On Zeta Elements for 𝔾_{m}

David Burns, Masato Kurihara, and Takamichi Sano

Received: November 18, 2015

Communicated by Otmar Venjakob

Abstract. In this paper, we present a unifying approach to the general theory of abelian Stark conjectures. To do so we define natural notions of ‘zeta element’, of ‘Weil-étale cohomology complexes’ and of ‘integral Selmer groups’ for the multiplicative group 𝔾_{m} over finite abelian extensions of number fields. We then conjecture a precise connection between zeta elements and Weil-étale cohomology complexes, we show this conjecture is equivalent to a special case of the equivariant Tamagawa number conjecture and we give an unconditional proof of the analogous statement for global function fields. We also show that the conjecture entails much detailed information about the arithmetic properties of generalized Stark elements including a new family of integral congruence relations between Rubin-Stark elements (that refines recent conjectures of Mazur and Rubin and of the third author) and explicit formulas in terms of these elements for the higher Fitting ideals of the integral Selmer groups of 𝔾_{m}, thereby obtaining a clear and very general approach to the theory of abelian Stark conjectures. As first applications of this approach, we derive, amongst other things, a proof of (a refinement of) a conjecture of Darmon concerning cyclotomic units, a proof of (a refinement of) Gross’s ‘Conjecture for Tori’ in the case that the base field is ℚ, explicit conjectural formulas for both annihilating elements and, in certain cases, the higher Fitting ideals (and hence explicit structures) of ideal class groups and a strong refinement of many previous results concerning abelian Stark conjectures.

Contents

1. Introduction 557 1.1. The leading term conjecture and Rubin-Stark elements 557 1.2. Refined class number formulas for 𝔾_{m} 558 1.3. Selmer groups and their higher Fitting ideals 560 1.4. Galois structures of ideal class groups 562 1.5. Annihilators and Fitting ideals of class groups for small Σ 564 1.6. New verifications of the leading term conjecture 567 1.7. Notation 568 2. Canonical Selmer groups and complexes for 𝔾_{m} 569 2.1. Integral dual Selmer groups 569 2.2. ‘Weil-étale cohomology’ complexes 571 2.3. Tate sequences 576 3. Zeta elements and the leading term conjecture 577 3.1. L-functions 577 3.2. The leading term lattice 578 3.3. Zeta elements 581 4. Preliminaries concerning exterior powers 581 4.1. Exterior powers 582 4.2. Rubin lattices 584 4.3. Homomorphisms between Rubin lattices 586 4.4. Congruences between exterior powers 588 5. Congruences for Rubin-Stark elements 590 5.1. The Rubin-Stark conjecture 590 5.2. Conventions for Rubin-Stark elements 592 5.3. Conjectures on Rubin-Stark elements 592 5.4. An explicit resolution 595 5.5. The equivalence of Conjectures 5.4 and 5.9 596 5.6. The leading term conjecture implies the Rubin-Stark conjecture 597 5.7. The leading term conjecture implies Conjecture 5.4 598 6. Conjectures of Darmon and of Gross 602 6.1. Darmon’s Conjecture 602 6.2. Gross’s conjecture for tori 605 7. Higher Fitting ideals of Selmer groups 607 7.1. Relative Fitting ideals 607 7.2. Statement of the conjecture 610 7.3. The leading term conjecture implies Conjecture 7.3 610 7.4. The proof of Theorem 1.10 611 7.5. The proof of Corollary 1.14 615 7.6. The higher relative Fitting ideals of the dual Selmer group 615 8. Higher Fitting ideals of character components of class groups 616 8.1. General abelian extensions 617 8.2. The order of character components in CM abelian extensions 618 8.3. The structure of the class group of a CM field 619 References 623

The second and the third authors are partially supported by JSPS Core-to-core program, “Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory and Geometry”.

On Zeta Elements for 𝔾_{m} 557

1. Introduction

The study of the special values of zeta functions and, more generally, of Lfunctions is a central theme in number theory that has a long tradition stretching back to Dirichlet and Kummer in the nineteenth century. In particular, much work has been done concerning the arithmetic properties of the special values of L-functions and their incarnations in appropriate arithmetic cohomology groups, or ‘zeta elements’ as they are commonly known. The aim of our project is to systematically study the fine arithmetic properties of such zeta elements and thereby to obtain both generalizations and refinements of a wide range of well-known results and conjectures in the area. In this first article we shall concentrate, for primarily pedagogical reasons, on the classical and very concrete case of the L-functions that are attached to the multiplicative group 𝔾_{m} over a finite abelian extension K/k of global fields. In subsequent articles we will then investigate the key Iwasawa-theoretic aspects of our approach (see [9]) and also explain how the conjectures and results presented here naturally extend both to the case of Galois extensions that are not abelian and to the case of the zeta elements that are associated (in general conjecturally) to a wide class of motives over number fields. The main results of the present article are given below as Theorems 1.1, 1.5 and 1.10. In the rest of this introduction we state these results and also discuss a selection of interesting consequences. To do this we fix a finite abelian extension of global fields K/k with Galois group G = Gal(K/k). We then fix a finite non-empty set of places S of k containing both the set S_{ram}(K/k) of places which ramify in K/k and the set S_{∞}(k) of archimedean places (if any). Lastly we fix an auxiliary finite non-empty set of places T of k which is disjoint from S and such that the group O_{K,S,T}^{×} of S-units of K that are endowed with a trivialization at each place of K above a place in T is ℤ-torsion-free (for the precise definition of O_{K,S,T}^{×}, see §1.7).

1.1. The leading term conjecture and Rubin-Stark elements. As a first step we shall define a canonical ‘T -modified Weil-étale cohomology’ complex for 𝔾_{m} and then formulate (as Conjecture 3.6) a precise ‘leading term conjecture’ LTC(K/k) for the extension K/k. This conjecture predicts that the canonical zeta element z_{K/k,S,T} interpolating the leading terms at s = 0 of the (S-truncated T -modified) L-functions L_{k,S,T} (χ, s) generates the determinant module over G of the T -modified Weil-étale cohomology complex for 𝔾_{m} over K. The main result of the first author in [5] implies that LTC(K/k) is valid if k is a global function field. In the number field case our formulation of LTC(K/k) is motivated by the ‘Tamagawa Number Conjecture’ formulated by Bloch and Kato in [1] and by the ‘generalized Iwasawa main conjecture’ studied by Kato in [24] and [25]. In particular, we shall show that for extensions K/k of number fields LTC(K/k) is equivalent to the relevant special case of the ‘equivariant Tamagawa number conjecture’ formulated in the article [7] of Flach and the first author. Taken in conjunction with previous work of several authors, this fact implies that LTC(K/k) is also unconditionally valid for several important families of number fields. We assume now that S contains a subset V = {v_{1}, …, v_{r}} of places which split completely in K. In this context, one can use the values at s = 0 of the r-th derivatives of S-truncated T -modified L-functions to define a canonical element

ϵ_{K/k,S,T}^{V}

in the exterior power module ⋀_{ℤ[G]}^{r} O_{K,S,T}^{×} ⊗ ℝ (for the precise definition see §5.1). As a natural generalization of a classical conjecture of Stark (dealing with the case r = 1) Rubin conjectured in [45] that the elements ϵ_{K/k,S,T}^{V} should always satisfy certain precise integrality conditions (for more details see Remark 1.6). As is now common in the literature, in the sequel we shall refer to ϵ_{K/k,S,T}^{V} as the ‘Rubin-Stark element’ (relative to the given data) and to the central conjecture of Rubin in [45] as the ‘Rubin-Stark Conjecture’. In some very special cases ϵ_{K/k,S,T}^{V} can be explicitly computed and the Rubin-Stark Conjecture verified. For example, this is the case if r = 0 (so V = ∅) when ϵ_{K/k,S,T}^{V} can be described in terms of Stickelberger elements and if k = ℚ and V = {∞} when ϵ_{K/k,S,T}^{V} can be described in terms of cyclotomic units. As a key step in our approach we show that in all cases the validity of LTC(K/k) implies that ϵ_{K/k,S,T}^{V} can be computed as ‘the canonical projection’ of the zeta element z_{K/k,S,T}. This precise result is stated as Theorem 5.14 and its proof will also incidentally show that LTC(K/k) implies the validity of the Rubin-Stark conjecture for K/k. The latter implication was in fact already observed by the first author in [3] (and the techniques developed in loc. cit. have since been used by several other authors) but we would like to point out that the proof presented here is very much simpler than that given in [3] and is therefore much more amenable to subsequent generalization.

1.2. Refined class number formulas for 𝔾_{m}. The first consequence of Theorem 5.14 that we record here concerns a refined version of a conjecture that was recently formulated independently by Mazur and Rubin in [37] (where it is referred to as a ‘refined class number formula for 𝔾_{m}’) and by the third author in [46]. To discuss this we fix an intermediate field L of K/k and a subset V′ = {v_{1}, …, v_{r′}} of S which contains V and is such that every place in V′ splits completely in L. In this context it is known that the elements ϵ_{K/k,S,T}^{V} naturally constitute an Euler system of rank r and the elements ϵ_{L/k,S,T}^{V′} an Euler system of rank r′. If r < r′, then the image of ϵ_{K/k,S,T}^{V} under the map induced by the field theoretic norm K^{×} → L^{×} vanishes. However, in this case Mazur and Rubin (see [37, Conj. 5.2]) and the third author (see [46, Conj. 3]) independently observed that the reciprocity maps of local class field theory lead to an important conjectural relationship between the elements ϵ_{K/k,S,T}^{V} and ϵ_{L/k,S,T}^{V′}. We shall here formulate an interesting refinement MRS(K/L/k, S, T) of the central conjectures of [37] and [46] (see Conjecture 5.4 and the discussion of Remark 5.7) and we shall then prove the following result.

Theorem 1.1. LTC(K/k) implies the validity of MRS(K/L/k, S, T).

This result is both a generalization and strengthening of the main result of the third author in [46, Th. 3.22] and provides strong evidence for MRS(K/L/k, S, T). As already remarked earlier, if k is a global function field, then the validity of LTC(K/k) is a consequence of the main result of [5]. In addition, if k = ℚ, then the validity of LTC(K/k) follows from the work of Greither and the first author in [8] and of Flach in [14]. Theorem 1.1 therefore has the following consequence.

Corollary 1.2. MRS(K/L/k, S, T) is valid if k = ℚ or if k is a global function field.

This result is of particular interest since it verifies the conjectures of Mazur and Rubin [37] and of the third author [46] even in cases for which one has r > 1. In a sequel [9] to this article we will also prove a partial converse to Theorem 1.1 and show that this converse can be used to derive significant new evidence in support of the conjecture LTC(K/k) (for more details see §1.6 below). Next we recall that in [12] Darmon used the theory of cyclotomic units to formulate a refined version of the class number formula for the class groups of real quadratic fields. We further recall that Mazur and Rubin in [36], and later the third author in [46], have proved the validity of the central conjecture of [12] but only after inverting the prime 2. We shall formulate in §6 a natural refinement of Darmon’s conjecture. By using Corollary 1.2 we shall then give a full proof of our refined version of Darmon’s conjecture, thereby obtaining the following result (for a precise version of which see Theorem 6.1).

Corollary 1.3. A natural refinement of Darmon’s conjecture in [12] is valid.

Let now K/k be an abelian extension as above and choose intermediate fields L and L̃ with [L : k] = 2, L ∩ L̃ = k and K = LL̃. In this context Gross has formulated in [21] a ‘conjecture for tori’ regarding the value of the canonical Stickelberger element associated to K/k modulo a certain ideal constructed from class numbers and a canonical integral regulator map. This conjecture has been widely studied in the literature, perhaps most notably by Hayward in [22] and by Greither and Kučera in [16, 17].

We shall formulate (as Conjecture 6.3) a natural refinement of Gross’s conjecture for tori and we shall then prove (in Theorem 6.5) that the validity of this refinement is a consequence of MRS(K/L/k, S, T). As a consequence of Corollary 1.2 we shall therefore obtain the following result.

Corollary 1.4. A natural refinement of Gross’s conjecture for tori is valid if k = ℚ or if k is a global function field.

This result is a significant improvement of the main results of Greither and Kučera in [16, 17]. In particular, whilst the latter articles only study the case that k = ℚ, L is an imaginary quadratic field, and L̃/ℚ is an abelian extension satisfying several technical conditions (see Remark 6.6), Corollary 1.4 now proves Gross’s conjecture completely in the case k = ℚ and with no assumption on either L or L̃.

1.3. Selmer groups and their higher Fitting ideals. In order to state our second main result, we introduce two new Galois modules which are each finitely generated abelian groups and will play a key role in the arithmetic theory of zeta elements. The first of these is a canonical ‘(Σ-truncated T -modified) integral dual Selmer group’ S_{Σ,T} (𝔾_{m/K}) for the multiplicative group over K for each finite nonempty set of places Σ of K that contains S_{∞}(K) and each finite set of places T of K that is disjoint from Σ. If Σ = S_{∞}(K) and T is empty, then S_{Σ,T} (𝔾_{m/K}) is simply defined to be the cokernel of the map

∏_{w} ℤ ⟶ Hom_{ℤ}(K^{×}, ℤ), (x_{w})_{w} ↦ (a ↦ ∑_{w} ord_{w}(a)x_{w}),

where in the product and sum w runs over all finite places of K, and in this case constitutes a canonical integral structure on the Pontryagin dual of the Bloch-Kato Selmer group H_{f}^{1}(K, ℚ/ℤ(1)) (see Remark 2.3(i)). In general, the group S_{Σ,T} (𝔾_{m/K}) is defined to be a natural analogue for 𝔾_{m} of the ‘integral Selmer group’ that was introduced for abelian varieties by Mazur and Tate in [38] and, in particular, lies in a canonical exact sequence of Gmodules of the form (1)

0 ⟶ Hom_{ℤ}(Cl_{Σ}^{T}(K), ℚ/ℤ) ⟶ S_{Σ,T} (𝔾_{m/K}) ⟶ Hom_{ℤ}(O_{K,Σ,T}^{×} , ℤ) ⟶ 0

where Cl_{Σ}^{T}(K) is the ray class group of O_{K,Σ} modulo the product of all places of K above T (see §1.7). This Selmer group is also philosophically related to the theory of Weil-étale cohomology that is conjectured to exist by Lichtenbaum in [34], and in this direction we show that in all cases there is a natural identification

S_{Σ,T} (𝔾_{m/K}) = H_{c,T}^{2} ((O_{K,Σ})_{W}, ℤ)

where the right hand group denotes the cohomology in degree two of a canonical ‘T -modified compactly supported Weil-étale cohomology complex’ that we introduce in §2.2.

The second module S_{Σ,T}^{tr} (𝔾_{m/K}) that we introduce is a canonical ‘transpose’ (in the sense of Jannsen’s homotopy theory of modules [23]) for S_{Σ,T} (𝔾_{m/K}). In terms of the complexes introduced in §2.2 this module can be described as a certain ‘T -modified Weil-étale cohomology group’ of 𝔾_{m}

S_{Σ,T}^{tr} (𝔾_{m/K}) = H_{T}^{1} ((O_{K,Σ})_{W}, 𝔾_{m})

and can also be shown to lie in a canonical exact sequence of G-modules of the form

(2)  0 ⟶ Cl_{Σ}^{T}(K) ⟶ S_{Σ,T}^{tr} (𝔾_{m/K}) ⟶ X_{K,Σ} ⟶ 0.

Here X_{K,Σ} denotes the subgroup of the free abelian group on the set Σ_{K} of places of K above Σ comprising elements whose coefficients sum to zero. We can now state our second main result. In this result we write Fitt_{G}^{r}(M) for the r-th Fitting ideal of a finitely generated G-module M, though the usual notation is Fitt_{r,ℤ[G]}(M), in order to make the notation consistent with the exterior power ⋀_{ℤ[G]}^{r} M. Note that we will review the definition of higher Fitting ideals in §7.1 and also introduce there for each finitely generated G-module M and each pair of non-negative integers r and i a natural notion of ‘higher relative Fitting ideal’

Fitt_{G}^{(r,i)}(M) = Fitt_{G}^{(r,i)}(M, M_{tors}).

We write x ↦ x^{#} for the ℂ-linear involution of ℂ[G] which inverts elements of G.

Theorem 1.5. Let K/k, S, T, V and r be as above, and assume that LTC(K/k) is valid. Then all of the following claims are also valid.

(i) One has

Fitt_{G}^{r}(S_{S,T} (𝔾_{m/K})) = {Φ(ϵ_{K/k,S,T}^{V})^{#} : Φ ∈ ⋀_{ℤ[G]}^{r} Hom_{ℤ[G]}(O_{K,S,T}^{×} , ℤ[G])}.

(ii) Let P_{k}(K) be the set of all places which split completely in K. Fix a non-negative integer i and set

V_{i} = {V′ ⊂ P_{k}(K) : |V′| = i and V′ ∩ (S ∪ T) = ∅}.

Then one has

Fitt_{G}^{(r,i)}(S_{S,T}^{tr} (𝔾_{m/K}))

= {Φ(ϵ_{K/k,S∪V′,T}^{V ∪V′}) : V′ ∈ V_{i} and Φ ∈ ⋀_{ℤ[G]}^{r+i} Hom_{ℤ[G]}(O_{K,S∪V′,T}^{×} , ℤ[G])}.

In particular, if i = 0, then one has

Fitt_{G}^{r}(S_{S,T}^{tr} (𝔾_{m/K})) = {Φ(ϵ_{K/k,S,T}^{V}) : Φ ∈ ⋀_{ℤ[G]}^{r} Hom_{ℤ[G]}(O_{K,S,T}^{×} , ℤ[G])}.

Remark 1.6. In terms of the notation of Theorem 1.5, the Rubin-Stark Conjecture asserts that Φ(ϵ_{K/k,S,T}^{V}) belongs to ℤ[G] for every Φ in ⋀_{ℤ[G]}^{r} Hom_{ℤ[G]}(O_{K,S,T}^{×} , ℤ[G]). The property described in Theorem 1.5 is deeper in that it shows the ideal generated by Φ(ϵ_{K/k,S,T}^{V}) as Φ runs over ⋀_{ℤ[G]}^{r} Hom_{ℤ[G]}(O_{K,S,T}^{×} , ℤ[G]) should encode significant arithmetic information relating to integral Selmer groups. (See also Remark 5.13 in this regard.)

1.4. Galois structures of ideal class groups. In this subsection, in order to better understand the content of Theorem 1.5, we discuss several interesting consequences concerning the explicit Galois structure of ideal class groups. To do this we fix an odd prime p and suppose that K/k is any finite abelian extension of global fields. We write L for the (unique) intermediate field of K/k such that K/L is a p-extension and [L : k] is prime to p. Then the group Gal(K/k) decomposes as a direct product Gal(L/k) × Gal(K/L) and we fix a non-trivial faithful character χ of Gal(L/k). We set Cl^{T} (K) := Cl_{∅}^{T} (K) and define its ‘(p, χ)-component’ by setting

A^{T} (K)^{χ} := (Cl^{T} (K) ⊗ ℤ_{p}) ⊗_{ℤ_{p}[Gal(L/k)]} O_{χ}.

Here we write O_{χ} for the module ℤ_{p}[im(χ)] upon which Gal(L/k) acts via χ so that A^{T} (K)^{χ} has an induced action of the group ring R_{K}^{χ} := O_{χ}[Gal(K/L)]. Then in Theorem 8.1 we shall derive the following results about the structure of A^{T} (K)^{χ} from the final assertion of Theorem 1.5(ii). In this result we write ‘χ(v) ≠ 1’ to mean that the decomposition group of v in Gal(L/k) is non-trivial.

Corollary 1.7. Let V be the set of archimedean places of k that split completely in K and set r := |V |. Assume that any ramifying place v of k in K satisfies χ(v) ≠ 1. Assume also that the equality of LTC(K/k) is valid after applying the functor − ⊗_{ℤ_{p}[Gal(L/k)]} O_{χ}. Then for any non-negative integer i one has an equality

Fitt_{R_{K}^{χ}}^{i} (A^{T} (K)^{χ}) = {Φ(ϵ_{K/k,S∪V′,T}^{V ∪V′,χ}) : V′ ∈ V_{i} and Φ ∈ ⋀_{R_{K}^{χ}}^{r+i} H_{χ}}

where we set S := S_{∞}(k) ∪ S_{ram}(K/k) and H_{χ} := Hom_{R_{K}^{χ}} ((O_{K,S∪V′,T}^{×} ⊗ ℤ_{p})^{χ}, R_{K}^{χ}).

We remark that Corollary 1.7 specializes to give refinements of several results in the literature. For example, if k = ℚ and K is equal to the maximal totally real subfield ℚ(ζ_{m})^{+} of ℚ(ζ_{m}) where ζ_{m} is a fixed choice of primitive m-th root of unity for some natural number m, then LTC(K/k) is known to be valid and so Corollary 1.7 gives an explicit description of the higher Fitting ideals of ideal class groups in terms of cyclotomic units (which are the relevant Rubin-Stark elements in this case). In particular, if m = p^{n} for any non-negative integer n, then the necessary condition on χ is satisfied for all non-trivial χ and Corollary 1.7 gives a strong refinement of Ohshita’s theorem in [41] for the field K = ℚ(ζ_{p^{n}})^{+}. The result is also stronger than that of Mazur and Rubin in [35, Th. 4.5.9] since the latter describes structures over a discrete valuation ring whilst Corollary 1.7 describes structures over the group ring R_{K}^{χ}. In addition, if K is a CM extension of a totally real field k, then Corollary 1.7 constitutes a generalization of the main results of the second author in both [28] and [30]. To explain this we suppose that K/k is a CM-extension and that χ is an odd character. Then classical Stickelberger elements can be used to define for each non-negative integer i a ‘higher Stickelberger ideal’

Θ^{i}(K/k) ⊆ ℤ_{p}[Gal(K/k)]

(for details see §8.3). By taking T to be empty we can consider the (p, χ)- component of the usual ideal class group

A(K)^{χ} := (Cl^{T} (K) ⊗ ℤ_{p}) ⊗_{ℤ_{p}[Gal(L/k)]} O_{χ}.

Then, by using both Theorem 1.1 and Corollary 1.7 we shall derive the following result as a consequence of the more general Theorem 8.6. In this result we write ω for the Teichmüller character giving the Galois action on the group of p-th roots of unity.

Corollary 1.8. Let K be a CM-field, k totally real, and χ an odd character with χ ≠ ω. We assume that any ramifying place v of k in K satisfies χ(v) ≠ 1 and that LTC(F/k) is valid for certain extensions F of K (see Theorem 8.6 for the precise conditions on F). Then for any non-negative integer i one has an equality

Fitt_{R_{K}^{χ}}^{i} (A(K)^{χ}) = Θ^{i}(K/k)^{χ}.

In the notation of Corollary 1.8 suppose that K is the n-th layer of the cyclotomic ℤ_{p}-extension of L for some non-negative integer n and that every place 𝔭 above p satisfies χ(𝔭) ≠ 1. Then the conditions on χ(v) and LTC(F/k) that are stated in Corollary 1.8 are automatically satisfied and so Corollary 1.8 generalizes the main results of the second author in [30]. To get a better feeling for Corollary 1.8, consider the simple case that [K : k] is prime to p. In this case K = L, the ring ℤ_{p}[Gal(K/k)] is semi-local and A(K)^{χ} is a module over the discrete valuation ring O_{χ} = R_{K}^{χ}. Then the conclusion in Corollary 1.8 in the case i = 0 implies that

(3)  |A(K)^{χ}| = |O_{χ}/L_{k}(χ^{−1}, 0)|

where L_{k}(χ^{−1}, s) is the usual Artin L-function. If every place 𝔭 above p satisfies χ(𝔭) ≠ 1, then this equality is known to be a consequence of the main conjecture for totally real fields proved by Wiles [54]. However, without any such restriction on the values χ(𝔭), the equality (3) is as yet unproved. In addition, in this case the result of Corollary 1.8 is much stronger than (3) in that it shows the explicit structure of A(K)^{χ} as a Galois module to be completely determined (conjecturally at least) by Stickelberger elements by using the obvious (non-canonical) isomorphism of O_{χ}-modules

A(K)^{χ} ≃ ⨁_{i≥1} Fitt_{O_{χ}}^{i}(A(K)^{χ})/ Fitt_{O_{χ}}^{i−1}(A(K)^{χ}) = ⨁_{i≥1} Θ^{i}(K/k)^{χ}/Θ^{i−1}(K/k)^{χ}.

Next we note that the proof of Corollary 1.8 also combines with the result of Theorem 1.16 below to give the following result (which does not itself assume the validity of LTC(K/k)). This result will be proved in Corollaries 8.4 and 8.8. In it we write μ_{p^{∞}}(k(ζ_{p})) for the p-torsion subgroup of k(ζ_{p})^{×}.

Corollary 1.9. Assume that K/k is a CM-extension, that the degree [K : k] is prime to p, and that χ is an odd character of G such that there is at most one p-adic place 𝔭 of k with χ(𝔭) = 1. Assume also that the p-adic μ-invariant of K_{∞}/K vanishes. Then one has both an equality

|A(K)^{χ}| = \{ \begin{matrix}|O_{χ}/L_{k}(χ^{−1}, 0)| & if χ ≠ ω, \\ |O_{χ}/(|μ_{p^{∞}}(k(ζ_{p}))| · L_{k}(χ^{−1}, 0))| & if χ = ω \end{matrix}

and a (non-canonical) isomorphism of O_{χ}-modules

A(K)^{χ} ≃ ⨁_{i≥1} Θ^{i}(K/k)^{χ}/Θ^{i−1}(K/k)^{χ}.

This result is a generalization of the main theorem of the second author in [28] where it is assumed that χ(𝔭) ≠ 1 for all p-adic places 𝔭. It also generalizes the main result of Rubin in [44] which deals only with the special case K = ℚ(μ_{p}) and k = ℚ. To end this subsection we note Remark 1.13 below explains why Theorem 1.5(ii) also generalizes and refines the main result of Cornacchia and Greither in [10].

1.5. Annihilators and Fitting ideals of class groups for small Σ. In this subsection we discuss further connections between Rubin-Stark elements and the structure of class groups of the form Cl_{Σ}^{T}(K) for ‘small’ sets Σ which do not follow from Theorem 1.5. In particular, we do not assume here that Σ contains S_{ram}(K/k). To do this we denote the annihilator ideal of a G-module M by Ann_{G}(M).

Theorem 1.10. Assume LTC(K/k) is valid. Fix Φ in ⋀_{ℤ[G]}^{r}Hom_{ℤ[G]}(O_{K,S,T}^{×} , ℤ[G]) and any place v in S \ V. Then one has

Φ(ϵ_{K/k,S,T}^{V}) ∈ Ann_{G}(Cl_{V ∪{v}}^{T}(K))

and, if G is cyclic, also

Φ(ϵ_{K/k,S,T}^{V}) ∈ Fitt_{G}^{0}(Cl_{V ∪{v}}^{T}(K)).

Remark 1.11. The first assertion of Theorem 1.10 provides a common refinement and wide-ranging generalization (to L-series of arbitrary order of vanishing) of several well-known conjectures and results in the literature. To discuss this we write Cl^{T} (K) for the full ray class group modulo T (namely, Cl^{T} (K) = Cl_{∅}^{T} (K), see §1.7).

(i) We first assume that r = 0 (so V is empty) and that k is a number field. Then, without loss of generality (for our purposes), we can assume that k is totally real and K is a CM field. In this case ϵ_{K/k,S,T}^{∅} is the Stickelberger element θ_{K/k,S,T} (0) of the extension K/k (see §3.1). We take v to be an archimedean place in S. Then Cl_{{v}}^{T}(K) = Cl^{T} (K) and so the first assertion of Theorem 1.10 shows that LTC(K/k) implies the classical Brumer-Stark Conjecture,

θ_{K/k,S,T} (0) · Cl^{T} (K) = 0.

(ii) We next consider the case that K is totally real and take V to be S_{∞}(k) so that r = |V | = [k : ℚ]. In this case Corollary 1.10 implies that for any non-archimedean place v in S, any element σ_{v} of the decomposition subgroup G_{v} of v in G and any element Φ of ⋀_{ℤ[G]}^{[k:ℚ]} Hom_{ℤ[G]}(O_{K,S,T}^{×} , ℤ[G]), one has

(4)  (1 − σ_{v}) · Φ(ϵ_{K/k,S,T}^{S_{∞}}) ∈ Ann_{G}(Cl^{T} (K)).

