates-P. Schneider-R. Sujatha (the general part, [CSS]). In this paper we will calculate the ranks of some Iwasawa modules arising from cohomology groups and determine their torsion submodules (see Theorems 3.2, 4.1, 4.3 etc.). We will also calculate the projective dimensions of certain Selmer groups (see Proposition 5.2 and Theorem 7.5) under the basic conjecture on the rank of the Selmer group mentioned above. In section 4 we refine Greenberg’s results on the structure of Galois cohomology groups in local ℤ_{p}-extensions (Theorem 4.3).

Iwasawa’s initial work studied certain Galois groups as Γ-modules, where Γ is a Galois group isomorphic to ℤ_{p} as a topological group. The section 4 of this paper extends this case to the cases of general p-adic Lie extensions, and with natural assumptions we prove analogous results as Iwasawa’s. Some of the results in the section were obtained by Harris but ours are slightly more general and we hope that our proofs are more structural by using Poitou-Tate duality.

This article will end with applications to the case of abelian varieties which was our main motivation of this work.

We will make a full use of the diagram in [OV, Lemma 4.5] , and the language of homotopy theory of modules, which we will review in the next section. This is a quite different approach from Harris’ one that uses the common descent diagram [Ma, p. 231] and an asymptotic formula ([Ha1, Theorem 1.10 and Lemma 3.4.1], but [Ha2, Theorem 1.10] is better), neither of which shall appear in this paper.

2. Prerequisites

Let K be a finite extension of ℚ and p an odd prime number. Let V be a p-adic Galois representation of G_{K}, which is unramified except a finite set of primes of K. For instance V = V_{p}E := T_{p}E ⊗_{ℤ_{p}} ℚ_{p} where T_{p}E denotes the Tate module. For such a V we can take a finite set S of places of K containing each prime above p and every archimedean place and every place whose inertia group acts nontrivially on V. S_{f}(K) denotes a subset of S(K) which consists of all finite places in S(K) and S_{p}(K) is {v ∈ S_{f}(K) : v | p}.

We say that an extension K_{∞}/K is a p-adic Lie extension if it is a Galois extension and the Galois group Gal(K_{∞}/K) is a compact p-adic Lie group of positive dimension. If it is a pro-p Lie group, we say K_{∞}/K is a pro-p Lie extension. Throughout this paper, whenever we consider a p-adic Lie extension K_{∞} of a number field K, we always assume that K_{∞} is contained in K_{S} for some S = S(K) associated to V. The GL_{2}-extension in the introduction and ℤ_{p}^{r}-extensions are examples of p-adic Lie extensions.

Let us fix V of dimension dim_{ℚ_{p}}V = r, take a Galois-invariant ℤ_{p}-lattice T of V and put A = V/T. Let K_{∞} be a p-adic Lie extension of K with Galois group G = G(K_{∞}/K) contained in Ω, where Ω denotes the maximal S-ramified p-extension of K(A), and K(A) denotes the extension of K defined by the kernel of the representation ρ : G_{K} → Aut(T). Put H = G(Ω/K_{∞}) and G = G(Ω/K). Then we have established the following ( [OV, 4.5]):

Diagram 2.1.

0  0

[figure: J_{A,K_{∞}}; =; J_{A,K_{∞}}]

[figure: 0; H^{2}(G_{S,∞}, A)^{∨}; (N^{ab}(p)^{#})_{H}; Λ(G)^{rd}; Y_{A,K_{∞}}; 0]
∥
[figure: 0; H^{2}(G_{S,∞}, A)^{∨}; (H^{1}(N^{ab}(p), A)^{H})^{∨}; ≅; H^{1}(R, A); H^{1}(G_{S,∞}, A)^{∨}; 0]

[figure: 0; 0]

Furthermore, N^{ab}(p)^{#} is a projective Λ(G)-module, hence (N^{ab}(p)^{#})_{H} a projective Λ(G)-module, and the latter module is isomorphic to (N^{ab}(p)_{H})^{#} if the action of H on A is trivial.

When K is a finite extension of ℚ_{l} (including the case l = p), we have a similar diagram for any p-adic Lie extension K_{∞}/K inside an algebraic closure Ω of K.

We will explain the notation as much as necessary for our applications.

(i) For a finitely generated Λ(G)-module M, we defined

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

We recall that Λ[A] is a free Λ-module of rank r. (ii) X = X_{A,K_{∞}} = H_{1}(H, A^{∨}). (iii) Y = Y_{A,K_{∞}} = (I(G)^{#})_{H}, which has projective dimension ≤ 1 if H^{2}(G_{S}(K_{∞}), A) =

0 since (N^{ab}(p)^{#})_{H} is a projective Λ(G)-module. (iv) J = J_{A,K_{∞}} = ker((Λ(G)^{#})_{H} → (A^{∨})_{H}), which has no nonzero torsion submod-

ule.

(v) d = dim_{𝔽_{p}}H^{1}(G, 𝔽_{p}) ([NSW, 3.9.1]).

Next we recall the language of homotopy theory of modules. Three functors are defined on the homotopy (stable) category Ho(Λ) of finitely generated Λ-modules, the two of them we will use here: the loop space functor Ω and the transpose functor D. The transpose functor satisfies D^{2} = 1. When two finitely generated Λ-modules M and N are isomorphic in the category (homotopically isomorphic), we write M ≃ N.

There is the following important exact sequence:

(5)  0 → E^{1}(DM) → M →^{ϕ_{M}} M^{++} → E^{2}(DM) → 0.

A finitely generated Λ-module M is called reflexive if the map ϕ_{M} in the sequence (5) is an isomorphism. It is proven that if Λ has no zero divisors and M is any finitely generated Λ-module, the set of all torsion elements in M forms a Λ-submodule of M, and it is isomorphic to E^{1}(DM) =: tor_{Λ}(M). The following result of Jannsen ([Ja1, 2.6]) is of frequent use in the sequel. We simplify the statements for our purpose.

Proposition 2.2. Let G be a p-adic Lie group of dimension = cd_{p}(G) = n and M a finitely generated Λ(G)-module, which is finitely generated as a ℤ_{p}-module of rank r.

(1) If M is finite and nonzero, then E^{i}(M) = 0 except i ≠ n + 1 and E^{n+1}(M) ≠ 0.

(2) If M is a free ℤ_{p}-module, then E^{i}(M) = 0 unless i = n and E^{n}(M) is a free ℤ_{p}-module of rank r and there is an isomorphism E^{n}(M)^{∨} ≅ M ⊗ D_{n}^{(p)}, where D_{n}^{(p)} denotes the p-dualizing module of G ([NSW, p.149]).

Further Notation and Convention: We follow the notations in the paper [OV]. However the assumption on the prime number p was unnecessarily too restrictive there. At least as long as abelian varieties are treated, the case p = 2 is OK too. In any case p = 3 has no problem. In general the “Assumption I” put in the next section will be enough for our applications. For more convenience of the reader, we put a list of the notations.

(1) For an abelian group A, A[p^{m}] := ker(A →^{×p^{m}} A) and A[p^{∞}] := ⋃_{n≥0} A[p^{n}]. μ_{p^{∞}} = 𝔾_{m}[p^{∞}] where 𝔾_{m} denotes the multiplicative group scheme over ℚ̄.

(2) For a ℤ_{p}-module N, N^{∨} = Hom_{ℤ_{p},cont}(N, ℚ_{p} / ℤ_{p}), is the Pontryagin dual of N, and for a p-divisible ℤ_{p}-module, N^{∗} = \underrightarrow{lim}_{i} Hom(_{p^{i}}N, μ_{p^{∞}}) = T (N)^{∨}(1), where T (N) = Hom(ℚ_{p} / ℤ_{p}, N) = \underleftarrow{lim}_{p^{i}}N.

