An Equivariant Main Conjecture in Iwasawa Theory

and the Coates-Sinnott Conjecture

Reza Taleb

Received: December 12, 2012 Revised: April 4, 2013

Communicated by Otmar Venjakob

Abstract. We formulate and prove an Equivariant Main Conjecture (EMC) for all prime numbers p under the assumptions μ = 0 and the validity of the 2-adic Main Conjecture in Iwasawa theory [47]. This equivariant version coincides with the version, which Ritter and Weiss formulated and proved for odd p under the assumption μ = 0 in [35]. Our proof combines the approach of Ritter and Weiss with ideas and techniques used by Greither and Popescu in [15] in a recent proof of an equivalent formulation of the above EMC under the same assumptions (p odd and μ = 0) as in [35]. As an application of the EMC we prove the Coates-Sinnott Conjecture, again assuming μ = 0 and the 2-adic Main Conjecture.

2010 Mathematics Subject Classification: 11R23, 11R42, 14F42, 11R70, 11R33, 11R34 Keywords and Phrases: Iwasawa theory, global and p-adic Lfunctions, motivic cohomology, algebraic K-theory, Fitting ideals

Contents

1 Introduction 750

2 Algebraic Construction 756

3 Analytic Construction 761 4 An Equivariant Main Conjecture in Iwasawa Theory 764

5 The Coates-Sinnott Conjecture as an application 772

1 Introduction

One of the most fascinating discoveries in Arithmetic Algebraic Geometry is the still mysterious relationship between certain algebraic and analytic data attached to a given arithmetic object. Classical examples include the Conjecture of Birch and Swinnerton-Dyer, which conjecturally relates the order of vanishing of the L-function attached to an elliptic curve at 1 to the rank of the algebraically defined Mordell-Weil group of the curve, and Dirichlets Analytic Class Number Formula, which gives a precise algebraic interpretation of the residue of the zeta-function of a number field at 1. This last connection has been generalised to yield interpretations of special values of zeta-functions at arbitrary negative integers in terms of algebraic K-theory and motivic cohomology. One of the main tool to understand the deep relations between algebraic and analytic objects is Iwasawa Theory. In this theory a precise formulation of such a relationship is called the Main Conjecture. We first recall the formulation of the Main Conjecture in Iwasawa theory in the classical form:

Let p be a prime number, let F be a totally real number field, and let ψ be a 1-dimensional p-adic Artin character for F with F_{ψ} totally real, where F_{ψ} denotes the fixed field of the kernel of ψ. Let F_{∞} denote the cyclotomic ℤ_{p}- extension of F. We recall Greenberg’s terminology about the different types of the characters ψ: ψ is of type S, if F_{ψ} ∩ F_{∞} = F, and ψ is of type W, if F_{ψ} ⊆ F_{∞}. Let O_{ψ} denote the ring obtained by adjoining all ψ-values to the ring ℤ_{p}. Let F_{ψ,∞} be the cyclotomic ℤ_{p}-extension of F_{ψ} with Galois group Γ over F_{ψ}. Throughout we fix a topological generator γ of Γ. We denote by S a finite set of primes of F containing the set S_{p} of primes above p and the set of the infinite primes, and by S_{f} the set of finite primes in S. Deligne and Ribet [8] and independently Cassou-Noguès [4] showed the existence of a p-adic Lfunction for the character ψ, which is continuous for s ∈ ℤ_{p} \ {1}, and even at s = 1, if ψ is not trivial. This satisfies the following interpolation property for any integer n ≥ 1:

L_{p}(1 − n, ψ) = L(1 − n, ψω^{−n}) ∏_{𝔭∈S_{p}}(1 − ψω^{−n}(𝔭)Nm(𝔭)^{1−n}).

Here L(1−n, ψω^{−n}) is the usual Artin L-function with respect to the character ψω^{−n}, where ω : F (μ_{2p}) → ℤ_{p}^{×} is the Teichmüller character. Let H_{ψ} ∈ O_{ψ}[T] be defined as ψ(γ)(T + 1) − 1 if ψ is of type W, and 1 otherwise. Deligne and Ribet showed that there exists a power series G_{ψ,S}(T) ∈ O_{ψ}[[T]] so that

L_{p}^{S}(1 − s, ψ) = \frac{G_{ψ,S}(κ(γ)^{s} − 1)}{H_{ψ}(κ(γ)^{s} − 1)},  (1.1)

Documenta Mathematica 18 (2013) 749–783

where L_{p}^{S}(1 − s, ψ) denotes the p-adic L-function with Euler factors removed at the primes in S, and κ is the restriction of the cyclotomic character to Γ. By the Weierstrass Preparation Theorem (cf. §7.1 in [16]) we have the following decomposition:

G_{ψ,S}(T) = π^{μ(G_{ψ,S})}g_{ψ,S}^{∗}(T)u_{ψ,S}(T),

where g_{ψ,S}^{∗}(T) is a distinguished polynomial in O_{ψ}[T], and u_{ψ,S}(T) is a unit power series in O_{ψ}[[T]]. This power series represents the analytic object in the Main Conjecture.

For the cyclotomic ℤ_{p}-extension F_{ψ,∞} of F_{ψ} let M_{ψ,∞}^{S} be the maximal abelian pro-p-extension of F_{ψ,∞}, which is unramified outside the primes in S, with Galois group 𝔛_{∞}^{S} := Gal(M_{ψ,∞}^{S}/F_{ψ,∞}). The pro-p-group 𝔛_{∞}^{S} is equipped with a (torsion) O_{ψ}[[Γ]]-module structure, as well as a Gal(F_{ψ}/F)-action given by inner automorphisms. Serre showed that the completed group ring O_{ψ}[[Γ]] can be identified with the one variable power series O_{ψ}[[T]], by mapping γ − 1 to T. By the Structure Theorem of Iwasawa theory (cf. §13.2 in [46]) for O_{ψ}[[T]]-modules, the ψ-eigenspace

𝔛_{∞}^{S,ψ} := {x ∈ 𝔛_{∞}^{S} ⊗_{ℤ_{p}} O_{ψ}|σ(x) = ψ(σ)x for all σ ∈ Gal(F_{ψ}/F)}

of 𝔛_{∞}^{S} is pseudo-isomorphic, as an O_{ψ}[[T]]-module, to a unique O_{ψ}[[T]]- module of the form

⨁_{i=1}^{m} O_{ψ}[[T]]/𝔭_{𝔦}^{n_{i}}

for m ≥ 1 and n_{i} ≥ 1. Here 𝔭_{𝔦} is the ideal generated by either a fixed uniformizer π ∈ O_{ψ} or a monic irreducible polynomial in O_{ψ}[T]. We call the ideal ∏_{i=1}^{m} 𝔭_{𝔦}^{n_{i}} the characteristic ideal. By the Weierstrass Preparation Theorem (cf. §7.1 in [16]) we can choose a unique generator for the characteristic ideal of the following form

F_{ψ,S}(T) = π^{μ(F_{ψ,S})}f_{ψ,S}^{∗}(T),

where f_{ψ,S}^{∗}(T) is a distinguished polynomial in O_{ψ}[T]. The polynomial F_{ψ,S}(T) is called the characteristic polynomial of 𝔛_{∞}^{S,ψ}. The classical Main Conjecture in Iwasawa theory is formulated as follows: If the character ψ is of type S, then

f_{ψ,S}^{∗}(T) = g_{ψ,S}^{∗}(T).

This was proved by Wiles in [47] for any totally real number field F and an odd prime p. He also proved the conjecture for the prime 2 and the character ψ provided F_{ψ} is an abelian extension of ℚ. He also showed the equality of the μ-invariants μ(F_{ψ,S}) = μ(G_{ψ,S}) for odd primes p and characters ψ. For odd primes p, both invariants μ(F_{ψ,S}) and μ(G_{ψ,S}) are known to be zero in the case F_{ψ}/ℚ is abelian (cf. [10]), and are conjectured to be zero in general.

The O_{ψ}[[T]]-torsion module 𝔛_{∞}^{S,ψ} is of projective dimension at most one and has a principal Fitting ideal generated by the characteristic polynomial F_{ψ,S}(T) (see for instance Lemma 2.4 in [32]). Therefore another formulation of the Main Conjecture for odd primes reads as follows: If the character ψ is of type S, then

F itt_{O_{ψ}[[T]]}(𝔛_{∞}^{S,ψ}) = (G_{ψ,S}(T)).

To obtain a similar formulation of the Main Conjecture in terms of ideals for the prime 2, we replace 𝔛_{∞} by 𝔛_{∞}^{f}, where 𝔛_{∞}^{f} := 𝔛_{∞}^{S_{f}} is the Galois group of the abelian pro-p-extension of F_{ψ,∞} unramified outside the primes in S_{f}, over F_{ψ,∞}. For an odd prime p the two O_{ψ}[[T]]-modules are the same, since infinite primes are unramified in p-extensions for p odd. However, for p = 2 the two modules are related by Lemma 5.9, which shows that they may differ in their μ-invariants. If we assume that 𝔛_{∞}^{f} has trivial μ-invariant, then the analogous formulation of the Main Conjecture in terms of ideals for the prime 2 reads as follows:

F itt_{O_{ψ}[[T]]}(𝔛_{∞}^{f,ψ}) = (G_{ψ,S}^{∗}(T)),  (1.2)

where G_{ψ,S}^{∗}(T) = g_{ψ,S}^{∗}(T)u_{ψ,S}(T). We note that for odd primes p this version is the same as before provided the μ-invariant of 𝔛_{∞}^{f} vanishes.

As we see, for any character ψ of an abelian extension E/F of totally real number fields, the Main Conjecture gives an equality of ideals over the power series ring O_{ψ}[[T]] in the form (1.2)under the assumption that the μ-invariant is vanishing. If we denote by G the Galois group of E/F, then the G-equivariant formulation over the Iwasawa algebra ℤ_{p}[G][[T]] is the so-called Equivariant Main Conjecture in Iwasawa theory. Ritter and Weiss formulated such a conjecture in [35] for any odd prime p, and proved it under the assumption of the vanishing of a certain Iwasawa μ-invariant. To explain this, we need the following set-up:

Let E/F be an abelian extension of totally real number fields with Galois group G, and let E_{∞} be the cyclotomic ℤ_{p}-extension of E. Let S be a finite set of primes of F containing the primes which ramify in E_{∞}, and infinite primes, and let S_{f} be the set of finite primes of S as before. Let M_{∞}^{S} denote the maximal abelian pro-p-extension of E_{∞}, unramified outside the primes in S, and let 𝔛_{∞} := 𝔛_{∞}^{S} = Gal(M_{∞}^{S}/E_{∞}). We denote by G_{∞} the Galois group of E_{∞}/F, by H the Galois group of E_{∞}/F_{∞}, and by 𝔸 the completed group ring ℤ_{p}[[G_{∞}]]. In this article the assumption μ = 0 refers to the following assumption:

μ = 0 : The μ-invariant of 𝔛_{∞}^{f} := 𝔛_{∞}^{S_{f}} is zero, i.e. the ℤ_{p}-module

𝔛_{∞}^{f} is finitely generated.

(1.3)

We note that 𝔛_{∞}^{f} maps by Galois restriction to the Galois group Gal(M_{ψ,∞}^{S_{f}}/F_{ψ,∞}), where M_{ψ,∞}^{S_{f}} is the maximal abelian pro-p-extension of F_{ψ,∞} unramified outside the primes in S_{f}, with a finite cokernel for any character ψ of G. Here the cokernel is Gal(E_{∞} ∩ M_{ψ,∞}^{S_{f}}/F_{ψ,∞}) which is a quotient of a subgroup of H, and whence finite. As a result the assumption μ = 0 implies that the μ-invariant of Gal(M_{ψ,∞}^{S_{f}}/F_{ψ,∞}) is zero, and therefore the μ-invariant of 𝔛_{∞}^{f,ψ} vanishes for any (even) character ψ of G.