To make this containment even more explicit we further specialize to the case that k = ℚ and that K is equal to ℚ(ζ_{m})^{+} for some natural number m. We recall that LTC(K/k) has been verified in this case. We take S to be the set comprising the (unique) archimedean place ∞ and all prime divisors of m, and V to be S_{∞} = {∞} (so r = 1). For a set T which contains an odd prime, we set δ_{T} := ∏_{v∈T} (1 − NvFr_{v}^{−1}), where Fr_{v} ∈ G denotes the Frobenius automorphism at a place of K above v. In this case, we have

ϵ_{K/k,S,T}^{{∞}} = ϵ_{m,T} := (1 − ζ_{m})^{δ_{T}} ∈ O_{K,S,T}^{×}

(see, for example, [50, p.79] or [42, §4.2]) and so (4) implies that for any σ_{v} ∈ G_{v} and any Φ ∈ Hom_{ℤ[G]}(O_{K,S,T}^{×} , ℤ[G]) one has

(1 − σ_{v}) · Φ(ϵ_{m,T}) ∈ Ann_{G}(Cl^{T} (K)).

Now the group G is generated by the decomposition subgroups G_{v} of each prime divisor v of m, and so for any σ ∈ G one has an equality σ − 1 = Σ_{v|m}x_{v} for suitable elements x_{v} of the ideal I(G_{v}) of ℤ[G] that is generated by {σ_{v} − 1 : σ_{v} ∈ G_{v}}. Hence, since ϵ_{m,T}^{σ−1} belongs to O_{K}^{×} one finds that for any φ ∈ Hom_{ℤ[G]}(O_{K}^{×}, ℤ[G]) one has φ(ϵ_{m,T}^{σ−1}) = Σ_{v|m}x_{v}φ̃(ϵ_{m,T}) where φ̃ is any lift of φ to Hom_{ℤ[G]}(O_{K,S,T}^{×} , ℤ[G]). Therefore, for any φ in Hom_{ℤ[G]}(O_{K}^{×}, ℤ[G]) and any σ in G, one has

(5)  φ(ϵ_{m,T}^{σ−1}) ∈ Ann_{G}(Cl^{T} (K)).

This containment is actually finer than the annihilation result proved by Rubin in [43, Th. 2.2 and the following Remark] since it deals with the group Cl^{T} (K) rather than Cl(K).

Remark 1.12. We next consider the case that K/k is a cyclic CM-extension and V is empty. As remarked above, in this case the Rubin-Stark element ϵ_{K/k,S,T}^{∅} coincides with the Stickelberger element θ_{K/k,S,T} (0). The second assertion of Theorem 1.10 therefore combines with the argument in Remark 1.11(i) to show that LTC(K/k) implies a containment

θ_{K/k,S,T} (0) ∈ Fitt_{G}^{0}(Cl^{T} (K)).

This is a strong refinement of the Brumer-Stark conjecture. To see this note that Cl^{T} (K) is equal to the ideal class group Cl(K) of K when T is empty. The above containment thus combines with [50, Chap. IV, Lem. 1.1] to imply that if G is cyclic, then one has

θ_{K/k,S}(0) · Ann_{G}(μ(K)) ⊂ Fitt_{G}^{0}(Cl(K))

where μ(K) denotes the group of roots of unity in K. It is known that this inclusion is not in general valid without the hypothesis that G is cyclic (see [18]). The possibility of such an explicit refinement of Brumer’s Conjecture was discussed by the second author in [29] and [31]. In fact, in the terminology of [29], the above argument shows that both properties (SB) and (DSB) follow from LTC(K/k) whenever G is cyclic. For further results in the case that G is cyclic see Corollary 7.10.

Remark 1.13. Following the discussion of Remark 1.11(ii) we can also now consider Theorem 1.5 further in the case that k = ℚ, K = ℚ(ζ_{p^{n}})^{+} for an odd prime p and natural number n and S = {∞, p}. In this case the G-module X_{K,S} is free of rank one and so the exact sequence (2) combines with the final assertion of Theorem 1.5(ii) (with r = 1) to give equalities

Fitt_{G}^{0}(Cl_{S}^{T} (K)) = Fitt_{G}^{1}(S_{S,T}^{tr} (𝔾_{m/K}))
= {Φ(ϵ_{p^{n},T}) | Φ ∈ Hom_{G}(O_{K,S,T}^{×} , ℤ[G])}
= Fitt_{G}^{0}(O_{K,S,T}^{×} /(ℤ[G] · ϵ_{p^{n},T}))

where the last equality follows from the fact that G is cyclic. Since (in this case) Cl_{S}(K) = Cl(K) a standard argument shows that the above displayed equality implies Fitt_{G}^{0}(Cl(K)) = Fitt_{G}^{0}(O_{K}^{×}/C_{K}) with C_{K} denoting the group ℤ[G] · {1 − ζ_{p^{n}}, ζ_{p^{n}}} ∩ O_{K}^{×} of cyclotomic units of K, and this is the main result of Cornacchia and Greither in [10]. Our results therefore constitute an extension of the main result in [10] for K = ℚ(ζ_{p^{n}})^{+}.

For any finite group Γ and any Γ-module M we write M^{∨} for its Pontryagin dual Hom_{ℤ}(M, ℚ/ℤ), endowed with the natural contragredient action of Γ. In §7.5 we show that the proof of Theorem 1.10 also implies the following result. In this result we fix an odd prime p and set Cl^{T} (K)_{p}^{∨} := Cl^{T} (K)^{∨} ⊗ ℤ_{p}.

Corollary 1.14. Let K/k be any finite abelian CM-extension and p any odd prime. If LTC(K/k) is valid, then one has a containment

θ_{K/k,S,T} (0)^{#} ∈ Fitt_{ℤ_{p}[G]}^{0}(Cl^{T} (K)_{p}^{∨}).

Remark 1.15.

(i) In [19] Greither and Popescu prove the validity of the displayed containment in Corollary 1.14 under the hypotheses that S contains all p-adic places of k (so that the Stickelberger element θ_{K/k,S,T} (0) is in general ‘imprimitive’) and that the p-adic μ-invariant of K vanishes. In [9, §3.5] we give a new proof of their result by using the natural Selmer modules for 𝔾_{m} defined in §2 below in place of the Galois modules ‘related to 1-motives’ that are explicitly constructed for this purpose in [19]. In addition, by combining Corollary 1.14 with the result of Theorem 1.16 below we can also now prove the containment in Corollary 1.14, both unconditionally and without the assumption that S contains all p-adic places, for important families of examples. For more details see [9, §3.5 and §5]. (ii) For any odd prime p the group Cl(K)_{p}^{∨} := Cl(K)^{∨} ⊗ ℤ_{p} is not a quotient of Cl^{T} (K)_{p}^{∨} and so Corollary 1.14 does not imply that θ_{K/k,S,T} (0) belongs to Fitt_{ℤ_{p}[G]}^{0}(Cl(K)_{p}^{∨}). (iii) For any odd prime p write Cl(K)_{p}^{∨,−} for the submodule of Cl(K) ⊗ ℤ_{p} upon which complex conjugation acts as multiplication by −1. Then, under a certain technical hypothesis on μ(K), the main result of Greither in [15] shows that LTC(K/k) also implies an explicit description of the Fitting ideal Fitt_{ℤ_{p}[G]}^{0}(Cl(K)_{p}^{∨,−}) in terms of suitably normalized Stickelberger elements. By replacing the role of ‘Tate sequences for small S’ in the argument of Greither by the ‘T -modified Weil-étale cohomology’ complexes that we introduce in §2.2 one can in fact prove the same sort of result without any hypothesis on μ(K).

1.6. New verifications of the leading term conjecture. In a sequel [9] to this article we investigate the natural Iwasawa-theoretic aspects of our general approach. In particular, we show in [9, Th. 5.2] that, without any restriction to CM extensions (or to the ‘minus parts’ of conjectures), under the assumed validity of a natural main conjecture of higher rank Iwasawa theory, the validity of the p-part of MRS(L/K/k, S, T) for all finite abelian extensions L/k implies the validity of the p-part of LTC(K/k). Such a result provides an important partial converse to Theorem 1.1 and can also be used to derive new evidence in support of LTC(K/k). For example, in [9, Th. 4.9] we show that, in all relevant cases, the validity of MRS(K/L/k, S, T) is implied by a well-known leading term formula for padic L-series that has been conjectured by Gross (the ‘Gross-Stark conjecture’ [20]). By combining this observation with significant recent work of Darmon, Dasgupta and Pollack and of Ventullo concerning the Gross-Stark conjecture we are then able to give (in [9, Cor. 5.8]) the following new evidence in support of the conjectures LTC(K/k) and MRS(K/L/k, S, T).

Theorem 1.16. Assume that k is a totally real field, that K is an abelian CM extension of k (with maximal totally real subfield K^{+}) and that p is an odd prime. If the p-adic Iwasawa μ-invariant of K vanishes and at most one padic place of k splits in K/K^{+}, then for any finite subextension K′/K of the cyclotomic ℤ_{p}-extension of K the minus parts of the p-parts of both LTC(K′/k) and MRS(K′/K/k, S, T) are valid.

For examples of explicit families of extensions K/k that satisfy all of the hypotheses of Theorem 1.16 with respect to any given odd prime p see [9, Examples 5.9].

1.7. Notation. In this final subsection of the Introduction we collect together some important notation which will be used in the article. For an abelian group G, a ℤ[G]-module is simply called a G-module. Tensor products, Hom, exterior powers, and determinant modules over ℤ[G] are denoted by ⊗_{G}, Hom_{G}, ⋀_{G}, and det_{G}, respectively. We use similar notation for Ext-groups, Fitting ideals, etc. The augmentation ideal of ℤ[G] is denoted by I(G). For any G-module M and any subgroup H ⊂ G, we denote M^{H} for the submodule of M comprising elements fixed by H. The norm element of H is denoted by N_{H}, namely,

N_{H} = ∑_{σ∈H} σ ∈ ℤ[G].

Let E denote either ℚ, ℝ or ℂ. For an abelian group A, we denote E ⊗_{ℤ} A by EA. The maximal ℤ-torsion subgroup of A is denoted by A_{tors}. We write A/A_{tors} by A_{tf}. The Pontryagin dual Hom_{ℤ}(A, ℚ/ℤ) of A is denoted by A^{∨} for discrete A. Fix an algebraic closure \overline{ℚ} of ℚ. For a positive integer n, we denote by μ_{n} the group of n-th roots of unity in \overline{ℚ}^{×}. Let k be a global field. The set of all infinite places of k is denoted by S_{∞}(k) or simply by S_{∞} when k is clear from the context. (If k is a function field, then S_{∞}(k) is empty.) Consider a finite Galois extension K/k, and denote its Galois group by G. The set of all places of k which ramify in K is denoted by S_{ram}(K/k) or simply by S_{ram} when K/k is clear from the context. For any non-empty finite set S of places of k, we denote by S_{K} the set of places of K lying above places in S. The ring of S-integers of K is defined by

O_{K,S} := {a ∈ K : ord_{w}(a) ≥ 0 for all finite places w of K not contained in S_{K}},

where ord_{w} denotes the normalized additive valuation at w. The unit group of O_{K,S} is called the S-unit group of K. Let T be a finite set of finite places of k, which is disjoint from S. The (S, T)-unit group of K is defined by

O_{K,S,T}^{×} := {a ∈ O_{K,S}^{×} : a ≡ 1 (mod w) for all w ∈ T_{K}}.

The ideal class group of O_{K,S} is denoted by Cl_{S}(K). This is called the S-class group of K. The (S, T)-class group of K, which we denote by Cl_{S}^{T} (K), is defined to be the ray class group of O_{K,S} modulo ∏_{w∈T_{K}} w (namely, the quotient of the group of fractional ideals whose supports are coprime to all places above S ∪ T by the subgroup of principal ideals with a generator congruent to 1 modulo all places in T_{K}). When S ⊂ S_{∞}, we omit S and write Cl^{T} (K) for Cl_{S}^{T} (K). When S ⊂ S_{∞} and T = ∅, we write Cl(K) which is the class group of the integer ring O_{K}. We denote by X_{K,S} the augmentation kernel of the divisor group Y_{K,S} := ⨁_{w∈S_{K}} ℤw. If S contains S_{∞}(k), then the Dirichlet regulator map

λ_{K,S} : ℝO_{K,S}^{×} ⟶ ℝX_{K,S},

defined by λ_{K,S}(a) := − ∑_{w∈S_{K}} log |a|_{w}w, is an isomorphism. For a place w of K, the decomposition subgroup of w in G is denoted by G_{w}. If w is finite, the residue field of w is denoted by κ(w). The cardinality of κ(w) is denoted by Nw. If the place v of k lying under w is unramified in K, then the Frobenius automorphism at w is denoted by Fr_{w} ∈ G_{w}. When G is abelian, then G_{w} and Fr_{w} depend only on v, so in this case we often denote them by G_{v} and Fr_{v} respectively. The ℂ-linear involution ℂ[G] → ℂ[G] induced by σ ↦ σ^{−1} with σ ∈ G is denoted by x ↦ x^{#}. A complex of G-modules is said to be ‘perfect’ if it is quasi-isomorphic to a bounded complex of finitely generated projective G-modules. We denote by D(ℤ[G]) the derived category of G-modules, and by D^{p}(ℤ[G]) the full subcategory of D(ℤ[G]) consisting of perfect complexes.

2. Canonical Selmer groups and complexes for 𝔾_{m}

In this section, we give a definition of ‘integral dual Selmer groups for 𝔾_{m}’, as analogues of Mazur-Tate’s ‘integral Selmer groups’ defined for abelian varieties in [38]. We shall also review the construction of certain natural arithmetic complexes, which are used for the formulation of the leading term conjecture.

2.1. Integral dual Selmer groups. Let K/k be a finite Galois extension of global fields with Galois group G. Let S be a non-empty finite set of places which contains S_{∞}(k). Let T be a finite set of places of k which is disjoint from S.

Definition 2.1. We define the ‘(S-relative T -trivialized) integral dual Selmer group for 𝔾_{m}’ by setting

S_{S,T} (𝔾_{m/K}) := coker( ∏_{w∉S_{K}∪T_{K}} ℤ ⟶ Hom_{ℤ}(K_{T}^{×}, ℤ)),

where K_{T}^{×} is the subgroup of K^{×} defined by

K_{T}^{×} := {a ∈ K^{×} : ord_{w}(a − 1) > 0 for all w ∈ T_{K}},

and the homomorphism on the right hand side is defined by

(x_{w})_{w} ↦ (a ↦ ∑_{w∉S_{K}∪T_{K}} ord_{w}(a)x_{w}).

When T is empty, we omit the subscript T from this notation.

By the following proposition we see that our integral dual Selmer groups are actually like usual dual Selmer groups (see also Remark 2.3 below).

Proposition 2.2. There is a canonical exact sequence

0 ⟶ Cl_{S}^{T} (K)^{∨} ⟶ S_{S,T} (𝔾_{m/K}) ⟶ Hom_{ℤ}(O_{K,S,T}^{×} , ℤ) ⟶ 0

of the form (1) in §1.

Proof. Consider the commutative diagram

\xymatrix{0 \ar[r] & ∏_{w∉S_{K}∪T_{K}} ℤ \ar[r] \ar[d] & ∏_{w∉S_{K}∪T_{K}} ℚ \ar[r] \ar[d] & ∏_{w∉S_{K}∪T_{K}} ℚ/ℤ \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Hom_{ℤ}(K_{T}^{×}, ℤ) \ar[r] & Hom_{ℤ}(K_{T}^{×}, ℚ) \ar[r] & (K_{T}^{×})^{∨},}

where each row is the natural exact sequence, and each vertical arrow is given by (x_{w})_{w} ↦ (a ↦ ∑_{w∉S_{K}∪T_{K}} ord_{w}(a)x_{w}). Using the exact sequence

0 ⟶ O_{K,S,T}^{×} ⟶ K_{T}^{×} \underrightarrow{⨁ ord_{w}} ⨁_{w∉S_{K}∪T_{K}} ℤ ⟶ Cl_{S}^{T} (K) ⟶ 0

and applying the snake lemma to the above commutative diagram, we obtain the exact sequence

0 ⟶ Cl_{S}^{T} (K)^{∨} ⟶ S_{S,T} (𝔾_{m/K}) ⟶ Hom_{ℤ}(O_{K,S,T}^{×} , ℚ) ⟶ (O_{K,S,T}^{×})^{∨}.

Since the kernel of the last map is Hom_{ℤ}(O_{K,S,T}^{×} , ℤ), we obtain the desired conclusion. □

Remark 2.3.

(i) The Bloch-Kato Selmer group H_{f}^{1}(K, ℚ/ℤ(1)) is defined to be the kernel of the diagonal map

K^{×} ⊗ ℚ/ℤ ⟶ ⨁_{w} K_{w}^{×}/O_{K_{w}}^{×} ⊗ ℚ/ℤ

where w runs over all finite places, and so lies in a canonical exact sequence

0 ⟶ O_{K}^{×} ⊗ ℚ/ℤ ⟶ H_{f}^{1}(K, ℚ/ℤ(1)) ⟶ Cl(K) ⟶ 0.

Its Pontryagin dual H_{f}^{1}(K, ℚ/ℤ(1))^{∨} is a finitely generated ℤ̂-module and our integral dual Selmer group S_{S_{∞}}(𝔾_{m/K}) provides a canonical finitely generated ℤ-structure on H_{f}^{1}(K, ℚ/ℤ(1))^{∨}. (ii) In general, the exact sequence (1) also means that S_{S,T} (𝔾_{m/K}) is a natural analogue (relative to S and T) for 𝔾_{m} over K of the ‘integral Selmer group’ that is defined for abelian varieties by Mazur and Tate in [38, p.720].

In the next subsection we shall give a natural cohomological interpretation of the group S_{S,T} (𝔾_{m/K}) (see Proposition 2.4(iii)) and also show that it has a canonical ‘transpose’ (see Definition 2.6).

2.2. ‘Weil-étale cohomology’ complexes. In the following, we construct two canonical complexes of G-modules, and use them to show that there exists a canonical transpose of the module S_{S,T} (𝔾_{m/K}). The motivation for our choice of notation (and terminology) for these complexes is explained in Remark 2.5 below. We fix data K/k, G, S, T as in the previous subsection. We also write 𝔽_{T_{K}}^{×} for the direct sum ⨁_{w∈T_{K}} κ(w)^{×} of the multiplicative groups of the residue fields of all places in T_{K}.

Proposition 2.4. There exist canonical complexes of G-modules RΓ_{c}((O_{K,S})_{W}, ℤ) and RΓ_{c,T} ((O_{K,S})_{W}, ℤ) which satisfy all of the following conditions.

(i) There exists a canonical commutative diagram of exact triangles in D(ℤ[G])

(6)

Hom_{ℤ}(O_{K,S}^{×}, ℚ)[−3]  ⟶^{θ} RΓ_{c}(O_{K,S}, ℤ) ⟶ RΓ_{c}((O_{K,S})_{W}, ℤ) ⟶
↓  ∥  ↓
(Hom_{ℤ}(O_{K,S}^{×}, ℚ) ⊕ (𝔽_{T_{K}}^{×})^{∨})[−3] ⟶^{θ′} RΓ_{c}(O_{K,S}, ℤ) ⟶ RΓ_{c,T} ((O_{K,S})_{W}, ℤ) ⟶
↓  ↓
(𝔽_{T_{K}}^{×})^{∨}[−3]  (𝔽_{T_{K}}^{×})^{∨}[−2]
↓  ↓_{θ′′}

in which the first column is induced by the obvious exact sequence

0 ⟶ Hom_{ℤ}(O_{K,S}^{×}, ℚ) ⟶ Hom_{ℤ}(O_{K,S}^{×}, ℚ) ⊕ (𝔽_{T_{K}}^{×})^{∨} ⟶ (𝔽_{T_{K}}^{×})^{∨} ⟶ 0

and H^{2}(θ′′) is the Pontryagin dual of the natural injective homomorphism

H^{3}(RΓ_{c}((O_{K,S})_{W}, ℤ))^{∨} = O_{K,tors}^{×} ⟶ 𝔽_{T_{K}}^{×} .

(ii) If S′ is a set of places of k which contains S and is disjoint from T, then there is a canonical exact triangle in D(ℤ[G])

RΓ_{c,T} ((O_{K,S′})_{W}, ℤ) ⟶ RΓ_{c,T} ((O_{K,S})_{W}, ℤ) ⟶ ⨁_{w∈S′_{K}\S_{K}} RΓ((κ(w))_{W}, ℤ),

where RΓ((κ(w))_{W}, ℤ) is the complex of G_{w}-modules which lies in the exact triangle

ℚ[−2] ⟶ RΓ(κ(w), ℤ) ⟶ RΓ((κ(w))_{W}, ℤ) ⟶,

where the H^{2} of the first arrow is the natural map

ℚ ⟶ ℚ/ℤ = H^{2}(κ(w), ℤ).

(iii) The complex RΓ_{c,T} ((O_{K,S})_{W}, ℤ) is acyclic outside degrees one, two and three, and there are canonical isomorphisms

H^{i}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ)) ≃ \begin{cases}Y_{K,S}/Δ_{S}(ℤ) & if i = 1, \\ S_{S,T} (𝔾_{m/K}) & if i = 2, \\ (K_{T,tors}^{×})^{∨} & if i = 3, \end{cases}

where Δ_{S} is the natural diagonal map. (iv) If S contains S_{ram}(K/k), then RΓ_{c}((O_{K,S})_{W}, ℤ) and

RΓ_{c,T} ((O_{K,S})_{W}, ℤ) are both perfect complexes of G-modules.

Proof. In this argument we use the fact that morphisms in D(ℤ[G]) between bounded above complexes K_{1}^{•} and K_{2}^{•} can be computed by means of the spectral sequence

(7) E_{2}^{p,q} = ∏_{a∈ℤ} Ext_{G}^{p}(H^{a}(K_{1}^{•}), H^{q+a}(K_{2}^{•})) ⇒ H^{p+q}(R Hom_{G}(K_{1}^{•}, K_{2}^{•}))

constructed by Verdier in [53, III, 4.6.10]. Set C^{•} = C_{S}^{•} := RΓ_{c}(O_{K,S}, ℤ) and W := Hom_{ℤ}(O_{K,S}^{×}, ℚ) for simplicity. Then we recall first that C^{•} is acyclic outside degrees one, two and three, that there are canonical isomorphisms

(8)  H^{i}(C^{•}) ≅ \begin{cases}Y_{K,S}/Δ_{S}(ℤ) & if i = 1, \\ Cl_{S}(K)^{∨} & if i = 2, \\ (O_{K,S}^{×})^{∨} & if i = 3, \end{cases}

where Δ_{S} is the map that occurs in the statement of claim (iii) and that, when S contains S_{ram}(K/k), C^{•} is isomorphic to a bounded complex of cohomologically-trivial G-modules. It is not difficult to see that the groups Ext_{G}^{i}(W, H^{3−i}(C^{•})) vanish for all i > 0, and so the spectral sequence (7) implies that the ‘passage to cohomology’ homomorphism

H^{0}(R Hom_{G}(W [−3], C^{•})) = Hom_{D(ℤ[G])}(W [−3], C^{•}) ⟶ Hom_{G}(W, (O_{K,S}^{×})^{∨})

is bijective. We may therefore define θ to be the unique morphism in D(ℤ[G]) for which H^{3}(θ) is equal to the natural map

W = Hom_{ℤ}(O_{K,S}^{×}, ℚ) ⟶ Hom_{ℤ}(O_{K,S}^{×}, ℚ/ℤ) = (O_{K,S}^{×})^{∨}

and then take C_{W}^{•} := RΓ_{c}((O_{K,S})_{W}, ℤ) to be any complex which lies in an exact triangle of the form that occurs in the upper row of (6). An analysis of the long exact cohomology sequence of this triangle then shows that C_{W}^{•} is acyclic outside degrees one, two and three, that H^{1}(C_{W}^{•}) = H^{1}(C^{•}), that H^{2}(C_{W}^{•})_{tors} = H^{2}(C^{•}), that H^{2}(C_{W}^{•})_{tf} = Hom_{ℤ}(O_{K,S}^{×}, ℤ) and that H^{3}(C_{W}^{•}) = (O_{K,tors}^{×})^{∨}. In particular, when S contains S_{ram}(K/k), since each of these groups is finitely generated and both of the complexes W [−3] and C^{•} are represented by bounded complexes of cohomologically-trivial G-modules, this implies that C_{W}^{•} is perfect.

To define the morphism θ′ we first choose a finite set S′′ of places of k which is disjoint from S ∪ T and such that Cl_{S′}(K) vanishes for S′ := S ∪ S′′. Note that (8) with S replaced by S′ implies C_{S′}^{•} is acyclic outside degrees one and three. We also note that, since each place in T is unramified in K/k, there is also an exact sequence of G-modules

(9)  0 ⟶ ⨁_{v∈T} ℤ[G] ⟶^{(1−NvFr_{w})_{v}} ⨁_{v∈T} ℤ[G] ⟶ (𝔽_{T_{K}}^{×})^{∨} ⟶ 0

where w is any choice of place of K above v. This sequence shows both that (𝔽_{T_{K}}^{×})^{∨}[−3] is a perfect complex of G-modules and also that the functor Ext_{G}^{i}((𝔽_{T_{K}}^{×})^{∨}, −) vanishes for all i > 1. In particular, the spectral sequence (7) implies that in this case the passage to cohomology homomorphism

Hom_{D(ℤ[G])}((𝔽_{T_{K}}^{×})^{∨}[−3], C_{S′}^{•}) ⟶ Hom_{G}((𝔽_{T_{K}}^{×})^{∨}, (O_{K,S′}^{×})^{∨})

is bijective. We may therefore define θ′ to be the morphism which restricts on W [−3] to give θ and on (𝔽_{T_{K}}^{×})^{∨}[−3] to give the composite morphism

(𝔽_{T_{K}}^{×})^{∨}[−3] ⟶^{θ′_{1}} RΓ_{c}(O_{K,S′}, ℤ) ⟶^{θ′_{2}} RΓ_{c}(O_{K,S}, ℤ)

where θ′_{1} is the unique morphism for which H^{3}(θ′_{1}) is the Pontryagin dual of the natural map O_{K,S′}^{×} → 𝔽_{T_{K}}^{×} and θ′_{2} occurs in the canonical exact triangle

(10)  RΓ_{c}(O_{K,S′}, ℤ) ⟶^{θ′_{2}} RΓ_{c}(O_{K,S}, ℤ) ⟶ ⨁_{w∈S′′_{K}} RΓ(κ(w), ℤ) ⟶

constructed by Milne in [39, Chap. II, Prop. 2.3 (d)]. We now take C_{W,T}^{•} := RΓ_{c,T} ((O_{K,S})_{W}, ℤ) to be any complex which lies in an exact triangle of the form that occurs in the second row of (6) and then, just as above, an analysis of this triangle shows that C_{W,T}^{•} is a perfect complex of G-modules when S contains S_{ram}(K/k). Note also that since for this choice of θ′ the upper left hand square of (6) commutes the diagram can then be completed to give the right hand vertical exact triangle. The claim (ii) follows easily from the above constructions. It only remains to prove claim (iii). It is easy to see that the groups H^{i}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ)) for i = 1 and 3 are as described in claim (iii), so we need only prove that there is a natural isomorphism

H^{2}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ)) ≃ S_{S,T} (𝔾_{m/K}).

To do this we first apply claim (ii) for a set S′ that is large enough to ensure that Cl_{S′}^{T}(K) vanishes. Since in this case

H^{2}(RΓ_{c,T} ((O_{K,S′})_{W}, ℤ)) = Hom_{ℤ}(O_{K,S′,T}^{×} , ℤ),

we obtain in this way a canonical isomorphism

(11) H^{2}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ)) ≃ coker( ⨁_{w∈S′_{K}\S_{K}} ℤ ⟶ Hom_{ℤ}(O_{K,S′,T}^{×} , ℤ)).