(3) By a Noetherian ring, we mean a left and right Noetherian ring (with a multiplicative unit). By pd_{Λ}(M) we denote the projective dimension of M. The global dimension of Λ is denoted by pd(Λ).

(4) Let G be a profinite group and H a closed subgroup of G. For a Λ(H)-module M, we define Ind_{G}^{H} M := M ˆ⊗_{Λ(H)}Λ(G) (compact induction), where ˆ⊗ denotes completed tensor product. Also Coind_{H}^{G} M := Hom_{Λ(H)}(M, Λ(G)).

(5) Let K be a field of characteristic 0. We write G_{K} for the absolute Galois group G(K̄/K). For a G_{K}-module A, we use the notation A(K) := H^{0}(G(K̄/K), A).

(6) Let Λ be a ring, and M a Λ-module. Then E^{i}(M) := Ext_{Λ}^{i}(M, Λ), and in the case i = 0, we also write M^{+} := E^{0}(M).

(7) Galois cohomology is always continuous one.

(8) The G-module ℚ_{p} / ℤ_{p} denotes an abelian group ℚ_{p} / ℤ_{p} on which G acts trivially.

(9) For a finitely generated abelian p-primary group A we denote by A_{div} the quotient of A by its maximal p-divisible subgroup.

3. The Global Case

Let us fix a prime number p, a number field K, V, T and S = S(K) as in the previous section. For a p-adic Lie extension K_{∞}/K, we write X_{S} = H^{1}(G_{S}(K_{∞}), A)^{∨} and U_{v} = H^{1}(K_{∞,v}, A)^{∨}.

Proposition 3.1. For any p-adic Lie extension K_{∞}/K, X_{S} and ⨁_{v∈S(K)} Ind_{G}^{G_{v}} U_{v} are finitely generated over Λ(G).

Proof: This is immediate from the diagram. Firstly Y is finitely generated since there is a surjective morphism from a free Λ-module of finite rank. As Λ is Noetherian, X_{S} is also finitely generated. In the same way U_{v} is finitely generated over Λ(G_{v}) and hence there is a surjective map Λ(G_{v})^{k} → U_{v}. Therefore there is also a surjective map Λ(G)^{k} = Λ(G) ⊗_{Λ(G_{v})} Λ(G_{v})^{k} → Λ(G) ⊗_{Λ(G_{v})} U_{v}. It should be now obvious that the Λ(G)-module ⨁_{v∈S(K)} Ind_{G}^{G_{v}}U_{v} = ⨁_{v∈S(K)}(Λ(G) ⊗_{Λ(G_{v})} U_{v}) is finitely generated over Λ(G). □

Let us recall the fact that if G is any pro-p group and has no element of order p, then Λ(G) has no divisors of zero (see [DSMS]). It follows that Λ(G) has a unique quotient field Q(Λ) (division ring). For any finitely generated module M over such a Λ(G), we define the rank of M, denoted by rk_{Λ}(M), to be dim_{Q(Λ)}(M ⊗_{Λ} Q(Λ)).

Now we make the following assumption, which we will have to assume at many places.

Assumption I: The prime number p is odd or if p = 2, then all global fields considered are totally imaginary.

The following is a generalization of the one in the ℤ_{p}-case ([Gr1, Prop. 3]) to general pro-p Lie extensions, assuming the above assumption.

Theorem 3.2. Assume the Assumption I. Let K be a number field and A, S as above. For a pro-p Lie extension K_{∞}/K such that Λ is an integral domain, i.e., has no zero divisor, we have

(6)  rk_{Λ}(H^{1}(G_{S}(K_{∞}), A)^{∨}) − rk_{Λ}(H^{2}(G_{S}(K_{∞}), A)^{∨}) = r_{2}(K) × r

where r is the ℤ_{p}-corank of A and r_{2}(K) is the number of complex places of K.

Proof: By [Ja1, 5.4],

(7)  (N^{ab}(p))_{H} ≅ ⨁_{v∈S′_{∞}(K)} Ind_{G}^{G_{v}} ℤ_{p} ⨁ Λ(G)^{d−r′_{1}−r_{2}−1}

where S′_{∞}(K) is the set of real places of K which ramify in K_{∞} and r′_{1} = #S′_{∞}(K). Since G is a pro-p group, Λ(G) is a local ring and any projective module is free. Hence N^{ab}(p) is a free Λ(G)-module: N^{ab}(p) ≅ Λ(G)^{s} for some s. Then (N^{ab}(p)[A])_{H} ≅ (Λ(G)^{s}[A])_{H} = Λ(G)^{rs} = (N^{ab}(p))_{H}[A]. Hence from the isomorphism (7), with the Assumption I, s = d − r_{2} − 1. From the Diagram 2.1, the result follows. Note rk_{Λ}(J_{A,K_{∞}}) = r. □

We should also like to consider the inverse limit version of these modules. We write

(8)  H_{Iw}^{i}(K_{∞}, T_{p}A) := \underleftarrow{lim}_{n,m} H^{i}(G_{S}(K_{n}),_{p^{m}}A).

Under the Assumption I, H_{Iw}^{i}(K_{∞}, T_{p}A) = 0 for i > 2 since cd_{p}(G_{S}(K_{n})) ≤ 2.

Theorem 3.3. The Λ-modules H_{Iw}^{i}(K_{∞}, T_{p}A) are finitely generated and

(9)  rk_{Λ}(H_{Iw}^{i}(K_{∞}, T_{p}A)) = rk_{Λ}(H^{i}(G_{S}(K_{∞}), A)^{∨}).

Proof: There is the following spectral sequence due to Jannsen ([Ja2]):

(10)  E_{2}^{p,q} = E^{p}(H^{q}(G_{S}(K_{∞}), A)^{∨}) ⟹ H_{Iw}^{p+q}(K_{∞}, T_{p}A).

This gives the following exact sequence:

(11) E^{1}(A(K_{∞})^{∨}) → H_{Iw}^{1}(K_{∞}, T_{p}A) → E^{0}(H^{1}(G_{S}(K_{∞}), A)^{∨}) → E^{2}(A(K_{∞})^{∨})

But E^{i}(M) are all torsion modules for i ≥ 1 and any finitely generated Λ-module M (cf. [OV]). Therefore we obtain

rk_{Λ}(H^{1}(G_{S}(K_{∞}), A)^{∨}) = rk_{Λ}(E^{0}(H^{1}(G_{S}(K_{∞}), A)^{∨})) = rk_{Λ}(H_{Iw}^{1}(K_{∞}, T_{p}A)).

There is a filtration E_{2}^{2} ⊂ E_{1}^{2} ⊂ E_{0}^{2} = E^{2} such that E^{2} = E_{0}^{2}/E_{1}^{2} ≅ E^{0,2}, E_{1}^{2}/E_{2}^{2} ≅ E^{1,1} and E_{2}^{2} ≅ E^{1,0}. This shows that rk_{Λ}(H^{2}(G_{S}(K_{∞}), A)^{∨}) = rk_{Λ}(H_{Iw}^{2}(K_{∞}, T_{p}A)). It is clear that H_{Iw}^{i}(K_{∞}, T_{p}A) are finitely generated over Λ. □

Remark: The spectral sequence of Jannsen and Proposition 3.3 also holds for any ℓ-adic local field, hence so does Theorem 3.3 (see Theorem 4.1 below).