The pro-p group 𝔛_{∞} has a (torsion) 𝔸-module structure, whose projective dimension is not necessarily at most one. However to formulate an Equivariant Main Conjecture, similar to the classical Main Conjecture, one needs a finitely generated 𝔸-torsion module of projective dimension at most one. Let d_{∞} be a non-zero divisor of the augmentation ideal ΔG_{∞} of 𝔸, let c_{∞} be an invertible element of the total ring of fraction of 𝔸 so that d_{∞} = c_{∞}((γ − 1)e + (1 − e)), where e is the idempotent attached to the trivial character of H. We denote by L the fixed field of E/F under the action of the p-Sylow subgroup of G, by G the Galois group of the maximal algebraic extension Ω_{L}^{S} of L unramified outside the primes in S, over F, and by H the Galois group of Ω_{L}^{S}/E_{∞}. There is a commutative diagram of 𝔸-modules

0  0
↓  ↓
𝔸 = 𝔸
↓ Ψ  ↓ ψ
0 → 𝔛_{∞} → Y_{∞} → ΔG_{∞} → 0
|| ↓ ↓
0 → 𝔛_{∞} → Z_{∞} → z_{∞} → 0
↓  ↓
0  0,

where ψ maps 1 to d_{∞}, Ψ maps 1 to a pre-image y_{∞} of d_{∞}, and Y_{∞} = H_{0}(H, ΔG). Here ΔG denotes the augmentation ideal of ℤ_{p}[[G]]. We will see that this definition of Y_{∞} is the same as the definition of Ritter-Weiss in [35]. The 𝔸-torsion module Z_{∞} in the diagram above, whose projective dimension is at most one, shows up as the algebraic object in the Equivariant Main Conjecture of Ritter-Weiss. We note that the construction of Z_{∞} depends on the choice of d_{∞}. Before stating the algebraic object we remark that there exists a subgroup Γ ≤ G_{∞}, topologically generated by γ, so that G_{∞} = H × Γ for the abelian group G_{∞}. The analytic object is defined as follows:

G_{S} := ∑_{ψ∈Ĥ} G_{ψ,S}(γ − 1) · e_{ψ} ∈ \frac{1}{|H|}O[H][[Γ]],

where e_{ψ} is the idempotent attached to the character ψ of H, i.e.

e_{ψ} := \frac{1}{| H |} ∑_{σ∈G} ψ(σ)σ^{−1}.

One version of the Equivariant Main Conjecture of Ritter-Weiss [35] for odd primes is as follows (cf. [28], §2, (CPE2)):

F itt_{𝔸}(Z_{∞}) = (c_{∞}G_{S}),

which was verified under the assumption of the vanishing of the μ-invariant of 𝔛_{∞}, i.e. assuming that 𝔛_{∞} is a finitely generated ℤ_{p}-module. It is worth mentioning that they have generalized and proved their Equivariant Main Conjecture in the non-commutative case, still assuming the vanishing of the μ-invariant of a certain Iwasawa module (see [36]). In [15], Greither and Popescu have recently formulated and proved an Equivariant Main Conjecture in Iwasawa theory in the abelian case in terms of the Tate module of a certain Iwasawa-theoretic abstract 1-motive again under the assumptions μ = 0 and p odd. More recently, Nickel [29] showed that this formulation is equivalent to the formulation of Ritter-Weiss.

We now describe our Equivariant Main Conjecture for an arbitrary prime p. For an abelian extension E/F, by applying the algebraic construction of the Equivariant Main Conjecture of Ritter-Weiss to the set S_{f} of finite primes in S, we construct the 𝔸-torsion module Z_{∞}^{f}, which is of projective dimension at most one. We show that it satisfies the following exact sequence:

0 → ℤ_{p} → (𝔸/d_{∞}𝔸)^{#} → α(Z_{∞}^{f}) → α(𝔛_{∞}^{f}) → 0,

in which

pd_{𝔸}((𝔸/d_{∞}𝔸)^{#}) ≤ 1 and pd_{𝔸}(α(Z_{∞}^{f})) ≤ 1.

Here α is the adjoint functor in Iwasawa theory with the contravariant action and M^{#}, for any 𝔸-module M, denotes the same underlying module but with g acting as g^{−1} for any g ∈ G_{∞}. The Equivariant Main Conjecture is then formulated as follows (cf. Conjecture 4.1):

F itt_{𝔸}(Z_{∞}^{f}) = (c_{∞}G_{S}^{∗}),

where

G_{S}^{∗} := ∑_{ψ∈Ĥ} G_{ψ,S}^{∗}(γ − 1) · e_{ψ} ∈ \frac{1}{|H|}O[H][[Γ]].

Here we recall that G_{ψ,S}^{∗}(T) = g_{ψ,S}^{∗}(T)u_{ψ,S}(T). In Section 4, we prove that this conjecture follows from the classical Main Conjecture under the assumption μ = 0 by taking advantage of the idea of determinantal ideals used by Greither and Popescu [15].

In the last section we show that the Coates-Sinnott Conjecture follows from the Equivariant Main Conjecture assuming μ = 0 (cf. Theorem 5.10). After some fundamental work of Coates-Sinnott in [6] and more recent results by Ritter-Weiss, Nguyen Quang Do, Burns-Greither, Greither-Popescu et al. the Coates-Sinnott Conjecture is completely known up to powers of 2, assuming μ = 0. However, the 2-primary information was neglected more or less completely due to various technical problems. For example, there was no formulation of an Equivariant Main Conjecture in Iwasawa theory for the prime 2 at the time.

The Coates-Sinnott Conjecture is a generalization of the classical Stickelberger Theorem, which provides elements annihilating the class group of a cyclotomic field, using special values of analytic functions. To make it more precise, let E/F be an abelian extension with Galois group G, and let S be a finite set of primes in F containing the primes ramified in E and the infinite primes. Let

Θ_{E/F}^{S} (s) := ∑_{χ∈Ĝ} L_{E/F}^{S} (s, χ^{−1}) · e_{χ} ∈ ℂ[G]

be the S-incomplete equivariant L-function, where e_{χ} is the idempotent attached to any character χ of G. Deligne and Ribet [8] and independently Cassou-Noguès [4] proved that

Ann_{ℤ[G]}(H^{0}(E, ℚ/ℤ(n))) · Θ_{E/F}^{S} (1 − n) ⊂ ℤ[G]

for any integer n ≥ 1. Stickelberger’s Theorem shows that the following analytic object is in the annihilator ideal of the class group Cl(O_{E}) of the field E in the case F = ℚ:

Ann_{ℤ[G]}(H^{0}(E, ℚ/ℤ(1))) · Θ_{E/F}^{S} (0) ⊆ Ann_{ℤ[G]}(Cl(O_{E})).

This setup has been generalized in two directions: First of all one looks at an arbitrary relative abelian extension E/F of number fields. Here the analogue of Stickelberger’s theorem (Brumer’s Conjecture) is still not completely known. In a different direction one replaces the class group by algebraic K-groups or motivic cohomology groups and studies annihilators of these groups as Galois modules for relative abelian extensions. In [6], Coates and Sinnott formulated the relevant conjecture in terms of higher Quillen Kgroups as

Ann_{ℤ[G]}(H^{0}(E, ℚ/ℤ(n))) · Θ_{E/F}^{S} (1 − n) ⊆ Ann_{ℤ[G]}(K_{2n−2}(O_{E}))

for any integer n ≥ 2. As a result of the recent work of Voevodsky in [45] the relation between algebraic K-theory, étale cohomology for all prime numbers and motivic cohomology is known. This yields the motivic formulation of the Coates-Sinnott Conjecture, which implies the K-theoretic version. Moreover, it enables us to study each p-primary part of the conjecture separately for any prime number p as follows:

Ann_{ℤ_{p}[G]}(H^{0}(E, ℚ_{p}/ℤ_{p}(n))) · Θ_{E/F}^{S} (1 − n) ⊆ Ann_{ℤ_{p}[G]}(H_{ét}^{2}(O′_{E}, ℤ_{p}(n)))

for any integer n ≥ 2, where H_{ét}^{2}(O′_{E}, ℤ_{p}(n)) = \underleftarrow{lim}_{m} H_{ét}^{2}(O′_{E}, μ_{p^{m}}^{⊗n}) and H_{ét}^{∗}(O′_{E}, μ_{p^{m}}^{⊗n}) refers to the étale cohomology of the scheme Spec(O_{E}[1/p]) with values in the étale sheaf μ_{p^{m}}^{⊗n}. In the last section we complete the proof of the Coates-Sinnott Conjecture under the assumption μ = 0 by proving it for the prime 2.

Acknowledgement. The material in this article forms a part of the author’s Ph.D. thesis [43] at McMaster university. The author would like to express his gratitude to his Ph.D. supervisor Manfred Kolster for introducing these subjects, for his encouragement and helpful discussions and suggestions. The author would also like to thank the referee for his detailed comments and suggestions.

2 Algebraic Construction

Let E/F be a finite abelian extension of totally real number fields with Galois group G, and let p be an arbitrary prime. Let E_{∞} (resp. F_{∞}) be the cyclotomic ℤ_{p}-extension of E (resp. F). We denote the multiplicative group Gal(E_{∞}/E) (resp. Gal(F_{∞}/F)) by Γ_{E} (resp. Γ_{F}). Let H denote the Galois group of the finite abelian extension E_{∞}/F_{∞}. We denote by G_{∞} := Gal(E_{∞}/F) the Galois group of the abelian extension E_{∞}/F. We let S denote a finite set of primes of F which ramify in E_{∞}, and the infinite primes. In particular S contains the set S_{p} of the primes above p. The set of finite primes in S is also denoted by S_{f}. We use the same notations for the set of primes above the primes in S and S_{f}, respectively, in any intermediate field of E_{∞}/F. Since Γ_{F} is topologically generated by one element, the exact sequence

0 → H → G_{∞} ⇆ Γ_{F} → 0  (2.1)

splits. We denote by Γ ≤ G_{∞} the image of Γ_{F}, so that G_{∞} ≃ H × Γ, and by Λ the completed group ring ℤ_{p}[[Γ]]. Let E′ be the fixed field of E_{∞} under the action of the closed subgroup Γ. Then E′ ∩ F_{∞} = F, E_{∞} = E′ · F_{∞}, Gal(E′/F) ≃ H and E_{∞}/E′ is also a cyclotomic ℤ_{p}-extension.

Let M_{∞}^{S} and M_{∞}^{S_{f}} be the maximal abelian pro-p-extensions of E_{∞} unramified outside the primes in S and S_{f}, respectively. We recall that 𝔛_{∞} = 𝔛_{∞}^{S} and 𝔛_{∞}^{f} := 𝔛_{∞}^{S_{f}} denote the Galois group of the extensions M_{∞}^{S}/E_{∞} and M_{∞}^{S_{f}} /E_{∞}, respectively. Since E is totally real, the Λ-module 𝔛_{∞} is a torsion module with no non-trivial finite submodule by Propositions 10.3.22 and 10.2.25 in [31]. The Λ-module 𝔛_{∞}^{f}, which is a quotient of 𝔛_{∞}, is also torsion and has no nontrivial finite submodule (cf. [38], §6.4). Finally, we set 𝔸 := ℤ_{p}[[G_{∞}]] and we freely use the identification

𝔸 ≃ ℤ_{p}[H][[T]],  (2.2)

which is given by mapping the topological generator γ of Γ to 1 + T.

Remark 2.1. For an odd prime p, infinite primes of F are unramified in a pro-p-extension. Hence M_{∞}^{S} and M_{∞}^{S_{f}} coincide and therefore, 𝔛_{∞} = 𝔛_{∞}^{f} for odd primes.