Consider next the commutative diagram

[figure: 0; ∏_{w∉S′_{K}∪T_{K}} ℤ; ∏_{w∉S_{K}∪T_{K}} ℤ; ⨁_{w∈S′_{K}\S_{K}} ℤ; 0; 0; ∏_{w∉S′_{K}∪T_{K}} ℤ; Hom_{ℤ}(K_{T}^{×}, ℤ); Hom_{ℤ}(O_{K,S′,T}^{×} , ℤ); 0]

with exact rows, where the first exact row is the obvious one, the second is the dual of the exact sequence

0 ⟶ O_{K,S′,T}^{×} ⟶ K_{T}^{×} \underrightarrow{⨁ ord_{w}} ⨁_{w∉S′_{K}∪T_{K}} ℤ ⟶ 0,

and the vertical arrows are given by (x_{w})_{w} ↦ (a ↦ ∑_{w} ord_{w}(a)x_{w}). From this we have the canonical isomorphism

(12) S_{S,T} (𝔾_{m/K}) ≃ coker( ⨁_{w∈S′_{K}\S_{K}} ℤ ⟶ Hom_{ℤ}(O_{K,S′,T}^{×} , ℤ)).

From (11) and (12) our claim follows. □

Given the constructions in Proposition 2.4, in each degree i we set

H_{c,T}^{i} ((O_{K,S})_{W}, ℤ) := H^{i}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ)).

We also define a complex

RΓ_{T} ((O_{K,S})_{W}, 𝔾_{m}) := R Hom_{ℤ}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ), ℤ)[−2].

We endow this complex with the natural contragredient action of G and then in each degree i set

H_{T}^{i} ((O_{K,S})_{W}, 𝔾_{m}) := H^{i}(RΓ_{T} ((O_{K,S})_{W}, 𝔾_{m})).

Remark 2.5. Our notation for the above cohomology groups and complexes is motivated by the following facts.

(i) Assume that k is a function field. Write C_{k} for the corresponding smooth projective curve, C_{k,W ét} for the Weil-étale site on C_{k} that is defined by Lichtenbaum in [33, §2] and j for the open immersion Spec(O_{k,S}) ⟶ C_{k}. Then the group H_{c}^{i}((O_{K,S})_{W}, ℤ) defined above is canonically isomorphic to the Weil-étale cohomology group H^{i}(C_{k,W ét}, j_{!}ℤ). (ii) Assume that k is a number field. In this case there has as yet been no construction of a ‘Weil-étale topology’ for Y_{S} := Spec(O_{K,S}) with all of the properties that are conjectured by Lichtenbaum in [34]. However, if \overline{Y_{S}} is a compactification of Y_{S} and ϕ is the natural inclusion Y_{S} ⊂ \overline{Y_{S}}, then the approach of [4] can be used to show that, should such a topology exist with all of the expected properties, then the groups H_{c}^{i}((O_{K,S})_{W}, ℤ) defined above would be canonically isomorphic to the group H_{c}^{i}(Y_{S}, ℤ) := H^{i}(\overline{Y_{S}}, ϕ_{!}ℤ) that is discussed in [34]. (iii) The definition of RΓ_{T} ((O_{K,S})_{W}, 𝔾_{m}) as the (shifted) linear dual of the complex RΓ_{c,T} ((O_{K,S})_{W}, ℤ) is motivated by [4, Rem. 3.8] and hence by the duality theorem in Weil-étale cohomology for curves over finite fields that is proved by Lichtenbaum in [33].

An analysis of the complex RΓ_{c,T} ((O_{K,S})_{W}, ℤ) as in the proof of Lemma 2.8 below then leads us to give the following definition. In this definition we use the notion of ‘transpose’ in the sense of Jannsen’s homotopy theory of modules [23].

Definition 2.6. The ‘transpose’ of S_{S,T} (𝔾_{m/K}) is the group

S_{S,T}^{tr} (𝔾_{m/K}) := H_{T}^{1} ((O_{K,S})_{W}, 𝔾_{m}) = H^{−1}(R Hom_{ℤ}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ), ℤ)).

When T is empty, we omit the subscript T from this notation.

Remark 2.7. By using the spectral sequence

E_{2}^{p,q} = Ext_{ℤ}^{p}(H_{c,T}^{−q} ((O_{K,S})_{W}, ℤ), ℤ) ⇒ H_{T}^{p+q+2}((O_{K,S})_{W}, 𝔾_{m}),

which is obtained from (7), one can check that RΓ_{T} ((O_{K,S})_{W}, 𝔾_{m}) is acyclic outside degrees zero and one, that there is a canonical isomorphism

H_{T}^{0} ((O_{K,S})_{W}, 𝔾_{m}) ≃ O_{K,S,T}^{×} ,

and that there is a canonical exact sequence

0 ⟶ Cl_{S}^{T} (K) ⟶ S_{S,T}^{tr} (𝔾_{m/K}) ⟶ X_{K,S} ⟶ 0

of the form (2) in §1.

In the sequel we shall say that a G-module M has a ‘locally-quadratic presentation’ if it lies in an exact sequence of finitely generated G-modules of the form

P → P′ → M → 0

in which P and P′ are projective and the ℚ[G]-modules ℚP and ℚP′ are isomorphic.

Lemma 2.8. Assume that G is abelian, that S contains S_{∞}(k) ∪ S_{ram}(K/k), and that O_{K,S,T}^{×} is ℤ-torsion-free. Then each of the groups S_{S,T} (𝔾_{m/K}) and S_{S,T}^{tr} (𝔾_{m/K}) have locally-quadratic presentations, and for each non-negative integer i one has an equality

Fitt_{G}^{i}(S_{S,T}^{tr} (𝔾_{m/K})) = Fitt_{G}^{i}(S_{S,T} (𝔾_{m/K}))^{#}.

Proof. Set

C^{•} := RΓ_{c,T} ((O_{K,S})_{W}, ℤ)

and

C^{•,∗} := R Hom_{ℤ}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ), ℤ).

From Proposition 2.4 we also know that C^{•} is a perfect complex of G-modules that is acyclic outside degree one and two and ℤ-torsion-free in degree one. This implies, by a standard argument, that C^{•} can be represented by a complex P ⟶^{δ} P′ of G-modules, where P and P′ are finitely generated and projective and the first term is placed in degree one, and hence that there is a tautological exact sequence of G-modules

(13)  0 ⟶ H^{1}(C^{•}) ⟶ P ⟶^{δ} P′ ⟶ H^{2}(C^{•}) ⟶ 0.

The descriptions in Proposition 2.4(iii) imply that the linear dual of the Dirichlet regulator map λ_{K,S} induces an isomorphism of ℝ[G]-modules

(14)  λ_{K,S}^{∗} : ℝH^{1}(C^{•}) ≅ ℝH^{2}(C^{•}).

Taken in conjunction with the sequence (13) this isomorphism implies that the ℚ[G]-modules ℚP and ℚP′ are isomorphic and hence that S_{S,T} (𝔾_{m/K}) = H^{2}(C^{•}) has a locally-quadratic presentation, as claimed. The complex C^{•,∗}[−2] is represented by Hom_{ℤ}(P′, ℤ) →^{δ^{∗}} Hom_{ℤ}(P, ℤ) where the linear duals are endowed with contragredient action of G, the first term is placed in degree zero and δ^{∗} is the map induced by δ. There is therefore a tautological exact sequence

(15)

0 ⟶ H^{0}(C^{•,∗}[−2]) ⟶ Hom_{ℤ}(P′, ℤ) ⟶^{δ^{∗}} Hom_{ℤ}(P, ℤ) ⟶ H^{1}(C^{•,∗}[−2]) ⟶ 0,

and, since the above observations imply that Hom_{ℤ}(P′, ℤ) and Hom_{ℤ}(P, ℤ) are projective G-modules that span isomorphic ℚ[G]-spaces, this sequence implies that the module S_{S,T}^{tr} (𝔾_{m/K}) = H^{1}(C^{•,∗}[−2]) has a locally-quadratic presentation. It now only remains to prove the final claim and it is enough to prove this after completion at each prime p. We shall denote for any abelian group A the pcompletion A⊗ℤ_{p} of A by A_{p}. By Swan’s Theorem (cf. [11, (32.1)]) one knows that for each prime p the ℤ_{p}[G]-modules P_{p} and P′_{p} are both free of rank, d say, that is independent of p. In particular, after fixing bases of P_{p} and P′_{p} the homo-

morphism P_{p} →^{δ} P′_{p} corresponds to a matrix A_{δ,p} in M_{d}(ℤ_{p}[G]) and the sequence (13) implies that the ideal Fitt_{G}^{i}(H^{2}(C^{•}))_{p} is generated over ℤ_{p}[G] by the determinants of all (d − i) × (d − i) minors of A_{δ,p}. The corresponding dual bases induce identifications of both Hom_{ℤ}(P′, ℤ)_{p} and Hom_{ℤ}(P, ℤ)_{p} with ℤ_{p}[G]^{⊕d},

with respect to which the homomorphism Hom_{ℤ}(P′, ℤ)_{p} →^{δ^{∗}} Hom_{ℤ}(P, ℤ)_{p} is represented by the matrix A_{δ,p}^{tr,#} that is obtained by applying the involution # to each entry of the transpose of A_{δ,p}. The exact sequence (15) therefore implies that Fitt_{G}^{i}(H^{1}(C^{•,∗}[−2]))_{p} is generated over ℤ_{p}[G] by the determinants of all (d − i) × (d − i) minors of A_{δ,p}^{tr,#}. Hence one has an equality

Fitt_{G}^{i}(H^{2}(C^{•}))_{p} = Fitt_{G}^{i}(H^{1}(C^{•,∗}[−2]))_{p}^{#},

as required. □

2.3. Tate sequences. In this subsection we review the construction of Tate’s exact sequence, which is used in the formulation of the leading term conjecture in the next section. Let K/k, G, S be as in the previous subsection. We assume that S_{ram}(K/k) ⊂ S. We assume only in this subsection that S is large enough so that Cl_{S}(K) vanishes.

In this setting, Tate constructed a ‘fundamental class’ τ_{K/k,S} ∈ Ext_{G}^{2}(X_{K,S}, O_{K,S}^{×}) using the class field theory [49]. This class τ_{K/k,S} has the following property: if we regard τ_{K/k,S} as an element of H^{2}(G, Hom_{ℤ}(X_{K,S}, O_{K,S}^{×})) via the canonical isomorphism

Ext_{G}^{2}(X_{K,S}, O_{K,S}^{×}) ≃ Ext_{G}^{2}(ℤ, Hom_{ℤ}(X_{K,S}, O_{K,S}^{×}))

= H^{2}(G, Hom_{ℤ}(X_{K,S}, O_{K,S}^{×})),

then, for every integer i, the map between Tate cohomology groups

Ĥ^{i}(G, X_{K,S}) ⟶^{∼} Ĥ^{i+2}(G, O_{K,S}^{×})

that is defined by taking cup product with τ_{K/k,S} is bijective. The Yoneda extension class of τ_{K/k,S} is therefore represented by an exact sequence of the following sort:

0 ⟶ O_{K,S}^{×} ⟶ A ⟶ B ⟶ X_{K,S} ⟶ 0, (16)

where A and B are finitely generated cohomologically-trivial G-modules (see [50, Chap. II, Th. 5.1]). We call this sequence a ‘Tate sequence’ for K/k.

Proposition 2.9. The complex RΓ((O_{K,S})_{W}, 𝔾_{m}) defines an element of

Ext_{G}^{2}(S_{S}^{tr}(𝔾_{m/K}), O_{K,S}^{×}).

This element is equal to Tate’s fundamental class τ_{K/k,S}.

Proof. The first assertion follows directly from the discussion of Remark 2.7. The assumed vanishing of Cl_{S}(K) combines with the exact sequence (2) to imply that S_{S}^{tr}(𝔾_{m/K}) = X_{K,S}. Given this, the second claim is proved by the first author in [4, Prop. 3.5(f)] □

3. Zeta elements and the leading term conjecture

In this section, we suppose that K/k is a finite abelian extension of global fields with Galois group G. We fix a finite non-empty set of places S of k which contains both S_{∞}(k) and S_{ram}(K/k) and an auxiliary finite set of places T of k that is disjoint from S.

3.1. L-functions. We recall the definition of (abelian) L-functions of global fields. For any linear character χ ∈ Ĝ := Hom(G, ℂ^{×}), we define the Struncated T -modified L-function for K/k and χ by setting

L_{k,S,T} (χ, s) := ∏_{v∈T}(1 − χ(Fr_{v})Nv^{1−s}) ∏_{v∉S}(1 − χ(Fr_{v})Nv^{−s})^{−1}.

This is a complex function defined on Re(s) > 1 and is well-known to have a meromorphic continuation on ℂ and to be holomorphic at s = 0. We denote by r_{χ,S} the order of vanishing of L_{k,S,T} (χ, s) at s = 0 (this clearly does not depend on T). We denote the leading coefficient of the Taylor expansion of L_{k,S,T} (χ, s) at s = 0 by

L_{k,S,T}^{∗} (χ, 0) := lim_{s→0} s^{−r_{χ,S}}L_{k,S,T} (χ, s).

We then define the S-truncated T -modified equivariant L-function for K/k by setting

θ_{K/k,S,T} (s) := ∑_{χ∈Ĝ} L_{k,S,T} (χ^{−1}, s)e_{χ},

where e_{χ} := \frac{1}{|G|} ∑_{σ∈G} χ(σ)σ^{−1}, and we define its leading term to be

θ_{K/k,S,T}^{∗} (0) := ∑_{χ∈Ĝ} L_{k,S,T}^{∗} (χ^{−1}, 0)e_{χ}.

It is then easy to see that θ_{K/k,S,T}^{∗} (0) belongs to ℝ[G]^{×}. When T = ∅, we simply denote L_{k,S,∅}(χ, s), θ_{K/k,S,∅}(s), etc., by L_{k,S}(χ, s), θ_{K/k,S}(s), etc., respectively, and refer to them as the S-truncated L-function for K/k, S-truncated equivariant L-function for K/k, etc., respectively.

3.2. The leading term lattice. In this section we recall the explicit formulation of a conjectural description of the lattice θ_{K/k,S,T}^{∗} (0)·ℤ[G] which involves Tate sequences. In particular, up until Remark 3.3, we always assume (without further explicit comment) that S is large enough to ensure the group Cl_{S}(K) vanishes. At the outset we also note that, as observed by Knudsen and Mumford in [27], to avoid certain technical difficulties regarding signs, determinant modules must be regarded as graded invertible modules. Nevertheless, for simplicity of notation, in the following we have preferred to omit explicit reference to the grading of any graded invertible modules. Thus, for a finitely generated projective G-module P, we have abbreviated the graded invertible G-module (det_{G}(P), rk_{G}(P)) to det_{G}(P), where rk_{G}(P) is the rank of P. Since the notation det_{G}(P) explicitly indicates P, which in turn determines rk_{G}(P), we feel that this abbreviation should not cause difficulties. We shall also use the following general notation. Suppose that we have a perfect complex C^{•} of G-modules, which is concentrated in degree i and i+1 with some integer i, and an isomorphism λ : ℝH^{i}(C^{•}) →^{∼} ℝH^{i+1}(C^{•}). Then we define an isomorphism

ϑ_{λ} : ℝdet_{G}(C^{•}) ⟶^{∼} ℝ[G]

as follows:

ℝdet_{G}(C^{•}) ⟶^{∼} ⨂_{j∈ℤ} det_{ℝ[G]}^{(−1)^{j}}(ℝC^{j})
⟶^{∼} ⨂_{j∈ℤ} det_{ℝ[G]}^{(−1)^{j}}(ℝH^{j}(C^{•}))
= det_{ℝ[G]}^{(−1)^{i}}(ℝH^{i}(C^{•})) ⊗_{ℝ[G]} det_{ℝ[G]}^{(−1)^{i+1}}(ℝH^{i+1}(C^{•}))
⟶^{∼} det_{ℝ[G]}^{(−1)^{i}}(ℝH^{i+1}(C^{•})) ⊗_{ℝ[G]} det_{ℝ[G]}^{(−1)^{i+1}}(ℝH^{i+1}(C^{•}))
⟶^{∼} ℝ[G],

where the fourth isomorphism is induced by λ^{(−1)^{i}}. Let A and B be the G-modules which appear in the Tate sequence (16). Since we have the regulator isomorphism

λ_{K,S} : ℝO_{K,S}^{×} ⟶^{∼} ℝX_{K,S},

the above construction for C^{•} = (A → B), where A is placed in degree 0, gives the isomorphism

ϑ_{λ_{K,S}} : ℝdet_{G}(A) ⊗_{ℝ[G]} ℝdet_{G}^{−1}(B) ⟶^{∼} ℝ[G].

We study the following conjecture.

Conjecture 3.1. In ℝ[G] one has

ϑ_{λ_{K,S}}(det_{G}(A) ⊗_{G} det_{G}^{−1}(B)) = θ_{K/k,S}^{∗}(0) · ℤ[G].

Remark 3.2. This conjecture coincides with the conjecture C(K/k) stated in [3, §6.3]. The observations made in [3, Rem. 6.2] therefore imply that Conjecture 3.1 is equivalent in the number field case to the ‘equivariant Tamagawa number conjecture’ [7, Conj. 4 (iv)] for the pair (h^{0}(Spec K), ℤ[G]), that the validity of Conjecture 3.1 is independent of S and of the choice of Tate sequence and that its validity for the extension K/k implies its validity for all extensions F/E with k ⊆ E ⊆ F ⊆ K.

Remark 3.3. Conjecture 3.1 is known to be valid in each of the following cases:

(i) K is an abelian extension of ℚ (by Greither and the first author [8] and Flach [14]),

(ii) k is a global function field (by the first author [5]), (iii) [K : k] ≤ 2 (by Kim [26, §2.4, Rem. i)]).

In the following result we do not assume that the group Cl_{S}(K) vanishes and we interpret the validity of Conjecture 3.1 in terms of the ‘Weil-étale cohomology’ complexes RΓ_{c,T} ((O_{K,S})_{W}, ℤ) and RΓ_{T} ((O_{K,S})_{W}, 𝔾_{m}) defined in §2.2. We note at the outset that RΓ_{c,T} ((O_{K,S})_{W}, ℤ) (resp. RΓ_{T} ((O_{K,S})_{W}, 𝔾_{m})) is represented by a complex which is concentrated in degrees one and two (resp. zero and one), and so we can define the isomorphism

ϑ_{λ_{K,S}^{∗}} : ℝdet_{G}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ)) ⟶^{∼} ℝ[G]

(resp. ϑ_{λ_{K,S}} : ℝdet_{G}(RΓ_{T} ((O_{K,S})_{W}, 𝔾_{m})) ⟶^{∼} ℝ[G]).

Proposition 3.4. Let S be any finite non-empty set of places of k containing both S_{∞}(k) and S_{ram}(K/k) and let T be any finite set of places of k that is disjoint from S. Then the following conditions on K/k are equivalent.

(i) Conjecture 3.1 is valid. (ii) In ℝ[G] one has an equality

ϑ_{λ_{K,S}^{∗}}(det_{G}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ))) = θ_{K/k,S,T}^{∗} (0)^{−1}^{#} · ℤ[G].

(iii) In ℝ[G] one has an equality

ϑ_{λ_{K,S}}(det_{G}(RΓ_{T} ((O_{K,S})_{W}, 𝔾_{m}))) = θ_{K/k,S,T}^{∗} (0) · ℤ[G].

Proof. For any finitely generated projective G-module P of (constant) rank d there is a natural identification

⋀_{G}^{d} Hom_{ℤ}(P, ℤ) ≅ ⋀_{G}^{d} Hom_{G}(P, ℤ[G])^{#} ≅ Hom_{G}(⋀_{G}^{d} P, ℤ[G])^{#},

where G acts on Hom_{ℤ}(P, ℤ) contragrediently and on Hom_{G}(P, ℤ[G]) via right multiplication. The equivalence of the equalities in claims (ii) and (iii) is therefore a consequence of the fact that for any element Δ of the mutliplicative group of invertible ℤ[G]-lattices in ℝ[G] the evaluation pairing identifies Hom_{G}(Δ, ℤ[G])^{#} with the image under the involution # of the inverse lattice Δ^{−1}. To relate the equalities in claims (ii) and (iii) to Conjecture 3.1 we note first that the third column of (6) implies that

ϑ_{λ_{K,S}^{∗}}(det_{G}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ)))
= det_{G}((𝔽_{T_{K}}^{×})^{∨}[−2]) · ϑ_{λ_{K,S}^{∗}}(det_{G}(RΓ_{c}((O_{K,S})_{W}, ℤ))),

whilst the resolution (9) implies that

det_{G}((𝔽_{T_{K}}^{×})^{∨}[−2]) = ( ∏_{v∈T}(1 − NvFr_{w}))^{−1} · ℤ[G]
= (θ_{K/k,S,T}^{∗} (0)/θ_{K/k,S}^{∗}(0))^{−1}^{#} · ℤ[G].

The equality in claim (ii) is therefore equivalent to an equality

(17)  ϑ_{λ_{K,S}^{∗}}(det_{G}(RΓ_{c}((O_{K,S})_{W}, ℤ))) = θ_{K/k,S}^{∗}(0)^{−1}^{#} · ℤ[G].

We now choose an auxiliary set of places S′′ as in the proof of Proposition 2.4 and set S′ := S ∪ S′′. By Chebotarev density theorem we can even assume that all places in S′′ split completely in K/k and, for simplicity, this is what we shall do. Then, in this case, the exact triangle (10) combines with the upper triangle in (6) to give an exact triangle in D(ℤ[G]) of the form (18)

Y_{K,S′′}[−1] ⊕ Y_{K,S′′}[−2] ⟶^{α} RΓ_{c}((O_{K,S′})_{W}, ℤ) ⟶^{β} RΓ_{c}((O_{K,S})_{W}, ℤ) ⟶ .

After identifying the cohomology groups of the second and the third occurring complexes by using Proposition 2.4(iii) the long exact cohomology sequence of this triangle induces (after scaler extension) the sequence

0 ⟶ ℚY_{K,S′′} ⟶ ℚY_{K,S′}/Δ_{S′}(ℚ) ⟶ ℚY_{K,S}/Δ_{S}(ℚ)

⟶^{0} ℚY_{K,S′′} ⟶^{ord_{S′′}^{∗}} Hom_{ℤ}(O_{K,S′}^{×}, ℚ) ⟶^{π_{S′′}} Hom_{ℤ}(O_{K,S}^{×}, ℚ) ⟶ 0.

Here ord_{S′′}^{∗} is induced by the linear dual of the map O_{K,S′}^{×} → Y_{K,S′′} induced by taking valuations at each place in S′′_{K} and π_{S′′} by the linear dual of the inclusion O_{K,S}^{×} ⊆ O_{K,S′}^{×} and all other maps are obvious. This sequence implies that there is an exact commutative diagram

0 ⟶ ℝY_{K,S′′} ⟶^{H^{1}(α)} ℝH_{c}^{1}((O_{K,S′})_{W}, ℤ) ⟶^{H^{1}(β)} ℝH_{c}^{1}((O_{K,S})_{W}, ℤ) ⟶ 0
η_{S′′} ↓  ^{λ_{K,S′}^{∗}} ↓  ^{λ_{K,S}^{∗}} ↓
0 ⟶ ℝY_{K,S′′} ⟶^{H^{2}(α)} ℝH_{c}^{2}((O_{K,S′})_{W}, ℤ) ⟶^{H^{2}(β)} ℝH_{c}^{2}((O_{K,S})_{W}, ℤ) ⟶ 0

where η_{S′′} sends each sum ∑_{v∈S′′} ∑_{w|v} x_{w}w to ∑_{v∈S′′} ∑_{w|v} log(Nv)x_{w}w. This diagram combines with the triangle (18) to imply that

ϑ_{λ_{K,S′}^{∗}} (det_{G}(RΓ_{c}((O_{K,S′})_{W}, ℤ)))

= det_{ℝ[G]}(η_{S′′})^{−1}ϑ_{λ_{K,S}^{∗}}(det_{G}(RΓ_{c}((O_{K,S})_{W}, ℤ)))

= ( ∏_{v∈S′′} log(Nv))^{−1}ϑ_{λ_{K,S}^{∗}}(det_{G}(RΓ_{c}((O_{K,S})_{W}, ℤ))).

Since θ_{K/k,S′}^{∗}(0) = (∏_{v∈S′′} log(Nv))θ_{K/k,S}^{∗}(0) this equality shows that (after changing S if necessary) we may assume that Cl_{S}(K) vanishes when verifying (17). Given this, the proposition follows from Proposition 2.9. □

3.3. Zeta elements. We now use the above results to reinterpret Conjecture 3.1 in terms of the existence of a canonical ‘zeta element’. This interpretation will then play a key role in the proofs of Theorem 5.12, 5.16 and 7.5 given below. The following definition of zeta element is in the same spirit as that used by Kato in [24] and [25].

Definition 3.5. The ‘zeta element’ z_{K/k,S,T} of 𝔾_{m} relative to the data K/k, S and T is the unique element of

ℝdet_{G}(RΓ_{T} ((O_{K,S})_{W}, 𝔾_{m})) ≅ det_{ℝ[G]}(ℝO_{K,S}^{×}) ⊗_{ℝ[G]} det_{ℝ[G]}^{−1}(ℝX_{K,S})

which satisfies ϑ_{λ_{K,S}}(z_{K/k,S,T}) = θ_{K/k,S,T}^{∗} (0).

The following ‘leading term conjecture’ is then our main object of study.

Conjecture 3.6 (LTC(K/k)). In ℝdet_{G}(RΓ_{T} ((O_{K,S})_{W}, 𝔾_{m})) one has an equality

ℤ[G] · z_{K/k,S,T} = det_{G}(RΓ_{T} ((O_{K,S})_{W}, 𝔾_{m})).

Given the definition of z_{K/k,S,T}, Proposition 3.4 implies immediately that this conjecture is equivalent to Conjecture 3.1 and hence is independent of the choices of S and T.

4. Preliminaries concerning exterior powers

In this section, we recall certain useful constructions concerning exterior powers and also prove algebraic results that are to be used in later sections.

4.1. Exterior powers. Let G be a finite abelian group. For a G-module M and f ∈ Hom_{G}(M, ℤ[G]), there is a G-homomorphism

⋀_{G}^{r} M ⟶ ⋀_{G}^{r−1} M

for all r ∈ ℤ_{≥1}, defined by

m_{1} ∧ ⋯ ∧ m_{r} ↦ ∑_{i=1}^{r}(−1)^{i−1}f(m_{i})m_{1} ∧ ⋯ ∧ m_{i−1} ∧ m_{i+1} ∧ ⋯ ∧ m_{r}.

This morphism is also denoted by f. This construction gives a homomorphism

⋀_{G}^{s} Hom_{G}(M, ℤ[G]) ⟶ Hom_{G}(⋀_{G}^{r} M, ⋀_{G}^{r−s} M) (19)

for all r, s ∈ ℤ_{≥0} such that r ≥ s, defined by

f_{1} ∧ ⋯ ∧ f_{s} ↦ (m ↦ f_{s} ∘ ⋯ ∘ f_{1}(m)).

By using this homomorphism we often regard an element of ⋀_{G}^{s} Hom_{G}(M, ℤ[G]) as an element of Hom_{G}(⋀_{G}^{r} M, ⋀_{G}^{r−s} M). For a G-algebra Q and a homomorphism f in Hom_{G}(M, Q), there is a Ghomomorphism

⋀_{G}^{r} M ⟶ ( ⋀_{G}^{r−1} M) ⊗_{G} Q

defined by

m_{1} ∧ ⋯ ∧ m_{r} ↦ ∑_{i=1}^{r}(−1)^{i−1}m_{1} ∧ ⋯ ∧ m_{i−1} ∧ m_{i+1} ∧ ⋯ ∧ m_{r} ⊗ f(m_{i}).

By the same method as the construction of (19), we have a homomorphism

⋀_{G}^{s} Hom_{G}(M, Q) ⟶ Hom_{G}(⋀_{G}^{r} M, ( ⋀_{G}^{r−s} M) ⊗_{G} Q). (20)

In the sequel we will find an explicit description of this homomorphism to be useful. This description is well-known and given by the following proposition, the proof of which we omit.