By Propositions 3.2 and 3.3, we have

(12) rk_{Λ}(H_{Iw}^{1}(K_{∞}, T_{p}A)) − rk_{Λ}(H_{Iw}^{2}(K_{∞}, T_{p}A)) = r_{2}(K) × ℤ_{p}-corank(A).

From Proposition 2.2 and the long exact sequence (11), if dim(G(K_{∞}/k)) ≥ 2, then H_{Iw}^{1}(K_{∞}, T_{p}A) has no nonzero torsion submodule. But more can be said:

Proposition 3.4. Let K be a number field, and K_{∞}/K a p-adic Lie extension such that dim(G(K_{∞}/K)) ≥ 3. Then H_{Iw}^{1}(K_{∞}, T_{p}A) is reflexive.

Proof: Put X = H^{1}(G_{S}(K_{∞}), A)^{∨} and H^{1} = H_{Iw}^{1}(K_{∞}, T_{p}A). Then from the sequence (11), Proposition 2.2 and the assumption, we have H^{1} = X^{+}. Hence the sequence (5) becomes

(13)  0 → E^{1}(DX) → X →^{ϕ} (H^{1})^{+} → E^{2}(DX) → 0.

Put M = Coker(ϕ). Then the exact sequence 0 → M → (H^{1})^{+} → E^{2}(DX) → 0 yields 0 → (H^{1})^{++} → M^{+} → E^{1}E^{2}(DX). But E^{1}E^{2}(DX) = 0 according to [OV, 2.4.3.]. Therefore (H^{1})^{++} = M^{+}. Another sequence 0 → E^{1}(DX) → X → M → 0 shows M^{+} = X^{+} = H^{1}. Hence H^{1} = (H^{1})^{++}. □

Proposition 3.5. Let K be a number field, and K_{∞}/K a p-adic Lie extension such that dim(G(K_{∞}/K)) ≥ 2 and the Iwasawa algebra Λ is an integral domain. Then

(14)  tor_{Λ}(H^{1}(G_{S}(K_{∞}), A)^{∨}) ≅ E^{1}(H_{Iw}^{2}(K_{∞}, T_{p}A)).

Proof: Since J has no torsion part (see Section 2), tor_{Λ}(X) = tor_{Λ}(Y). But [OV, 4.10] says DY ≃ H_{Iw}^{2}(K_{∞}, T_{p}A). Hence E^{1}(DY) = E^{1}(H_{Iw}^{2}(K_{∞}, T_{p}A)). □

Because of the isomorphism (14), the structure of H_{Iw}^{2}(K_{∞}, T_{p}A) is more interesting but nothing is known except that if H^{2}(G_{S}(K_{∞}), A) = 0, it is torsion by Theorem 3.3. The structure of H^{i}(G_{S}(K_{∞}), A)^{∨} in the ℤ_{p} case, for i = 1, 2, is known as follows. Let us prove those from the results above.

Proposition 3.6 ([Gr1]). Let K_{∞}/K be a ℤ_{p}-extension and p odd. Then

(1) H^{2}(G_{S}(K_{∞}), A)^{∨} is a free Λ-module.

(2) If H^{2}(G_{S}(K_{∞}), A)^{∨} = 0, then H^{1}(G_{S}(K_{∞}), A)^{∨} has no finite submodule.

Proof: Since (N^{ab}(p)^{#})_{H} is free and pd(Λ) = 2 in this case, H^{2}(G_{S}(K_{∞}), A)^{∨} must be a projective Λ-module, and hence a free Λ-module as Λ is a local ring. If H^{2}(G_{S}(K_{∞}), A)^{∨} = 0, then pd_{Λ}(Y) ≤ 1, and hence it has no finite submodule ([NSW, 5.5.3, iv]), and therefore X = H^{1}(G_{S}(K_{∞}), A)^{∨} has no finite submodule. □

For general p-adic Lie extensions, the first statement of the above proposition may not be true in general, though in many interesting cases it is predicted to be zero (see below “Assumption II”). At least it is torsion-free and H^{2}(G_{S}(K_{∞}), A)^{∨} ≃ Ω^{2}Y. The second statement of the proposition has been extended to higher dimensional pro-p Lie extensions, by Greenberg and Nguyen-Quang-Do (see [Ng]) in the commutative cases, and by the second author in the noncommutative cases (see [Ve1, 3.0.3] or [OV, 4.6]).

4. Local Results

Throughout this section K denotes a finite extension of ℚ_{ℓ}. Let A be as above and K_{∞}/K a p-adic Lie extension. First let us calculate a rank. We must assume that Λ = Λ(G) is an integral domain. We begin by noting that H^{2}(K_{∞}, A) = 0, because G_{L} has p-cohomological dimension ≤ 1 for any infinite Galois extension L of K such that G(L/K) contains an infinite pro-p subgroup ([NSW, 7.1.8.i]).

Theorem 4.1. Let K_{∞}/K be a pro-p Lie extension and denote the ℤ_{p}-corank of A by r. Then

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

The proof is similar to the one of Proposition 3.2, which will use the structure of N^{ab}(p)_{H} in the local case ([Ja1, 5.1.c] in the case ℓ = p). In the case ℓ ≠ p, we have N^{ab}(p)_{H} = Λ and d = 2 if K contains μ_{p} and N^{ab}(p)_{H} = 0 and d = 1 otherwise. For the calculation of d = dim_{𝔽_{p}}H^{1}(K, ℤ /p ℤ) see [NSW, 7.3.9]. The proof of the structure of N^{ab}(p)_{H} is the same as the one for the case ℓ = p, ([Ja1] or [NSW, 7.4.1]). Let us dare to have proof here as it is rather interesting. Firstly we look at the exact sequence of Galois groups 1 → N → R → G_{K_{n}} → 1 (see the diagram just before Lemma 4.5 in [OV]; here we take H = G(Ω/K_{n})). The inflation-restriction homology sequence with this sequence is

H^{2}(K_{n}, ℚ_{p} / ℤ_{p})^{∨} → H_{1}(N , ℤ_{p})_{G_{Kn}} → H_{1}(R, ℤ_{p}) → H_{1}(K_{n}, ℤ_{p}) → 0.

But H^{2}(K_{n}, ℚ_{p} / ℤ_{p}) = 0 (use Tate’s local duality). Hence we get

(15)  0 → N^{ab}(p)_{G_{Kn}} → R^{ab}(p) → G_{K_{n}}^{ab}(p) → 0.

In the Grothendieck group K_{0}(ℚ_{p}[G_{n}]), where G_{n} = Gal(K_{n}/K), we have

(16)  [N^{ab}(p)_{G_{Kn}} ⊗ ℚ_{p}] = [R^{ab}(p) ⊗ ℚ_{p}] − [G_{K_{n}}^{ab}(p) ⊗ ℚ_{p}].

To calculate [R^{ab}(p) ⊗ ℚ_{p}], we use the following exact sequence (“Fox-Lyndon resolution”, [OV, Section 4.1]):

(17)  0 → R^{ab}(p) → ℚ_{p}[G_{n}]^{d} → ℚ_{p}[G_{n}] → ℤ_{p} → 0

This sequence implies

[R^{ab}(p) ⊗ ℚ_{p}] = [ℚ_{p}[G_{n}]^{d}] − [ℚ_{p}[G_{n}]] + [ℚ_{p}] = [ℚ_{p}[G_{n}]^{d−1}] − [ℚ_{p}].

By local class field theory,

(18)  G_{K_{n}}^{ab}(p) ≅ \underleftarrow{lim}_{m} K_{n}^{×}/(K_{n}^{×})^{p^{m}} ≅ H^{1}(K_{n}, ℤ_{p}(1)).