The following diagram illustrates the situation:

[figure: M_{∞}^{S_{f}}; 𝔛_{∞}^{f}; E_{∞}; Γ_{E}; Γ; H; G_{∞}; F_{∞}; Γ_{F}; E; G; E′; H; F]

We recall that the Main Conjecture in Iwasawa theory for a character ψ of G can be written in the form (1.2), assuming μ = 0. Here the Fitting ideal of the (finitely generated) O_{ψ}[[T]]-torsion module 𝔛^{f,ψ} is principal, because it has projective dimension at most one, and is generated by the p-adic Lfunction associated to ψ. Hence to formulate an Equivariant Main Conjecture we construct an appropriate (finitely generated) 𝔸-torsion module of projective dimension at most one. The resulting Fitting ideal is then principal, and conjecturally generated by an equivariant p-adic L-function. This was done by Ritter-Weiss for odd primes in [35]. The strategy of this part is as follows: Since the 𝔸-torsion module 𝔛_{∞}^{f} is not necessarily of projective dimension at most one, we first construct an 𝔸-module Y_{∞}^{f} of projective dimension at most one. Since this module is not necessarily 𝔸-torsion, we pass to a quotient Z_{∞}^{f} of Y_{∞}^{f}, which is then shown to be a (finitely generated) 𝔸-torsion module of projective dimension at most one, whose principal Fitting ideal is conjecturally generated by an equivariant L-function.

Let P be the p-Sylow subgroup of G and let L be the fixed field of E under the action of P with Galois group Q over F. Let Ω_{L}^{S_{f}} be the maximal algebraic pro-p-extension of L, which is unramified outside the primes in S_{f}. We denote by H the Galois group of Ω_{L}^{S_{f}} over E_{∞}, and by G the Galois group of Ω_{L}^{S_{f}} over F. The finitely generated group G has a presentation of the form G ≃ F/W, where F is an appropriate free profinite group of rank d and W is a relation subgroup of F of rank r. For a certain relation subgroup R of F we then have an isomorphism G_{∞} ≃ F/R. The following diagram illustrates the situation:

[figure: Ω_{L}^{S_{f}}; H; G; E_{∞}; Γ_{E}; G_{∞}; E; P; L; Q; F]

We apply Proposition 5.6.7 in [31] (see also Lemma 4.3 in [17]) to the profinite groups in the commutative diagram

1  1
↓  ↓
W = W
↓  ↓
1 → R → F → G_{∞} → 1
↓ ↓ ||
1 → H → G → G_{∞} → 1
↓  ↓
1  1,

and obtain a commutative diagram:

0  0
↓  ↓
0 → H_{2}(H, ℤ_{p}) → H_{0}(H, W^{ab}) → R^{ab}(p) → 𝔛_{∞}^{f} → 0
|| || ↓ ↓
0 → H_{2}(H, ℤ_{p}) → H_{0}(H, W^{ab}) → 𝔸^{d} → Y_{∞}^{f} → 0
↓  ↓
ΔG_{∞} = ΔG_{∞}
↓  ↓
0  0,

(2.3)

where ΔG_{∞} denotes the augmentation ideal of 𝔸. Here Y_{∞}^{f} := H_{0}(H, ΔG) and ΔG and ΔG_{∞} denote the augmentation ideals of G and G_{∞}, respectively.

Remark 2.2. The same construction leads to a similar diagram for an arbitrary intermediate field of Ω_{L}^{S_{f}} /L.

Since the cyclotomic ℤ_{p}-extension E_{∞}/E satisfies the weak Leopoldt Conjecture by Proposition 10.3.25 in [31], the group H_{2}(H, ℤ_{p}) in diagram (2.3)vanishes (cf. Proposition 10.3.22 in [31]). Moreover we have the following proposition:

Proposition 2.3. The 𝔸-module H_{0}(H, W^{ab}) is projective.

Proof. Since |Q| is prime to p, we have the equality of cohomological dimensions

cd_{p}(G) = cd_{p}(Gal(Ω_{L}^{S_{f}} /L)).

The p-cohomological dimension of the pro-p group Gal(Ω_{L}^{S_{f}} /L) is at most 2 by Proposition 8.3.17 in [31] for odd primes p (note that infinite primes are unramified in any p-extension for p odd), and by Theorem 1 in [38] for p = 2, i.e.

cd_{p}(G) ≤ 2

for any prime p. Now Proposition 5.6.7 in [31] completes the proof.

Let χ be a ℂ_{p}-valued character of the group Q, let

e_{χ} := \frac{1}{| Q |} ∑_{σ∈Q} χ(σ)σ^{−1}

be the idempotent of Q attached to the character χ, and let 𝔸_{χ} := O_{χ}[[G_{∞}(p)]], where O_{χ} is the ring obtained by adjoining all character values of χ to ℤ_{p}. Since G_{∞}(p) ≃ P × Γ is a pro-p group, 𝔸_{χ} is a local ring and therefore, e_{χ}H_{0}(H, W^{ab}) is a free 𝔸_{χ}-module of rank r_{χ}:

e_{χ}H_{0}(H, W^{ab}) ≃ 𝔸_{χ}^{r_{χ}}.  (2.4)

Now by applying e_{χ} to the exact sequence in the second row of diagram (2.3) we obtain:

0 → 𝔸_{χ}^{r_{χ}} → 𝔸_{χ}^{d} → e_{χ}Y_{∞}^{f} → 0.

From the last column of diagram (2.3)we have

0 → e_{χ}𝔛_{∞}^{f} → e_{χ}Y_{∞}^{f} → 𝔸_{χ} → e_{χ}ℤ_{p} → 0.

This implies that e_{χ}Y_{∞}^{f} has rank one and as a result r_{χ} = d−1 for any character χ. Now by taking the direct sum over all characters of Q in equality (2.4)we obtain:

H_{0}(H, W^{ab}) ≃ 𝔸^{r}

for r = d − 1. Therefore, diagram (2.3)can be rewritten as

\begin{aligned}& & & & 0 & & 0 \\ & & & & ↓ & & ↓ \\ 0 & → & 𝔸^{r} & →^{f} & R^{ab}(p) & → & 𝔛_{∞}^{f} & → & 0 \\ & & || & & ↓ & & ↓ \\ 0 & → & 𝔸^{r} & →^{f} & 𝔸^{r+1} & → & Y_{∞}^{f} & → & 0 \\ & & & & ↓ & & ↓ \\ & & & & ΔG_{∞} & = & ΔG_{∞} \\ & & & & ↓ & & ↓ \\ & & & & 0 & & 0. \end{aligned}  (2.5)

So far we have constructed the module Y_{∞}^{f}, which fits into the exact sequence

0 → 𝔛_{∞}^{f} → Y_{∞}^{f} → ΔG_{∞} → 0.  (2.6)

The second row in diagram (2.5)implies that the 𝔸-module Y_{∞}^{f} is of projective dimension at most one. To replace it by a torsion 𝔸-module we now take a quotient of Y_{∞}^{f} by a certain submodule as follows:

Let d_{∞} ∈ ΔG_{∞} be a non-zero divisor in the augmentation ideal of 𝔸 and let c_{∞} be an invertible element in Q(𝔸) such that

d_{∞} = c_{∞}((γ − 1)e + (1 − e)),  (2.7)

where γ is the fixed (topological) generator of Γ ≤ G_{∞} and e = \frac{1}{|H|} ∑_{h∈H} h is the idempotent of ℚ_{p}[H] attached to the trivial character of H. Here we note that γ −1 and 1−e generate ΔG_{∞} ⊗ℚ_{p}, and that γ −1 and 1−e can be written in the form (2.7)as follows:

γ − 1 = (e + (γ − 1)(1 − e))((γ − 1)e + (1 − e)),
1 − e = (1 − e)((γ − 1)e + (1 − e)).

Let y_{∞} be a pre-image of d_{∞} in Y_{∞}^{f} in diagram (2.5). We have the following diagram:

\begin{aligned}& & & & 0 & & 0 \\ & & & & ↓ & & ↓ \\ & & & & 𝔸 & = & 𝔸 \\ & & & & ↓ Φ & & ↓ ϕ \\ 0 & → & 𝔛_{∞}^{f} & → & Y_{∞}^{f} & → & ΔG_{∞} & → & 0 \\ & & || & & ↓ & & ↓ \\ 0 & → & 𝔛_{∞}^{f} & → & Z_{∞}^{f} & → & z_{∞}^{f} & → & 0 \\ & & & & ↓ & & ↓ \\ & & & & 0 & & 0, \end{aligned}  (2.8)

where Φ and ϕ are defined by mapping 1 ∈ 𝔸 to y_{∞} and to d_{∞}, respectively, and Z_{∞}^{f} and z_{∞}^{f} are the quotients of Y_{∞}^{f} and ΔG_{∞} by the images of Φ and ϕ, respectively. We note that the vertical maps are injective since d_{∞} ∈ 𝔸 is a non-zero divisor. By a diagram chase in the diagram

𝔸 = 𝔸
↓ ϕ  ↓ ϕ
0 → ΔG_{∞} → 𝔸 → ℤ_{p} → 0,

we obtain:

Lemma 2.4. The sequence

0 → z_{∞}^{f} → 𝔸/d_{∞}𝔸 → ℤ_{p} → 0  (2.9)

is exact, where the middle term is of projective dimension one and

F itt_{𝔸}(𝔸/d_{∞}𝔸) = (d_{∞}).

By using the middle column of diagram (2.8)and the first row of diagram (2.5) we also obtain the commutative diagram

\begin{aligned}& & 0 & & 0 & & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & 𝔸^{r} & → & 𝔸^{r+1} & → & 𝔸 & → & 0 \\ & & ↓ & & ↓ Ψ & & ↓ ϕ \\ 0 & → & R^{ab}(p) & → & 𝔸^{r+1} & → & ΔG_{∞} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & 𝔛_{∞}^{f} & → & Z_{∞}^{f} & → & z_{∞}^{f} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ & & 0 & & 0 & & 0, \end{aligned}  (2.10)

which implies the following proposition:

Proposition 2.5. Z_{∞}^{f} is a finitely generated 𝔸-torsion module of projective dimension at most one:

pd_{𝔸}(Z_{∞}^{f}) ≤ 1.

3 Analytic Construction

Again let E/F be an abelian extension of totally real fields, let S be a finite set of primes in F containing the primes above p, the primes ramified in E and the infinite primes, and let S_{f} denote the set of finite primes in S. As before, we denote by E_{∞} and F_{∞} the cyclotomic ℤ_{p}-extensions of E and F, respectively, by H the Galois group of E_{∞}/F_{∞}, and by G_{∞} ≃ H × Γ the Galois group of E_{∞}/F. We define equivariant versions of G_{ψ,S} and H_{ψ} as follows (cf. [34], Proposition 5.4): For a character ψ of G_{∞}, let G_{ψ,S}(T), H_{ψ}(T) ∈ O_{ψ}[[T]] be the power series defined in (1.1). Let

\begin{matrix}G_{S} := ∑_{ψ∈Ĥ} G_{ψ,S}(γ − 1) · e_{ψ} ∈ \frac{1}{|H|}O[H][[Γ]] \\ H_{S} := ∑_{ψ∈Ĥ} H_{ψ}(γ − 1) · e_{ψ} ∈ \frac{1}{|H|}O[H][[Γ]] \end{matrix}  (3.1)

be the equivariant versions of G_{ψ,S} and H_{ψ}. For any character χ of G_{∞}, they satisfy the following:

χ(G_{S}) = G_{χ,S}(0) , χ(H_{S}) = H_{χ}(0).

We recall that for any character ψ of G one has

G_{ψ,S}(T) = π^{μ(G_{ψ,S})} · g_{ψ,S}^{∗}(T) · u_{ψ,S}(T)

by the Weierstrass Preparation Theorem, where π is a fixed uniformizer in O_{ψ}, g_{ψ,S}^{∗}(T) ∈ O_{ψ}[T] is a distinguished polynomial, and u_{ψ,S}(T) ∈ O_{ψ}[[T]] is a unit. The modified equivariant L-function G_{S}^{∗} is now defined as follows:

G_{S}^{∗} := ∑_{ψ∈Ĥ} G_{ψ,S}^{∗}(γ − 1) · e_{ψ} ∈ \frac{1}{|H|}O[H][[Γ]],  (3.2)

where G_{ψ,S}^{∗}(T) = g_{ψ,S}^{∗}(T)u_{ψ,S}(T). The following lemma relates G_{S} and G_{S}^{∗}, assuming μ = 0 (cf. (1.3)):

Lemma 3.1. Under the assumption μ = 0 we have the following equalities:

1. G_{S} = G_{S}^{∗} for any odd prime p.

2. G_{S} = 2^{r_{1}(F)}G_{S}^{∗} for p = 2.

Proof. Part 1 follows from the result of Wiles [47] that μ(G_{ψ,S}) is the same as the Iwasawa μ-invariant of 𝔛_{∞}^{f,ψ} for all odd primes p. Part 2 follows from the fact that π^{μ(G_{ψ,S})} = 2^{r_{1}(F)} under the assumption μ = 0, where π ∈ O_{ψ} is a uniformizer, for any character ψ of G (see [12], pages 82 and 87).