Proposition 4.1. Let m_{1}, …, m_{r} ∈ M and f_{1}, …, f_{s} ∈ Hom_{G}(M, Q). Then we have

(f_{1} ∧ ⋯ ∧ f_{s})(m_{1} ∧ ⋯ ∧ m_{r})

= ∑_{σ∈𝔖_{r,s}} sgn(σ)m_{σ(s+1)} ∧ ⋯ ∧ m_{σ(r)} ⊗ det(f_{i}(m_{σ(j)}))_{1≤i,j≤s},

where

𝔖_{r,s} := {σ ∈ 𝔖_{r} : σ(1) < ⋯ < σ(s) and σ(s + 1) < ⋯ < σ(r)}.

In particular, if r = s, then we have

(f_{1} ∧ ⋯ ∧ f_{r})(m_{1} ∧ ⋯ ∧ m_{r}) = det(f_{i}(m_{j}))_{1≤i,j≤r}.

We will also find the technical observations that are contained in the next two results to be very useful.

Lemma 4.2. Let E be a field and A an n-dimensional E-vector space. If we have an E-linear map

Ψ : A ⟶ E^{⊕m},

where Ψ = ⨁_{i=1}^{m} ψ_{i} with ψ_{1}, …, ψ_{m} ∈ Hom_{E}(A, E) (m ≤ n), then we have

im( ⋀_{1≤i≤m} ψ_{i} : ⋀_{E}^{n} A ⟶ ⋀_{E}^{n−m} A) = \begin{cases}⋀_{E}^{n−m} ker(Ψ), & if Ψ is surjective, \\ 0, & if Ψ is not surjective. \end{cases}

Proof. Suppose first that Ψ is surjective. Then there exists a subspace B ⊂ A such that A = ker(Ψ) ⊕ B and Ψ maps B isomorphically onto E^{⊕m}. We see that ⋀_{1≤i≤m} ψ_{i} induces an isomorphism

⋀_{E}^{m} B ⟶^{∼} E.

Hence we have an isomorphism

⋀_{1≤i≤m} ψ_{i} : ⋀_{E}^{n} A = ⋀_{E}^{n−m} ker(Ψ) ⊗_{E} ⋀_{E}^{m} B ⟶^{∼} ⋀_{E}^{n−m} ker(Ψ).

In particular, we have

im( ⋀_{1≤i≤m} ψ_{i} : ⋀_{E}^{n} A ⟶ ⋀_{E}^{n−m} A) = ⋀_{E}^{n−m} ker(Ψ).

Next, suppose that Ψ is not surjective. Then ψ_{1}, …, ψ_{m} ∈ Hom_{E}(A, E) are linearly dependent. In fact, since each ψ_{i} is contained in Hom_{E}(A/ ker(Ψ), E), we have

dim_{E}(⟨ψ_{1}, …, ψ_{m}⟩) ≤ dim_{E}(A/ ker(Ψ)) = dim_{E}(im(Ψ)),

so dim_{E}(⟨ψ_{1}, …, ψ_{m}⟩) < m if dim_{E}(im(Ψ)) < m. This shows that the element ⋀_{1≤i≤m} ψ_{i} vanishes, as required. □

Using the same notation as in Lemma 4.2, we now consider an endomorphism ψ ∈ End_{E}(A). We write r_{ψ} for the dimension over E of ker(ψ) and consider the composite isomorphism

F_{ψ} : ⋀_{E}^{n} A ⊗_{E} ⋀_{E}^{n} Hom_{E}(A, E) ≃ det_{E}(A) ⊗_{E} det_{E}^{−1}(A)
⟶^{∼} det_{E}(ker(ψ)) ⊗_{E} det_{E}^{−1}(coker(ψ))
≃ ⋀_{E}^{r_{ψ}} ker(ψ) ⊗_{E} ⋀_{E}^{r_{ψ}} Hom_{E}(coker(ψ), E),

where the second isomorphism is induced by the tautological exact sequence

0 ⟶ ker(ψ) ⟶ A ⟶^{ψ} A ⟶ coker(ψ) ⟶ 0.

Then the proof of Lemma 4.2 leads directly to the following useful description of this isomorphism F_{ψ}.

Lemma 4.3. With E, A and ψ as above, we fix an E-basis {b_{1}, …, b_{n}} of A so that im(ψ) = ⟨b_{r_{ψ}+1}, …, b_{n}⟩ and write {b_{1}^{∗}, …, b_{n}^{∗}} for the corresponding dual basis of Hom_{E}(A, E). For each index i we also set ψ_{i} := b_{i}^{∗} ∘ ψ. Then for every a in ⋀_{E}^{n} A the element (⋀_{r_{ψ}<i≤n} ψ_{i})(a) belongs to ⋀_{E}^{r_{ψ}} ker(ψ) and one has

F_{ψ}(a ⊗ (b_{1}^{∗} ∧ ⋯ ∧ b_{n}^{∗})) = (−1)^{r_{ψ}(n−r_{ψ})}( ⋀_{r_{ψ}<i≤n} ψ_{i})(a) ⊗ (b_{1}^{∗} ∧ ⋯ ∧ b_{r_{ψ}}^{∗}).

Here, on the right hand side of the equation, we use the equality im(ψ) = ⟨b_{r_{ψ}+1}, …, b_{n}⟩ to regard b_{i}^{∗} for each i with 1 ≤ i ≤ r_{ψ} as an element of Hom_{E}(coker(ψ), E).

4.2. Rubin lattices. The following definition is due to Rubin [45, §1.2]. We adopt the notation in [46] for the lattice. Note in particular that the notation ‘⋂’ does not refer to an intersection.

Definition 4.4. For a finitely generated G-module M and a non-negative integer r we define the ‘r-th Rubin lattice’ by setting

⋂_{G}^{r} M = {m ∈ ℚ ⋀_{G}^{r} M : Φ(m) ∈ ℤ[G] for all Φ ∈ ⋀_{G}^{r} Hom_{G}(M, ℤ[G])}.

In particular, one has ⋂_{G}^{0} M = ℤ[G].

Remark 4.5. We define the homomorphism ι : ⋀_{G}^{r} Hom_{G}(M, ℤ[G]) → Hom_{G}(⋀_{G}^{r} M, ℤ[G]) by sending each element φ_{1} ∧ ⋯ ∧ φ_{r} to φ_{r} ∘ ⋯ ∘ φ_{1} (see (19)). Then it is not difficult to see that the map

⋂_{G}^{r} M ⟶^{∼} Hom_{G}(im(ι), ℤ[G]); m ↦ (Φ ↦ Φ(m))

is an isomorphism (see [45, §1.2]).

By this remark, one obtains the following result.

Proposition 4.6. Let P be a finitely generated projective G-module. Then we have

⋂_{G}^{r} P = ⋀_{G}^{r} P

for all non-negative integers r.

Lemma 4.7. Let M be a G-module. Suppose that there is a finitely generated projective G-module P and an injection j : M ↪ P whose cokernel is ℤtorsion-free.

(i) The map

Hom_{G}(P, ℤ[G]) ⟶ Hom_{G}(M, ℤ[G])

induced by j is surjective. (ii) If we regard M as a submodule of P via j, then we have

⋂_{G}^{r} M = (ℚ ⋀_{G}^{r} M) ∩ ⋀_{G}^{r} P.

Proof. The assertion (i) follows from [45, Prop. 1.1 (ii)]. Note that

⋀_{G}^{r} Hom_{G}(P, ℤ[G]) ⟶ ⋀_{G}^{r} Hom_{G}(M, ℤ[G])

is also surjective. This induces a surjection

im(ι_{P}) ⟶ im(ι_{M}),

where ι_{P} and ι_{M} denote the maps defined in Remark 4.5 for P and M, respectively. Hence, taking the dual, we have an injection

⋂_{G}^{r} M ≃ Hom_{G}(im(ι_{M}), ℤ[G]) ⟶ Hom_{G}(im(ι_{P}), ℤ[G]) ≃ ⋂_{G}^{r} P.

Since P is projective, we have ⋂_{G}^{r} P = ⋀_{G}^{r} P by Proposition 4.6. Hence we have

⋂_{G}^{r} M ⊂ ⋀_{G}^{r} P.

Next, we show the reverse inclusion ‘⊃’. To do this we fix a in (ℚ ⋀_{G}^{r} M)∩⋀_{G}^{r} P and Φ in ⋀_{G}^{r} Hom_{G}(M, ℤ[G]). By (i), we can take a lift Φ̃ ∈ ⋀_{G}^{r} Hom_{G}(P, ℤ[G]) of Φ. Since a ∈ ⋀_{G}^{r} P, we have

Φ(a) = Φ̃(a) ∈ ℤ[G].

This shows that a belongs to ⋂_{G}^{r} M, as required. □

Remark 4.8. The proof of Lemma 4.7 shows that the cokernel of the injection

⋂_{G}^{r} M ⟶ ⋀_{G}^{r} P

is ℤ-torsion-free. This implies that for any abelian group A, the map

(⋂_{G}^{r} M) ⊗_{ℤ} A ⟶ (⋀_{G}^{r} P) ⊗_{ℤ} A

is injective.

4.3. Homomorphisms between Rubin lattices. In the sequel we fix a subgroup H of G and an ideal J of ℤ[H]. Recall that we denote the augmentation ideal of ℤ[H] by I(H). Put J_{H} := J/I(H)J. We also put J := Jℤ[G], and J_{H} := J /I(H)J .

Proposition 4.9. We have a natural isomorphism of G/H-modules

J_{H} ≃ ℤ[G/H] ⊗_{ℤ} J_{H}.

Proof. Define a homomorphism

ℤ[G/H] ⊗_{ℤ} J_{H} ⟶ J_{H}

by τ ⊗ \overline{a} ↦ \overline{τ̃a}, where τ ∈ G/H, a ∈ J, and τ̃ ∈ G is a lift of τ. One can easily check that this homomorphism is well-defined, and bijective. □

Definition 4.10. Let M be a G-module. For φ ∈ Hom_{G}(M, ℤ[G]), we define φ^{H} ∈ Hom_{G/H}(M^{H}, ℤ[G/H]) by

M^{H} ⟶^{φ} ℤ[G]^{H} ≃ ℤ[G/H],

where the last isomorphism is given by N_{H} = ∑_{σ∈H} σ ↦ 1. Let r be a non-negative integer. For Φ ∈ ⋀_{G}^{r} Hom_{G}(M, ℤ[G]), we define Φ^{H} ∈ ⋀_{G/H}^{r} Hom_{G/H}(M^{H}, ℤ[G/H]) to be the image of Φ under the map

φ_{1} ∧ ⋯ ∧ φ_{r} ↦ φ_{1}^{H} ∧ ⋯ ∧ φ_{r}^{H}.

For convention, if r = 0, then we define Φ^{H} ∈ ℤ[G/H] to be the image of Φ ∈ ℤ[G] under the natural map : ℤ[G] ⟶ ℤ[G/H].

Proposition 4.11. Let M be a G-module and r ∈ ℤ_{≥0}. For any m ∈ ℚ ⋀_{G}^{r} M and Φ ∈ ⋀_{G}^{r} Hom_{G}(M, ℤ[G]), we have

Φ(m) = Φ^{H}(N_{H}^{r} m) in ℚ[G/H],

where N_{H}^{r} denote the map ℚ ⋀_{G}^{r} M → ℚ ⋀_{G/H}^{r} M^{H} induced by N_{H} : M → M^{H}.

Proof. This follows directly from the definition of Φ^{H}. □

We consider the canonical map

ν : ⋂_{G/H}^{r} M^{H} ⟶ ⋂_{G}^{r} M

which is defined as follows. Let

ι : ⋀_{G}^{r} Hom_{G}(M, ℤ[G]) ⟶ Hom_{G}(⋀_{G}^{r} M, ℤ[G])

and

ι_{H} : ⋀_{G/H}^{r} Hom_{G/H}(M^{H}, ℤ[G/H]) ⟶ Hom_{G/H}( ⋀_{G/H}^{r} M^{H}, ℤ[G/H])

be the homomorphisms defined in Remark 4.5. The map

im(ι) ⟶ im(ι_{H}); ι(Φ) ↦ ι_{H}(Φ^{H})

induces a map

α : Hom_{G}(im(ι_{H}), ℤ[G]) ⟶ Hom_{G}(im(ι), ℤ[G]) ≃ ⋂_{G}^{r} M.

Note that we have a canonical isomorphism

β : Hom_{G}(im(ι_{H}), ℤ[G]) ⟶^{∼} Hom_{G/H}(im(ι_{H}), ℤ[G/H]) ≃ ⋂_{G/H}^{r} M^{H}; φ ↦ φ^{H}.

We define a map ν by

ν := α ∘ β^{−1} : ⋂_{G/H}^{r} M^{H} ⟶ ⋂_{G}^{r} M.

Proposition 4.12. Let M be a finitely generated G-module which is ℤ-torsion-free. For any r ∈ ℤ_{≥0}, the map ν : ⋂_{G/H}^{r} M^{H} → ⋂_{G}^{r} M is injective. Furthermore, the maps

( ⋂_{G/H}^{r} M^{H}) ⊗_{ℤ} J_{H} ⟶ (⋂_{G}^{r} M) ⊗_{ℤ} J_{H} ⟶ (⋂_{G}^{r} M) ⊗_{ℤ} ℤ[H]/I(H)J

are both injective, where the first map is induced by ν, and the second by inclusion J_{H} ↪ ℤ[H]/I(H)J.

Proof. The proof is the same as [46, Lem. 2.11], so we omit it. □

Remark 4.13. The inclusion M^{H} ⊂ M induces a map

ξ : ⋂_{G/H}^{r} M^{H} ⟶ ⋂_{G}^{r} M.

We note that this map does not coincide with the above map ν if r > 1. Indeed, one can check that im(ξ) ⊂ |H|^{max{0,r−1}} ⋂_{G}^{r} M (see [37, Lem. 4.8]), and

ν = |H|^{−max{0,r−1}}ξ.

Remark 4.14. Let P be a finitely generated projective G-module. Then, any element of P^{H} is written as N_{H} a with some a ∈ P, since P is cohomologically trivial. One can check that, if r > 0 (resp. r = 0), then the map ν : ⋀_{G/H}^{r} P^{H} → ⋀_{G}^{r} P constructed above coincides with the map

N_{H} a_{1} ∧ ⋯ ∧ N_{H} a_{r} ↦ N_{H} a_{1} ∧ ⋯ ∧ a_{r}
(resp. ℤ[G/H] ≃ ℤ[G]^{H} ↪ ℤ[G]).

In particular, we know that im(ν) = N_{H} ⋀_{G}^{r} P.

Proposition 4.15. Let M be a finitely generated G-module which is ℤ-torsion-free, and r ∈ ℤ_{≥0}. Then the map

( ⋂_{G/H}^{r} M^{H}) ⊗_{ℤ} J_{H} ⟶ Hom_{G}(⋀_{G}^{r} Hom_{G}(M, ℤ[G]), J_{H}); α ↦ (Φ ↦ Φ^{H}(α))

is injective. (We regard Φ^{H}(α) ∈ ℤ[G/H] ⊗_{ℤ} J_{H} as an element of J_{H} via the isomorphism J_{H} ≃ ℤ[G/H] ⊗_{ℤ} J_{H} in Proposition 4.9.)

Proof. The proof is the same as [46, Th. 2.12]. □

4.4. Congruences between exterior powers. The following definition is originally due to Darmon [12], and used in [46, Def. 2.13] and [37, Def. 5.1].

Definition 4.16. Let M be a G-module. For m ∈ M, define

N_{H}(m) = ∑_{σ∈H} σm ⊗ σ^{−1} ∈ M ⊗_{ℤ} ℤ[H]/I(H)J.

The following proposition is an improvement of the result of the third author in [46, Prop. 2.15].

Proposition 4.17. Let P be a finitely generated projective G-module, r ∈ ℤ_{≥0}, and

ν : ( ⋀_{G/H}^{r} P^{H}) ⊗_{ℤ} J_{H} ⟶ (⋀_{G}^{r} P) ⊗_{ℤ} ℤ[H]/I(H)J

be the injection in Proposition 4.12. For an element a ∈ ⋀_{G}^{r} P, the following are equivalent.

(i) a ∈ J · ⋀_{G}^{r} P, (ii) N_{H}(a) ∈ im(ν), (iii) Φ(a) ∈ J for every Φ ∈ ⋀_{G}^{r} Hom_{G}(P, ℤ[G]).

Furthermore, if the above equivalent conditions are satisfied, then for every Φ ∈ ⋀_{G}^{r} Hom_{G}(P, ℤ[G]) we have

Φ(a) = Φ^{H}(ν^{−1}(N_{H}(a))) in J_{H},

where we regard Φ^{H}(ν^{−1}(N_{H}(a))) ∈ ℤ[G/H] ⊗_{ℤ} J_{H} as an element of J_{H} via the isomorphism J_{H} ≃ ℤ[G/H] ⊗_{ℤ} J_{H} in Proposition 4.9.

Proof. By Swan’s Theorem (see [11, (32.1)]), for every prime p, P_{p} is a free ℤ_{p}[G]-module of rank, d say, independent of p. Considering locally, we may assume that P is a free G-module of rank d. We may assume r ≤ d. Clearly,

(i) implies (iii). We shall show that (iii) implies (ii). Suppose Φ(a) ∈ J for all Φ ∈ ⋀_{G}^{r} Hom_{G}(P, ℤ[G]). Fix a basis {b_{1}, …, b_{d}} of P. Write

a = ∑_{μ∈𝔖_{d,r}} x_{μ}b_{μ(1)} ∧ ⋯ ∧ b_{μ(r)},

with some x_{μ} ∈ ℤ[G]. For each μ, by Proposition 4.1, we have

x_{μ} = (b_{μ(1)}^{∗} ∧ ⋯ ∧ b_{μ(r)}^{∗})(a) ∈ J ,

where b_{i}^{∗} ∈ Hom_{G}(P, ℤ[G]) is the dual basis of b_{i}. For each τ ∈ G/H, fix a lift τ̃ ∈ G. Note that we have a direct sum decomposition

J = ⨁_{τ∈G/H} Jτ̃.

Therefore, we can write each x_{μ} as follows:

x_{μ} = ∑_{τ∈G/H} y_{τμ}τ̃,

where y_{τμ} ∈ J. Hence we have

N_{H}(a) = ∑_{σ∈H} ∑_{μ∈𝔖_{d,r}} ∑_{τ∈G/H} σy_{τμ}τ̃b_{μ(1)} ∧ ⋯ ∧ b_{μ(r)} ⊗ σ^{−1}
= ∑_{σ∈H} ∑_{μ∈𝔖_{d,r}} ∑_{τ∈G/H} στ̃b_{μ(1)} ∧ ⋯ ∧ b_{μ(r)} ⊗ σ^{−1}y_{τμ}
= ∑_{μ∈𝔖_{d,r}} ∑_{τ∈G/H} N_{H} τ̃b_{μ(1)} ∧ ⋯ ∧ b_{μ(r)} ⊗ y_{τμ}

∈ N_{H} ⋀_{G}^{r} P ⊗_{ℤ} J_{H} = im(ν)

(see Remark 4.14). This shows (ii). We also see by Remark 4.14 that

ν^{−1}(N_{H}(a)) = ∑_{μ∈𝔖_{d,r}} ∑_{τ∈G/H} τ N_{H} b_{μ(1)}∧⋯∧N_{H} b_{μ(r)}⊗y_{τμ} ∈ ( ⋀_{G/H}^{r} P^{H})⊗_{ℤ}J_{H}.

Hence, by Proposition 4.11, we have

Φ(a) = Φ^{H}(ν^{−1}(N_{H}(a))) in J_{H}

for all Φ ∈ ⋀_{G}^{r} Hom_{G}(P, ℤ[G]). Finally, we show that (ii) implies (i). Suppose N_{H}(a) ∈ im(ν) = (N_{H} ⋀_{G}^{r} P)⊗_{ℤ} J_{H}. As before, we write

a = ∑_{μ∈𝔖_{d,r}} ∑_{τ∈G/H} y_{τμ}τ̃b_{μ(1)} ∧ ⋯ ∧ b_{μ(r)}

with y_{τμ} ∈ ℤ[H]. We have

N_{H}(a) = ∑_{σ∈H} ∑_{μ∈𝔖_{d,r}} ∑_{τ∈G/H} στ̃b_{μ(1)} ∧ ⋯ ∧ b_{μ(r)} ⊗ σ^{−1}y_{τμ} ∈ (N_{H} ⋀_{G}^{r} P) ⊗_{ℤ} J_{H}.

Since (⋀_{G}^{r} P) ⊗_{ℤ} ℤ[H]/I(H)J ≃ ⨁_{σ,μ,τ} ℤ[H]/I(H)J as abelian groups, we must have y_{τμ} ∈ J. This shows that a ∈ J · ⋀_{G}^{r} P. □

5. Congruences for Rubin-Stark elements

For a finite abelian extension K/k, and an intermediate field L, a conjecture which describes a congruence relation between two Rubin-Stark elements for K/k and L/k was formulated by the third author in [46, Conj. 3]. Mazur and Rubin also formulated in [37, Conj. 5.2] essentially the same conjecture. In this section, we formulate a refined version (see Conjecture 5.4) of these conjectures. We also recall a conjecture formulated by the first author, which was studied in [22], [3], [16], [17], [48], and [46] (see Conjecture 5.9). In [46, Th. 3.15], the third author proved a link between Conjecture 5.4 and Conjecture 5.9. We now improve the argument given there to show that Conjecture 5.4 and Conjecture 5.9 are in fact equivalent (see Theorem 5.10). Finally we prove that the natural equivariant leading term conjecture (Conjecture 3.6) implies both Conjecture 5.4 and Conjecture 5.9 (see Theorem 5.16).

5.1. The Rubin-Stark conjecture. In this subsection, we recall the formulation of the Rubin-Stark conjecture [45, Conj. B′]. Let K/k, G, S, T be as in §3, namely, K/k is a finite abelian extension of global fields, G is its Galois group, S is a non-empty finite set of places of k such that S_{∞}(k) ∪ S_{ram}(K/k) ⊂ S, and T is a finite set of places of k which is disjoint from S. In this section, we assume that O_{K,S,T}^{×} is ℤ-torsion-free. Following Rubin [45, Hyp. 2.1] we assume that S satisfies the following hypothesis with respect to some chosen integer r with 0 ≤ r < |S|: there exists a subset V ⊂ S of order r such that each place in V splits completely in K/k. Recall that for any χ ∈ Ĝ we denote by r_{χ,S} the order of vanishing of L_{k,S,T} (χ, s) at s = 0. We know by [50, Chap. I, Prop. 3.4] that

r_{χ,S} = dim_{ℂ}(e_{χ}ℂX_{K,S}) = \begin{cases}|\{v ∈ S : χ(G_{v}) = 1\}| & if χ ≠ 1, \\ |S| − 1 & if χ = 1. \end{cases} (21)

Therefore, the existence of V ensures that r ≤ r_{χ,S} for every χ and hence the function s^{−r}L_{k,S,T} (χ, s) is holomorphic at s = 0. We define the ‘r-th order Stickelberger element’ by

θ_{K/k,S,T}^{(r)} := lim_{s→0} ∑_{χ∈Ĝ} s^{−r}L_{k,S,T} (χ^{−1}, s)e_{χ} ∈ ℝ[G].

Note that the 0-th order Stickelberger element θ_{K/k,S,T}^{(0)} (= θ_{K/k,S,T} (0)) is the usual Stickelberger element. Recall that we have the regulator isomorphism

λ_{K,S} : ℝO_{K,S,T}^{×} ⟶^{∼} ℝX_{K,S}

defined by

λ_{K,S}(a) = − ∑_{w∈S_{K}} log|a|_{w}w.

This map λ_{K,S} induces the isomorphism

⋀_{ℝ[G]}^{r} ℝO_{K,S,T}^{×} ⟶^{∼} ⋀_{ℝ[G]}^{r} ℝX_{K,S},

which we also denote by λ_{K,S}. For each place v ∈ S, fix a place w of K lying above v. Take any v_{0} ∈ S\V, and define the ‘(r-th order) Rubin-Stark element’

ϵ_{K/k,S,T}^{V} ∈ ⋀_{ℝ[G]}^{r} ℝO_{K,S,T}^{×} = ℝ ⋀_{G}^{r} O_{K,S,T}^{×} .

by

(22)  λ_{K,S}(ϵ_{K/k,S,T}^{V}) = θ_{K/k,S,T}^{(r)} ⋀_{v∈V} (w − w_{0}),

where ⋀_{v∈V} (w − w_{0}) is arranged by some chosen order of the elements in V. One can show that the Rubin-Stark element ϵ_{K/k,S,T}^{V} does not depend on the choice of v_{0} ∈ S \ V. We consider the Rubin lattice

⋂_{G}^{r} O_{K,S,T}^{×} ⊂ ℚ ⋀_{G}^{r} O_{K,S,T}^{×}

(see Definition 4.4). The Rubin-Stark conjecture claims

Conjecture 5.1 (The Rubin-Stark conjecture for (K/k, S, T, V)). One has

ϵ_{K/k,S,T}^{V} ∈ ⋂_{G}^{r} O_{K,S,T}^{×} .

Remark 5.2. One can check that the above Rubin-Stark conjecture is equivalent to [45, Conj. B′] for the data (K/k, S, T, V), and that our Rubin-Stark element ϵ_{K/k,S,T}^{V} coincides with the unique element predicted by [45, Conj. B′]. This shows, in particular, that the validity of the conjecture does not depend on the choice of the places lying above v ∈ S or on the ordering of the elements in V.

Remark 5.3. The Rubin-Stark conjecture for (K/k, S, T, V) is known to be true in the following cases:

(i) r = 0. In this case ϵ_{K/k,S,T}^{∅} = θ_{K/k,S,T}^{(0)} = θ_{K/k,S,T} (0) ∈ ℝ[G] so the Rubin-Stark conjecture claims only that θ_{K/k,S,T} (0) ∈ ℤ[G] which is a celebrated result of Deligne-Ribet, Cassou-Noguès, and Barsky.

(ii) [K : k] ≤ 2. This is due to Rubin [45, Cor. 3.2 and Th. 3.5].

(iii) K is an abelian extension over ℚ. This is due to the first author [3,

Th. A]. (iv) k is a global function field. This is due to the first author [3, Th. A].

5.2. Conventions for Rubin-Stark elements. The notation ϵ_{K/k,S,T}^{V} has some ambiguities, since ϵ_{K/k,S,T}^{V} depends on the choice of the places lying above v ∈ S, and on the choice of the order of the elements in V. To avoid this ambiguity, we use the following convention: when we consider the Rubin-Stark element ϵ_{K/k,S,T}^{V}, we always fix a place w of K lying above each v ∈ S, and label the elements of S as

S = {v_{0}, v_{1}, …, v_{n}}

so that V = {v_{1}, …, v_{r}}, and thus we fix the order of the elements in V. So, under this convention, the Rubin-Stark element ϵ_{K/k,S,T}^{V} is the element characterized by

λ_{K,S}(ϵ_{K/k,S,T}^{V}) = θ_{K/k,S,T}^{(r)} ⋀_{1≤i≤r}(w_{i} − w_{0}).

5.3. Conjectures on Rubin-Stark elements. In this subsection, we give a refinement of the conjecture formulated by the third author [46, Conj. 3], and Mazur and Rubin [37, Conj. 5.2]. Let K/k, G, S, T be as before, and we assume that, for a non-negative integer r, there exists a subset V ⊂ S of order r such that each place in V splits completely in K. We fix a subgroup H of G for which, for some integer r′ with r′ ≥ r, there exists a subset V′ ⊂ S of order r′, which contains V, and satisfies that each place in V′ splits completely in the field L := K^{H}. Following the convention in §5.2, we fix, for each place v ∈ S, a place w of K lying above v, and label the elements of S as S = {v_{0}, …, v_{n}} so that V = {v_{1}, …, v_{r}} and V′ = {v_{1}, …, v_{r′}}. We consider the Rubin-Stark elements ϵ_{K/k,S,T}^{V} and ϵ_{L/k,S,T}^{V′} characterized by

λ_{K,S}(ϵ_{K/k,S,T}^{V}) = θ_{K/k,S,T}^{(r)} ⋀_{1≤i≤r}(w_{i} − w_{0})

and

λ_{L,S}(ϵ_{L/k,S,T}^{V′}) = θ_{L/k,S,T}^{(r′)} ⋀_{1≤i≤r′}(w_{i} − w_{0})

respectively, where we denote the place of L lying under w also by w. For each integer i with 1 ≤ i ≤ n we write G_{i} for the decomposition group of v_{i} in G. For any subgroup U ⊂ G, recall that the augmentation ideal of ℤ[U] is denoted by I(U). Put I_{i} := I(G_{i})ℤ[G] and I_{H} := I(H)ℤ[G]. We define

Rec_{i} : O_{L,S,T}^{×} ⟶ (I_{i})_{H} = I_{i}/I_{H}I_{i}

by

Rec_{i}(a) = ∑_{τ∈G/H} τ^{−1}(rec_{w_{i}}(τa) − 1).