Since ℓ ≠ p, [H^{1}(K_{n}, ℤ_{p}(1)) ⊗ ℚ_{p}] = [ℚ_{p}] ([NSW, 7.3.10]). In conclusion we have obtained [N^{ab}(p)_{G_{Kn}} ⊗ ℚ_{p}] = [ℚ_{p}[G_{n}]^{d−1}]. Since N^{ab}(p)_{H} is a projective Λ-module, N^{ab}(p)_{H} ≃ Λ^{d−1}. By (5.6.10) in [NSW] we get the result as required. Now the structure of N^{ab}(p)_{H} is determined in both cases ℓ = p and ℓ ≠ p, we obtain Theorem 4.1 in the same way as Theorem 3.2.

We now assume K_{∞}/K is a ℤ_{p}-extension. In this case Greenberg has proved the following basic results on the structure of the cohomology group H^{1}(K_{∞}, A)^{∨}.

Proposition 4.2 ([Gr1]). Let K_{∞}/K be a ℤ_{p}-extension and r = ℤ_{p} -corank of A. Put m = r[K : ℚ_{p}].

(1) Assume ℓ = p. Then H^{1}(K_{∞}, A)^{∨} ∼ Λ^{m} ⨁ T_{p}A^{∗}(K_{∞}), where ∼ means pseudoisomorphism.

(2) Assume ℓ = p. If A^{∗}(K_{∞}) is finite, then X = H^{1}(K_{∞}, A)^{∨} is isomorphic to a submodule of Λ^{m} of finite index. The quotient Λ^{m}/X is isomorphic to a submodule of A^{∗}(K_{∞}).

(3) Assume ℓ = p. If A^{∗}(K_{∞}) = 0, then H^{1}(K_{∞}, A)^{∨} is a free Λ-module of rank m.

(4) Assume ℓ ≠ p. Then H^{1}(K_{∞}, A)^{∨} ∼ T_{p}A^{∗}(K_{∞}). If A^{∗}(K_{∞}) is finite, then H^{1}(K_{∞}, A) = 0.

We shall extend this result to extensions the Galois group of which is the semidirect product of Γ ≅ ℤ_{p} with a finite group of order prime to p. Moreover, we get canonical exact sequences instead of just pseudo-isomorphisms.

Theorem 4.3. Let K_{∞}/K be a Galois extension with the 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}^{∗}. We write χ = ω^{−1} for the inverse of the character (of G) which determines the action on the p-dualizing module of G and we set X = H^{1}(K_{∞}, A^{∗})^{∨} ≅ H_{Iw}^{1}(K_{∞}, T_{p}A). Let r be the ℤ_{p}-codimension of A^{∗}.

(1) If l = p, then the Λ(G)-module X fits into the canonical exact sequence

0 → T_{p}A(K_{∞}) → X → P → N → 0

where P is a projective Λ(G)-module of rk_{Λ(Γ)}P = rt[K : ℚ_{p}] and the finite module N is determined by the exact sequence

0 → N → A(K_{∞})_{div}(χ) → tor_{ℤ_{p}}(A^{∗}(K_{∞})^{∨}).

Furthermore,

(a) if A(K_{∞}) is finite, then T_{p}A(K_{∞})(χ) = 0. If, in addition, A^{∗}(K_{∞})^{∨} is ℤ_{p}-free, then N ≅ A(K_{∞}).

(b) if A(K_{∞})^{∨} is ℤ_{p}-free, then X ≅ P ⊕ T_{p}A(K_{∞})(χ). In particular, X is

projective, if A(K_{∞}) = 0.

(2) If l ≠ p, then the Λ(G)-module X is isomorphic to

X ≅ T_{p}A(K_{∞})(χ).

Proof: Recall the basic sequence of homotopy theory of modules:

(19)  0 → E^{1}(DX) → X → X^{++} → E^{2}(DX) → 0.

By [OV, Lemma 4.10 ,Proposition 4.11 ], we have

E^{1}(DY) = E^{1}(A(K_{∞})^{∨}) = T_{p}A(K_{∞})(χ),

where the last equality follows from Proposition 2.2 and the fact that D_{1}^{(p)}(Γ) = ℚ_{p} / ℤ_{p}(ω). Since J has no torsion, E^{1}(DX) = tor_{Λ}(X) = tor_{Λ}(Y) = E^{1}(DY) = T_{p}A(K_{∞})(χ).

From the exact sequence

0 → X → Y → J → 0,

we get an exact sequence

E^{1}(J) → E^{1}(X) → E^{1}(Y) → E^{2}(J) = 0,

since J has no nonzero finite submodule([Ja1, 3.1.d]). From the exact sequence

0 → J → Λ[A^{∗}] → A^{∗}(K_{∞})^{∨} → 0,

we get E^{1}(J) = E^{2}(A^{∗}(K_{∞})^{∨}) = A^{∗}(K_{∞})_{div}(χ). Hence we have obtained the exact sequence

A^{∗}(k_{∞})_{div}(χ) →^{ϕ} E^{1}(Y) → E^{1}(X) → 0.

Denote C = Coker(ϕ) and decompose this sequence into

0 → C → E^{1}(Y) → E^{1}(X) → 0

and

E^{1}(J) →^{f} C → 0.

From the first one, we get

E^{1}(C) → E^{2}(DX) → E^{2}(DY) → E^{2}(C) → 0,

while from the second one

0 → E^{1}(C) → E^{1}E^{1}(J) = E^{1}(E^{2}(A^{∗}(K_{∞})^{∨})) = 0,

since E^{2}(A^{∗}(K_{∞})^{∨}) is finite ([Ja1, 3.1 (d)], and Proposition 2.2). Hence E^{1}(C) = 0 and E^{2}(DX) is a submodule of E^{2}(DY) = E^{2}(A(K_{∞}))^{∨} = A^{∗}(K_{∞})_{div}(χ) while it is easily seen that E^{2}(C) ⊆ tor_{ℤ_{p}}(A^{∗}(K_{∞})^{∨}) holds.

The facts that pd(Λ) = 2 and Λ is local imply that X^{++} is a projective Λ-module. Hence from all the things shown above, we obtain the following exact sequence

0 → T_{p}A(K_{∞}) → X → P → N → 0

where N is a submodule of the finite module A^{∗}(K_{∞})_{div}(χ).

As we have already calculated, the rank of X, rk_{Λ(Γ)}(X) = rt[K : ℚ_{p}] if ℓ = p and 0 otherwise, and obviously rk(X) = rk(X^{++}). □

This theorem gives another proof of the following

Corollary 4.4 ([Gr2, 2.3]). Let V be a one-dimensional p-adic representation of G_{K} and A = V/T with any G_{K}-invariant ℤ_{p}-lattice T in V. Then the ℤ_{p}-corank of H^{1}(K, A) is [K : ℚ_{p}] + δ, where δ = 1 if A = ℚ_{p} / ℤ_{p} or A = μ_{p^{∞}}, and δ = 0 otherwise.

Proof: Let K_{∞}/K be the cyclotomic ℤ_{p}-extension with the Galois group Γ. Suppose A ≠ μ_{p^{∞}}. Since there is no proper submodule of ℚ_{p} / ℤ_{p} of finite index, A^{∗}(K_{∞}) is finite. Put H^{1} = H_{Iw}^{1}(K_{∞}, T_{p}A^{∗}). We have the following exact short sequence

(20)  0 → H^{1} → Λ^{[K:ℚ_{p}]} → N → 0

with some finite submodule N. Taking Γ-homology, we know the ℤ_{p}-rank of H_{Γ}^{1} is [K : ℚ_{p}]. Hence by the local duality, the ℤ_{p}-rank of (H^{1}(K_{∞}, A)^{∨})_{Γ} is also [K : ℚ_{p}]. Next we look at the exact inflation-restriction sequence:

(21)  0 → H^{1}(Γ, A(K_{∞})) → H^{1}(K, A) → H^{1}(K_{∞}, A)^{Γ} → 0.