We note that Lemma 3.1 holds unconditionally for abelian extensions E of ℚ, since in this case μ = 0 (cf. [10]).

To prove the next lemma we briefly review the definition of a p-adic pseudomeasure on a certain Galois group and its relation to the p-adic L-function. For more properties one can consult [39]. For a commutative profinite group G, λ_{S} ∈ Q(ℤ_{p}[[G]]) is called a pseudo-measure on G if (g−1)λ_{S} is a measure, i.e. (g − 1)λ_{S} ∈ ℤ_{p}[[G]], for any g ∈ G, where Q(ℤ_{p}[[G]]) denotes the quotient ring of ℤ_{p}[[G]]. Let 𝔛 denote the Galois group of the maximal abelian extension of F unramified outside the primes in S_{f}, over F. By a theorem of Deligne and Ribet there is a unique pseudo-measure on 𝔛 denoted by λ_{S} ∈ Q(ℤ_{p}[[𝔛]]), which satisfies the following relation for any finite order character χ of 𝔛:

L_{p,S}(1 − s, χ) =< χκ^{s}, λ_{S} > .

For the definition of this inner product see [39]. Equivalently, if we let ε : 𝔛 → ℤ_{p} be the locally constant function defined by ε(g) = 1 if g has image 1 in H = Gal(E_{∞}/F_{∞}), and zero otherwise, then

ζ_{p}^{S}(ε_{h}, 1 − n) =< ε_{h}ρ^{n}, λ_{S} > .

Here ρ is the cyclotomic character, ε_{h} is the locally constant function satisfying ε_{h}(x) = ε(hx) and ζ_{p}^{S}(ε_{h}, s) is the S-incomplete p-adic partial zeta function associated to ε_{h}. The image of λ_{S} under the natural surjection π : 𝔛 ↠ G_{∞} is a p-adic pseudo-measure on G_{∞} which is denoted by θ_{S} ∈ Q(𝔸). So if γ̂ ∈ 𝔛 denotes a pre-image of γ ∈ G_{∞} under the surjection above, then

Z_{S} := π((γ̂ − 1)λ_{S}) ∈ 𝔸.

In fact θ_{S} = (γ − 1)^{−1}Z_{S} ∈ Q(𝔸). With notations as above we have the following lemma:

Lemma 3.2. Let d_{∞} be a non-zero divisor in the augmentation ideal ΔG_{∞}, and let c_{∞} be an invertible element in Q(𝔸) so that d_{∞} = c_{∞}((γ − 1)e + (1 − e)) ∈ ΔG_{∞}, where e is the idempotent associated to the trivial character of H. Then

c_{∞}G_{S} = d_{∞}θ_{S} ∈ 𝔸

Proof. A calculation in Proposition 12 in [35], which works for any character χ of G_{∞} satisfying χ(γ) = 1 and for any prime p,provides the following equality:

\frac{G_{χ,S}(T)}{T^{<χ,1>}} = ∑_{h∈H} χ(h)\frac{Z_{S}(h, T)}{T}

Here Z_{S}(h, T) is given by the relation Z_{S} = ∑_{h∈H} Z_{S}(h, γ − 1)h ∈ 𝔸. As a result,

G_{S}/H_{S} = ∑_{χ∈Ĥ} \frac{G_{χ,S}(T)}{T^{<χ,1>}}e_{χ} = θ_{S},

for any prime p. Since H_{S} = (γ−1)e+(1−e), we obtain c_{∞}G_{S} = d_{∞}θ_{S}. Hence, for the p-adic pseudo-measure θ_{S} on G_{∞}, we have c_{∞}G_{S} = d_{∞}θ_{S} ∈ 𝔸.

Let Ẽ = E(ζ_{2p}) be the field obtained by adjoining a primitive 2p-th root of unity ζ_{2p} to E, and let Ẽ_{∞} := E_{∞}(ζ_{2p}) = E(μ_{p^{∞}}) be the ℤ_{p}-cyclotomic extension of Ẽ, where μ_{p^{∞}} is the group of all p-power roots of unity. We denote by G̃_{∞} the Galois group of Ẽ_{∞}/F. Since Ẽ_{∞} contains all p-power roots of unity, we have the cyclotomic character

ρ : G̃_{∞} → ℤ_{p}^{∗} = Aut(μ_{p^{∞}})

of G̃_{∞}. We extend the definitions of a Tate twisted module and an inverse module to the following:

• Let t_{n} be the unique continuous isomorphism of O-algebras

t_{n} : O[[G̃_{∞}]] → O[[G̃_{∞}]],  (3.3)

which satisfies t_{n}(g) = ρ(g)^{n} · g for all g ∈ G̃_{∞} and n ∈ ℤ. For a O[[G̃_{∞}]]- module M let the Tate twisted module M(n) be the same underlying group M with a new O[[G̃_{∞}]]-action given by σ ∗_{n} m := t_{n}(σ)m for σ ∈ O[[G̃_{∞}]] and m ∈ M. For even n, we note that it is enough to replace Ẽ_{∞} by its maximal real subfield Ẽ_{∞}^{+} to have the isomorphisms t_{n}.

• Let ι be the unique continuous isomorphism of O-algebras

ι : O[[G_{∞}]] → O[[G_{∞}]],  (3.4)

which satisfies ι(g) = g^{−1} for all g ∈ G_{∞}. For a O[[G_{∞}]]-module M, let the inverse module M^{#} be the same underlying group M with a new O[[G_{∞}]]-action given by σ ∗ m := ι(σ)m for σ ∈ O[[G_{∞}]] and m ∈ M. In the following we mean by the ideal generated by m^{#}, for any m ∈ O[[G_{∞}]], the inverse ideal (m)^{#} of (m).

Lemma 3.3. Assume that E is the maximal real subfield of Ẽ = E(ζ_{2p}) where ζ_{2p} is a primitive 2p-th root of unity. For all even n we have:

\begin{matrix}(ι ∘ t_{n})(G_{S}) = ∑_{ψ∈Ĥ} G_{ψ^{−1}ω^{n},S}(u^{n}(γ)^{−1} − 1) · e_{ψ}, \\ (ι ∘ t_{n})(H_{S}) = ∑_{ψ∈Ĥ} H_{ψ^{−1}ω^{n},S}(u^{n}(γ)^{−1} − 1) · e_{ψ}, \end{matrix}

where u = κ(γ).

Proof. Under the assumptions of the lemma we first note that E_{∞} is the maximal real subfield of the field E_{∞}(ζ_{2p}) which contains all p-power roots of unity. So the actions of t_{n} on G_{S} and H_{S} are defined for all even n. Now it suffices to observe that (ι ∘ t_{n})(γ − 1) = u^{n}γ^{−1} − 1 and (ι ∘ t_{n})(e_{ψ}) = e_{ψ^{−1}ω^{n}}.

This lemma yields the following equality:

(π ∘ ι ∘ t_{n})G_{S}/H_{S} = ∑_{χ∈Ĝ} \frac{G_{χ^{−1}ω^{n},S}(u^{n} − 1)}{H_{χ^{−1}ω^{n},S}(u^{n} − 1)} · e_{χ},

where π : 𝔸 → ℤ_{p}[G] is the projection mapping γ − 1 to zero, and u = κ(γ). Therefore, we obtain:

Corollary 3.4. If we assume that E is the maximal real subfield of Ẽ = E(ζ_{2p}) where ζ_{2p} is a primitive 2p-th root of unity, and that n is even, then

(π ∘ ι ∘ t_{n})G_{S}/H_{S} = Θ_{E/F}^{S} (1 − n)

Remark 3.5. We note that in the non-dyadic L-functions we have defined, the set S can be replaced by S_{f}, since infinite primes have no influence on the definitions.

4 An Equivariant Main Conjecture in Iwasawa Theory

We recall that for the abelian extension E/F of totally real number fields, E_{∞} (resp. F_{∞}) is the cyclotomic ℤ_{p}-extension of E (resp. F), H = Gal(E_{∞}/F_{∞}), and G_{∞} = Gal(E_{∞}/F), which is abelian and hence of the form G_{∞} = H × Γ for Γ ≃ ℤ_{p}. We also recall that d_{∞} = c_{∞}((γ − 1)e + (1 − e)) is a non-zero divisor in the augmentation ideal of 𝔸 = ℤ_{p}[[G_{∞}]] so that 𝔸/d_{∞}𝔸 is a finitely generated ℤ_{p}-free module, e.g. d_{∞} = γ − 1.

Conjecture 4.1. (The Equivariant Main Conjecture). With notations as above, we have the following equality of ideals in 𝔸:

F itt_{𝔸}(Z_{∞}^{f}) = (c_{∞}G_{S}^{∗})

Remark 4.2. For any odd prime p, the formulation of the Equivariant Main Conjecture 4.1 is equivalent to the formulation of Ritter-Weiss in [35] (cf. [28], §2, (CPE2)). We note that Ritter-Weiss use the translation functor to define Y_{∞}, which equals to Y_{∞} = \frac{ΔG}{Δ(G,H)ΔG}. Here Δ(G, H) = ker(ℤ_{p}[[G]] ↠ ℤ_{p}[[G_{∞}]]), which is the same as (ΔH)ℤ_{p}[[G]] (cf. for example page 275 in [31]). As a result, Y_{∞} = \frac{ΔG}{ΔHΔG} = H_{0}(H, ΔG), which is the same as the definition used in this article.

Under the assumption μ = 0 we prove that Conjecture 4.1 follows from the classical Main Conjecture in Iwasawa theory [47]. For some technical reasons, we need to apply the contravariant functors E^{i}(−) := Ext_{𝔸}^{i}(−, 𝔸) to Z_{∞}^{f}, for i = 0, 1. We will see that E^{1}(Z_{∞}^{f}) is a finitely generated 𝔸-torsion module of projective dimension at most one, whose Fitting ideal is generated by the modified equivariant L-function. First, we recall the definition and some basic properties of Fitting ideals (see [30] for more properties).

For a commutative ring R with identity, the Fitting ideal F itt_{R}(M) of a finitely presented R-module M is defined as follows: Given a presentation of M as

R^{a} →^{h} R^{b} → M → 0,

let A be the matrix associated to the map h. The (initial) Fitting ideal of M is defined to be the ideal of R generated by all b-minors of A if a ≥ b, and (0) otherwise. Here are some properties:

1. F itt_{R}(M) is a finitely generated ideal of R satisfying

(Ann_{R}(M))^{b} ⊆ F itt_{R}(M) ⊆ Ann_{R}(M),

where Ann_{R}(M) is the annihilator ideal of M and b is an integer so that M can be generated by b elements as a R-module.

2. If M ↠ M′ is a surjective map of finitely presented R-modules, then

F itt_{R}(M) ⊆ F itt_{R}(M′).

3. If 0 → M′ → M → M′′ → 0 is an exact sequence of finitely presented R-modules, then

F itt_{R}(M′) · F itt_{R}(M′′) ⊆ F itt_{R}(M).

Moreover, we have equality if the exact sequence splits, i.e.

F itt_{R}(M′ ⊕ M′′) = F itt_{R}(M′) · F itt_{R}(M′′).

4. If M ≃ R/𝔞 is a cyclic module, then

F itt_{R}(M) = Ann_{R}(M) = 𝔞.

More generally, if we apply the previous property to a direct sum of n cyclic R-modules

M ≃ R/𝔞_{1} ⊕ R/𝔞_{2} ⊕ ⋯ ⊕ R/𝔞_{n},

then we obtain

F itt_{R}(M) = 𝔞_{1}𝔞_{2} ⋯ 𝔞_{n}.