Here, rec_{w_{i}} is the reciprocity map L_{w_{i}}^{×} → G_{i} at w_{i}. Note that τ^{−1}(rec_{w_{i}}(τa)−1) is well-defined for τ ∈ G/H in (I_{i})_{H}.

We put W := V′ \ V = {v_{r+1}, …, v_{r′}}. We define an ideal J_{W} of ℤ[H] by

J_{W} := \begin{cases}(∏_{r<i≤r′} I(G_{i}))ℤ[H], & if W ≠ ∅, \\ ℤ[H], & if W = ∅, \end{cases}

and put (J_{W})_{H} := J_{W} /I(H)J_{W}. We also define an ideal J_{W} of ℤ[G] by

J_{W} := \begin{cases}∏_{r<i≤r′} I_{i}, & if W ≠ ∅, \\ ℤ[G], & if W = ∅, \end{cases}

and put (J_{W})_{H} := J_{W} /I_{H}J_{W}. Note that J_{W} = J_{W} ℤ[G]. By Proposition 4.9, we have a natural isomorphism of G/H-modules ℤ[G/H] ⊗_{ℤ} (J_{W})_{H} ≃ (J_{W})_{H}. We consider the graded G/H-algebra

Q_{W} := ⨁_{a_{1},...,a_{r′−r}∈ℤ_{≥0}}(I_{r+1}^{a_{1}} ⋯ I_{r′}^{a_{r′−r}})_{H},

where

(I_{r+1}^{a_{1}} ⋯ I_{r′}^{a_{r′−r}})_{H} := I_{r+1}^{a_{1}} ⋯ I_{r′}^{a_{r′−r}}/I_{H}I_{r+1}^{a_{1}} ⋯ I_{r′}^{a_{r′−r}},

and we define the 0-th power of any ideal of ℤ[G] to be ℤ[G]. For any integer i with r < i ≤ r′ we regard Rec_{i} as an element of Hom_{G/H}(O_{L,S,T}^{×} , Q_{W}) via the natural embedding (I_{i})_{H} ↪ Q_{W}. Then by the same method as in [46, Prop. 2.7] (or [37, Cor. 2.1]), one shows that ⋀_{r<i≤r′} Rec_{i} ∈ ⋀_{G/H}^{r′−r} Hom_{G/H}(O_{L,S,T}^{×} , Q_{W}) induces the map

(23) ⋂_{G/H}^{r′} O_{L,S,T}^{×} ⟶ ( ⋂_{G/H}^{r} O_{L,S,T}^{×}) ⊗_{G/H} (J_{W})_{H} ≃ ( ⋂_{G/H}^{r} O_{L,S,T}^{×}) ⊗_{ℤ} (J_{W})_{H},

which we denote by Rec_{W}. Following Definition 4.16, we define

N_{H} : ⋂_{G}^{r} O_{K,S,T}^{×} ⟶ (⋂_{G}^{r} O_{K,S,T}^{×}) ⊗_{ℤ} ℤ[H]/I(H)J_{W}

by N_{H}(a) = ∑_{σ∈H} σa ⊗ σ^{−1}. Note that since (O_{K,S,T}^{×})^{H} = O_{L,S,T}^{×}, there is a natural injective homomorphism

ν : ( ⋂_{G/H}^{r} O_{L,S,T}^{×}) ⊗_{ℤ} (J_{W})_{H} ⟶ (⋂_{G}^{r} O_{K,S,T}^{×}) ⊗_{ℤ} ℤ[H]/I(H)J_{W}

by Proposition 4.12.

To state the following conjecture we assume the validity of the Rubin-Stark conjecture (Conjecture 5.1) for both (K/k, S, T, V) and (L/k, S, T, V′).

Conjecture 5.4 (MRS(K/L/k, S, T, V, V′)). The element N_{H}(ϵ_{K/k,S,T}^{V}) belongs to im(ν), and satisfies

N_{H}(ϵ_{K/k,S,T}^{V}) = (−1)^{r(r′−r)} · ν(Rec_{W} (ϵ_{L/k,S,T}^{V′})).

Remark 5.5. In this article we write ‘MRS(K/L/k, S, T) is valid’ to mean that the statement of Conjecture 5.4 is valid for all possible choices of V and V′.

Remark 5.6. In §6 we show that Conjecture 5.4 constitutes a natural refinement and generalization of both a conjecture of Darmon from [12] and of several conjectures of Gross from [21]. In addition, in a subsequent article [9] we will show that the validity of Conjecture 5.4 also implies the ‘Gross-Stark conjecture’ formulated by Gross in [20] and a refinement of the main result of Solomon in [47] concerning the ‘wild Euler system’ that he constructs in loc. cit.

Remark 5.7. One has I(G_{i})ℤ[H] ⊂ I(H), so J_{W} ⊆ I(H)^{e} where e := r′ − r ≥ 0. Thus there is a natural homomorphism

(⋂_{G}^{r} O_{K,S,T}^{×}) ⊗_{ℤ} (J_{W})_{H} ⟶ (⋂_{G}^{r} O_{K,S,T}^{×}) ⊗_{ℤ} I(H)^{e}/I(H)^{e+1}.

Conjecture 5.4 is therefore a strengthening of the central conjecture of the third author in [46, Conj. 3] and of the conjecture formulated by Mazur and Rubin in [37, Conj. 5.2], both of which claim only that the given equality is valid after projection to the group (⋂_{G}^{r}O_{K,S,T}^{×}) ⊗_{ℤ} I(H)^{e}/I(H)^{e+1}. This refinement is in the same spirit as Tate’s strengthening in [51] of the ‘refined class number formula’ formulated by Gross in [21].

Remark 5.8. Note that, when r = 0, following [46, Def. 2.13] N_{H} would be defined to be the natural map ℤ[G] → ℤ[G]/I_{H}J_{W}, but this does not make any change because of the observation of Mazur and Rubin in [37, Lem. 5.6 (iv)]. Note also that, by Remark 4.13, the map j_{L/K} in [37, Lem. 4.9] (where our K/L is denoted by L/K) is essentially the same as our homomorphism ν. Finally we note that Mazur and Rubin do not use the fact that j_{L/K} is injective, so the formulation of [46, Conj. 3] is slightly stronger than the conjecture [37, Conj. 5.2].

We next state a refinement of a conjecture that was formulated by the first author in [3] (the original version of which has been studied in many subsequent articles of different authors including [22], [16], [17], [48], and [46]).

Conjecture 5.9. (B(K/L/k, S, T, V, V′)). For every

Φ ∈ ⋀_{G}^{r} Hom_{G}(O_{K,S,T}^{×} , ℤ[G]),

we have

Φ(ϵ_{K/k,S,T}^{V}) ∈ J_{W}

and an equality

Φ(ϵ_{K/k,S,T}^{V}) = (−1)^{r(r′−r)}Φ^{H}(Rec_{W} (ϵ_{L/k,S,T}^{V′})) in (J_{W})_{H}.

In this article we improve an argument of the third author in [46] to prove the following result.

Theorem 5.10. The conjectures

MRS(K/L/k, S, T, V, V′) and B(K/L/k, S, T, V, V′)

are equivalent.

The proof of this result will be given in §5.5.

5.4. An explicit resolution. As a preliminary step, we choose a useful representative of the complex

D_{K,S,T}^{•} := RΓ_{T} ((O_{K,S})_{W}, 𝔾_{m}) ∈ D^{p}(ℤ[G]).

To do this we follow the method used in [3, §7]. Let d be a sufficiently large integer, and F be a free G-module of rank d with basis b = {b_{i}}_{1≤i≤d}. We define a surjection

π : F ⟶ S_{S,T}^{tr} (𝔾_{m/K})(= H^{1}(D_{K,S,T}^{•}))

as follows. Recall that S = {v_{0}, …, v_{n}}. Let F_{≤n} be a free ℤ[G]-module generated by {b_{i}}_{1≤i≤n}. First, choose a homomorphism

π_{1} : F_{≤n} ⟶ S_{S,T}^{tr} (𝔾_{m/K})

such that the composition map

F_{≤n} ⟶^{π_{1}} S_{S,T}^{tr} (𝔾_{m/K}) ⟶ X_{K,S}

sends b_{i} to w_{i} − w_{0}. (Such a homomorphism exists since F_{≤n} is free.) Next, let A denote the kernel of the composition map

S_{S,T}^{tr} (𝔾_{m/K}) ⟶ X_{K,S} ⟶ Y_{K,S\{v_{0}}},

where the last map sends the places above v_{0} to 0. Since d is sufficiently large, we can choose a surjection

π_{2} : F_{>n} ⟶ A,

where F_{>n} is the free ℤ[G]-module generated by {b_{i}}_{n<i≤d}. Define

π := π_{1} ⊕ π_{2} : F = F_{≤n} ⊕ F_{>n} ⟶ S_{S,T}^{tr} (𝔾_{m/K}).

One can easily show that π is surjective. D_{K,S,T}^{•} defines a Yoneda extension class in Ext_{G}^{2}(S_{S,T}^{tr} (𝔾_{m/K}), O_{K,S,T}^{×}). Since D_{K,S,T}^{•} is perfect, this class is represented by an exact sequence of the following form:

0 ⟶ O_{K,S,T}^{×} ⟶ P ⟶^{ψ} F ⟶^{π} S_{S,T}^{tr} (𝔾_{m/K}) ⟶ 0, (24)

where π is the above map and P is a cohomologically-trivial G-module. Since O_{K,S,T}^{×} is ℤ-torsion-free, it follows that P is also ℤ-torsion-free. Hence, P is projective. Note that the complex

P ⟶^{ψ} F,

where P is placed in degree 0, is quasi-isomorphic to D_{K,S,T}^{•}. Hence we have an isomorphism

(25) det_{G}(D_{K,S,T}^{•}) ≃ det_{G}(P) ⊗_{G} det_{G}^{−1}(F).

For each 1 ≤ i ≤ d, we define

ψ_{i} := b_{i}^{∗} ∘ ψ ∈ Hom_{G}(P, ℤ[G]),

where b_{i}^{∗} ∈ Hom_{G}(F, ℤ[G]) is the dual basis of b_{i} ∈ F.

5.5. The equivalence of Conjectures 5.4 and 5.9. In this subsection, we prove Theorem 5.10.

Proof of Theorem 5.10. We regard O_{K,S,T}^{×} ⊂ P by the exact sequence (24). Note that, since P/O_{K,S,T}^{×} ≃ im(ψ) ⊂ F is ℤ-torsion-free, we can apply Lemma 4.7 and Remark 4.8 for M = O_{K,S,T}^{×}. If N_{H}(ϵ_{K/k,S,T}^{V}) ∈ im(ν), then we have

Φ(ϵ_{K/k,S,T}^{V}) = Φ^{H}(ν^{−1}(N_{H}(ϵ_{K/k,S,T}^{V}))) in (J_{W})_{H} (26)

for every Φ ∈ ⋀_{G}^{r} Hom_{G}(O_{K,S,T}^{×} , ℤ[G]), by Proposition 4.17. Hence Conjecture 5.4 implies Conjecture 5.9. Conversely, suppose that Conjecture 5.9 is valid. Then we have Φ(ϵ_{K/k,S,T}^{V}) ∈ J_{W} for every Φ ∈ ⋀_{G}^{r} Hom_{G}(O_{K,S,T}^{×} , ℤ[G]), so again we use Proposition 4.17 to deduce that

N_{H}(ϵ_{K/k,S,T}^{V}) ∈ im (ν : ( ⋀_{G/H}^{r} P^{H}) ⊗_{ℤ} (J_{W})_{H} → (⋀_{G}^{r} P) ⊗_{ℤ} ℤ[H]/I(H)J_{W} ) ,

and that the equality (26) holds for every Φ ∈ ⋀_{G}^{r} Hom_{G}(P, ℤ[G]). By Proposition 4.15, we see that the equality

ν^{−1}(N_{H}(ϵ_{K/k,S,T}^{V})) = (−1)^{r(r′−r)}Rec_{W} (ϵ_{L/k,S,T}^{V′})

holds in (⋀_{G/H}^{r} P^{H}) ⊗_{ℤ} (J_{W})_{H}. Since the natural map

( ⋂_{G/H}^{r} O_{L,S,T}^{×}) ⊗_{ℤ} (J_{W})_{H} ⟶ ( ⋀_{G/H}^{r} P^{H}) ⊗_{ℤ} (J_{W})_{H}

is injective by Remark 4.8, we see that the above equality holds in (⋂_{G/H}^{r} O_{L,S,T}^{×}) ⊗_{ℤ} (J_{W})_{H}. Thus Conjecture 5.9 implies Conjecture 5.4. □

Remark 5.11. Although in the proof of Theorem 5.10 we used the exact sequence (24) to verify the existence of a finitely generated projective G-module P and an injection O_{K,S,T}^{×} ↪ P whose cokernel is ℤ-torsion-free, the referee pointed out that it is unnecessary to use (24) at this point. Indeed, choosing a projective module P′ and a surjection f : P′ → Hom_{ℤ}(O_{K,S,T}^{×} , ℤ), we have an embedding O_{K,S,T}^{×} ↪ P := Hom_{ℤ}(P′, ℤ) by dualizing f, whose cokernel is ℤ-torsion-free.

5.6. The leading term conjecture implies the Rubin-Stark conjecture. The following result was first proved by the first author in [3, Cor. 4.1] but the proof given here is very much simpler than that given in loc. cit.

Theorem 5.12. LTC(K/k) implies the Rubin-Stark conjecture for both (K/k, S, T, V) and (L/k, S, T, V′).

Proof. Assume that LTC(K/k) is valid so the zeta element z_{K/k,S,T} is a ℤ[G]- basis of det_{G}(D_{K,S,T}^{•}). In this case one also knows that P must be free of rank d and we define z_{b} ∈ ⋀_{G}^{d} P to be the element corresponding to the zeta element z_{K/k,S,T} ∈ det_{G}(D_{K,S,T}^{•}) via the isomorphism

⋀_{G}^{d} P ⟶^{∼} ⋀_{G}^{d} P ⊗ ⋀_{G}^{d} Hom_{G}(F, ℤ[G]) ≃ det_{G}(D_{K,S,T}^{•}),

where the first isomorphism is defined by

a ↦ a ⊗ ⋀_{1≤i≤d} b_{i}^{∗},

and the second isomorphism is given by (25). Then Theorem 5.12 follows immediately from the next theorem (see also Corollary 5.15 below for (L/k, S, T, V′)). □

Remark 5.13. In [52] Vallières closely follows the proof of [3, Cor. 4.1] to show that Conjecture 3.1 (and hence also LTC(K/k) by virtue of Proposition 3.4) implies the extension of the Rubin-Stark Conjecture formulated by Emmons and Popescu in [13]. The arguments used here can be used to show that LTC(K/k) implies a refinement of the main result of Vallières, and hence also of the conjecture of Emmons and Popescu, that is in the spirit of Theorem 1.5. This result is to be explained in forthcoming work of Livingstone-Boomla.

The following theorem was essentially obtained in [3] by the first author. This theorem describes the Rubin-Stark element in terms of the zeta elements. It is a key to prove Theorem 5.12, and also plays important roles in the proofs of Theorem 5.16 and Theorem 7.5 given below.

Theorem 5.14. Assume that LTC(K/k) holds. Then, regarding O_{K,S,T}^{×} as a submodule of P, one has

( ⋀_{r<i≤d} ψ_{i})(z_{b}) ∈ ⋂_{G}^{r} O_{K,S,T}^{×} (⊂ ⋀_{G}^{r} P)

(see Lemma 4.7 (ii)) and also

(−1)^{r(d−r)}( ⋀_{r<i≤d} ψ_{i})(z_{b}) = ϵ_{K/k,S,T}^{V} .

Proof. Take any χ ∈ Ĝ. Recall from (21) that

r_{χ,S} = dim_{ℂ}(e_{χ}ℂX_{K,S}) = dim_{ℂ}(e_{χ}ℂO_{K,S,T}^{×})

(the last equality follows from ℂO_{K,S,T}^{×} ≃ ℂX_{K,S}). Consider the map

Ψ := ⨁_{r<i≤d} ψ_{i} : e_{χ}ℂP ⟶ e_{χ}ℂ[G]^{⊕(d−r)}.

This map is surjective if and only if r_{χ,S} = r. Indeed, if r_{χ,S} = r, then {e_{χ}(w_{i} − w_{0})}_{1≤i≤r} is a ℂ-basis of e_{χ}ℂX_{K,S}, so e_{χ}ℂ im(ψ) = e_{χ}ℂ ker(π) = ⨁_{r<i≤d} e_{χ}ℂ[G]b_{i}. In this case, Ψ is surjective. If r_{χ,S} > r, then dim_{ℂ}(e_{χ}ℂ im(ψ)) = d − r_{χ,S} < d − r, so Ψ is not surjective. Applying Lemma 4.2, we have

e_{χ}( ⋀_{r<i≤d} ψ_{i})(z_{b}) \begin{cases}∈ e_{χ}ℂ ⋀_{G}^{r} O_{K,S,T}^{×} , & if r_{χ,S} = r, \\ = 0, & if r_{χ,S} > r. \end{cases}

From this and Lemma 4.7 (ii), we have

( ⋀_{r<i≤d} ψ_{i})(z_{b}) ∈ (ℚ ⋀_{G}^{r} O_{K,S,T}^{×}) ∩ ⋀_{G}^{r} P = ⋂_{G}^{r} O_{K,S,T}^{×} .

By Lemma 4.3 and the definition of z_{b}, we have

λ_{K,S}((−1)^{r(d−r)}( ⋀_{r<i≤d} ψ_{i})(z_{b})) = θ_{K/k,S,T}^{(r)} ⋀_{1≤i≤r}(w_{i} − w_{0}).

By the characterization of the Rubin-Stark element, we have

(−1)^{r(d−r)}( ⋀_{r<i≤d} ψ_{i})(z_{b}) = ϵ_{K/k,S,T}^{V} .

This completes the proof. □

By the same argument as above, one obtains the following result.

Corollary 5.15. Assume that LTC(K/k) holds. Then we have an equality

(−1)^{r′(d−r′)}( ⋀_{r′<i≤d} ψ_{i}^{H})(N_{H}^{d} z_{b}) = ϵ_{L/k,S,T}^{V′}

in ⋂_{G/H}^{r′} O_{L,S,T}^{×}.

5.7. The leading term conjecture implies Conjecture 5.4. In this subsection we prove the following result.

Theorem 5.16. LTC(K/k) implies MRS(K/L/k, S, T, V, V′).

By Remark 3.3, this directly implies the following result.

Corollary 5.17. MRS(K/L/k, S, T, V, V′) is valid if K is an abelian extension over ℚ or if k is a function field.

Remark 5.18. Theorem 5.16 is an improvement of the main result in [46, Th. 3.22] by the third author, which asserts that under some hypotheses LTC(K/k) implies most of Conjecture 5.4. In [3, Th. 3.1], the first author proved that LTC(K/k) implies most of Conjecture 5.9. Since we know by Theorem 5.10 that Conjecture 5.4 and Conjecture 5.9 are equivalent, Theorem 5.16 is also an improvement of [3, Th. 3.1].

Remark 5.19. In [46, §4], by using a weak version of Corollary 5.17, the third author gave another proof of the ‘except 2-part’ of Darmon’s conjecture on cyclotomic units [12], which was first proved by Mazur and Rubin in [36] via Kolyvagin systems. In §6, we shall use Corollary 5.17 to give a full proof of a refined version of Darmon’s conjecture, and also give a new evidence for Gross’s conjecture on tori [21], which was studied by Hayward [22], Greither and Kučera [16], [17].

We prove Theorem 5.16 after proving some lemmas. The following lemma is a restatement of [3, Lem. 7.4].

Lemma 5.20. If 1 ≤ i ≤ n, then we have an inclusion

im(ψ_{i}) ⊂ I_{i}.

In particular, ψ_{i} = 0 for 1 ≤ i ≤ r.

Proof. Take any a ∈ P. Write

ψ(a) = ∑_{j=1}^{d} x_{j}b_{j}

with some x_{j} ∈ ℤ[G]. For each i with 1 ≤ i ≤ n, we show that x_{i} ∈ I_{i}, or equivalently, N_{G_{i}}x_{i} = 0. Noting that F^{G_{i}} is a free G/G_{i}-module with basis {N_{G_{i}}b_{j}}_{1≤j≤d}, it is sufficient to show that

∑_{j=1}^{d} N_{G_{i}}x_{j}b_{j} ∈ ⟨N_{G_{i}}b_{j} : 1 ≤ j ≤ d, j ≠ i⟩_{G/G_{i}}.

The left hand side is equal to ψ(N_{G_{i}}a). By the exact sequence (24), this is contained in ker(π|_{F^{G_{i}}}). Note that we have a natural isomorphism

N_{G_{i}}X_{K,S} ≃ X_{K^{G_{i}} ,S}.

Since v_{i} splits completely in K^{G_{i}}, the G/G_{i}-submodule of N_{G_{i}}X_{K,S} generated by N_{G_{i}}(w_{i} − w_{0}) is isomorphic to ℤ[G/G_{i}]. This shows that

ker(π|_{F^{G_{i}}}) ⊂ ⟨N_{G_{i}}b_{j} : 1 ≤ j ≤ d, j ≠ i⟩_{G/G_{i}}.
□

For each integer i with r < i ≤ r′, we define a map

R̃ec_{i} : P^{H} ⟶ (I_{i})_{H}

as follows. For a ∈ P^{H}, take ã ∈ P such that N_{H} ã = a (this is possible since P is cohomologically-trivial). Define

R̃ec_{i}(a) := ψ_{i}(ã) mod I_{H}I_{i} ∈ (I_{i})_{H}.

(Note that im(ψ_{i}) ⊂ I_{i} by Lemma 5.20.) One can easily check that this is well-defined.

Lemma 5.21. On O_{L,S,T}^{×}, which we regard as a submodule of P^{H}, R̃ec_{i} coincides with the map Rec_{i}. In particular, by the construction of (20), we can extend the map

Rec_{W} : ⋂_{G/H}^{r′} O_{L,S,T}^{×} ⟶ ( ⋂_{G/H}^{r} O_{L,S,T}^{×}) ⊗_{ℤ} (J_{W})_{H}

to

R̃ec_{W} := ⋀_{r<i≤r′}R̃ec_{i} : ⋀_{G/H}^{r′} P^{H} → ( ⋀_{G/H}^{r} P^{H})⊗_{G/H} (J_{W})_{H} ≃ ( ⋀_{G/H}^{r} P^{H})⊗_{ℤ}(J_{W})_{H} .

Proof. The proof is essentially the same as [3, Prop. 10.1] and [2, Lem. 8]. For a ∈ O_{L,S,T}^{×}, take ã ∈ P such that N_{H} ã = a in P^{H}. For each τ ∈ G/H, fix a lift τ̃ ∈ G. Regard F as the free H-module with basis {τ̃b_{i}}_{i,τ}. It is sufficient to show that

(τ̃b_{i})^{∗} ∘ ψ(ã) = rec_{τ̃w_{i}}(a) − 1 = rec_{w_{i}}(τ^{−1}a) − 1, (27)

for every r < i ≤ r′, where (τ̃b_{i})^{∗} ∈ Hom_{H}(F, ℤ[H]) is the dual basis of F as a free H-module. Indeed, using

R̃ec_{i}(a) = ψ_{i}(ã) = ∑_{τ∈G/H} τ̃((τ̃b_{i})^{∗} ∘ ψ(ã)),

we know from (27) that

R̃ec_{i}(a) = ∑_{τ∈G/H} τ̃(rec_{w_{i}}(τ^{−1}a) − 1) = Rec_{i}(a).

We shall show (27). For simplicity, set w := τ̃w_{i} and b := τ̃b_{i}. We denote the decomposition group of w by G_{w}(= G_{i}). As in the proof of Proposition 2.4, one can show that there is a unique morphism

θ_{w} : ℚ[−2] ⟶ RΓ(K_{w}, 𝔾_{m})

in D(ℤ[G_{w}]) such that H^{2}(θ_{w}) is equal to the natural map

ℚ ⟶ ℚ/ℤ ≃ H^{2}(K_{w}, 𝔾_{m}),

where the last isomorphism is the invariant map in the local class field theory. We define the complex RΓ((K_{w})_{W}, 𝔾_{m}) by

RΓ((K_{w})_{W}, 𝔾_{m}) := Cone(θ_{w})

for the local field K_{w}. We have natural identifications H^{0}((K_{w})_{W}, 𝔾_{m}) = K_{w}^{×} and H^{1}((K_{w})_{W}, 𝔾_{m}) = ℤ. The complex RΓ((K_{w})_{W}, 𝔾_{m}) defines a Yoneda extension class τ_{w} in Ext_{G_{w}}^{2}(ℤ, K_{w}^{×}), and [6, Prop. 3.5(a)] shows that τ_{w} coincides with the local fundamental class in H^{2}(G_{w}, K_{w}^{×}). The class τ_{w} is represented by a 2-extension of the form

0 ⟶ K_{w}^{×} ⟶ P_{w} ⟶^{ψ_{w}} ℤ[G_{w}] ⟶ ℤ ⟶ 0,

where P_{w} is a cohomologically trivial G_{w}-module. Define

ρ_{w} : L_{w}^{×} ⟶ I(G_{w})/I(G_{w})^{2}

by ρ(x) := ψ_{w}(x̃), where x̃ ∈ P_{w} is taken so that N_{G_{w}} x̃ = x (note that ψ_{w}(x̃) is well-defined in I(G_{w})/I(G_{w})^{2}). Then, the existence of a natural localization morphism

D_{K,S,T}^{•} = RΓ_{T} ((O_{K,S})_{W}, 𝔾_{m}) ⟶ ℤ[H] ⊗_{ℤ[G_{w}]}^{𝕃} RΓ((K_{w})_{W}, 𝔾_{m})

and our choice of a representative of D_{K,S,T}^{•} implies

b^{∗} ∘ ψ(ã) = ρ_{w}(a) in I(G_{w})/I(H)I(G_{w}).

Hence, (27) is reduced to the equality

ρ_{w} = rec_{w} − 1.

Consider the map

(28) I(G_{w})/I(G_{w})^{2} = Ĥ^{−1}(G_{w}, I(G_{w})) ≃ Ĥ^{−2}(G_{w}, ℤ) ⟶^{∼} Ĥ^{0}(G_{w}, K_{w}^{×}),

where the first isomorphism is the connecting homomorphism with respect to the short exact sequence

0 ⟶ I(G_{w}) ⟶ ℤ[G_{w}] ⟶ ℤ ⟶ 0,

and the last is given by the cup product with τ_{w}. The map (28) is the inverse of rec_{w} −1 by definition. One can also check that (28) coincides with the δ-map of the snake lemma applied to the diagram

\xymatrix{ & (K_{w}^{×})_{G_{w}} \ar[r] \ar[d]^{_{N_{Gw}}} & (P_{w})_{G_{w}} \ar[r]^{ψ_{w}} \ar[d]^{_{N_{Gw}}} & I(G_{w})/I(G_{w})^{2} \ar[r] \ar[d]^{0} & 0 \\ 0 \ar[r] & L_{w}^{×} \ar[r] & P_{w}^{G_{w}} \ar[r]^{ψ_{w}} & I(G_{w})^{G_{w}},}

i.e. the inverse of ρ_{w}. Thus we have ρ_{w} = rec_{w} − 1, which completes the proof. □

Note that, by Lemma 5.20, ⋀_{r<i≤d} ψ_{i} defines a map

⋀_{G}^{d} P ⟶ J_{W} ⋀_{G}^{r} P.