Now assume moreover A ≠ ℚ_{p} / ℤ_{p}. Then H^{1}(Γ, A(K_{∞})) is finite, hence the ℤ_{p}-corank of H^{1}(K, A) is [K : ℚ_{p}]. If A = ℚ_{p} / ℤ_{p}, then the ℤ_{p}-corank of H^{1}(K, ℚ_{p} / ℤ_{p}) is [K : ℚ_{p}] + 1.

Finally let us consider the case A = μ_{p^{∞}}. In this case we will get similarly an exact sequence

0 → ℤ_{p} → H^{1} →^{f} Λ^{[K:ℚ_{p}]} → W → 0

with some finite W. Put Z = Coker(f). Then H_{1}(Γ, Z) = H_{2}(Γ, W) is finite, hence the ℤ_{p}-rank of H_{Γ}^{1} = [K : ℚ_{p}] + 1. Since H^{1}(Γ, A(K_{∞})) finite, we get the ℤ_{p}-corank of H^{1}(K, μ_{p^{∞}}) = [K : ℚ_{p}] + 1. □

In the higher dimensional case the results are not as fine as above but still we obtain the following

Proposition 4.5. Let K_{∞}/K be an p-adic Lie extension with Galois group G, such that dim(G) ≥ 2. Then the following hold:

(1) If p ≠ l, then H^{1}(K_{∞}, A)^{∨} = 0.

(2) If p = l, then H^{1}(K_{∞}, A)^{∨} is a torsion-free Λ(G)-module, which is even reflexive in case cd_{p} ≥ 3.

The proof is completely analogous to that of theorem 4.3 using again 2.2 and [OV, 4.10,4.11], see also [OV, 5.4]. Note that in the case p ≠ l 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 proper quotient G of dim(G) ≥ 2.

5. Selmer Groups

Let K be a number field and V and T as in Section 2. The Bloch-Kato Selmer group Sel^{BK}(K, A) is defined as follows ([BK]):

Sel^{BK}(K, V/T) := ker(H^{1}(K, V/T) → ∏ H̃^{1}(K_{v}, V/T).

here, the definition of H̃^{1} is the following:

H̃^{1}(K_{v}, V/T) = H^{1}(K_{v}, V/T)/Im(H_{f}^{1}(K_{v}, V/T))

where Im(H_{f}^{1}(K_{v}, V/T)) is the image of H_{f}^{1}(K_{v}, V) under the natural map V → V/T and H_{f}^{1}(K_{v}, V) is defined as follows:

If v does not divide p, then

H_{f}^{1}(K_{v}, V) := ker(H^{1}(K_{v}, V) → ∏ H^{1}(K_{v}^{nr}, V))

where K_{v}^{nr} denotes the maximal unramified extension of K_{v};

If v divides p, then

H_{f}^{1}(K_{v}, V) := ker(H^{1}(K_{v}, V) → ∏ H^{1}(K_{v}, V ⊗_{ℚ_{p}} B_{crys}))

where B_{crys} is the period ring defined by Fontaine ([Fo]).

It is shown in [BK] that for V_{p}(E), where E is an elliptic curve over K, H_{f}^{1}(K_{v}, V) = E(K_{v}) ⊗_{ℤ_{p}} ℚ_{p} and Sel^{BK}(K, V/T) = Sel^{BK}(K, E[p^{∞}]) = Sel(E/K){p}.

On the other hand Greenberg has defined a Selmer group for V which satisfies the so-called Panchishkin condition ([Gr3]). Let us recall the definition: V is said to satisfy the Panchishkin condition at the place v if

• V is de Rham.

• There is an exact sequence of p-adic representations

(22)  0 → V̂ → V → Ṽ → 0

where V̂ and Ṽ satisfy V̂ ⊗ ℂ_{p} ≅ ⨁_{i>0} ℂ_{p}(i)^{⊕k_{i}} and Ṽ ⊗ ℂ_{p} ≅ ⨁_{i≤0} ℂ_{p}(i)^{⊕k_{i}} as G_{K_{v}}-modules.

If V satisfies the Panchishkin condition, then it satisfies the ordinary condition in the sense of [Gr1]. For a p-adic representation satisfying the Panchishkin condition, the Selmer group is defined in the following way: (23)

Sel^{Gr}(K, A) := ker(H^{1}(G_{S}(K), A) →^{ϕ} ∏_{v|p} H^{1}(K_{v}^{nr}, A/F_{v}^{+}A) ⊕ ∏_{v∤p,v∈S} H^{1}(K_{v}^{nr}, A)),

where F_{v}^{+}A is a submodule of A defined by V̂ in the Panchishkin filtration (F_{v}^{+}A is the image of V̂ in A under the natural map), and S is a finite set of primes of K satisfying the condition as before.

We have the following long exact sequence deduced from Poitou-Tate’s global duality and Tate’s local duality:

0 → Sel^{BK}(K, V/T) → H^{1}(G_{S}(K), V/T) → ⨁_{v∈S(K)} H̃^{1}(K_{v}, V/T)

→^{ψ} Sel^{BK}(K, T^{∗})^{∨} → H^{2}(G_{S}(K), V/T).

Here Sel(K, T^{∗}) is defined as follows: Put B_{v} = ImH_{f}^{1}(K_{v}, V). Let C_{v} be the orthogonal complement of B_{v} under the perfect paring of Tate’s duality theorem. Then we define the Selmer group Sel^{BK}(K, T^{∗}) as follows:

(24)  Sel^{BK}(K, T^{∗}) := ker(H^{1}(G_{S}(K), T^{∗}) → ⨁_{v∈S(K)} H^{1}(K_{v}, T^{∗})/C_{v}).

The map ψ in the long exact sequence is obtained as follows. Put A = V/T as before. Let us make the following notation: Write for any i ≥ 0

(25)  � ^{i}(G_{S}(K), A) := ker(H^{i}(G_{S}(K), A) → ⨁_{S} H^{i}(K_{v}, A)).

By Poitou-Tate global duality we have the following commutative diagram:

0 → � ^{1}(G_{S}(K), A^{∗})^{∨} ≅� ^{2}(G_{S}(K), A) →H^{2}(G_{S}(K), A)(global duality)
↑
H^{1}(G_{S}(K), A^{∗})
↑
⨁_{S} H^{1}(K_{v}, A^{∗})^{∨} ≅ ⨁_{S} H^{1}(K_{v}, T)  (local duality)
∪  ∪
⨁_{S}(H^{1}(K_{v}, A^{∗})/C_{v})^{∨}=  ⨁_{S} B_{v}  (by perfect paring)

Hence we get the following diagram commutative, which gives the long exact sequence:

0  0
↓  ↓
⨁_{S} B_{v}  =⨁_{S}(H^{1}(K_{v}, A^{∗})/C_{v})^{∨}

[figure: H^{1}(G_{S}(K), A); →; ⨁_{S} H^{1}(K_{v}, A); →; H^{1}(G_{S}(K), T^{∗})^{∨}; →; H^{2}(G_{S}(K), A)]
∥  ↓  ∥
H^{1}(G_{S}(K), A)→⨁_{S} H^{1}(K_{v}, A)/B_{v}→ Sel^{BK}(K, T^{∗})^{∨} →H^{2}(G_{S}(K), A)
↓  ↓
0  0

Over an infinite Galois extension K_{∞} of K we define the Selmer groups Sel^{BK}(K_{∞}, V/T) to be the direct limit of the Selmer group over each layer. The same with Sel^{Gr}(K_{∞}, V/T). We have similar long exact sequences as above for Sel^{BK}(K_{∞}, V/T), Sel^{Gr}(K, V/T), and Sel^{Gr}(K_{∞}, V/T). The finitely generatedness of H^{1}(G_{S}(K_{∞}), V/T)^{∨} implies the one of Sel^{?}(K_{∞}, V/T) for ? = BK or Gr.