5. Let M be a finite R-module for a group ring R of a finite abelian group with coefficients in ℤ_{p}. If M is cyclic as a ℤ_{p}-module, then

F itt_{R}(M^{∗}) = Ann_{R}(M^{∗}) ≃ Ann_{R}(M) = F itt_{R}(M),

where M^{∗} = Hom(M, ℚ_{p}/ℤ_{p}) is the Pontryagin dual equipped with the covariant action.

We recall that # denotes the inverse action defined in (3.4). In the next lemma we list some general properties of E^{i}(M) for an 𝔸-module M. For a proof see propositions 5.4.17, 5.5.6 and corollary 5.5.7 in [31], or [17].

Lemma 4.3. Let M be an 𝔸-module, let α(M) denote the adjoint of M with the contravariant action, and let M^{∨} = Hom_{ℤ_{p}}(M, ℤ_{p}) be the dual with the contravariant G_{∞}-action. Then

1. E^{i}(M) = Ext_{Λ}^{i}(M, Λ) as Λ-modules for any 𝔸-module M and i ≥ 0,

2. E^{1}(M)^{#} ≃ α(M) as Λ-modules, provided M is a finitely generated Λ-torsion module,

3. E^{1}(M)^{#} ≃ M^{∨} as Λ-modules, provided M is a Λ-torsion module with trivial μ-invariant, i.e. M is a finitely generated ℤ_{p}-module.

We list some results obtained by applying the contravariant functors E^{i}(−) for i = 1, 2, to some of the exact sequences arising from diagram (2.8). We first remark that E^{0}(−) = Hom_{𝔸}(−, 𝔸) is a left exact functor, and that E^{i}(𝔸) = 0 for i ≥ 1, since pd_{𝔸}(𝔸) = 0 (cf. Proposition 5.2.11 in [31]).

Lemma 4.4. The 𝔸-module E^{1}(Z_{∞}^{f})^{#} is of projective dimension at most one, and

F itt_{𝔸}(E^{1}(Z_{∞}^{f})^{#}) = F itt_{𝔸}(Z_{∞}^{f})^{#}.

Proof. This lemma is a consequence of Proposition 2 in [13] and Lemma 4.3. We give here a direct proof, since we need some of the methods in the proof of Lemma 4.5 below: We first apply E^{i}(−) to the last column of diagram (2.8). We observe that Hom_{𝔸}(z_{∞}^{f}, 𝔸) is the set of all morphisms in Hom_{𝔸}(ΔG_{∞}, 𝔸), whose restriction to d_{∞}𝔸 vanishes. This observation and the choice of d_{∞} ∈ ΔG_{∞} as a non-zero divisor imply that E^{0}(z_{∞}^{f}) = 0. By part 1 of Lemma 4.3 E^{0}(𝔛_{∞}^{f}) is also trivial for the Λ-torsion module 𝔛_{∞}^{f}. Hence by applying the contravariant functor E^{i}(−) to the last row of diagram (2.8), we obtain E^{0}(Z_{∞}^{f}) = 0. On the other hand E^{i}(𝔸) is trivial for i ≥ 1 as we mentioned before. Therefore, applying E^{i}(−) to the middle column of diagram (2.10) leads to the exact sequence

0 → E^{0}(𝔸)^{r+1} →^{E^{0}(Ψ)} E^{0}(𝔸)^{r+1} → E^{1}(Z_{∞}^{f}) → 0.

Here E^{0}(Ψ) is the transpose of Ψ. We now apply “#” to the exact sequence above to obtain:

0 → 𝔸^{r+1} →^{E^{0}(Ψ)^{#}} 𝔸^{r+1} → E^{1}(Z_{∞}^{f})^{#} → 0,

which shows that the projective dimension of the 𝔸-module E^{1}(Z_{∞}^{f})^{#} is at most one. To complete the proof it is enough to note that the Fitting ideal of the 𝔸-module E^{1}(Z_{∞}^{f})^{#} is given by the determinant of E^{0}(Ψ)^{#}, whereas the Fitting ideal of Z_{∞}^{f} is given by the determinant of the map Ψ defined in diagram (2.10).

Lemma 4.5. We have the following exact sequence of finitely generated 𝔸-torsion modules:

0 → ℤ_{p} → (𝔸/d_{∞}𝔸)^{#} → E^{1}(z_{∞}^{f})^{#} → 0.

Proof. We saw in the proof of Lemma 4.4 that E^{0}(z_{∞}^{f}) is trivial. As a consequence of Proposition 5.2.11 in [31] we obtain that E^{2}(ℤ_{p}) is trivial as well, since the projective dimension of the Λ-module ℤ_{p} is one. By applying E^{i}(−) to the exact sequence (2.9)we therefore obtain the exact sequence

0 → E^{1}(ℤ_{p}) → E^{1}(𝔸/d_{∞}𝔸) → E^{1}(z_{∞}^{f}) → 0.

By part 2 of Lemma 4.3 together with the fact that an elementary module is isomorphic to the inverse module of its adjoint (cf. 1.3 in [16]) we have E^{1}(ℤ_{p})^{#} ≃ ℤ_{p}, where ℤ_{p} has the trivial G_{∞}-action. We also have the isomorphism E^{1}(𝔸/d_{∞}𝔸)^{#} ≃ (𝔸/d_{∞}𝔸)^{#}, since pd_{𝔸}(𝔸) = 0. Therefore, by applying “#” to the exact sequence above, we obtain:

0 → ℤ_{p} → (𝔸/d_{∞}𝔸)^{#} → E^{1}(z_{∞}^{f})^{#} → 0.

Lemma 4.6. We have the following exact sequence of finitely generated 𝔸-torsion modules:

0 → E^{1}(z_{∞}^{f}) → E^{1}(Z_{∞}^{f}) → E^{1}(𝔛_{∞}^{f}) → 0.

Proof. First we observe that E^{2}(z_{∞}^{f}) is trivial by applying E^{i}(−) to the exact sequence (2.9)and by noting that the projective dimensions of ℤ_{p} and 𝔸/d_{∞}𝔸 are both one. Now we apply E^{i}(−) to the last row of diagram (2.8)to obtain the exact sequence above. We note that the surjectivity of the last map in the diagram follows from the observation that E^{2}(z_{∞}^{f}) = 0, and that the injectivity of the first map in the diagram is a consequence of the observation that E^{0}(𝔛_{∞}^{f}) = 0 in the proof of lemma (4.4).

We combine Lemmas 4.5 and 4.6 to obtain the following theorem:

Theorem 4.7. We have the following exact sequence of finitely generated 𝔸-torsion modules:

0 → ℤ_{p} → (𝔸/d_{∞}𝔸)^{#} → E^{1}(Z_{∞}^{f})^{#} → E^{1}(𝔛_{∞}^{f})^{#} → 0,

in which

pd_{𝔸}((𝔸/d_{∞}𝔸)^{#}) ≤ 1 and pd_{𝔸}(E^{1}(Z_{∞}^{f})^{#}) ≤ 1.

The Λ-module E^{1}(𝔛_{∞}^{f})^{#} is isomorphic to the adjoint of 𝔛_{∞}^{f} by part 2 of Lemma 4.3 and so it is a finitely generated ℤ_{p}-free module under the assumption μ = 0. Therefore we obtain from Theorem 4.7 the first statement in the following proposition:

Proposition 4.8. Let d_{∞} ∈ ΔG_{∞} be a non-zero divisor so that 𝔸/d_{∞}𝔸 is a finitely generated ℤ_{p}-free module. Then, under the assumption μ = 0, the sequence

0 → ℤ_{p} → (𝔸/d_{∞}𝔸)^{#} → E^{1}(Z_{∞}^{f})^{#} → E^{1}(𝔛_{∞}^{f})^{#} → 0

is an exact sequence of finitely generated ℤ_{p}-free modules. Moreover, if we consider this sequence as an exact sequence of ℤ_{p}[H]-modules, then under the assumptions of μ = 0 we have

pd_{ℤ_{p}[H]}((𝔸/d_{∞}𝔸)^{#}) = 0 and pd_{ℤ_{p}[H]}(E^{1}(Z_{∞}^{f})^{#}) = 0.

Proof. We only need to prove the second part. We first remark that 𝔸/d_{∞}𝔸 and E^{1}(Z_{∞}^{f}) are both H-cohomologically trivial by Proposition 2.2 in [32], since their projective dimensions are at most one as 𝔸-modules. By a classical theorem of Nakayama pd_{ℤ_{p}[H]}(M) = 0 if and only if M is ℤ_{p}-free and Hcohomologically trivial (p-adic version of Theorem 8 in [40], Chapter IX, §5). Therefore, both modules are of projective dimension zero as ℤ_{p}[H]-modules.

To finish the proof that the Equivariant Main Conjecture follows from the classical Main Conjecture under the assumption μ = 0, we first review the definition of the determinantal ideal, which plays a role similar to that of the characteristic ideal for some Λ-modules with an extra group action: For a commutative ring R with identity, a finitely generated projective R-module P and f ∈ End_{R}(P), the determinant of f is defined as

det_{R}(f | P) := det_{R}(f ⊕ id_{Q} | P ⊕ Q),

where Q is a complement of P, i.e. P ⊕ Q is free. One can check that the definition is independent of Q by using Schanuel’s lemma. By the same strategy, since P ⊗_{R} R[X] is a finitely generated projective R[X]-module, the monic polynomial det_{R}(X − f | P) ∈ R[X] is defined to be

det_{R}(X − f | P) := det_{R[X]}(id_{P} ⊗ X − f ⊗ 1 | P ⊗_{R} R[X])

for any projective R-module P. One can see that these definitions are wellbehaved under base-change, i.e.

\begin{matrix}det_{R}(f | P) = det_{R′}(f ⊗ id_{R′} | P ⊗ R′) \\ det_{R}(X − f | P) = det_{R′}(X − (f ⊗ id_{R′}) | P ⊗ R′), \end{matrix}  (4.1)

where R′ is any commutative R-algebra. We have the following general proposition:

Proposition 4.9. ([15], Proposition 7.2). Let R be a commutative, semi-local, compact topological ring and let Γ be a pro-cyclic group with topological generator γ. Let M be a topological R[[Γ]]-module, which is projective and finitely generated as an R-module. Let

F (X) := det_{R}(X − m_{γ} | M),

where m_{γ} is the R[[Γ]]-module automorphism of M given by multiplication by γ. Then the following holds:

1. M is finitely presented as an R[[Γ]]-module. If we let F (γ) be the image of F (X) via the R-algebra morphism R[X] → R[[Γ]] sending X to γ, we have an equality of R[[Γ]]-ideals

F itt_{R[[Γ]]}(M) = (F (γ)).

2. If we view M_{R}^{∨} = Hom_{R}(M, R) as a topological R[[Γ]]-module with the covariant Γ-action, then

F itt_{R[[Γ]]}(M) = F itt_{R[[Γ]]}(M_{R}^{∨}).

3. If we view M^{∨} = Hom_{ℤ_{p}}(M, ℤ_{p}) as a topological R[[Γ]]-module with the covariant Γ-action, where R = ℤ_{p}[G] and G is a finite abelian group, then

F itt_{R[[Γ]]}(M) = F itt_{R[[Γ]]}(M^{∨}).

By using Proposition 4.9 for the ring R = ℤ_{p}[H] and the finitely generated R-modules M = E^{1}(Z_{∞}^{f})^{#} and M = (𝔸/d_{∞}𝔸)^{#}, which are projective by Proposition 4.8, we obtain:

Lemma 4.10. If m_{γ} denotes the R[[Γ]]-module automorphism of M given by multiplication by γ, then

F itt_{𝔸}(E^{1}(Z_{∞}^{f})^{#}) = (det_{ℤ_{p}[H]}((T + 1) − m_{γ} | E^{1}(Z_{∞}^{f})^{#})),
F itt_{𝔸}((𝔸/d_{∞}𝔸)^{#}) = (det_{ℤ_{p}[H]}((T + 1) − m_{γ} | (𝔸/d_{∞}𝔸)^{#})).