Let ν be the injection

ν : ( ⋀_{G/H}^{r} P^{H}) ⊗_{ℤ} (J_{W})_{H} ⟶ (⋀_{G}^{r} P) ⊗_{ℤ} ℤ[H]/I(H)J_{W}

in Proposition 4.12. By Proposition 4.17, we have

N_{H}(J_{W} ⋀_{G}^{r} P) ⊂ im(ν),

so we can define a map

ν^{−1} ∘ N_{H} : J_{W} ⋀_{G}^{r} P ⟶ ( ⋀_{G/H}^{r} P^{H}) ⊗_{ℤ} (J_{W})_{H}.

Lemma 5.22. We have the following commutative diagram:

\xymatrix{⋀_{G}^{d} P \ar[d]_{_{N{d \atop H}}} \ar[r] & J_{W} ⋀_{G}^{r} P \ar[d]^{ν^{−1}∘N_{H}} \\ ⋀_{G/H}^{d} P^{H} \ar[r] & (⋀_{G/H}^{r} P^{H}) ⊗_{ℤ} (J_{W})_{H},}

where the top arrow is (−1)^{r(d−r)} ⋀_{r<i≤d} ψ_{i}, and the bottom arrow is the composition of (−1)^{r(r′−r)}R̃ec_{W} and (−1)^{r′(d−r′)} ⋀_{r′<i≤d} ψ_{i}^{H}.

Proof. We can prove this lemma by explicit computations, using Proposition 4.1, Proposition 4.11, and Remark 4.14. □

Proof of Theorem 5.16. By Remark 4.8 we may compute in (⋀_{G/H}^{r} P^{H}) ⊗_{ℤ} (J_{W})_{H}. Using Corollary 5.15, Lemma 5.21, Lemma 5.22, and Theorem 5.14 in this order, we compute

(−1)^{r(r′−r)}Rec_{W} (ϵ_{L/k,S,T}^{V′})
= (−1)^{r(r′−r)}R̃ec_{W} ((−1)^{r′(d−r′)}( ⋀_{r′<i≤d} ψ_{i}^{H})(N_{H}^{d} z_{b}))
= (−1)^{r(d−r)}ν^{−1}(N_{H}(( ⋀_{r<i≤d} ψ_{i})(z_{b})))
= ν^{−1}(N_{H}(ϵ_{K/k,S,T}^{V})).

This completes the proof of Theorem 5.16. □

6. Conjectures of Darmon and of Gross

In this section we use Corollary 5.17 to prove a refined version of the conjecture formulated by Darmon in [12] and to obtain important new evidence for a refined version of the ‘conjecture for tori’ formulated by Gross in [21].

6.1. Darmon’s Conjecture. We formulate a slightly modified and refined version of Damon’s conjecture ([12],[36]). Let L be a real quadratic field. Let f be the conductor of L. Let χ be the Dirichlet character defined by

χ : (ℤ/fℤ)^{×} = Gal(ℚ(μ_{f})/ℚ) ⟶ Gal(L/ℚ) ≃ {±1},

where the first map is the restriction map. Fix a square-free positive integer n which is coprime to f, and let K be the maximal real subfield of L(μ_{n}). Set G := Gal(K/ℚ) and H := Gal(K/L). Put n_{±} := ∏_{ℓ|n,χ(ℓ)=±1} ℓ, and ν_{±} := |{ℓ|n_{±}}| (in this section, ℓ always denotes a prime number). We fix an embedding \overline{ℚ} ↪ ℂ. Define a cyclotomic unit by

β_{n} := N_{L(μ_{n})/K}( ∏_{σ∈Gal(ℚ(μ_{nf})/ℚ(μ_{n}))} σ(1 − ζ_{nf})^{χ(σ)}) ∈ K^{×},

where ζ_{nf} = e^{\frac{2πi}{nf}}. Let τ be the generator of G/H = Gal(L/ℚ). Write n_{+} = ℓ_{1} ⋯ ℓ_{ν_{+}}. Note that (1−τ)O_{L}[1/n]^{×} is a free abelian group of rank ν_{+} +1 (see [36, Lem. 3.2 (ii)]). Take u_{0}, …, u_{ν_{+}} ∈ O_{L}[1/n]^{×} so that {u_{0}^{1−τ} , …, u_{ν_{+}}^{1−τ} } is a basis of (1 − τ)O_{L}[1/n]^{×} and that det(log |u_{i}^{1−τ} |_{λ_{j}})_{0≤i,j≤ν_{+}} > 0, where each λ_{j} (1 ≤ j ≤ ν_{+}) is a (fixed) place of L lying above ℓ_{j}, and λ_{0} is the infinite place of L determined by the embedding \overline{ℚ} ↪ ℂ fixed above. Define

R_{n} := ( ⋀_{1≤i≤ν_{+}}(rec_{λ_{i}} − 1))(u_{0}^{1−τ} ∧ ⋯ ∧ u_{ν_{+}}^{1−τ}) ∈ L^{×} ⊗_{ℤ} (J_{n_{+}})_{H},

where

J_{n_{+}} := \begin{cases}(∏_{i=1}^{i=ν_{+}} I(G_{ℓ_{i}}), & if ν_{+} ≠ 0, \\ ℤ[H], & if ν_{+} = 0, \end{cases}

where G_{ℓ_{i}} is the decomposition group of ℓ_{i} in G (note that since ℓ_{i} splits in L, we have G_{ℓ_{i}} ⊂ H), and (J_{n_{+}})_{H} := J_{n_{+}}/I(H)J_{n_{+}}. We set h_{n} := |Pic(O_{L}[\frac{1}{n}])|. For any element a ∈ K^{×}, following Definition 4.16 we define

N_{H}(a) := ∑_{σ∈H} σa ⊗ σ^{−1} ∈ K^{×} ⊗_{ℤ} ℤ[H]/I(H)J_{n_{+}}.

Note that, since K^{×}/L^{×} is ℤ-torsion-free, the natural map

(L^{×}/{±1}) ⊗_{ℤ} (J_{n_{+}})_{H} ⟶ (K^{×}/{±1}) ⊗_{ℤ} ℤ[H]I(H)J_{n_{+}}

is injective. Our refined Darmon’s conjecture is formulated as follows.

Theorem 6.1. One has

N_{H}(β_{n}) = −2^{ν_{−}}h_{n}R_{n} in (L^{×}/{±1}) ⊗_{ℤ} (J_{n_{+}})_{H}.

Remark 6.2. Let I_{n} be the augmentation ideal of ℤ[Gal(L(μ_{n})/L)]. Note that there is a natural isomorphism

I_{n}^{ν_{+}}/I_{n}^{ν_{+}+1} ⊗_{ℤ} ℤ[\frac{1}{2}] ⟶^{∼} I(H)^{ν_{+}}/I(H)^{ν_{+}+1} ⊗_{ℤ} ℤ[\frac{1}{2}].

It is not difficult to see that the following statement is equivalent to [36, Th. 3.9]:

N_{H}(β_{n}) = −2^{ν_{−}}h_{n}R_{n} in (L^{×}/\{±1\}) ⊗_{ℤ} I(H)^{ν_{+}}/I(H)^{ν_{+}+1} ⊗_{ℤ} ℤ[\frac{1}{2}]

(see [46, Lem. 4.7]). Since there is a natural map (J_{n_{+}})_{H} ⟶ I(H)^{ν_{+}}/I(H)^{ν_{+}+1}, Theorem 6.1 refines [36, Th. 3.9]. Note also that, in the original statement of Darmon’s conjecture, the cyclotomic unit is defined by

α_{n} := ∏_{σ∈Gal(ℚ(μ_{nf})/ℚ(μ_{n}))} σ(1 − ζ_{nf})^{χ(σ)},

whereas our cyclotomic unit is β_{n} = N_{L(μ_{n})/K}(α_{n}). Since cyclotomic units, as Stark elements, lie in real fields, it is natural to consider β_{n}. Thus, modifying the original statement of Darmon’s conjecture in the ‘2-part’, we obtained Theorem 6.1, which does not exclude the ‘2-part’.

Proof of Theorem 6.1. We show that Darmon’s conjecture is a consequence of Conjecture 5.4, and use Corollary 5.17 to prove it. We fit notation in this section into that in §5. Set S := {∞} ∪ {ℓ|nf}. Take a prime v_{0} of ℚ, which divides f. We denote by w_{1} the infinite place of K (and also L) which corresponds to the fixed embedding ℚ ↪ ℂ. For 2 ≤ i ≤ ν_{+} +1, set w_{i} := λ_{i−1}. Let T be a finite set of primes that is disjoint from S and satisfying that O_{K,S,T}^{×} is ℤ-torsion-free. (In the sequel, we refer such a set of primes as ‘T ’.) Since K and L are abelian over ℚ, the Rubin-Stark conjecture for K/ℚ and L/ℚ holds (see Remark 5.3 (iii)). Set V := {∞} and V′ := {∞, ℓ_{1}, …, ℓ_{ν_{+}}}. We denote ϵ_{K,T} = ϵ_{K/ℚ,S,T}^{V} ∈ O_{K,S,T}^{×} and ϵ_{L,T} = ϵ_{L/ℚ,S,T}^{V′} ∈ ⋂_{G/H}^{ν_{+}+1} O_{L,S,T}^{×} for the Rubin-Stark elements, characterized by

λ_{K,S}(ϵ_{K,T}) = θ_{K/ℚ,S,T}^{(1)} (w_{1} − w_{0}),

λ_{L,S}(ϵ_{L,T}) = θ_{L/ℚ,S,T}^{(ν_{+}+1)} ⋀_{1≤i≤ν_{+}+1}(w_{i} − w_{0}).

We take 𝒯 , a finite family of ‘T ’, such that

∑_{T ∈𝒯} a_{T} δ_{T} = 2

for some a_{T} ∈ ℤ[G], where δ_{T} := ∏_{ℓ∈T} (1 − ℓFr_{ℓ}^{−1}) (see [50, Chap. IV, Lem. 1.1]). By [46, Lem. 4.6], we have

(1 − τ) ∑_{T ∈𝒯} a_{T} ϵ_{K,T} = β_{n} in K^{×}/{±1},

(where τ ∈ Gal(L/ℚ) is regarded as an element of Gal(K/ℚ(μ_{n})^{+})) and

(1 − τ) ∑_{T ∈𝒯} a_{T} ϵ_{L,T} = (−1)^{ν_{+}+1}2^{ν_{−}}h_{n}(1 − τ)u_{0} ∧ ⋯ ∧ u_{ν_{+}} in ℚ ⋀_{G/H}^{ν_{+}+1} O_{L,S}^{×}.

As in §5.3, for 1 < i ≤ ν_{+} + 1 we denote by Rec_{i} the homomorphism

Rec_{i} : O_{L,S,T}^{×} ⟶ (J_{n_{+}})_{H} = J_{n_{+}}ℤ[G]/I_{H}J_{n_{+}}ℤ[G]

defined by

Rec_{i}(a) = rec_{λ_{i−1}}(a) − 1 + τ(rec_{λ_{i−1}}(τa) − 1).

⋀_{1<i≤ν_{+}+1} Rec_{i} induces a homomorphism

⋂_{G/H}^{ν_{+}+1} O_{L,S,T}^{×} ⟶ ( ⋂_{G/H}^{1} O_{L,S,T}^{×}) ⊗_{ℤ} (J_{n_{+}})_{H} = O_{L,S,T}^{×} ⊗_{ℤ} (J_{n_{+}})_{H},

which we denote by Rec_{n_{+}}. We compute

(1 − τ) ∑_{T ∈𝒯} a_{T} Rec_{n_{+}}(ϵ_{L,T})

= ∑_{T ∈𝒯} Rec_{n_{+}}(a_{T} (1 − τ)ϵ_{L,T})
= ∑_{T ∈𝒯} ( ⋀_{1≤i≤ν_{+}}(rec_{λ_{i}} − 1))((1 − τ)^{ν_{+}+1}a_{T} ϵ_{L,T})
= ( ⋀_{1≤i≤ν_{+}}(rec_{λ_{i}} − 1))((−1)^{ν_{+}+1}2^{ν_{−}}h_{n}u_{0}^{1−τ} ∧ ⋯ ∧ u_{ν_{+}}^{1−τ})
= (−1)^{ν_{+}+1}2^{ν_{−}}h_{n}R_{n}.

By Corollary 5.17, we have

N_{H}(ϵ_{K,T}) = (−1)^{ν_{+}}Rec_{n_{+}}(ϵ_{L,T})

(note that the map ν in Conjecture 5.4 is the natural inclusion map in this case.) Hence, we have

N_{H}(β_{n}) = (1 − τ) ∑_{T ∈𝒯} a_{T} N_{H}(ϵ_{K,T})
= (−1)^{ν_{+}}(1 − τ) ∑_{T ∈𝒯} a_{T} Rec_{n_{+}}(ϵ_{L,T})
= −2^{ν_{−}}h_{n}R_{n},

as required. □

6.2. Gross’s conjecture for tori. In this section we use Corollary 5.17 to obtain some new evidence in support of the ‘conjecture for tori’ formulated by Gross in [21]. We review the formulation of Gross’s conjecture for tori. We follow [22, Conj. 7.4]. Let k be a global field, and L/k be a quadratic extension. Let L̃/k be a finite abelian extension, which is disjoint to L, and set K := LL̃. Set G := Gal(K/k), and H := Gal(K/L) = Gal(L̃/k). Let τ be the generator of G/H = Gal(L/k). Let S be a non-empty finite set of places of k such that S_{∞}(k) ∪ S_{ram}(K/k) ⊂ S. Let T be a finite set of places of k that is disjoint from S and satisfies that O_{K,S,T}^{×} is ℤ-torsion-free. Let v_{1}, …, v_{r′} be all places in S which split in L. We assume r′ < |S|. Then, by [45, Lem. 3.4 (i)], we see that h_{k,S,T} := |Cl_{S}^{T} (k)| divides h_{L,S,T} := |Cl_{S}^{T} (L)|. Take u_{1}, …, u_{r′} ∈ O_{L,S,T}^{×} such that {u_{1}^{1−τ} , …, u_{r′}^{1−τ} } is a basis of (1 − τ)O_{L,S,T}^{×}, which is isomorphic to ℤ^{⊕r′}, and det(− log |u_{i}^{1−τ} |_{w_{j}})_{1≤i,j≤r′} > 0, where w_{j} is a (fixed) place of L lying above v_{j}. Put W := {v_{1}, …, v_{r′}}. As in §5.3, we define

J_{W} := \begin{cases}(∏_{0<i≤r′} I(G_{i}))ℤ[H], & if W ≠ ∅, \\ ℤ[H], & if W = ∅, \end{cases}

where G_{i} ⊂ H denotes the decomposition group of v_{i}, and I(G_{i}) is the augmentation ideal of ℤ[G_{i}]. Set

R_{S,T} := det(rec_{w_{j}}(u_{i}^{1−τ}) − 1)_{1≤i,j≤r′} ∈ (J_{W})_{H}.

Let χ be the non-trivial character of G/H. The map

ℤ[G] = ℤ[H × G/H] ⟶ ℤ[H]

induced by χ is also denoted by χ. Gross’s tori conjecture is formulated as follows.

Conjecture 6.3.

χ(θ_{K/k,S,T} (0)) = 2^{|S|−1−r′} \frac{h_{L,S,T}}{h_{k,S,T}} R_{S,T} in (J_{W})_{H}.

Remark 6.4. The statement that the equality of Conjecture 6.3 holds in ℤ[H]/I(H)^{r′+1} is equivalent to [22, Conj. 7.4] (if we neglect the sign). Indeed, we see that

R_{S,T} = ((O_{L,S,T}^{×})^{−} : (1 − τ)O_{L,S,T}^{×})R_{H}^{−},

where R_{H}^{−} is the ‘minus-unit regulator’ defined in [22, §7.2] (where our H is denoted by G). Since there is a natural map (J_{W})_{H} → ℤ[H]/I(H)^{r′+1}, Conjecture 6.3 refines [22, Conj. 7.4].

Theorem 6.5. Conjecture 5.4 implies Conjecture 6.3. In particular, Conjecture 6.3 is valid if K is an abelian extension over ℚ or k is a global function field.

Proof. First, note that the Rubin-Stark conjecture for (K/k, S, T, ∅) and (L/k, S, T, W) is true by Remark 5.3 (i) and (ii), respectively. By Conjecture 5.4, we have

θ_{K/k,S,T} (0) = Rec_{W} (ϵ_{L/k,S,T}^{W}) in (J_{W})_{H}(= ℤ[G/H] ⊗_{ℤ} (J_{W})_{H}).

(Note that ν^{−1}(N_{H}(θ_{K/k,S,T} (0))) = θ_{K/k,S,T} (0) in (J_{W})_{H} by [37, Lem. 5.6 (iv)].) Note that χ ∘ Rec_{i} = rec_{w_{i}}((1 − τ)(·)) − 1. So we have

χ(Rec_{W} (ϵ_{L/k,S,T}^{W})) = ( ⋀_{1≤i≤r′}(rec_{w_{i}} − 1))((1 − τ)^{r′}ϵ_{L/k,S,T}^{W}).

We know by the proof of [45, Th. 3.5] that

(1 − τ)^{r′}ϵ_{L/k,S,T}^{W} = 2^{|S|−1−r′} \frac{h_{L,S,T}}{h_{k,S,T}} u_{1}^{1−τ} ∧ ⋯ ∧ u_{r′}^{1−τ} .

Hence we have

χ(θ_{K/k,S,T} (0)) = χ(Rec_{W} (ϵ_{L/k,S,T}^{W}))
= ( ⋀_{1≤i≤r′}(rec_{w_{i}} − 1))((1 − τ)^{r′}ϵ_{L/k,S,T}^{W})

= 2^{|S|−1−r′} \frac{h_{L,S,T}}{h_{k,S,T}} ( ⋀_{1≤i≤r′}(rec_{w_{i}} − 1))(u_{1}^{1−τ} ∧ ⋯ ∧ u_{r′}^{1−τ})

= 2^{|S|−1−r′} \frac{h_{L,S,T}}{h_{k,S,T}} R_{S,T} ,

as required. Having now proved the first claim, the second claim follows directly from Corollary 5.17. □

Remark 6.6. The strongest previous evidence in favour of the conjecture for tori is that obtained by Greither and Kučera in [16, 17], in which it is referred to as the ‘Minus Conjecture’ and studied in a slightly weaker form in order to remove any occurence of the auxiliary set T. More precisely, by using rather different methods they were able to prove that this conjecture was valid in the case that k = ℚ, K = F K^{+} where F is imaginary quadratic of conductor f and class number h_{F} and K^{+}/ℚ is tamely ramified, abelian of exponent equal to an odd prime ℓ and ramified at precisely s primes {p_{i}}_{1≤i≤s} each of which splits in F/ℚ; further, any of the following conditions are satisfied

• s = 1 and ℓ ∤ f [16, Th. 8.8], or

• s = 2, ℓ ∤ fh_{F} and either K^{+}/ℚ is cyclic or p_{1} is congruent to an ℓ-th power modulo p_{2} [16, Th. 8.9], or

• ℓ ≥ 3(s + 1) and ℓ ∤ h_{F} [17, Th. 3.7]. It is straightforward to show that the conjecture for tori implies their ‘Minus conjecture’, using [50, Chap. IV, Lem. 1.1] to eliminate the dependence on ‘T ’ (just as in the proof of Theorem 6.1). The validity of the ‘Minus conjecture’ in the case k = ℚ is thus also a consequence of Theorem 6.5.

7. Higher Fitting ideals of Selmer groups

In this section, we introduce a natural notion of ‘higher relative Fitting ideals’ in §7.1, and then study the higher Fitting ideals of the transposed Selmer group S_{S,T}^{tr} (𝔾_{m/K}). In this way we prove Theorems 1.5 and 1.10 and Corollary 1.14.

7.1. Relative Fitting ideals. In this subsection, we recall the definition of Fitting ideals and also introduce a natural notion of ‘higher relative Fitting ideals’. Suppose that R is a noetherian ring and M is a finitely generated R-module. Take an exact sequence

R^{⊕m} →^{f} R^{⊕n} → M → 0,

and denote by A_{f} the matrix with n rows and m columns corresponding to f. Then for i ∈ ℤ_{≥0} the i-th Fitting ideal of M, denoted by Fitt_{R}^{i}(M), is defined to be the ideal generated by all (n − i) × (n − i) minors of A_{f} if 0 ≤ i < n and R if i ≥ n. In this situation we call A_{f} a relation matrix of M. These ideals do not depend on the choice of the above exact sequence (see [40, Chap. 3]). The usual notation is Fitt_{i,R}(M), but we use the above notation which is consistent with the exterior power ⋀_{R}^{i} M. If we can take a presentation

R^{⊕m} →^{f} R^{⊕n} → M → 0

of M with m = n, then we say M has a quadratic presentation. We now fix a submodule N of M, and non-negative integers a and b. We write ν for the minimal number of generators of N. If b > ν, then we simply set

Fitt_{R}^{(a,b)}(M, N) := Fitt_{R}^{a}(M/N).

However, if b ≤ ν then we consider a relation matrix for M of the form

A = ( \begin{matrix}A_{1} & A_{2} \\ 0 & A_{3} \end{matrix} )

where A_{1} is a relation matrix of N. We suppose that A_{1} is a matrix with n_{1} rows and m_{1} columns and A_{3} is a matrix with n_{2} rows and m_{2} columns. We remove b rows from among the first row to the n_{1}-th row of A to get a matrix A′, and remove a rows from A′ to get A′′. We denote by F (A′′) the ideal generated by all c × c minors of A′′ where c = n_{1} + n_{2} − a − b if c > 0 and F (A′′) = R otherwise. We consider all such A′′ obtained from A and then define the relative Fitting ideal by setting

Fitt_{R}^{(a,b)}(M, N) := ∑_{A′′} F (A′′).

By the standard method using the elementary operations of matrices (see the proof of [40, p.86, Th. 1]), one can show that this sum does not depend on the choice of relation matrix A. The following result gives an alternative characterization of this ideal.

Lemma 7.1. Let X be an R-submodule of M that is generated by (a+b) elements x_{1}, …, x_{a+b} such that the elements x_{1}, …, x_{b} belong to N. Let X be the set of such R-submodules of M. Then we have

Fitt_{R}^{(a,b)}(M, N) = ∑_{X∈X} Fitt_{R}^{0}(M/X).

Proof. If b > ν, both sides equal Fitt_{R}^{a}(M/N), so we may assume b ≤ ν. Let e_{1}, …, e_{n} be the generators of M corresponding to the above matrix A where n = n_{1} + n_{2}. Suppose that A′′ is obtained from A by removing (a + b) rows, from the i_{1}-th row to the i_{a+b}-th row with 1 ≤ i_{1}, …, i_{b} ≤ n_{1}. Let X be a submodule of M generated by e_{i_{1}}, …, e_{i_{a+b}}. Then by definitions X ∈ X and F (A′′) = Fitt_{R}^{0}(M/X). This shows that the left hand side of the equation in Lemma 7.1 is in the right hand side. On the other hand, suppose that X is in X and x_{1}, …, x_{a+b} are generators of X. Regarding e_{1}, …, e_{n_{1}}, x_{1}, …, x_{b}, e_{n_{1}+1}, …, e_{n}, x_{b+1}, …, x_{a+b} as generators of M, we have a relation matrix of M of the form

B = ( \begin{matrix}A_{1} & B_{1} & A_{2} & B_{2} \\ 0 & I_{b} & 0 & 0 \\ 0 & 0 & A_{3} & B_{3} \\ 0 & 0 & 0 & I_{a} \end{matrix} )

where I_{a}, I_{b} are the identity matrices of degree a, b, respectively. Then

C = ( \begin{matrix}A_{1} & B_{1} & A_{2} & B_{2} \\ 0 & 0 & A_{3} & B_{3} \end{matrix} )

is a relation matrix of M/X. Since C is obtained from B by removing (a + b) rows in the way of obtaining A′′ from A, it follows from the definition of the relative Fitting ideal that Fitt_{R}^{0}(M/X) ⊂ Fitt_{R}^{(a,b)}(M, N). This shows the other inclusion. □

In the next result we record some useful properties of higher relative Fitting ideals.

Lemma 7.2.

(i) Fitt_{R}^{(a,b)}(M, N) ⊂ Fitt_{R}^{a+b}(M). (ii) Fitt_{R}^{(a,0)}(M, N) = Fitt_{R}^{a}(M). (iii) Suppose that there exists an exact sequence 0 → M′ → M → R^{⊕r} → 0

of R-modules and that N ⊂ M′. Then one has

Fitt_{R}^{(a,b)}(M, N) = \begin{cases}Fitt_{R}^{(a−r,b)}(M′, N), & if a ≥ r, \\ 0, & otherwise. \end{cases}

(iv) Assume that M/N has a quadratic presentation. Then one has

Fitt_{R}^{(0,b)}(M, N) = Fitt_{R}^{b}(N) Fitt_{R}^{0}(M/N).

Proof. Claims (i), (ii) and (iii) follow directly from the definition of the higher relative Fitting ideal. To prove claim (iv), we consider a relation matrix

A = ( \begin{matrix}A_{1} & A_{2} \\ 0 & A_{3} \end{matrix} )

as above, where A_{1} is a matrix with n_{1} rows and A_{3} is a square matrix of n_{2} rows. Put n = n_{1} + n_{2}. Then a matrix A′′ obtained from A as above is of the form

A′′ = ( \begin{matrix}A′′_{1} & A′′_{2} \\ 0 & A_{3} \end{matrix} ) .

This is a matrix with (n − b) rows and so a nonzero (n − b) × (n − b) minor of A′′ must be det(A_{3}) times a (n_{1} − b) × (n_{1} − b) minor of A′′_{1}. This implies the required conclusion. □

7.2. Statement of the conjecture. Let K/k, G, S, T, V be as in §5.1. For the element ϵ_{K/k,S,T}^{V}, the Rubin-Stark conjecture asserts that Φ(ϵ_{K/k,S,T}^{V}) belongs to ℤ[G] for every Φ in ⋀_{G}^{r} Hom_{G}(O_{K,S,T}^{×} , ℤ[G]). We next formulate a much stronger conjecture which describes the arithmetic significance of the ideal generated by the elements Φ(ϵ_{K/k,S,T}^{V}) when Φ runs over ⋀_{G}^{r} Hom_{G}(O_{K,S,T}^{×} , ℤ[G]).

Conjecture 7.3. One has an equality

Fitt_{G}^{r}(S_{S,T} (𝔾_{m/K})) = {Φ(ϵ_{K/k,S,T}^{V})^{#} : Φ ∈ ⋀_{G}^{r} Hom_{G}(O_{K,S,T}^{×} , ℤ[G])},

or equivalently (by Lemma 2.8),

Fitt_{G}^{r}(S_{S,T}^{tr} (𝔾_{m/K})) = {Φ(ϵ_{K/k,S,T}^{V}) : Φ ∈ ⋀_{G}^{r} Hom_{G}(O_{K,S,T}^{×} , ℤ[G])}.

The following result shows that this conjecture refines the first half of the statement of Conjecture 5.9.

Proposition 7.4. For a finite set Σ of places, we put J_{Σ} = ∏_{v∈Σ} I(G_{v})ℤ[G]. Assume Conjecture 7.3 is valid. Then, for every Φ ∈ ⋀_{G}^{r} Hom_{G}(O_{K,S,T}^{×} , ℤ[G]) and v ∈ S \ V, one has

Φ(ϵ_{K/k,S,T}^{V}) ∈ J_{S\(V ∪{v})}.

Proof. It is sufficient to show that Fitt_{G}^{r}(S_{S,T}^{tr} (𝔾_{m/K})) ⊂ J_{S\(V ∪{v})}. Since there is a canonical surjective homomorphism

S_{S,T}^{tr} (𝔾_{m/K}) ⟶ X_{K,S} ≃ ℤ[G]^{⊕r} ⊕ X_{K,S\V} ,

we have

Fitt_{G}^{r}(S_{S,T}^{tr} (𝔾_{m/K})) ⊂ Fitt_{G}^{r}(X_{K,S}) = Fitt_{G}^{0}(X_{K,S\V}).