From now on we assume that V satisfies the Panchishkin condition, we consider the Greenberg’s Selmer group Sel^{Gr}(K, V/T) and omit Gr from the notation. Note that Sel^{BK}(K, V/T) is a submodule of Sel^{Gr}(K, V/T). We also assume the following

Assumption II: H^{2}(G_{S}(K_{∞}), A) = 0.

For instance if K_{∞} contains the cyclotomic ℤ_{p}-extension, then this is believed to hold ([Gr1, Conjecture 3]).

The structure of the Selmer group, in particular, in the case V = V_{p}(E), has been studied in [OV], where E/K is an elliptic curve without complex multiplication and has good ordinary reduction at any prime over p. There are also interesting results in other different cases (cf. [Gr2] and [Pe]). The following is easy.

Proposition 5.1. Assume the Assumptions I and II. Then we have inequalities

(26)  r · r_{2}(K) − Σ_{v∈S_{p}}[K_{v} : ℚ_{p}]codim(F_{v}^{+}V) ≤ rk_{Λ}(Sel(K_{∞}, A)^{∨}) ≤ r · r_{2}(K)

where r is the dimension of V.

Proof: By the Assumption II and Theorem 3.2, we have

rk_{Λ}(H^{1}(G_{S}(K_{∞}), A)^{∨}) = r · r_{2}(K).

Hence rk_{Λ}(Sel(K_{∞}, A)^{∨}) ≤ r · r_{2}(K). In the same way one calculates with the results above to obtain

rk_{Λ}(⨁_{S} H^{1}(K_{∞,v}, A/F_{v}^{+}A)) = Σ_{v∈S_{p}}[K_{v} : ℚ_{p}].

Hence from the exact sequence we get the other inequality

r · r_{2}(K) − Σ_{v∈S_{p}}[K_{v} : ℚ_{p}]codim(F_{v}^{+}V) ≤ rk_{Λ}(Sel(K_{∞}, A)^{∨}).
□

The proof of [OV, Proposition 5.3] was not given, so we shall give it here.

Proposition 5.2. Suppose that 3 ≤ cd_{p}(G) = dim(G) and dim(G) > cd_{p}(G_{v}) = dim(G_{v}) and dim(G_{v}) ≥ 2 for any v ∤ p, and assume the morphism ϕ in the sequence (23) is surjective and the Assumption II. Then if A(K_{∞})^{∨} has a nontrivial finite submodule, then pd_{Λ}(Sel(K_{∞}, A)^{∨}) = dim(G) − 1 and if A(K_{∞})^{∨} has no finite submodule, then pd_{Λ}(Sel(K_{∞}, A)^{∨}) = dim(G) − 2.

Proof: In this proof we also determine the projective dimension of X = H^{1}(G_{S}(K_{∞}), A)^{∨} under the same assumptions in the statement. Let M be a Λ-module such that it is a finitely generated ℤ_{p}-module. Then if it has no finite submodule, then pd_{Λ}(M) = dim(G), and if it has a nontrivial finite submodule, then pd_{Λ}(M) = dim(G) + 1. This follows from Proposition 2.2. Indeed putting n = dim(G) = cd_{p}(G). By Proposition 2.2, if it has no finite submodule, then E^{n}(M) ≠ 0, hence pd_{Λ}(M) ≥ dim(G) and if it has a nontrivial finite submodule, then E^{n+1}(M) ≠ 0 and pd_{Λ}(M) ≥ dim(G) + 1. But in any case pd_{Λ}(M) ≤ dim(G) + 1 ([Br, 4.1]).

Now we recall the well-known fact from homological algebra: Given an exact sequence 0 → N → M → L → 0, then we have

(27)  pd_{Λ}(M) ≤ max{pd_{Λ}(N), pd_{Λ}(L)}.

and if it is not equality, then pd_{Λ}(N) = pd_{Λ}(L) − 1. Now we show the claim in the case that A(K_{∞})^{∨} has no finite submodule. We have the exact sequence 0 → J → Λ(G)[A] → A(K_{∞})^{∨} → 0, and since pd_{Λ}(Λ(G)[A]) = 0, pd_{Λ}(J) = n − 1. Now let us look at another sequence 0 → X → Y → J → 0. By the assumption II, it is known from the Diagram 2.1 that pd_{Λ}(Y) ≤ 1. As n − 1 ≥ 2 by another assumption, pd_{Λ}(X) = n − 2.

Repeat the same thing with H^{1}(K_{v,∞}, A)^{∨}. Then with the assumption dim(G) > dim(G_{v}) we get pd_{Λ}(X_{v}) < n − 2. Since dim(G_{v}) ≥ 2, if v does not divide p, then H^{1}(K_{v,∞}, A)^{∨} = 0. Hence

pd_{Λ}(⨁_{S} Ind_{G}^{G_{v}} H^{1}(K_{v,∞}, A)^{∨}) = max ⨁_{S} pd_{Λ}(Ind_{G}^{G_{v}} H^{1}(K_{v,∞}, A)^{∨}) < n − 2.

Now from the long exact sequence above in the beginning of this section and (27), we get pd_{Λ}(Sel(K_{∞}, A)^{∨}) = dim(G) − 2.

The case that A(K_{∞})^{∨} contains a nontrivial finite submodule goes similarly. □

6. Galois Groups as Iwasawa Modules

In this section we consider the “original” case of Iwasawa theory, i.e., the case A = ℚ_{p} / ℤ_{p} with trivial Galois action. Let K be a global number field and S a finite set of places of K containing all primes above p and ∞. The dual of the cohomology groups are isomorphic to Galois groups as Λ-modules, that is, 𝔛 = Gal(L/K_{∞}) = Sel(K_{∞}, ℚ_{p} / ℤ_{p})^{∨} and X = Gal(M/K_{∞}) = H^{1}(G_{S}(K_{∞}), ℚ_{p} / ℤ_{p})^{∨}, where M is the maximal abelian pro-p S-ramified extension of K_{∞} and L is the maximal unramified extension of K_{∞}. We also consider the extension L′, the subextension of L, in which all primes of K_{∞} split completely. Write 𝔛′ = Gal(L′/K_{∞}). We assume K_{∞}/K is a pro-p Lie extension such that its associated Iwasawa algebra has no nonzero zero divisor. For the results in this section the following is crucial

Assumption III: No finite place of S_{f}(K) splits completely in K_{∞}.

The groups of global and local units are very important: In the following L denotes any finite extension of K contained in K_{∞}.

(1) E_{∞} = \underleftarrow{lim}_{L} E′(L), where E′(L) is p-adic completion of units of L: E′(L) =

O_{L}^{×} ⊗ ℤ_{p}.

(2) E_{S,∞} = \underleftarrow{lim}_{L} E′_{S}(L), where E′_{S}(L) is p-adic completion of S-units of L: E′(L) =

O_{L}^{×} ⊗ ℤ_{p}.