Let O be the ring of integers obtained by adjoining all character values of the characters of H to ℤ_{p}, let π be a fixed uniformizer in O and let Q(O) denote the field of fractions of O. We consider O and Q(O) as 𝔸-modules with trivial G_{∞}- action. We note that for the idempotent e attached to the trivial character of H we have H_{S}(T) = T · e + (1 − e) using the identification (2.2)(see (3.1)for the definition of H_{S}(T)). Therefore, using Lemma 4.10 we obtain the following lemma:

Lemma 4.11. We have the following equalities of ideals in Q(O)[H][[Γ]]:

(H_{S}(T)) = (det_{Q(O)[H]}((T + 1) − m_{γ} | Q(O))),
(d_{∞}^{#}) = (det_{Q(O)[H]}((T + 1) − m_{γ} | (Q(O)[H][[Γ]]/d_{∞})^{#})).

Remark 4.12. Any character χ of H can be extended to a Q(O)[X]-algebra homomorphism, for a variable X, and to a Q(O) ⊗_{O} O[[Γ]]-algebra homomorphism

χ : Q(O)[H][X] → Q(O)[X],
χ : Q(O) ⊗_{O} O[[G_{∞}]] → Q(O) ⊗_{O} O[[Γ]],

which maps h → χ(h) for h ∈ H.

Lemma 4.13. We have the following equality of ideals in O[[Γ]] under the assumption μ = 0:

\begin{aligned}(det_{Q(O)}((T + 1) − m_{γ} & | e_{ψ}(𝔛_{∞}^{f} ⊗_{ℤ_{p}} Q(O))))^{#} \\ & = (det_{Q(O)}((T + 1) − m_{γ} | e_{ψ}(E^{1}(𝔛_{∞}^{f})^{#} ⊗_{ℤ_{p}} Q(O))))). \end{aligned}

Proof. Since the R := O_{ψ}[[Γ]]-torsion module M_{1} := e_{ψ}(𝔛_{∞}^{f} ⊗_{ℤ_{p}} Q(O)) is of projective dimension at most one (cf. for example Lemma 2.4 in [32]), the Rtorsion module M_{2} := e_{ψ}(E^{1}(𝔛_{∞}^{f})^{#} ⊗_{ℤ_{p}} Q(O)) is also of projective dimension at most one. Hence by the first part of Proposition 4.9 it is enough to show that F itt_{R}(M_{1})^{#} = F itt_{R}(M_{2}). But this is a direct consequence of the second part of Lemma 4.3 and Proposition 2 in [13].

At this point we use the classical Main Conjecture in Iwasawa theory [47], i.e. the equality of fractional ideals

(G_{ψ,S}^{∗}(T)) = (det_{Q(O)}((T + 1) − m_{γ} | e_{ψ}(𝔛_{∞}^{f} ⊗_{ℤ_{p}} Q(O))))

in Q(O)[[T]] for any character ψ of H (see equality (1.2)and part 1 of Proposition 4.9). Here we note that the assumption μ = 0 implies the vanishing of the μ-invariants of 𝔛_{∞}^{f,ψ} for all (even) characters ψ of H. Hence, using Lemma 4.13, we obtain the following equality of fractional ideals in Q(O)[[T]]:

(ψ(G_{S}^{∗}(T)^{#})) = (G_{ψ,S}^{∗}(T)^{#})
= (det_{Q(O)}((T + 1) − m_{γ} | e_{ψ}(E^{1}(𝔛_{∞}^{f})^{#} ⊗_{ℤ_{p}} Q(O))))
= (ψ(det_{Q(O[H])}((T + 1) − m_{γ} | E^{1}(𝔛_{∞}^{f})^{#} ⊗_{ℤ_{p}} Q(O)))).

Consequently the exact sequence

0 → ℤ_{p} → (𝔸/d_{∞}𝔸)^{#} → E^{1}(Z_{∞}^{f})^{#} → E^{1}(𝔛_{∞}^{f})^{#} → 0

in Theorem 4.7, and Lemma 4.10 together with the base-change property of determinants (cf. (4.1)) imply the following equality of ideals in O[[T]]:

ψ((c_{∞}G_{S}^{∗}(T))^{#}) = (ψ(det_{Q(O[H])}((T + 1) − m_{γ} | E^{1}(Z_{∞}^{f})^{#} ⊗_{ℤ_{p}} Q(O)))). (4.2)

Before completing the proof we recall that the μ-invariant μ(F) of a power series F ∈ O[[T]] is the largest exponent μ ≥ 0 such that f ∈ (π^{μ})O[[T]]. For F ∈ 𝔸 we define the μ-invariant of F to be zero if μ(χ(F)) = 0 for any p-adic valued character χ of H.

Let F := det_{ℤ_{p}[H]}((T + 1) − m_{γ} | E^{1}(Z_{∞}^{f})^{#}) and G := (c_{∞}G_{S}^{∗}(T))^{#} in 𝔸 (cf. Lemma 3.2). Then

• μ(F) = 0, since the determinantal polynomial F ∈ ℤ_{p}[H][[T]] of the projective ℤ_{p}[H]-module E^{1}(Z_{∞}^{f})^{#} (cf. Proposition 4.8) is monic.

• μ(G) = 0, since for any p-adic character ψ of the group H, we have

μ(ψ(G_{S}^{∗})) = μ(g_{ψ,S}^{∗} · u_{ψ,S}) = 0 , μ(ψ(H_{S})) = 0 , μ(ψ(d_{∞})) = 0.

Here we note that the determinantal polynomials of the field Q(O) and the projective ℤ_{p}[H]-module (𝔸/d_{∞}𝔸)^{#} (cf. Proposition 4.8), which are generated by H_{S} and d_{∞}^{#}, respectively (cf. Lemma 4.11), are monic.

• (ψ(F)) = (ψ(G)), using equality (4.2).

In the terminology of [2], R := ℤ_{p}[H] is admissible for the abelian group H, i.e. R is a finite product of strictly admissible rings R_{i}, which means that each R_{i} is separated and complete in the rad(R_{i})-adic topology and also R_{i}/rad(R_{i}) is a skew field. Since the μ-invariants of F, G ∈ R[[T]] are both zero, Proposition 2.1 in [2] as an equivariant Weierstrass Preparation Theorem implies the existence of unique distinguished polynomials f^{∗}, g^{∗} ∈ R[T] and units u, v ∈ (R[[T]])^{×} such that

F = u · f^{∗} and G = v · g^{∗}.

We apply a p-adic character ψ of H to both sides, and note that ψ(f^{∗}) and ψ(g^{∗}) are both distinguished polynomials in O[T], and that ψ(u), ψ(v) ∈ O[[T]]^{×} are units. Hence the equality (ψ(F)) = (ψ(G)) together with the uniqueness of the Weierstrass decomposition yields

ψ(f^{∗}) = ψ(g^{∗})

for any p-adic character ψ of H. Therefore, f^{∗} = g^{∗} and F = uv^{−1}G. The equality (F) = (G) now implies the following:

(ι(c_{∞}G_{S}^{∗})) = (det_{ℤ_{p}[H]}((T + 1) − m_{γ} | E^{1}(Z_{∞}^{f})^{#}))
= F itt_{𝔸}(E^{1}(Z_{∞}^{f})^{#}) by Lemma 4.10
= F itt_{𝔸}(Z_{∞}^{f})^{#} by Lemma 4.4.

(4.3)

Consequently, the equality

F itt_{𝔸}(Z_{∞}^{f}) = (c_{∞}G_{S}^{∗})

holds, and this completes the proof of the Equivariant Main Conjecture 4.1 under the assumptions of the classical Main Conjecture and of μ = 0.

Theorem 4.14. The Equivariant Main Conjecture 4.1 follows from the classical Main Conjecture in Iwasawa theory under the assumption μ = 0.

Remark 4.15. For any odd prime p, or for the prime 2 if F is an absolutely abelian number field, the classical Main Conjecture holds, and hence the Equivariant Main Conjecture 4.1 is verified under the assumption μ = 0.

We recall that the assumption μ = 0 holds for any absolute abelian number field E, i.e. for any number field E whose Galois group over ℚ is abelian, by the result of Ferrero and Washington in [10]. Hence

Corollary 4.16. If E is an absolute abelian number field, the Equivariant Main Conjecture 4.1 holds unconditionally.

5 The Coates-Sinnott Conjecture as an application

Let E/F be an abelian extension of number fields with Galois group G, let n ≥ 2 be an integer, and let p be an arbitrary prime. Let S be a finite set of primes in F containing the primes above p, the primes ramified in E and the infinite primes, and let S_{f} denote the set of all finite primes in S. Let

Θ_{E/F}^{S} (s) = ∑_{χ∈Ĝ} L_{E/F}^{S} (s, χ^{−1}) · e_{χ}

be the G-equivariant S-incomplete L-function associated to E/F. We recall that for an integer n ≥ 1 by a result of Siegel [41]

Θ_{E/F}^{S} (1 − n) ∈ ℚ[G],

and furthermore, by Deligne and Ribet [8] or by Cassou-Noguès [4]

Ann_{ℤ[G]}(H^{0}(E, ℚ/ℤ(n))) · Θ_{E/F}^{S} (1 − n) ⊂ ℤ[G].

For n ≥ 1 the n-th higher Stickelberger ideal is defined as follows:

Stick_{E/F}^{S} (n) := Ann_{ℤ[G]}(H^{0}(E, ℚ/ℤ(n))) · Θ_{E/F}^{S} (1 − n) ⊂ ℤ[G].

Remark 5.1. The classical theorem of Stickelberger states that

Stick_{E/ℚ}^{S}(1) ⊆ Ann_{ℤ[G]}(Cl(O_{E}),

where Cl(O_{E}) denotes the class group of O_{E}. Brumer conjectured that the same holds for any abelian extension E/F of number fields.

The original formulation of the Coates-Sinnott Conjecture is as follows:

Conjecture 5.2. (The Coates-Sinnott Conjecture, K-theoretic version). Let E/F be an abelian Galois extension of number fields with Galois group G, and let n ≥ 2. Then

Stick_{E/F}^{S} (n) ⊆ Ann_{ℤ[G]}(K_{2n−2}(O_{E})).

As a consequence of the work of Voevodsky (cf. [45]) the Quillen-Lichtenbaum Conjecture holds, i.e. for all odd primes p the étale Chern characters defined by Soulé [42]

ch_{i,n}^{(p)} : K_{2n−i}(O_{F}) ⊗ ℤ_{p} → H_{ét}^{i}(O′_{F} , ℤ_{p}(n)) are isomorphisms for i = 1, 2, and all n ≥ 2. Here O′_{F} = O_{F} [1/p]. The surjectivity of the Chern characters was proved by Soulé for even n (see [42]) and by Dwyer and Friedlander in general (see [7]).

For the prime 2 the situation is in general different. The deviation between K_{2n−i}(O_{F}) ⊗ ℤ_{2} and H_{ét}^{i}(O′_{F} , ℤ_{2}(n)) has been determined by Rognes and Weibel [37]. In [21] it was suggested to replace the K-groups K_{2n−2}(O_{F}) by the motivic cohomology groups H_{M}^{2}(E, ℤ(n)), because the latter groups have the advantage that their p-parts are isomorphic to H_{ét}^{2}(O′_{E}, ℤ_{p}(n)) for all primes p ([19], Theorem 2.4). This leads to the following motivic version of the Coates-Sinnott Conjecture:

Conjecture 5.3. (The Coates-Sinnott Conjecture, motivic version). Let E/F be an abelian Galois extension of number fields with Galois group G, and let n ≥ 2. Then

Stick_{E/F}^{S} (n) ⊆ Ann_{ℤ[G]}(H_{M}^{2}(E, ℤ(n))).