The existence of the surjective homomorphism X_{K,S\V} → Y_{K,S\(V ∪{v})} implies that Fitt_{G}^{0}(X_{K,S\V}) ⊂ Fitt_{G}^{0}(Y_{K,S\(V ∪{v})}) = J_{S\(V ∪{v})}. This completes the proof. □

7.3. The leading term conjecture implies Conjecture 7.3. The following result combines with Lemma 2.8 to imply the statement of Theorem 1.5(i).

Theorem 7.5. LTC(K/k) implies Conjecture 7.3. In particular, Conjecture 7.3 is valid if either K is an abelian extension over ℚ or k is a function field or [K : k] ≤ 2.

Proof. The second claim is a consequence of Remark 3.3. To prove the first claim we assume the validity of LTC(K/k). Then the module P that occurs in the exact sequence (24) is free of rank d, as we noted before.

Hence we may assume P = F. Let z_{b} ∈ ⋀_{G}^{d} F be as in §5.6. By LTC(K/k), z_{b} is a G-basis of ⋀_{G}^{d} F. Write z_{b} as

z_{b} = x ⋀_{1≤i≤d} b_{i}

with some x ∈ ℤ[G]^{×}. By Theorem 5.14 and Proposition 4.1, we have

ϵ_{K/k,S,T}^{V} = ±x ∑_{σ∈𝔖_{d,r}} sgn(σ) det(ψ_{i}(b_{σ(j)}))_{r<i,j≤d}b_{σ(1)} ∧ ⋯ ∧ b_{σ(r)}.

Take Φ ∈ ⋀_{G}^{r} Hom_{G}(O_{K,S,T}^{×} , ℤ[G]). Since F/O_{K,S,T}^{×} ≃ im(ψ) ⊂ F is ℤ-torsion-free, we know by Lemma 4.7(ii) that the map

Hom_{G}(F, ℤ[G]) ⟶ Hom_{G}(O_{K,S,T}^{×} , ℤ[G])

induced by the inclusion O_{K,S,T}^{×} ⊂ F is surjective. Hence, we can take a lift Φ̃ of Φ to ⋀_{G}^{r} Hom_{G}(F, ℤ[G]). We have

Φ(ϵ_{K/k,S,T}^{V}) = ±x ∑_{σ∈𝔖_{d,r}} sgn(σ) det(ψ_{i}(b_{σ(j)}))_{r<i,j≤d}Φ̃(b_{σ(1)} ∧ ⋯ ∧ b_{σ(r)})
∈ ⟨det(ψ_{i}(b_{σ(j)}))_{r<i,j≤d} : σ ∈ 𝔖_{d,r}⟩_{G}.

We consider the matrix A corresponding to the presentation

F → F → S_{S,T}^{tr} (𝔾_{m/K}) → 0

which comes from the exact sequence (24). By Lemma 5.20, ψ_{i} = 0 for 1 ≤ i ≤ r. If we write elements in F as column vectors, this implies that the i-th row of A is zero for all i such that 1 ≤ i ≤ r. Hence we have

Fitt_{G}^{r}(S_{S,T}^{tr} (𝔾_{m/K})) = ⟨det(ψ_{i}(b_{σ(j)}))_{r<i,j≤d} : σ ∈ 𝔖_{d,r}⟩_{G}.

Therefore, we get an inclusion

{Φ(ϵ_{K/k,S,T}^{V}) : Φ ∈ ⋀_{G}^{r} Hom_{G}(O_{K,S,T}^{×} , ℤ[G])} ⊂ Fitt_{G}^{r}(S_{S,T}^{tr} (𝔾_{m/K})).

We obtain the reverse inclusion from

(b_{σ(1)}^{∗} ∧ ⋯ ∧ b_{σ(r)}^{∗})(ϵ_{K/k,S,T}^{V}) = ±x det(ψ_{i}(b_{σ(j)}))_{r<i,j≤d}

and the fact that x is a unit in ℤ[G]. □

7.4. The proof of Theorem 1.10. For any G-module M we write M^{∗} for the linear dual Hom_{ℤ}(M, ℤ) endowed with the natural contragredient action of G. We also set V′ := V ∪ {v}. We start with a useful technical observation.

Lemma 7.6. For each integer i with 1 ≤ i ≤ r let φ_{i} be an element of (O_{K,S,T}^{×})^{∗}. Then for any given integer N there is a subset {φ′_{i} : 1 ≤ i ≤ r} of (O_{K,S,T}^{×})^{∗} which satisfies the following properties.

(i) For each i one has φ′_{i} ≡ φ_{i} modulo N · (O_{K,S,T}^{×})^{∗}.

(ii) The image in (O_{K,V′,T}^{×})^{∗} of the submodule of (O_{K,S,T}^{×})^{∗} that is gener-

ated by the set {φ′_{i} : 1 ≤ i ≤ r} is free of rank r.

Proof. Our choice of V implies that we may choose a free G-submodule F of (O_{K,V′,T}^{×})^{∗} of rank r. We then choose a subset {f_{i} : 1 ≤ i ≤ r} of (O_{K,S,T}^{×})^{∗} which the natural surjection ρ : (O_{K,S,T}^{×})^{∗} → (O_{K,V′,T}^{×})^{∗} sends to a basis of F. For any integer m we set φ_{i,m} := φ_{i} + mNf_{i} and note it suffices to show that for any sufficiently large m the elements {ρ(φ_{i,m}) : 1 ≤ i ≤ r} are linearly independent over ℚ[G]. Consider the composite homomorphism of G-modules F → ℚ(O_{K,V′,T}^{×})^{∗} → ℚF where the first arrow sends each ρ(f_{i}) to ρ(φ_{i,m}) and the second is induced by a choice of ℚ[G]-equivariant section to the projection ℚ(O_{K,V′,T}^{×})^{∗} → ℚ((O_{K,V′,T}^{×})^{∗}/F). Then, with respect to the basis {ρ(f_{i}) : 1 ≤ i ≤ r}, this linear map is represented by a matrix of the form A + mNI_{r} for a matrix A in M_{r}(ℚ[G]) that is independent of m. In particular, if m is large enough to ensure that −mN is not an eigenvalue of e_{χ}A for any χ, then the composite homomorphism is injective and so the elements {ρ(φ_{i,m}) : 1 ≤ i ≤ r} are linearly independent over ℚ[G], as required. □

For each integer i with 1 ≤ i ≤ r let φ_{i} be an element of (O_{K,S,T}^{×})^{∗}. Then, for any non-zero integer N which belongs to Fitt_{G}^{0}(Cl(K)) we choose homomorphisms φ′_{i} as in Lemma 7.6. Then the congruences in Lemma 7.6(i) imply that

( ⋀_{1≤i≤r} φ_{i})(ϵ_{K/k,S,T}^{V}) ≡ ( ⋀_{1≤i≤r} φ′_{i})(ϵ_{K/k,S,T}^{V}) modulo N · ℤ[G].

Given this, Lemma 7.6(ii) implies that Theorem 1.10 is true provided that it is true for all Φ of the form ⋀_{1≤i≤r}φ_{i} where the images in (O_{K,V′,T}^{×})^{∗} of the homomorphisms φ_{i} span a free module of rank r. We shall therefore assume in the sequel that Φ is of this form. For each index i we now choose a lift φ̃_{i} of φ_{i} to S_{S,T} (𝔾_{m/K}) and then write E_{Φ} for the G-module that is generated by the set {φ̃_{i} : 1 ≤ i ≤ r}.

Proposition 7.7. If LTC(K/k) is valid, then for every Φ as above one has

Φ(ϵ_{K/k,S,T}^{V})^{#} ∈ Fitt_{G}^{0}(S_{S,T} (𝔾_{m/K})/E_{Φ}).

Proof. We use the existence of an exact triangle in D^{p}(ℤ[G]) of the form

(29)  ℤ[G]^{⊕r,•} ⟶^{θ} RΓ_{c,T} ((O_{K,S})_{W}, ℤ) ⟶^{θ′} C^{•} → ℤ[G]^{⊕r,•}[1].

Here ℤ[G]^{⊕r,•} denotes the complex ℤ[G]^{⊕r}[−1] ⊕ ℤ[G]^{⊕r}[−2] and, after choosing an ordering {v_{i} : 1 ≤ i ≤ r} of the places in V, the morphism θ is uniquely specified by the condition that H^{1}(θ) sends each element b_{i} of the canonical basis {b_{i} : 1 ≤ i ≤ r} of ℤ[G]^{⊕r} to w_{i}^{∗} in (Y_{K,V})^{∗} ⊂ (X_{K,S})^{∗} = H_{c,T}^{1} ((O_{K,S})_{W}, ℤ) and H^{2}(θ) sends each b_{i} to φ̃_{i} in S_{S,T} (𝔾_{m/K}). Note that the long exact cohomology sequence of this triangle implies C^{•} is acyclic outside degrees one and two and identifies H^{1}(C^{•}) and H^{2}(C^{•}) with (X_{K,S\V})^{∗} and S_{S,T} (𝔾_{m/K})/E_{Φ}, respectively.

In particular, if we now write e_{r} for the idempotent of ℚ[G] obtained as ∑_{r_{χ,S}=r} e_{χ}, then the space e_{r}ℚH^{i}(C^{•}) vanishes for both i = 1 and i = 2. We may therefore choose a commutative diagram of ℝ[G]-modules (30)

0 ⟶ ℝ[G]^{⊕r} ⟶^{H^{1}(θ)} ℝH_{c,T}^{1} ((O_{K,S})_{W}, ℤ) ⟶^{H^{1}(θ′)} ℝH^{1}(C^{•}) ⟶ 0
λ_{1}↓  ^{λ_{2}}↓  ^{λ_{3}}↓
0 ⟶ ℝ[G]^{⊕r} ⟶^{H^{2}(θ)} ℝH_{c,T}^{2} ((O_{K,S})_{W}, ℤ) ⟶^{H^{2}(θ′)} ℝH^{2}(C^{•}) ⟶ 0

such that e_{r}λ_{2} = e_{r}λ_{K,S}^{∗}. This diagram combines with the triangle (29) to imply that there is an equality of lattices

(31)  ϑ_{λ_{2}}(det_{G}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ)))^{−1} = det(λ_{1}) · ϑ_{λ_{3}}(det_{G}(C^{•}))^{−1}.

We now assume that the conjecture LTC(K/k) is valid. Then Proposition 3.4 implies that det_{G}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ))^{−1} is a free rank one ℤ[G]-module and further that if we choose any basis ξ for this module, then both e_{r}ξ and e_{r}θ_{K/k,S,T}^{∗} (0)^{#} = θ_{K/k,S,T}^{(r),#} are bases of the e_{r}ℤ[G]-module

e_{r}ϑ_{λ_{2}}(det_{G}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ)))^{−1} = e_{r}ϑ_{λ_{K,S}^{∗}}(det_{G}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ)))^{−1}.

Bass’s Theorem (cf. [32, Chap. 7, (20.9)]) implies that for each prime p the projection map ℤ_{(p)}[G]^{×} → e_{r}ℤ_{(p)}[G]^{×} is surjective. The above equality thus implies that the ℤ_{(p)}[G]-module ϑ_{λ_{2}}(det_{G}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ)))^{−1} ⊗_{ℤ} ℤ_{(p)} has a basis ξ_{p} for which one has e_{r}ξ_{p} = e_{r}θ_{K/k,S,T}^{∗} (0)^{#} = θ_{K/k,S,T}^{(r),#} . For each prime p the equality (31) therefore implies that

e_{r}ϑ_{λ_{3}}(det_{G}(C^{•}))^{−1} ⊗_{ℤ} ℤ_{(p)} (32)

= det(λ_{1})^{−1}e_{r}ϑ_{λ_{2}}(det_{G}(RΓ_{c,T} ((O_{K,S})_{W}, ℤ)))^{−1} ⊗_{ℤ} ℤ_{(p)}
= ℤ_{(p)}[G] · e_{r}det(λ_{1})^{−1}θ_{K/k,S,T}^{(r),#} .

Now the commutativity of (30) implies that e_{r}det(λ_{1}) is equal to the determinant of the matrix which represents e_{r}λ_{K,S}^{∗} with respect to the bases {e_{r}w_{i}^{∗} : 1 ≤ i ≤ r} and {e_{r}φ_{i} : 1 ≤ i ≤ r} and hence that

e_{r} ⋀_{1≤i≤r} λ_{K,S}^{∗}(w_{i}^{∗}) = e_{r}det(λ_{1})Φ.

Since the element ϵ_{K/k,S,T}^{V} is defined via the equality

θ_{K/k,S,T}^{(r)} ⋀_{1≤i≤r} λ_{K,S}^{−1}(w_{i} − w) = ϵ_{K/k,S,T}^{V} ,

one therefore has

Φ(ϵ_{K/k,S,T}^{V})^{#}
(33) = (e_{r}det(λ_{1}))^{−1} ⋀_{1≤i≤r} λ_{K,S}^{∗}(w_{i}^{∗})(θ_{K/k,S,T}^{(r),#} ( ⋀_{1≤i≤r} λ_{K,S}^{−1}(w_{i} − w)))
= (e_{r}det(λ_{1}))^{−1}θ_{K/k,S,T}^{(r),#}
∈ e_{r}ϑ_{λ_{3}}(det_{G}(C^{•}))^{−1} ⊗_{ℤ} ℤ_{(p)}

where the last containment follows from (32). Now by the same reasoning as used in the proof of Lemma 2.8, we know that the p-localized complex ℤ_{(p)} ⊗C^{•} is represented by a complex P ⟶^{δ} P, where P is a finitely generated free ℤ_{(p)}[G]-module and the first term is placed in degree one. In particular, since for any character χ of G the space e_{χ}ℂH^{1}(C^{•}) = e_{χ}ℂ ker(δ) does not vanish if e_{χ}e_{r} = 0, one has

(34) e_{r}ϑ_{λ_{3}}(det_{G}(C^{•}))_{(p)}^{−1} = ℤ_{(p)}[G]e_{r}det(δ)
= ℤ_{(p)}[G]det(δ)
⊂ Fitt_{ℤ_{(p)}[G]}^{0}((H_{c,T}^{2} ((O_{K,S})_{W}, ℤ)/E_{Φ}) ⊗_{ℤ} ℤ_{(p)})
= Fitt_{G}^{0}(H_{c,T}^{2} ((O_{K,S})_{W}, ℤ)/E_{Φ}) ⊗_{ℤ} ℤ_{(p)}.

The inclusion here follows from the tautological exact sequence

P ⟶^{δ} P ⟶ H^{2}(ℤ_{(p)} ⊗ C^{•}) ⟶ 0

and the identification H^{2}(ℤ_{(p)} ⊗ C^{•}) = (H_{c,T}^{2} ((O_{K,S})_{W}, ℤ)/E_{Φ}) ⊗_{ℤ} ℤ_{(p)}. The claimed result now follows from (33) and (34). □

Now we proceed to the proof of Theorem 1.10. The existence of a surjective homomorphism of G-modules f : S_{S,T} (𝔾_{m/K}) → S_{V′∪S_{∞},T} (𝔾_{m/K}) (see Proposition 2.4(ii)) combines with Proposition 7.7 to imply that

(35)  Φ(ϵ_{K/k,S,T}^{V})^{#} ∈ Fitt_{G}^{0}(S_{V′∪S_{∞},T} (𝔾_{m/K})/f(E_{Φ})).

This implies the first assertion of Theorem 1.10 since the natural map Cl_{V′}^{T}(K)^{∨} → S_{V′∪S_{∞},T} (𝔾_{m/K}) induces an injection

Cl_{V′}^{T}(K)^{∨} → S_{V′∪S_{∞},T} (𝔾_{m/K})/f(E_{Φ}).

In addition, if G is cyclic, then the latter injection combines with (35) to imply that

Φ(ϵ_{K/k,S,T}^{V}) ∈ Fitt_{G}^{0}(Cl_{V′}^{T}(K)^{∨})^{#} = Fitt_{G}^{0}(Cl_{V′}^{T}(K)),

as claimed by the second assertion of Theorem 1.10. This completes the proof of Theorem 1.10.

7.5. The proof of Corollary 1.14. Let K/k be a CM-extension, S = S_{∞}(k), and p an odd prime. For a ℤ_{p}[G]-module M, we denote by M^{−} the submodule on which the complex conjugation acts as −1. Then, since complex conjugation acts trivially on Hom_{ℤ}(O_{K,S,T}^{×} , ℤ) ⊗ ℤ_{p}, the exact sequence

0 ⟶ Cl^{T} (K)^{∨} ⟶ S_{S,T} (𝔾_{m/K}) ⟶ Hom_{ℤ}(O_{K,S,T}^{×} , ℤ) ⟶ 0,

implies that in this case there is an equality

((Cl^{T} (K) ⊗ ℤ_{p})^{∨})^{−} = (S_{S,T} (𝔾_{m/K}) ⊗ ℤ_{p})^{−}.

In addition, in this case the containment of Proposition 7.7 applies with V empty (so r = 0 and E_{Φ} vanishes) to imply that

θ_{K/k,S,T} (0)^{#} ∈ Fitt_{G}^{0}(S_{S,T} (𝔾_{m/K})),

and hence one has

θ_{K/k,S,T} (0)^{#} ∈ Fitt_{ℤ_{p}[G]}^{0}(((Cl^{T} (K) ⊗ ℤ_{p})^{∨})^{−}).

Since θ_{K/k,S,T} (0) lies in the minus component of ℤ_{p}[G], this is in turn equivalent to the required containment

θ_{K/k,S,T} (0)^{#} ∈ Fitt_{ℤ_{p}[G]}^{0}((Cl^{T} (K) ⊗ ℤ_{p})^{∨}).

This completes the proof of Corollary 1.14.

7.6. The higher relative Fitting ideals of the dual Selmer group. We write M_{tors} for the ℤ-torsion submodule of a G-module M and abbreviate the higher relative Fitting ideal Fitt_{ℤ[G]}^{(a,b)}(M, M_{tors}) to Fitt_{G}^{(a,b)}(M).

In this subsection, we study the ideals Fitt_{G}^{(r,i)}(S_{S,T}^{tr} (𝔾_{m/K})) and, in particular, prove Theorem 1.5(ii). We note that the exact sequence (2) identifies S_{S,T}^{tr} (𝔾_{m/K})_{tors} with the group Cl_{S}^{T} (K). For each non-negative integer i we define the set V_{i} as in Theorem 1.5(ii).

Conjecture 7.8. For each non-negative integer i one has an equality

Fitt_{G}^{(r,i)}(S_{S,T}^{tr} (𝔾_{m/K}))

= {Φ(ϵ_{K/k,S∪V′,T}^{V ∪V′}) : V′ ∈ V_{i} and Φ ∈ ⋀_{G}^{r+i} Hom_{G}(O_{K,S∪V′,T}^{×} , ℤ[G])}.

The following result is a generalization of Theorem 7.5 in §7.3.

Theorem 7.9. If LTC(K/k) is valid, then so is Conjecture 7.8.

Proof. We consider the composition of the two canonical homomorphisms

S_{S,T}^{tr} (𝔾_{m/K}) → X_{K,S} → Y_{K,V} ,

and denote its kernel by S_{S,T}^{tr} (𝔾_{m/K})′. By Lemma 7.2 (iii), we have

(36)  Fitt_{G}^{(r,i)}(S_{S,T}^{tr} (𝔾_{m/K})) = Fitt_{G}^{(0,i)}(S_{S,T}^{tr} (𝔾_{m/K})′).

We also note that the sequence (2) gives rise to an exact sequence of G-modules

(37)  0 ⟶ Cl_{S}^{T} (K) ⟶ S_{S,T}^{tr} (𝔾_{m/K})′ ⟶ X_{K,S\V} ⟶ 0.

For V′ ∈ V_{i}, we denote by S_{S∪V′,T}^{tr} (𝔾_{m/K})′ the kernel of the natural composition

S_{S∪V′,T}^{tr} (𝔾_{m/K}) → X_{K,S∪V′} → Y_{K,V ∪V′}

so that the following sequence is exact

0 ⟶ Cl_{S∪V′}^{T}(K) ⟶ S_{S∪V′,T}^{tr} (𝔾_{m/K})′ ⟶ X_{K,S\V} ⟶ 0.

Let X_{V′} be the subgroup of Cl_{S}^{T} (K) generated by the classes of places of K above V′ in Cl_{S}^{T} (K). Since Cl_{S}^{T} (K)/X_{V′} = Cl_{S∪V′}^{T}(K), there is an isomorphism S_{S,T}^{tr} (𝔾_{m/K})′/X_{V′} ≃ S_{S∪V′,T}^{tr} (𝔾_{m/K})′. By Chebotarev density theorem and Lemma 7.1, we obtain

Fitt_{G}^{(0,i)}(S_{S,T}^{tr} (𝔾_{m/K})′) = ∑_{V′∈V′_{i}} Fitt_{G}^{0}(S_{S∪V′,T}^{tr} (𝔾_{m/K})′)
(38)  = ∑_{V′∈V′_{i}} Fitt_{G}^{r+i}(S_{S∪V′,T}^{tr} (𝔾_{m/K}))

where we used Lemma 7.2 (iii) again to get the last equality. Now Theorem 7.9 follows from (36), (38) and Theorem 7.5. □

Corollary 7.10. We assume that LTC(K/k) is valid and that the group G = Gal(K/k) is cyclic. Then for each non-negative integer i one has an equality

Fitt_{G}^{i}(Cl_{S}^{T} (K)) Fitt_{G}^{0}(X_{K,S\V})

= {Φ(ϵ_{K/k,S∪V′,T}^{V ∪V′}) : V′ ∈ V_{i} and Φ ∈ ⋀_{G}^{r+i} Hom_{G}(O_{K,S∪V′,T}^{×} , ℤ[G])}.

Proof. Since G is cyclic, the G-module X_{K,S\V} has a quadratic presentation. We may therefore apply Lemma 7.2(iv) to the exact sequence (37) to obtain an equality

Fitt_{G}^{i}(Cl_{S}^{T} (K)) Fitt_{G}^{0}(X_{K,S\V}) = Fitt_{G}^{(0,i)}(S_{S,T}^{tr} (𝔾_{m/K})′).

Given this equality, the claimed result follows from Theorem 7.9 and the equality (36). □

An application of Theorem 7.9 to character components of ideal class groups will be given in §8.

8. Higher Fitting ideals of character components of class groups

In this section, as an application of Theorem 7.9, we study the higher Fitting ideals of character components of class groups.

8.1. General abelian extensions. We suppose that K/k is a finite abelian extension as in §7. We take and fix an odd prime p in this section. We put A_{S}^{T} (K) = Cl_{S}^{T} (K) ⊗ ℤ_{p}, A^{T} (K) = Cl^{T} (K) ⊗ ℤ_{p}, and A(K) = Cl(K) ⊗ ℤ_{p}. We take a character χ of G = Gal(K/k). Throughout this section, we assume that the order of χ is prime to p. We decompose G = Δ_{K} × Γ_{K} where |Δ_{K}| is prime to p and Γ_{K} is a p-group. By our assumption χ is regarded as a character of Δ_{K}. For any ℤ_{p}[Δ_{K}]-module M, we define the χ-component M^{χ} by setting

M^{χ} := M ⊗_{ℤ_{p}[Δ_{K}]} O_{χ}

where O_{χ} = ℤ_{p}[im(χ)] on which Δ_{K} acts via χ. This is an exact functor from the category of ℤ_{p}[G]-modules to that of O_{χ}[Γ_{K}]-modules. Let k_{χ} be the subfield of K corresponding to the kernel of χ, namely, χ induces a faithful character of Gal(k_{χ}/k). Put K_{(Δ)} := K^{Γ_{K}}, then k_{χ} ⊂ K_{(Δ)}. We also put Δ_{K,χ} := Gal(K_{(Δ)}/k_{χ}) which is a subgroup of Δ_{K}. We consider K(χ) := K^{Δ_{K,χ}}, then Gal(K(χ)/k_{χ}) = Γ_{K}. We consider A_{S}^{T} (K)^{χ} which we regard as an O_{χ}[Γ_{K}]-module. By the standard norm argument, we know the natural map A_{S}^{T} (K(χ))^{χ} → A_{S}^{T} (K)^{χ} is bijective, so when we consider the χcomponent A_{S}^{T} (K)^{χ}, we may assume that χ is a faithful character of Δ_{K} by replacing K with K(χ). In the following, we assume this. We write χ(v) ≠ 1 if the decomposition group of Δ_{K} at v is non-trivial. We denote the χ-component of ϵ_{K/k,S,T}^{V} by ϵ_{K/k,S,T}^{V,χ} ∈ ((⋂_{G}^{r} O_{K,S,T}^{×}) ⊗ ℤ_{p})^{χ}. Let V_{i} be the set as in Theorem 1.5(ii) for i ≥ 0. Finally we assume that the following condition is satisfied

(∗) any ramifying place v of k in K does not split completely in K_{(Δ)}.

Theorem 8.1. Let V be the set of the archimedean places of k that split completely in K and set r := |V |. We assume that χ ≠ 1 is a faithful character of Δ_{K}, and consider the χ-component of the class group A^{T} (K)^{χ} which is an O_{χ}[Γ_{K}]-module. We assume that the χ-component of LTC(K/k) is valid and that the condition (∗) is satisfied. Then for any non-negative integer i one has an equality

Fitt_{O_{χ}[Γ_{K}]}^{i}(A^{T} (K)^{χ}) = {Φ(ϵ_{K/k,S∪V′,T}^{V ∪V′,χ}) : V′ ∈ V_{i} and Φ ∈ ⋀_{O_{χ}[Γ_{K}]}^{r+i} H_{χ}}

where S = S_{∞}(k) ∪ S_{ram}(K/k) and H_{χ} = Hom_{O_{χ}[Γ_{K}]}((O_{K,S∪V′,T}^{×} ⊗ ℤ_{p})^{χ}, O_{χ}[Γ_{K}]).

Proof. Since v ∈ S_{ram}(K/k) does not split completely in K_{(Δ)}, one has χ(v) ≠ 1 and hence (Y_{K,S_{ram}} ⊗ ℤ_{p})^{χ} = 0. As χ ≠ 1, we therefore also have (X_{K,S_{ram}} ⊗ℤ_{p})^{χ} = (Y_{K,S_{ram}} ⊗ℤ_{p})^{χ} = 0. Hence (X_{K,S} ⊗ ℤ_{p})^{χ} = (Y_{K,S_{∞}} ⊗ ℤ_{p})^{χ} is isomorphic to O_{χ}[Γ_{K}]^{⊕r}. This implies that

Fitt_{O_{χ}[Γ_{K}]}^{(r,i)}((S_{S,T}^{tr} (𝔾_{m/K}) ⊗ ℤ_{p})^{χ}), A_{S}^{T} (K)^{χ}) = Fitt_{O_{χ}[Γ_{K}]}^{i}(A_{S}^{T} (K)^{χ})

and so the claim follows from Theorem 7.9. □

In the case K = k_{χ}, the condition (∗) is automatically satisfied. We denote the group A^{T} (k_{χ})^{χ} by (A^{T})^{χ}, which is determined only by χ.

Corollary 8.2. Let χ be a non-trivial linear character of k of order prime to p, and V the set of the archimedean places of k that split completely in k_{χ} and set r := |V |. We assume the χ-component of LTC(k_{χ}/k) to be valid. Then for any non-negative integer i one has an equality

Fitt_{O_{χ}}^{i}((A^{T})^{χ}) = {Φ(ϵ_{k_{χ}/k,S∪V′,T}^{V ∪V′,χ}) : V′ ∈ V_{i} and Φ ∈ ⋀_{O_{χ}}^{r+i} H_{χ}}

where S = S_{∞}(k) ∪ S_{ram}(k_{χ}/k) and H_{χ} = Hom_{O_{χ}}((O_{k_{χ},S∪V′,T}^{×} ⊗ ℤ_{p})^{χ}, O_{χ}).