(3) U_{S,∞} = ⨁_{v∈S} Ind_{G}^{G_{v}}U′_{v}(L_{v}), where U′_{v}((L) is p-adic completion of units of L_{v}.

(4) A_{S,∞} = ⨁_{v∈S} Ind_{G}^{G_{v}}A_{v}(L_{v}), where A_{v}(L_{v}) = \underleftarrow{lim} L^{×}/L^{×}^{p^{n}}.

The basic diagram connecting these modules is the following ([NSW, 11.3.10]):

[figure: 0; H^{2}(G_{S}(K_{∞}), ℚ_{p} / ℤ_{p})^{∨}; E_{S,∞}; A_{S,∞}; X_{S}; 𝔛′; 0]
∥  ∥
[figure: 0; H^{2}(G_{S}(K_{∞}), ℚ_{p} / ℤ_{p})^{∨}; E_{∞}; U_{S,∞}; X_{S}; 𝔛; 0]

There is also the following exact sequence:

(28)  0 → E_{∞} → E_{S,∞} → ⨁_{S_{un}} Ind_{G}^{G_{v}} ℤ_{p} → 𝔛 → 𝔛′ → 0,

where S_{un} := {ν ∈ 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.

Now by the Assumption III, ⨁_{S} Ind_{G}^{G_{v}} ℤ_{p} is torsion, so 𝔛′ is torsion if and only if 𝔛′ is torsion.

The main result of this section is the following

Theorem 6.1. 𝔛′ and 𝔛 are torsion Λ-modules under the Assumption III.

Lemma 6.2. Under the Assumption III, H^{2}(G_{S}(K_{∞}), A) = 0 if and only if � ^{1}(G_{S}(K_{∞}), A^{∗})^{∨} is Λ-torsion.

Proof: By [OV, Proposition 4.10 (ii) ], H^{2}(G_{S}(K_{∞}, A) = 0 if and only if Z_{A}(K_{∞}) (see loc. cit. for the definition) is Λ-torsion. From Poitou-Tate’s global duality, we have the following exact sequence

(29)  0 → � ^{1}(G_{S}(K_{∞}), A^{∗})^{∨} → Z_{A}(K_{∞}) → ⨁_{S}(A^{∗})^{∨} ⊗_{Λ(G_{v})} Λ(G).

By the Assumption III, ⨁_{S}(A^{∗})^{∨}⊗_{Λ(G_{v})}Λ(G) is Λ-torsion. Hence Z_{A}(K_{∞}) is Λ-torsion if and only if � ^{1}(K_{∞}, A^{∗})^{∨} is Λ-torsion. □

Lemma 6.3. Under the assumption III, H^{2}(G_{S}(K_{∞}), μ_{p^{∞}}) = 0.

Proof: Under the Assumption III,

H^{2}(G_{S}(K_{∞}), μ_{p^{∞}}) = � ^{2}(K_{∞}, μ_{p^{∞}}) = \underrightarrow{lim} � ^{1}(G_{S}(K_{n}), ℤ /p^{m} ℤ)

= \underrightarrow{lim}_{n,m} Cl_{S}(K_{n})/p^{m} = \underrightarrow{lim}_{n} Cl_{S}(K_{n}) ⊗_{ℤ} ℚ_{p} / ℤ_{p} = 0

since Cl_{S}(K_{n}) is finite for every n. □

Proof of Theorem 6.1:

It is easy to see that 𝔛′ ≅ � ^{1}(K_{∞}, ℚ_{p} / ℤ_{p})^{∨}. Hence from the two lemmas above, 𝔛′ is Λ-torsion under the assumption III. Hence so is 𝔛 since 𝔛′ is torsion if and only if 𝔛′ is torsion.

Remarks:

1. In the case of any ℤ_{p}-extension, Iwasawa proved that 𝔛 is a finitely generated Λtorsion module. This follows from our results immediately, as the Assumption III is satisfied if S_{f} is taken to be the set of all primes of K above p.

2. It was claimed in [Ha1] that for a pro-p Lie extension K_{∞}/K such that K contains μ_{p}, 𝔛 is torsion (Theorem 3.3). However, the proof in [Ha1] was incorrect (but Harris himself has given a correct proof recently in [Ha2]). Our proof is quite different, and was discovered independently.

Proposition 6.4. Assume the Assumption III and H^{2}(G_{S}(K_{∞}), ℚ_{p} / ℤ_{p}) = 0 (the weak Leopoldt conjecture). Then

(1) rk_{Λ}(X_{S}) = r_{2}(K).

(2) rk_{Λ}(A_{S,∞}) = rk_{Λ}(U_{S,∞}) = [K : ℚ].

(3) rk_{Λ}(E_{S,∞}) = rk_{Λ}(E_{∞}) = [K : ℚ] − r_{2}(K).

Proof: From the vanishing assumption and Proposition 3.2, we get rk_{Λ}(X_{S}) = r_{2}(K). There is an exact sequence (compare the sequence (28)):

(30)  0 → U_{S,∞} → A_{S,∞} → ⨁_{S_{un}} Ind_{G}^{G_{v}} ℤ_{p} → 0.

Hence the Assumption III says rk_{Λ}(A_{S,∞}) = rk_{Λ}(U_{S,∞}). By Kummer theory,

A_{v}(K_{∞}) = \underleftarrow{lim}_{i}(\underleftarrow{lim}_{n} K_{i}^{×}/K_{i}^{×}^{p^{n}}) = \underleftarrow{lim}_{i}(\underleftarrow{lim}_{n} H^{i}(K_{v,i}, ℤ_{p}(1))).

By Proposition 3.3,

rk_{Λ}(\underleftarrow{lim}_{n} H^{i}(K_{v,i}, ℤ_{p}(1))) = rk_{Λ}(H^{1}(K_{∞}, μ_{p^{∞}})).

But by Lemma 6.3, H^{2}(K_{∞}, μ_{p^{∞}}) = 0, hence by Proposition 4.1, consequently we have rk_{Λ(G_{v})}(A_{v}(K_{∞,v})) = [K_{v} : ℚ_{p}]. Therefore we have rk_{Λ(G)}(Λ(G) ⊗_{Λ(G_{v})} A_{v}(K_{∞,v})) = #{v ∈ S_{p}(K)} × [K_{v} : ℚ_{p}] = [K : ℚ].

As the sequence (28) shows, rk_{Λ}(E_{S,∞}) = rk_{Λ}(E_{∞}) under the Assumption III. But by now we know the ranks of 𝔛, X_{S}, U_{S,∞} and A_{S,∞}. Hence from the diagram above we get rk_{Λ}(E_{S,∞}) = rk_{Λ}(E_{∞}) = [K : ℚ] − r_{2}(K). □

7. Abelian Varieties

Let X be an abelian variety defined over a finite extension K of ℚ, and X′ denote the dual abelian variety of X. Throughout this section we put A = X[p^{∞}] and A^{∗} = X′[p^{∞}], and we consider the Selmer group Sel^{BK}(K_{∞}, A) = Sel(X/K_{∞}){p} over the field K_{∞} = K(X[p^{∞}]) and just write Sel(K_{∞}, A) omitting BK from the notation. Fix a finite set S of places of K which contains all primes above p and ∞ and all the primes where A has bad reduction. For an infinite extension K_{∞} of K, we have a long exact sequence, which is very basic for the Iwasawa theory of an abelian variety:

0 → Sel(K_{∞}, A) → H^{1}(G_{S}(K_{∞}), A) → ⨁_{v∈S(K)} Coind_{G_{v}}^{G} H^{1}(K_{∞,v}, X)(p)

→ (\underleftarrow{lim} Sel(K_{n}, T_{p}X′))^{∨} → H^{2}(G_{S}(K_{∞}), A).