The explicit results of Rognes-Weibel show that the motivic version implies the K-theoretic version. Moreover, the validity of the motivic version is equivalent to the validity of the following p-adic version for all primes p:

Conjecture 5.4. (The Coates-Sinnott Conjecture, p-adic version). Let E/F be an abelian Galois extension of number fields with Galois group G, let p be prime, and let n ≥ 2. Then

Ann_{ℤ_{p}[G]}(H^{0}(E, ℚ_{p}/ℤ_{p}(n))) · Θ_{E/F}^{S} (1 − n) ⊆ Ann_{ℤ_{p}[G]}(H_{ét}^{2}(O′_{E}, ℤ_{p}(n))).

We remark that by the functional equation of L-functions (see for example [1]) L_{E/F} (s, χ) vanishes at negative integers 1 − n for n ≥ 2, unless F is a totally real number field and χ(−1) = (−1)^{n}. Therefore only the following cases are of interest:

• E is a totally real number field and n ≥ 2 is even.

• E is a totally complex number field, F is totally real and n ≥ 2 is odd.

We will consider the first case, namely that E/F is an abeian extension of totally real fields with Galois group G, and that n ≥ 2 is even. We show that in this case the p-adic version of the Coates-Sinnott Conjecture follows from the Equivariant Main Conjecture for all primes p assuming μ = 0. For odd primes this has been done by Nguyen Quang Do [28] and independently by Greither-Popescu [15] (see also [2] for a slightly weaker result).

We first assume without loss of generality for the proof of the Coates-Sinnott Conjecture that E is the maximal real subfield of E(ζ_{2p}) (this assumption clearly holds for p = 2, and for odd primes p one can see for instance Lemma 6.14 in [15]). Here ζ_{2p} is a primitive 2p-th root of unity. We recall the set up from Section 2: Let E_{∞} (resp. F_{∞}) be the cyclotomic ℤ_{p}-extension of E (resp. F) with Galois group Γ_{E} (resp. Γ_{F}) over E (resp. over F). We denote by G_{∞} the Galois group of E_{∞}/F, by H the Galois group of E_{∞}/F_{∞}, and by Γ = \overline{< γ >} the image of Γ_{F} under the splitting map in (2.1). The following diagram illustrates the situation:

[figure: E_{∞}; Γ_{E}; H; G_{∞}; F_{∞}; Γ_{F}; E; G; F]

Since G_{∞} is abelian, G_{∞} = H × Γ and the completed group ring 𝔸 = ℤ_{p}[[G_{∞}]] is identified with ℤ_{p}[H][[T]] under the identification (2.2). We let d_{∞} ∈ ΔG_{∞} be a non-zero divisor so that 𝔸/d_{∞}𝔸 is a finitely generated ℤ_{p}-free module, e.g. d_{∞} = γ − 1. By Theorem 4.7 we obtain an exact sequence of finitely generated 𝔸-torsion modules

0 → ℤ_{p} → (𝔸/d_{∞}𝔸)^{#} → E^{1}(Z_{∞}^{f})^{#} → E^{1}(𝔛_{∞}^{f})^{#} → 0  (5.1)

in which the middle terms are of projective dimensions at most one. Using the equalities in (4.3)we also have

\begin{matrix}F itt_{𝔸}((𝔸/d_{∞}𝔸)^{#}(n)) = ((ι ∘ t_{n})(d_{∞})) \\ F itt_{𝔸}(E^{1}(Z_{∞}^{f})^{#}(n)) = ((ι ∘ t_{n})(c_{∞}G_{S}^{∗})). \end{matrix}  (5.2)

We note that the sequence (5.1)is also an exact sequence of finitely generated Λ-modules, where Λ = ℤ_{p}[[Γ]]. Moreover, the exact sequence (5.1)is an exact sequence of finitely generated ℤ_{p}-free modules since we have assumed μ = 0.

Lemma 5.5. Let G_{E}^{S_{f}} be the Galois group of the maximal algebraic pro-p-extension of E unramified outside the primes above S_{f}, over E. Under the assumption that E is the maximal real subfield of E(ζ_{2p}), we have the following:

1. H^{0}(E, ℚ_{p}/ℤ_{p}(n)) ≃ ℤ_{p}(n)_{Γ_{E}} for n ≥ 2

2. H^{2}(G_{E}^{S_{f}} , ℤ_{p}(n)) ≃ (E^{1}(𝔛_{∞}^{f})^{#}(n))_{Γ_{E}} for even n ≥ 2 (under the hypothesis μ = 0)

Proof. 1. It is enough to take the Γ_{E}-invariants and Γ_{E}-coinvariants of the exact sequence 0 → ℤ_{p}(n) → ℚ_{p}(n) → ℚ_{p}(n)/ℤ_{p}(n) → 0 to get a 6 term exact sequence in which ℚ_{p}(n)_{Γ_{E}} = ℚ_{p}(n)^{Γ_{E}} = 0 for n ≥ 2. We note that H^{0}(E, ℚ_{p}/ℤ_{p}(n)) = ℚ_{p}/ℤ_{p}(n)^{Γ_{E}}.

2. Let E_{∞} be the cyclotomic ℤ_{p}-extension of E, with Galois group Γ_{E}. As before we denote by Ω_{E}^{S_{f}} the maximal algebraic pro-p-extension of E unramified outside the primes above S_{f}. We have the following isomorphisms:

E^{1}(𝔛_{∞}^{f})^{#}(n)_{Γ_{E}} ≃ Hom(𝔛_{∞}^{f}, ℤ_{p})(n)_{Γ_{E}}  By part 3 in Lemma 4.3
≃ Hom(𝔛_{∞}^{f}(−n), ℤ_{p})_{Γ_{E}}
≃ Hom(𝔛_{∞}^{f}(−n)_{Γ_{E}}, ℚ_{p}/ℤ_{p}) By Lemma 5.18 in [14]
≃ Hom(𝔛_{∞}^{f}, ℚ_{p}/ℤ_{p}(n))^{Γ_{E}}
≃ Hom(Gal(Ω_{E}^{S_{f}} /E_{∞}), ℚ_{p}/ℤ_{p}(n))^{Γ_{E}}.

If we assume that

Hom(𝔛_{∞}^{f}, ℚ_{p}/ℤ_{p}(n)) ≃ H^{1}(Gal(Ω_{E}^{S_{f}} /E_{∞}), ℚ_{p}/ℤ_{p}(n)),  (5.3)

then - using the fact that cd_{p}(Γ_{E}) = 1 - we can continue the isomorphisms above as follows:

H^{1}(Gal(Ω_{E}^{S_{f}} /E_{∞}), ℚ_{p}/ℤ_{p}(n))^{Γ_{E}} ≃ H^{1}(Gal(Ω_{E}^{S_{f}} /E), ℚ_{p}/ℤ_{p}(n))
≃ H^{2}(G_{E}^{S_{f}} , ℤ_{p}(n)),

where the last isomorphism follows from the finiteness of the groups H^{1}(G_{E}^{S_{f}} , ℚ_{p}/ℤ_{p}(n)) and H^{2}(G_{E}^{S_{f}} , ℤ_{p}(n)) for even n and the totally real field E (cf. Corollary 2.5 in [20] and Proposition 2.3 in [44]).

Hence, to complete the proof it is enough to show that the claim (5.3)is true. Clearly

Hom(𝔛_{∞}^{f}, ℚ_{p}/ℤ_{p}(n)) ≃ Hom(Gal(Ω_{E}^{S_{f}} /E_{∞}), ℚ_{p}/ℤ_{p}(n))

since 𝔛_{∞}^{f} is the abelianization of Gal(Ω_{E}^{S_{f}} /E_{∞}). Now we notice that the Galois group Gal(Ω_{E}^{S_{f}} /E_{∞}) acts trivially on ℚ_{p}/ℤ_{p}(n), since n is even, and therefore

Hom(Gal(Ω_{E}^{S_{f}} /E_{∞}), ℚ_{p}/ℤ_{p}(n)) ≃ H^{1}(Gal(Ω_{E}^{S_{f}} /E_{∞}), ℚ_{p}/ℤ_{p}(n))

At this point we recall the following lemma from Iwasawa theory (See Lemma 6.3 in [5], where a special case is proved, or [31], Chapter V, §3, Ex. 3):

Lemma 5.6. Let M be an O[[Γ]]-torsion module, where O is a finite extension of ℤ_{p}, and let F (T) be the characteristic polynomial of M. The following are equivalent:

1. M^{Γ} is finite.

2. M_{Γ} is finite.

3. F (0) ≠ 0

If these conditions hold, then

\frac{|M^{Γ|}}{|M_{Γ}|} = |F (0)|_{v} = p^{−f·v(F (0))},

where v is the normalized valuation, i.e. v(π) = 1 for a uniformizer π ∈ O, and f is the residue degree of π over p.

As a consequence of Lemma 5.5 we see that (E^{1}(𝔛_{∞}^{f})^{#}(n))_{Γ_{E}} and ℤ_{p}(n)_{Γ_{E}} are both finite and so by Lemma 5.6 (E^{1}(𝔛_{∞}^{f})^{#}(n))^{Γ_{E}} and ℤ_{p}(n)^{Γ_{E}} are both trivial for even n ≥ 2. We note that both ℤ_{p}(n) and E^{1}(𝔛_{∞}^{f})^{#}(n) have no non-trivial finite Λ-submodules. For E^{1}(𝔛_{∞}^{f})^{#}(n) this follows from the fact that by Lemma 4.3 E^{1}(𝔛_{∞}^{f})^{#} is isomorphic to the adjoint of 𝔛_{∞}^{f} and as such it does not have any non-trivial finite Λ-submodules (see [16], Section 1.3). Moreover, ((𝔸/d_{∞}𝔸)^{#}(n))^{Γ_{E}} = 0, since for n ≥ 2 we have κ(γ)^{n} ≠ 1. Therefore, ((𝔸/d_{∞}𝔸)^{#}(n))_{Γ_{E}} is again finite by Lemma 5.6. As a result the Γ_{E}- coinvariants of E^{1}(Z_{∞}^{f})^{#}(n) are also finite and similarly (E^{1}(Z_{∞}^{f})^{#}(n))^{Γ_{E}} = 0, for any even n ≥ 2. Hence by taking the Γ_{E}-coinvariants of the exact sequence (5.1)we obtain the following exact sequence of finite ℤ_{p}[G]-modules for any even n ≥ 2:

\begin{aligned}0 → H^{0}(E, ℚ_{p}/ℤ_{p}(n)) → ((𝔸/d_{∞} & 𝔸)^{#}(n))_{Γ_{E}} → \\ & (E^{1}(Z_{∞}^{f})^{#}(n))_{Γ_{E}} → H^{2}(G_{E}^{S_{f}} , ℤ_{p}(n)) → 0, \end{aligned} ; (5.4)

where the two middle ℤ_{p}[G]-modules are of projective dimension at most one as a consequence of the last part of Theorem 4.7 and the facts that ℤ_{p}[G_{∞}]^{Γ_{E}} = 0 and ℤ_{p}[G_{∞}]_{Γ_{E}} = ℤ_{p}[G]. Furthermore, following the equalities in (5.2), we have

F itt_{ℤ_{p}[G]}(((𝔸/d_{∞}𝔸)^{#}(n))_{Γ_{E}}) = ((π ∘ ι ∘ t_{n})(d_{∞})),
F itt_{ℤ_{p}[G]}((E^{1}(Z_{∞}^{f})^{#}(n))_{Γ_{E}}) = ((π ∘ ι ∘ t_{n})(c_{∞}G_{S}^{∗}),

where π : 𝔸 → ℤ_{p}[G] is the projection mapping γ − 1 to 0.

Now we take advantage of the following Proposition due to Burns-Greither, which relates the Fitting ideals of the modules of a 4-term exact sequence under some assumptions:

Proposition 5.7. ([2], Lemma 5) Let R := ℤ_{p}[G], for a finite abelian group G and a prime number p. Assume that we have an exact sequence of finite R-modules

0 → A → P → P′ → A′ → 0.