8.2. The order of character components in CM abelian extensions. In this subsection, we assume that k is totally real, K is a CM-field, and χ is an odd character. In this case, we can compute the right hand side of Theorem 8.1 more explicitly. First of all, note that r = 0 in this case. We first consider the case K = k_{χ} and i = 0. When S = S_{∞}(k) ∪ S_{ram}(k_{χ}/k), we denote the L-function L_{k,S,T} (χ^{−1}, s) by L_{k}^{T} (χ^{−1}, s). When T is empty, we denote L_{k}^{T} (χ^{−1}, s) by L_{k}(χ^{−1}, s). In this case, we know

ϵ_{k_{χ}/k,S,T}^{∅,χ} = θ_{k_{χ}/k,S,T} (0)^{χ} = L_{k}^{T} (χ^{−1}, 0)

(see §5.3). Therefore, Corollary 8.2 with i = 0 implies

Corollary 8.3. Let k be totally real, and χ a one dimensional odd character of k of order prime to p. We assume the χ-component of LTC(k_{χ}/k) to be valid.

(i) One has |(A^{T})^{χ}| = |O_{χ}/L_{k}^{T} (χ^{−1}, 0)|.

(ii) Let Cl(k_{χ}) be the ideal class group of k_{χ}, A(k_{χ}) = Cl(k_{χ}) ⊗ ℤ_{p}, and A^{χ} = A(k_{χ})^{χ}. We denote by ω the Teichmüller character giving the Galois action on μ_{p}, the group of p-th roots of unity, and by μ_{p^{∞}}(k(μ_{p})) the group of roots of unity of p-power order in k(μ_{p}). Then one has

|A^{χ}| = \{ \begin{matrix}|O_{χ}/L_{k}(χ^{−1}, 0)| & if χ ≠ ω, \\ |O_{χ}/(|μ_{p^{∞}}(k(μ_{p}))|L_{k}(χ^{−1}, 0))| & if χ = ω. \end{matrix}

Proof. Claim (i) is an immediate consequence of Corollary 8.2 and a remark before this corollary. We shall now prove claim (ii). When χ ≠ ω, we take a finite place v such that v is prime to p and Nv ≢ χ(Fr_{v}) (mod p). We put T = {v}. Then (A^{T})^{χ} = A^{χ} and ord_{p}L_{k}^{T} (χ^{−1}, 0) = ord_{p}L_{k}(χ^{−1}, 0). Therefore, claim (i) implies the equality in claim (ii). When χ = ω, using Chebotarev density theorem we take a finite place v such that v splits completely in k_{χ} = k_{ω} = k(μ_{p}) and ord_{p}|μ_{p^{∞}}(k_{χ})| = ord_{p}(Nv −1). We take T = {v}, then we also have (A^{T})^{χ} = A^{χ} from the exact sequence

μ_{p^{∞}}(k_{χ}) ⟶ (⨁_{w|v} κ(w)^{×} ⊗ ℤ_{p})^{χ} ⟶ (A^{T})^{χ} ⟶ A^{χ} ⟶ 0

where w runs over all places of k_{χ} above v. Therefore, claim (ii) follows from claim (i) in this case, too. □

By combining the argument of Corollary 8.3 with the result of Theorem 1.16 one also directly obtains the following result.

Corollary 8.4. Assume that at most one p-adic place 𝔭 of k satisfies χ(𝔭) = 1. Then the same conclusion as Corollary 8.3 holds.

Remark 8.5. We note that the formula on A^{χ} in Corollary 8.3 has not yet been proved in general even in such a semi-simple case (namely the case that the order of χ is prime to p). If no p-adic place 𝔭 satisfies χ(𝔭) = 1, this is an immediate consequence of the main conjecture proved by Wiles [54]. Corollary 8.4 shows that this holds even if the set {𝔭 : p-adic place of k with χ(𝔭) = 1} has cardinality one.

8.3. The structure of the class group of a CM field. Now we consider a general CM-field K over a totally real number field k (in particular, we do not assume that K = k_{χ}). We assume the condition (∗) stated just prior to Theorem 8.1. We fix a strictly positive integer N. Suppose that v is a place of k such that v is prime to p, v splits completely in K and there is a cyclic extension F (v)/k of degree p^{N}, which is unramified outside v and in which v is totally ramified. (Note that F (v) is not unique.) We denote by S(K) the set of such places v and recall that S(K) is infinite (see [30, Lem. 3.1]). Suppose now that V = {v_{1}, …, v_{t}} is a subset of S(K) consisting of t distinct places. We take a cyclic extension F (v_{j})/k as above, and put F = F (v_{1}) ⋯ F (v_{t}) the compositum of fields F (v_{j}). In particular, F is totally real. We denote by F_{t,N} the set of all fields F constructed in this way. When t = 0, we define F_{0,N} = {k}. We set

H := Gal(KF/K) ≅ Gal(F/k) ≅ ∏_{j=1}^{t} Gal(F (v_{j})/k),

where the first (restriction) isomorphism is due to the fact that K ∩ F = k and the second to the fact that each extension F (v_{j})/k is totally ramified at v_{j} and unramified at all other places. We fix a generator σ_{j} of Gal(F (v_{j})/k) and set S_{j} := σ_{j} − 1 ∈ ℤ[Gal(KF/k)]. Noting that Gal(KF/k) = G × H where G = Gal(K/k), for each element x of ℤ[Gal(KF/k)] = ℤ[G][H] we write x = ∑ x_{n_{1},...,n_{t}}S_{1}^{n_{1}} ⋯ S_{t}^{n_{t}} where each x_{n_{1},...,n_{t}} belongs to ℤ[G]. We then define a map

φ_{V} : ℤ[Gal(KF/k)] → ℤ/p^{N}[G]

by sending x to x_{1,...,1} modulo p^{N} and we note that this map is a well-defined homomorphism of G-modules. We consider θ_{KF/k,S∪V,T} (0) ∈ ℤ[Gal(KF/k)]. We define Θ_{N,S,T}^{i} (K/k) to be the ideal of ℤ/p^{N}[G] generated by all φ_{V} (θ_{KF/k,S∪V,T} (0)) ∈ ℤ/p^{N}[G] where F runs over F_{t,N} such that t ≤ i. We note that we can compute θ_{KF/k,S∪V,T} (0), and hence also φ_{V} (θ_{KF/k,S∪V,T} (0)), numerically. Taking F = k, we know that θ_{K/k,S,T} (0) mod p^{N} is in Θ_{N,S,T}^{i} (K/k) for any i ≥ 0. We set F_{N} := ⋃_{t≥0} F_{t,N}. For any abelian extension M/k, if S = S_{∞}(k) ∪ S_{ram}(M/k) and T is the empty set, we write θ_{M/k}(0) for θ_{M/k,S,T} (0). We take a character χ of Δ_{K} such that χ ≠ ω, at first. We take S = S_{∞}(k) ∪ S_{ram}(K/k) and T = ∅. In this case, we know that the χ-component θ_{KF/k}(0)^{χ} is integral, namely is in O_{χ}[Γ_{K} × H]. We simply denote the χ-component Θ_{N,S,∅}^{i}(K/k)^{χ} by Θ_{N}^{i}(K/k)^{χ} (⊂ O_{χ}[Γ_{K}]). This ideal Θ_{N}^{i}(K/k)^{χ} coincides

with the higher Stickelberger ideal Θ_{i,K}^{(δ,N),χ} defined in [30, §8.1]. When χ = ω, we assume that K = k(μ_{p^{m}}) for some m ≥ 1. By using the Chebotarev density theorem we can choose a place v which satisfies all of the following conditions

(i) v splits completely in k(μ_{p})/k,

(ii) each place above v of k(μ_{p}) is inert in K/k(μ_{p}), and

(iii) each place w of K above v satisfies ord_{p}|μ_{p^{∞}}(K)| = ord_{p}(Nw − 1).

We set T := {v}. We consider the ω-component Θ_{N,S,{v}}^{i}(K/k)^{ω}, which we denoted by Θ_{N,{v}}^{i}(K/k)^{ω}

Theorem 8.6. Let K/k be a finite abelian extension, K a CM-field, and k totally real. Suppose that χ is an odd faithful character of Δ_{K}, and consider the χ-component of the class group A(K)^{χ} which is an O_{χ}[Γ_{K}]-module. We assume the condition (∗) stated just prior to Theorem 8.1 and the validity of the χ-component of LTC(F K/k) for every field F in F_{N}.

(i) Suppose that χ ≠ ω. For any integer i ≥ 0, we have

Fitt_{O_{χ}[Γ_{K}]/p^{N}}^{i} (A(K)^{χ} ⊗ ℤ/p^{N}) = Θ_{N}^{i}(K/k)^{χ}.

(ii) We assume that K = k(μ_{p^{m}}) for some m ≥ 1. For χ = ω, using a place v as above, we have

Fitt_{O_{ω}[Γ_{K}]/p^{N}}^{i} (A(K)^{ω} ⊗ ℤ/p^{N}) = Θ_{N,{v}}^{i}(K/k)^{ω}

for any i ≥ 0.

Proof. We first prove claim (i). Since the image of θ_{KF/k}(0) in ℤ[G] is a multiple of θ_{K/k}(0), Θ_{N}^{0}(K/k)^{χ} is a principal ideal generated by θ_{K/k}(0)^{χ}. Therefore, this theorem for i = 0 follows from Theorem 8.1. Now suppose that i > 0. For a place v ∈ S(K), we take a place w of K above v. Put H(v) = Gal(F (v)K/K) = Gal(F (v)/k) ≃ ℤ/p^{N}. We take a generator σ_{v} of H(v) and fix it. We define ϕ_{v} by

ϕ_{v} : K^{×} ⟶^{Rec_{v}} (I(H(v))ℤ[Gal(F (v)K/k)]/I(H(v))^{2}ℤ[Gal(F (v)K/k)])
= ℤ[G] ⊗_{ℤ} I(H(v))/I(H(v))^{2} ≃ ℤ/p^{N}[G]

Here, the last isomorphism is defined by σ_{v} − 1 ↦ 1, and Rec_{v} is defined by

Rec_{v}(a) = ∑_{τ∈G} τ^{−1}(rec_{w}(τa) − 1)

as in §5.3 by using the reciprocity map rec_{w} : K_{w}^{×} → H(v) at w. Taking the χ-component of ϕ_{v}, we obtain

ϕ_{v} : (K^{×} ⊗ ℤ/p^{N})^{χ} ⟶ O_{χ}[Γ_{K}]/p^{N},

which we also denote by ϕ_{v}. We take S = S_{∞}(k) ∪ S_{ram}(K/k), T = ∅, and V = {v_{1}, …, v_{i}} ∈ V_{i}. Suppose that Φ = φ_{1} ∧ ⋯ ∧ φ_{i} where

φ_{j} ∈ H_{χ} = Hom_{O_{χ}[Γ_{K}]}((O_{K,S∪V}^{×} ⊗ ℤ_{p})^{χ}, O_{χ}[Γ_{K}]/p^{N})

for j = 1, …, i. We take a place w_{j} of K above v_{j} for j such that 1 ≤ j ≤ i. We denote by [w_{j}] the class of w_{j} in A(K)^{χ}. By [30, Lem. 10.1], for each integer j = 1,...,i we can choose a place v′_{j} ∈ S(K) that satisfies all of the following conditions;

(a) [w′_{j}] = [w_{j}] in A(K)^{χ} where w′_{j} is a place of K above v′_{j},

(b) φ_{j}(x) = ϕ_{v′_{j}}(x) for any x ∈ (O_{K,S∪V}^{×} ⊗ ℤ/p^{N})^{χ},

Here, we used the fact that the natural map (O_{K,S∪V}^{×} ⊗ ℤ/p^{N})^{χ} → (K^{×} ⊗ ℤ/p^{N})^{χ} is injective. Set V′ = {v′_{1}, …, v′_{i}}. By property (b), we have

Φ(ϵ_{K/k,S∪V,∅}^{V,χ}) = (ϕ_{v′_{1}} ∧ ⋯ ∧ ϕ_{v′_{i}})(ϵ_{K/k,S∪V,∅}^{V,χ}).

By property (a), there exists an x_{j} in O_{K,S∪V ∪V′}^{×} whose prime decomposition is (x_{j}) = w_{j}(w′_{j})^{−1} for any j such that 1 ≤ j ≤ i. Put V_{i−1} = {v_{1}, ..., v_{i−1}} and V′_{i} = {v_{1}, ..., v_{i−1}, v′_{i}}. Then

ϵ_{K/k,S∪V,∅}^{V,χ} = ϵ_{K/k,S∪V′_{i},∅}^{V′_{i},χ} + ϵ_{K/k,S∪V_{i−1},∅}^{V_{i−1},χ} ∧ x_{i}

and by using this kind of equation recursively, one deduces that ϵ_{K/k,S∪V,∅}^{V,χ} −

ϵ_{K/k,S∪V′,∅}^{V′,χ} is a sum of elements of the form ϵ_{K/k,S∪W,∅}^{W,χ} ∧ b_{j} with |W | = i − 1.

Now, by induction on i, we know Ψ(ϵ_{K/k,S∪W,∅}^{W,χ}) is in Θ_{N}^{i−1}(K/k)^{χ} for any Ψ in Hom_{O_{χ}[Γ_{K}]}((O_{K,S∪W}^{×} ⊗ ℤ_{p})^{χ}, O_{χ}[Γ_{K}]/p^{N}). Therefore we have

(ϕ_{v′_{1}} ∧⋯∧ϕ_{v′_{i}})(ϵ_{K/k,S∪V,∅}^{V,χ}) ≡ (ϕ_{v′_{1}} ∧⋯∧ϕ_{v′_{i}})(ϵ_{K/k,S∪V′,∅}^{V′,χ}) (mod Θ_{N}^{i−1}(K/k)^{χ}).

Set F = F (v′_{1}) ⋯ F (v′_{i}) and H = Gal(F K/K) = Gal(F/k). Then as in §5.3 we can define Rec_{V′}(ϵ_{K/k,S∪V′,∅}^{V′,χ}) ∈ ℤ[G]⊗(J_{V′})_{H}. Let φ_{V′} : ℤ[G×H] → ℤ/p^{N}[G] be the homomorphism defined before Theorem 8.6 by using the generators σ_{v′_{i}} we fixed. This φ_{V′} induces a homomorphism

ℤ[G × H]/I(H)^{i+1}ℤ[G × H] = ℤ[G] ⊗ ℤ[H]/I(H)^{i+1} → ℤ/p^{N}[G]

and we also denote the composite homomorphism

ℤ[G] ⊗ (J_{V′})_{H} → ℤ[G] ⊗ ℤ[H]/I(H)^{i+1} →^{φ_{V ′}} ℤ/p^{N}[G]

by φ_{V′}. Then by the definitions of these homomorphisms, we have

(ϕ_{v′_{1}} ∧ ⋯ ∧ ϕ_{v′_{i}})(ϵ_{K/k,S∪V′,∅}^{V′,χ}) = φ_{V′}(Rec_{V′}(ϵ_{K/k,S∪V′,∅}^{V′,χ})).

By Conjecture 5.4 which is a theorem under our assumptions (Theorem 5.16), we get

φ_{V′}(Rec_{V′}(ϵ_{K/k,S∪V′,∅}^{V′,χ})) = φ_{V′}(θ_{KF/k}(0)^{χ}).

Combining the above equations, we get

Φ(ϵ_{K/k,S∪V,∅}^{V,χ}) ≡ φ_{V′}(θ_{KF/k}(0)^{χ}) (mod Θ_{N}^{i−1}(K/k)^{χ}).

Since φ_{V′}(θ_{KF/k}(0)^{χ}), Θ_{N}^{i−1}(K/k)^{χ} are in Θ_{N}^{i}(K/k)^{χ}, we get Φ(ϵ_{K/k,S∪V,∅}^{V,χ}) ∈ Θ_{N}^{i}(K/k)^{χ}. It follows from Theorem 8.1 that the left hand side of the equation in Theorem 8.6 (i) is in the right hand side. On the other hand, suppose that F is in F_{t,N} with t ≤ i, and that V = {v_{1}, …, v_{t}} is the set of ramifying place in F/k. As above, by Theorem 5.16 we have

φ_{V} (θ_{KF/k}(0)^{χ}) = φ_{V} (Rec_{V} (ϵ_{K/k,S∪V,∅}^{V,χ})) = (ϕ_{v_{1}} ∧ ⋯ ∧ ϕ_{v_{t}})(ϵ_{K/k,S∪V,∅}^{V,χ}).

Therefore, by Theorem 8.1 we have

φ_{V} (θ_{KF/k}(0)^{χ}) ∈ Fitt_{O_{χ}[Γ_{K}]/p^{N}}^{t} (A(K)^{χ} ⊗ ℤ/p^{N}).

Since Fitt_{O_{χ}[Γ_{K}]/p^{N}}^{t} (A(K)^{χ} ⊗ ℤ/p^{N}) ⊂ Fitt_{O_{χ}[Γ_{K}]/p^{N}}^{i} (A(K)^{χ} ⊗ ℤ/p^{N}), we get

φ_{V} (θ_{KF/k}(0)^{χ}) ∈ Fitt_{O_{χ}[Γ_{K}]/p^{N}}^{i} (A(K)^{χ} ⊗ ℤ/p^{N}).

Thus, the right hand side of the equation in Theorem 8.6 (i) is in the left hand side. We can prove claim (ii) by the same method. The condition on v is used to show the injectivity of the natural homomorphism (O_{K,S∪V,T}^{×} ⊗ ℤ/p^{N})^{ω} → (K^{×} ⊗ ℤ/p^{N})^{ω} with T = {v}. □

Corollary 8.7. Let K/k and χ be as in Theorem 8.6. We assume the condition (∗) stated just prior to Theorem 8.1 and that there is at most one place 𝔭 of k above p such that χ(𝔭) = 1. Then the same conclusion as in Theorem 8.6 holds.

Proof. It suffices to note that, under the stated conditions, Theorem 1.16 implies that the χ-component of LTC(F K/k) is valid. □

To give an example of Corollary 8.7 we suppose that K is the m-th layer of the cyclotomic ℤ_{p}-extension of K_{(Δ)} for some strictly positive integer m, and assume that χ(𝔭) ≠ 1 for any 𝔭 | p. Then this assumption implies that the condition (∗) is satisfied and so all of the assumptions in Corollary 8.7 are satisfied. Therefore, by taking the projective limit of the conclusion, Corollary 8.7 implies the result of the second author in [30, Th. 2.1]. In this sense, Corollary 8.7 is a natural generalization of the main result in [30].

To state our final result we now set

Θ^{i}(K/k)^{χ} = lim_{⟵}_{N}Θ_{N}^{i}(K/k)^{χ} ⊆ O_{χ}[Γ_{K}].

Then Theorem 8.6 implies that Fitt_{O_{χ}[Γ]}^{i}(A(K)^{χ}) = Θ^{i}(K/k)^{χ}.

Let k_{χ} be the field corresponding to the kernel of χ as in Corollary 8.3. We denote Θ^{i}(k_{χ}/k)^{χ} by Θ^{i,χ}, which is an ideal of O_{χ}. For χ = ω, we denote

lim_{⟵}_{N}Θ_{N,{v}}^{i}(k_{χ}/k)^{ω} by Θ^{i,ω}.

Then Corollary 8.7 implies the following result, which is a generalization of the main result of the second author in [28].

Corollary 8.8. Set A^{χ} := (Cl(k_{χ}) ⊗ ℤ_{p})^{χ} as in Corollary 8.3. Assume that there is at most one p-adic place 𝔭 of k such that χ(𝔭) = 1 and that the p-adic Iwasawa μ-invariant of K vanishes. Then there is an isomorphism of O_{χ}-modules of the form A^{χ} ≃ ⨁_{i≥1} Θ^{i,χ}/Θ^{i−1,χ}.

Acknowledgements It is a pleasure for the first author to thank Dick Gross for much encouragement at an early stage of this general project and, in addition, to thank Cornelius Greither for stimulating discussions. The second author would like to thank Cornelius Greither for discussions with him on various topics related to the subjects in this paper. He also thanks Kazuya Kato and Karl Rubin for their beautiful ideas to define several zeta elements, and for stimulating conversations and discussions with them. The third author would like to thank Kazuya Kato for his interest in the works of the third author and for encouragement.

References

[1] S. Bloch, K. Kato, L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift Vol I, Progress in Math. Vol 86, Birkhäuser (1990), 333-400.

[2] D. Burns, On the values of equivariant Zeta functions of curves over finite fields, Doc. Math., J. DMV 9 (2004) 357-399.

[3] D. Burns, Congruences between derivatives of abelian L-functions at s = 0, Invent. math. 169 (2007) 451-499.

[4] D. Burns, Perfecting the nearly perfect, Pure and Applied Math. Q. 4 (2008) 1041-1058 (Jean-Pierre Serre Special Issue, Part I).

[5] D. Burns, Congruences between derivatives of geometric L-functions (with an appendix by D. Burns, K. F. Lai and K-S. Tan), Invent. math. 184 (2011) 221-256.

[6] D. Burns, M. Flach, On Galois structure invariants associated to Tate motives, Amer. J. Math. 120 (1998) 1343-1397.

[7] D. Burns, M. Flach, Tamagawa numbers for motives with (noncommutative) coefficients, Doc. Math. 6 (2001) 501-570.

[8] D. Burns, C. Greither, On the Equivariant Tamagawa Number Conjecture for Tate motives, Invent. math. 153 (2003) 303-359.

[9] D. Burns, M. Kurihara, T. Sano, Iwasawa theory and zeta elements for 𝔾_{m}, preprint (2015) arXiv:1506.07935v3.

[10] P. Cornacchia, C. Greither, Fitting ideals of class groups of real fields with prime power conductor, J. Number Theory 73 (1998) 459-471.

[11] C. W. Curtis, I. Reiner, Methods of Representation Theory, Vol. I, John Wiley and Sons, New York, 1987.

[12] H. Darmon, Thaine’s method for circular units and a conjecture of Gross, Canadian J. Math., 47 (1995) 302-317.

[13] C. Emmons, C. Popescu, Special Values of Abelian L-functions at s = 0, J. Number Theory 129 (2009) 1350-1365.

[14] M. Flach, On the cyclotomic main conjecture for the prime 2, J. reine angew. Math. 661 (2011) 1-36.

[15] C. Greither, Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture, Compositio Math. 143 (2007) 1399-1426.

[16] C. Greither, R. Kučera, On a conjecture concerning minus parts in the style of Gross, Acta Arith. 132 (2008) 1-48.

[17] C. Greither, R. Kučera, The Minus Conjecture revisited, J. reine angew. Math. 632 (2009) 127-142.

[18] C. Greither, M. Kurihara, Stickelberger elements, Fitting ideals of class groups of CM fields, and dualisation, Math. Zeitschrift 260 (2008), 905-930.

[19] C. Greither, C. Popescu, An Equivariant Main Conjecture in Iwasawa Theory and Applications, J. Algebraic Geom. 24 (2015) 629-692.

[20] B. H. Gross, On p-adic L-series at s = 0, J. Fac. Sci. Univ. Tokyo, Sect. IA, 28 (1982) 979-994.

[21] B. H. Gross, On the value of abelian L-functions at s = 0, J. Fac. Sci. Univ. Tokyo, Sect. IA, 35 (1988) 177-197.

[22] A. Hayward, A class number formula for higher derivatives of abelian L-functions, Compositio Math. 140 (2004) 99-129.

[23] U. Jannsen, Iwasawa modules up to isomorphism, Advanced Studies in Pure Mathematics 17 (1989) 171-207.

[24] K. Kato, Iwasawa theory and p-adic Hodge theory, Kodai Math. J. 16 no 1 (1993) 1-31.

[25] K. Kato, Lectures on the approach to Iwasawa theory of Hasse-Weil Lfunctions via B_{dR}, Part I, In: Arithmetical Algebraic Geometry (ed. E. Ballico), Lecture Notes in Math. 1553 (1993) 50-163, Springer, New York, 1993.

[26] S. Y. Kim, On the Equivariant Tamagawa Number Conjecture for Quaternion Fields, Ph. D. thesis, King’s College London, 2003.

[27] F. Knudsen, D. Mumford, The projectivity of the moduli space of stable curves I: Preliminaries on ‘det’ and ‘Div’, Math. Scand. 39 (1976) 19-55.

[28] M. Kurihara, On the structure of ideal class groups of CM-fields, Documenta Mathematica, Extra Volume Kato (2003), 539-563.

[29] M. Kurihara, On stronger versions of Brumer’s conjecture, Tokyo Journal of Math. 34 (2011) 407-428.

[30] M. Kurihara, Refined Iwasawa theory and Kolyvagin systems of Gauss sum type, Proceedings of the London Mathematical Society 104 (2012), 728-769.

[31] M. Kurihara, T. Miura, Ideal class groups of CM-fields with non-cyclic Galois action, Tokyo Journal of Math. 35 (2012) 411-439.

[32] T. Y. Lam, A First Course in Noncommutative Rings (Second Edn.), Grad. Text Math. 131, Springer-Verlag 2001.

[33] S. Lichtenbaum, The Weil-étale topology on schemes over finite fields, Compositio Math. 141 (2005) 689-702.

[34] S. Lichtenbaum, The Weil-étale topology for number rings, Ann. Math. 170 (2009), 657-683.

[35] B. Mazur, K. Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 Number 799 (2004).

[36] B. Mazur, K. Rubin, Refined class number formulas and Kolyvagin systems, Compos. Math. 147 (2011) 56-74.

[37] B. Mazur, K. Rubin, Refined class number formulas for 𝔾_{m}, preprint (2013) arXiv:1312.4053v1.

[38] B. Mazur, J. Tate, Refined Conjectures of the Birch and Swinnerton-Dyer Type, Duke Math. J. 54 (1987) 711-750.

[39] J.S. Milne, Arithmetic Duality Theorems, Perspectives in Mathematics 1, Academic Press, 1986.

[40] D. G. Northcott, Finite free resolutions, Cambridge Univ. Press, Cambridge New York 1976.

[41] T. Ohshita, On the higher Fitting ideals of Iwasawa modules of ideal class groups over real abelian fields, J. Number Theory 135 (2014) 67-138.

[42] C. D. Popescu, Integral and p-adic refinements of the Abelian Stark conjecture, In: Arithmetic of L-functions by C. Popescu, K. Rubin, and A. Silverberg, IAS/Park City Mathematics Series 18, AMS (2011) 45-101.

[43] K. Rubin, Global units and ideal class groups, Invent. math. 89 (1987) 511-526.

[44] K. Rubin, Kolyvagin’s system of Gauss sums, In: Arithmetic Algebraic Geometry, G. van der Geer et al eds, Progress in Math 89 (1991) 309-324.

[45] K. Rubin, A Stark Conjecture ‘over ℤ’ for abelian L-functions with multiple zeros, Ann. Inst. Fourier 46 (1996) 33-62.

[46] T. Sano, Refined abelian Stark conjectures and the equivariant leading term conjecture of Burns, Compositio Math. 150 (2014) 1809-1835.

[47] D. Solomon, On a construction of p-units in abelian fields, Invent. Math. 109 (1992) 329-350.

[48] K.-S. Tan, Generalized Stark formulae over function fields, Trans. Am. Math. Soc. 361 (2009) 2277-2304.

[49] J. Tate, The cohomology groups of tori in finite Galois extensions of number fields, Nagoya Math. J., 27 (1966) 709-719.

[50] J. Tate, Les Conjectures de Stark sur les Fonctions L d’Artin en s = 0 (notes par D. Bernardi et N. Schappacher), Progress in Math., 47, Birkhäuser, Boston, 1984.

[51] J. Tate, Refining Gross’s conjecture on the values of abelian L-functions, Contemp. Math. 358 (2004) 189-192.

[52] D. Vallières, The equivariant Tamagawa number conjecture and the extended abelian Stark conjecture, to appear in J. reine u. angew. Math.

[53] J. L. Verdier, Des Catégories Dérivées des Catégories Abéliennes, Astérisque 239, Soc. Math. France, 1996.

[54] A. Wiles, The Iwasawa conjecture for totally real fields, Ann. Math. 131 (1990) 493-540.

David Burns King’s College London Department of Mathematics London WC2R 2LS, U.K. david.burns@kcl.ac.uk Masato Kurihara Keio University Department of Mathematics 3-14-1 Hiyoshi Kohoku-ku, Yokohama, 223-8522 Japan kurihara@math.keio.ac.jp

Takamichi Sano Osaka University Department of Mathematics 1-1 Machikaneyama-cho Toyonaka, Osaka, 560-0043 Japan tkmc310@gmail.com