In the classical Iwasawa theory this sequence is considered over a ℤ_{p}-extension of K. In particular, when X is an elliptic curve over ℚ and for the cyclotomic ℤ_{p}-extension of ℚ, then the structure of all these Λ-modules has been determined due to the recent work of K. Kato ([Ka]). In our case we can calculate the ranks of these modules, except the one of the Selmer group, relatively more easily than in ℤ_{p}-cases. One reason is the following

Proposition 7.1. (cf. [OV, 4.7]) Let K_{∞} be as above. Then H^{2}(G_{S}(K_{∞}), A) = 0.

Another is the work of Coates and Greenberg on the Kummer theory for Selmer groups ([CG]). From now on we assume that X has good reduction at all primes above p. Then according to [CG], K_{v,∞} = K_{v}(X[p^{∞}]) is a deeply ramified field ([CG] for the definition) where v divides p, since it contains K_{v}(μ_{p^{∞}}) by the Weil pairing. Then there is the following isomorphism:

(31)  H^{1}(K_{v,∞}, X)(p) ≅ H^{1}(K_{v,∞}, X̃[p^{∞}])

where X̃ is mod-p reduction of X, which is an abelian variety over an extension of 𝔽_{p} by assumption. From these we obtain the following

Proposition 7.2. Let K be a finite extension of ℚ_{ℓ}, X an abelian variety over K of dimension g, and K_{∞} = K(A). Assume that X has good reduction. Put G = Gal(K_{∞}/K_{0}) and Λ = Λ(G), where K_{0} = K(X[p]). Let r be the p-rank of X̃. First suppose ℓ = p.

(1) The Λ-rank of H^{1}(K_{∞}, A)^{∨} is 2g[K_{0} : ℚ_{p}].

(2) The Λ-rank of H^{1}(K_{∞}, X)(p)^{∨} is r[K_{0} : ℚ_{p}].

(3) The Λ-rank of (X(K_{∞}) ⊗ ℚ_{p} / ℤ_{p})^{∨} is (2g − r)[K_{0} : ℚ_{p}].

(4) Now suppose ℓ ≠ p. Then H^{1}(K_{∞}, A) = H^{1}(K_{∞}, X)(p) is Λ-torsion.

Proof: The first and second statements follow from Theorem 4.1 and the isomorphism (31). For 3, use the well-known exact sequence:

0 → X(K_{∞}) ⊗ ℚ_{p} / ℤ_{p} → H^{1}(K_{∞}, A) → H^{1}(K_{∞}, X)(p) → 0.

and the isomorphism (31). Note if ℓ ≠ p, X(K_{∞}) ⊗ ℚ_{p} / ℤ_{p} = 0. The last statement follows from Proposition 4.5.

□

Remark: The reason why we take K_{0} as the base field is that the Galois group G = Gal(K_{∞}/K_{0}) is a uniform group ([DSMS]) and therefore Λ(G) is an integral domain.

The results in the global situation are the following. First let us prepare the following notation (as above, r_{v} denotes the p-rank):

α_{v} = r_{v}[K_{0,v} : ℚ_{p}] for v ∈ S_{p};
α_{p} = Σ_{v∈S_{p}}α_{v};

β_{v} = (g − r_{v})[K_{0,v} : ℚ_{p}] for v ∈ S_{p};
β_{p} = Σ_{v∈S_{p}}β_{v}.

Proposition 7.3. Let K be a finite extension of ℚ, X an abelian variety over K of dimension g, and K_{∞} = K(A). Put G = Gal(K_{∞}/K_{0}) and Λ = Λ(G). Then

(1) The Λ-rank of H^{1}(G_{S}(K_{∞}), A)^{∨} is g[K_{0} : ℚ_{p}].

(2) H_{Iw}^{2}(K_{∞}, T_{p}X) is Λ-torsion. The rank of H_{Iw}^{1}(K_{∞}, T_{p}X) is also g[K_{0} : ℚ], which is torsion-free.

(3) The Λ-rank of Λ(G) ⊗_{Λ(G_{v})} H^{1}(K_{∞,v}, X)(p)^{∨} is α_{v} for any v ∈ S_{p}.

(4) rk_{Λ}(Sel(K_{∞}, A)^{∨}) − rk_{Λ}(\underleftarrow{lim} Sel(K_{n}, T_{p}X′))^{∨} = β_{p}. In particular,

rk_{Λ}(Sel(K_{∞}, A)^{∨}) ≥ Σ_{v∈S_{p}}(g − r_{v})[K_{0,v} : ℚ_{p}].

Proof: The first statement is obtained by using Theorem 3.2 with Proposition 7.1. This time we use the long exact sequence which is in the beginning of this section. H_{Iw}^{2}(K_{∞}, T_{p}A) is Λ-torsion because of Proposition 7.1 and [OV, 4.10]. The rank of H_{Iw}^{1}(K_{∞}, T_{p}A) is r_{2}(K_{0}) by Theorem 3.3 and 1. The third statement follows from Proposition 7.2. The last statement follows from the long exact sequence in the beginning of this section and Proposition 5.1. □

The following is the fundamental conjecture on the Selmer group:

Conjecture 7.4.

rk_{Λ}(Sel(K_{∞}, A)^{∨}) = Σ_{v∈S_{p}}(g − r_{v})[K_{0,v} : ℚ_{p}].

This would follow if the similar conjectures due to Mazur (ordinary case) and Schneider in the cyclotomic ℤ_{p}-case are proven for any finite extension of ℚ ([Sc2], also [CH, 2.4]). We end this paper with the projective dimension of the Selmer group of an elliptic curve without complex multiplication.

Theorem 7.5. Assume E is an elliptic curve without complex multiplication, has good reduction at any prime over p, and that the above rank conjecture holds. Let K_{∞} be K(E[p^{∞}]). Then the projective dimension of Sel(K_{∞}, E[p^{∞}])^{∨} is 2.

Proof: This follows from Proposition 5.2 and Serre’s calculation of the dimension of the Galois groups (see [Se2] and [CH]) and a result in [ST], which says that there is no bad reduction of additive type over K_{0}, and hence dim(G_{v}) = 2 or 3 maybe except the case E has supersingular reduction at v. But this case may be excluded thanks to the isomorphism (31). Hence we can say G and G_{v} satisfy the condition in Proposition 5.2 (dim(G) = 4 and 4 > dim(G_{v}) ≥ 2 for any v). Proposition 7.3.4 shows that the Conjecture implies the surjection of the map ϕ in the sequence (23), because the Conjecture with the vanishing of H^{2}(G_{S}(K_{∞}), E[p^{∞}]) implies that \underleftarrow{lim}_{n} Sel(K_{n}, T_{p}E) is torsion. But ⨁_{v∈S(K_{0})} H^{1}(K_{∞,v}, E[p^{∞}]) has no nontrivial torsion submodule by Proposition 4.5. Hence \underleftarrow{lim}_{n} Sel(K_{n}, T_{p}E) = 0 and therefore ϕ is surjective. □

Acknowledgements. We thank John Coates and Susan Howson for helpful discussions. Howson has independently obtained similar results which will appear in one of her forthcoming papers (see [Ho1] for the case of elliptic curves). Her results are based on an Euler characteristic formula (“homological rank”, see [Ho2]).

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Korea Institute for Advanced Study (KIAS), 207-43 Cheongryangri-dong, Dongdaemungu, Seoul 130-012, South Korea

E-mail address: ochi@kias.re.kr

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/