Further, assume that pd_{ℤ_{p}[G]}P ≤ 1 and pd_{ℤ_{p}[G]}P′ ≤ 1. Then, we have

F itt_{R}(A^{∗}) · F itt_{R}(P′) = F itt_{R}(A′) · F itt_{R}(P),

where the Pontryagin dual A^{∗} := Hom(A, ℚ_{p}/ℤ_{p}) is endowed with the covariant G-action.

Applying Proposition 5.7 to the exact sequence (5.4)of finite ℤ_{p}[G]-modules yields the following equality:

\begin{matrix}F itt_{ℤ_{p}[G]}(H^{0}(E, ℚ_{p}/ & ℤ_{p}(n))^{∗})F itt_{ℤ_{p}[G]}((E^{1}(Z_{∞}^{f})^{#}(n))_{Γ_{E}}) \\ = F itt_{ℤ_{p}[G]}((𝔸/d_{∞}𝔸)^{#}(n)_{Γ_{E}})F itt_{ℤ_{p}[G]}(H^{2}(G_{E}^{S_{f}} , ℤ_{p}(n))). \end{matrix}

Property 5 of Fitting ideals (section 4) shows that the Fitting ideal of H^{0}(E, ℚ_{p}/ℤ_{p}(n))^{∗} is the same as the annihilator ideal of H^{0}(E, ℚ_{p}/ℤ_{p}(n)). This, together with (5.2), yields the following equality of fractional ideals in ℤ_{p}[G]:

\begin{matrix}F itt_{ℤ_{p}[G]}(H^{2}(G_{E}^{S_{f}} , & ℤ_{p}(n))) \\ = Ann_{ℤ_{p}[G]}(H^{0}(E, ℚ_{p}/ℤ_{p}(n)))((π ∘ ι ∘ t_{n})(c_{∞}/d_{∞} · G_{S}^{∗})). \end{matrix}

Finally we have d_{∞}/c_{∞} = (γ − 1)e + (1 − e), which can be identified by (2.2) with T e + 1 − e = H_{S}(T), and obtain the following theorem:

Theorem 5.8. We have the following equality of ideals of ℤ_{p}[G]:

F itt_{ℤ_{p}[G]}(H^{2}(G_{E}^{S_{f}} , ℤ_{p}(n))) = Ann_{ℤ_{p}[G]}(H^{0}(E, ℚ_{p}/ℤ_{p}(n)))((π ∘ ι ∘ t_{n})(\frac{G_{S}^{∗}}{H_{S}})).

Now let p be an odd prime. In this case we have an equality of the Galois groups G_{E}^{S} and G_{E}^{S_{f}}, as we noticed before. Furthermore, by Lemma 3.1 we have G_{S}^{∗} = G_{S} under the assumption μ = 0. On the other hand for odd primes p, since E is the maximal real subfield of E(ζ_{2p}), we have (π ∘ι∘t_{n})(G_{S}/H_{S}) = Θ_{E/F}^{S} (1 − n) for any even n by Corollary 3.4. Therefore from Theorem 5.8 we obtain

F itt_{ℤ_{p}[G]}(H_{ét}^{2}(O_{E}^{S}, ℤ_{p}(n))) = Ann_{ℤ_{p}[G]}(H^{0}(E, ℚ_{p}/ℤ_{p}(n))) · Θ_{E/F}^{S} (n).

This implies the p-adic version of the Coates-Sinnott Conjecture for odd primes. Here we note that H_{ét}^{2}(O′_{E}, ℤ_{p}(n)) ⊆ H_{ét}^{2}(O_{E}^{S}, ℤ_{p}(n)).

For p = 2, we use the following lemma:

Lemma 5.9. Let E/F be an abelian extension of totally real fields with Galois group G, and let n ≥ 2 be an integer. Let r_{1}(F) = [F : ℚ]. Then we have the following exact sequence of 𝔸-modules for 𝔸 = Λ[H]:

0 → (𝔸/2𝔸)^{r_{1}(F)} → 𝔛_{∞}^{S} → 𝔛_{∞}^{f} → 0.

Proof. We have the following commutative diagram by class field theory (cf. [18]) for the finite sets S and S_{f} of primes in F:

D_{E} ↣ Û_{E}^{S} → ∏_{v∈S} ∏_{w|v} Ê_{w} → Gal(M_{E}^{S}/H_{E}^{S}) → 0
↓ ↓ ↓ ↓
D′_{E} ↣ Û_{E}^{S_{f}} → ∏_{v∈S_{f}} ∏_{w|v} Ê_{w} → Gal(M_{E}^{S_{f}} /H_{E}^{S_{f}}) → 0,

where D_{E} and D′_{E} are the kernels of the corresponding maps and are bounded by the Leopoldt defect δ_{E}, Û_{E}^{S} (resp. Û_{E}^{S_{f}}) is the p-adification of the S-unit (resp. S_{f}-unit) group of the ring of integers of E, Ê_{w} is the p-adic completion of the local field E_{w}, M_{E}^{S} (resp. M_{E}^{S_{f}}) is the maximal abelian pro-p-extension of E unramified outside the primes in S (resp. in S_{f}), and H_{E}^{S} (resp. H_{E}^{S_{f}}) is the Hilbert S-class (S_{f}-class) field of E. Here the S_{f}-unit group means the group of the totally real S-units. Since Ê_{w} is isomorphic to ℤ/2ℤ for a real prime w, and G acts transitively on the set

{Ê_{w} | w : infinite primes of E lying above v }

for any infinite prime v of F, we obtain the following exact sequence of Λ[H]- modules:

0 → D_{E} → D′_{E} → ∏_{w∈S\S_{f}} ℤ[G]/2 → Gal(M_{E}^{S}/M_{E}^{S_{f}}) → 0,

where ∏_{w∈S\S_{f}} ℤ[G]/2 ≃ (ℤ[G]/2)^{r_{1}(F)}. Since the real primes are unramified in E_{∞}/E, we can write the exact sequence above for the unique intermediate fields E_{n} of E_{∞}/E with G_{n} := Gal(E_{n}/E) ≃ ℤ/2^{n}ℤ for all n ≥ 0, as follows:

0 → D_{E_{n}} → D′_{E_{n}} → (ℤ/2ℤ[G_{n}])^{r_{1}(F)} → Gal(M_{E_{n}}^{S}/M_{E_{n}}^{S_{f}}) → 0.

Now the claim is that we have the isomorphism

\underleftarrow{lim}(ℤ/2ℤ[G_{n}])^{r_{1}(F)} ≃ (𝔸/2𝔸)^{r_{1}(F)}

for 𝔸 = Λ[H]. Since Λ/2Λ ≃ \underleftarrow{lim} ℤ/2ℤ[T]/T^{2^{n}}, it suffices to show for a fixed real prime v of F that the inverse limits of {∏_{v_{n}|v} ℤ/2ℤ} and {ℤ/2ℤ[T]/T^{2^{n}}} are isomorphic. For this we inductively define an isomorphism

f_{n} : ∏_{v_{n}|v} ℤ/2ℤ → ℤ/2ℤ[T]/T^{2^{n}}

compatible with the norm maps as follow: Let f_{0} be the identity and assume we have defined the isomorphisms f_{m} compatible with the norm maps for all m ≤ n. Let v_{n+1} and v′_{n+1} be the extensions of v_{n} to F_{n+1}. We define

f_{n+1} : ∏_{v_{n+1}} ℤ/2ℤ · ∏_{v′_{n+1}} ℤ/2ℤ → ℤ/2ℤ[T]/T^{2^{n+1}}

as follows:

f_{n+1}(a_{1}, ⋯, a_{2^{n}}, b_{1}, ⋯, b_{2_{n}}) = f_{n}(a_{1}+b_{1}, ⋯, a_{2^{n}} +b_{2^{n}})+T^{2^{n}}f_{n}(a_{1}, ⋯, a_{2^{n}}).

Now we have the commutative diagram

∏_{v_{n}|v} ℤ/2ℤ →^{f_{n}} ℤ/2ℤ[T]/T^{2^{n}}
↓  ↓
∏_{v_{n+1}|v} ℤ/2ℤ →^{f_{n+1}} ℤ/2ℤ[T]/T^{2^{n+1}}

for any n ≥ 0, and hence

\underleftarrow{lim}_{n} ∏_{v_{n}|v} ℤ/2ℤ ≃ Λ/2Λ.

This completes the proof of the claim. Now the exact sequence

0 → Gal(M_{E}^{S}/M_{E}^{S_{f}}) → 𝔛_{∞}^{S} → 𝔛_{∞}^{f} → 0

yields an exact sequence

(𝔸/2𝔸)^{r_{1}(F)} → 𝔛_{∞}^{S} → 𝔛_{∞}^{f} → 0.

We note that under the assumption of the weak Leopoldt Conjecture for E_{∞}/E the cokernels of D_{E_{n}} → D′_{E_{n}}, for all n ≥ 0, are finite elementary 2groups of order bounded independent of n (see for example [31], Chapter X, §3). Finally, since Λ/2Λ has no non-trivial finite submodules, we obtain the following exact sequence of Λ[H]-modules:

0 → (𝔸/2𝔸)^{r_{1}(F)} → 𝔛_{∞}^{S} → 𝔛_{∞}^{f} → 0.

We now take Γ_{E}-coinvariants and then the Pontryagin dual of the exact sequence of Lemma 5.9. The same calculation as in the proof of the second part of Lemma 5.5 leads to the exact sequence

0 → H^{2}(G_{E}^{S_{f}} , ℤ_{2}(n)) → H^{2}(G_{E}^{S}, ℤ_{2}(n)) → (ℤ/2ℤ[G])^{r_{1}(F)}(n)

of ℤ_{2}[G]-modules. This yields

2^{r_{1}(F)}Ann_{ℤ_{2}[G]}(H^{2}(G_{E}^{S_{f}} , ℤ_{2}(n))) ⊆ Ann_{ℤ_{2}[G]}(H^{2}(G_{E}^{S}, ℤ_{2}(n))).

Consequently by Theorem 5.8 we obtain

\begin{matrix}2^{r_{1}(F)}Ann_{ℤ_{p}[G]}(H^{0}(E, ℚ_{p}/ℤ_{p}(n)))((π ∘ ι ∘ t_{n}) & (\frac{G_{S}^{∗}}{H_{S}})) \\ ⊆ Ann_{ℤ_{2}[G]}(H_{ét}^{2}(O_{E}^{S}, ℤ_{2}(n))). \end{matrix}

Under the assumption μ = 0, we have G_{S} = 2^{r_{1}(F)} · G_{S}^{∗} for the prime 2 (cf. Lemma 3.1). Consequently Θ_{E/F}^{S} (1 − n) = 2^{r_{1}(F)}((π ∘ ι ∘ t_{n})(\frac{G_{S}^{∗}}{H_{S}})) (cf. 3.4) for any even integer n ≥ 2 and as a result,

Ann_{ℤ_{2}[G]}(H^{0}(E, ℚ_{2}/ℤ_{2}(n))) · Θ_{E}^{S}(1 − n) ⊆ Ann_{ℤ_{2}[G]}(H_{ét}^{2}(O_{E}^{S}, ℤ_{2}(n))).

Finally, we note that H_{ét}^{2}(O′_{E}, ℤ_{2}(n)) ⊆ H_{ét}^{2}(O_{E}^{S}, ℤ_{2}(n)). Hence, the 2-adic version of the Coates-Sinnott Conjecture 5.4 holds. This finishes the proof of the following result:

Theorem 5.10. Let E/F be an abelian extension of totally real number fields with Galois group G, and let n ≥ 2 be an even integer. Then the motivic version - and therefore the original version - of the Coates-Sinnott Conjecture holds under the assumptions that μ = 0 and that the 2-primary part of the classical Main Conjecture in Iwasawa theory is valid.

We note that both assumptions are true if E is abelian over ℚ, and therefore we obtain the following unconditional result:

Corollary 5.11. Let E be a totally real absolute abelian field. For an abelian extension E/F with Galois group G and even n ≥ 2, the Coates-Sinnott Conjecture 5.3 holds.

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Reza Taleb Department of Mathematics McMaster University Hamilton, Ontario Canada L8S 4K1 talebr@math.mcmaster.ca 784

Documenta Mathematica 18 (2013)