Equivariant Weierstrass Preparation

and Values of L-functions at Negative Integers

Dedicated to Professor Kazuya Kato

David Burns and Cornelius Greither

Received: September 30, 2002 Revised: March 18, 2003

Abstract. We apply an equivariant version of the p-adic Weierstrass Preparation Theorem in the context of possible non-commutative generalizations of the power series of Deligne and Ribet. We then consider CM abelian extensions of totally real fields and by combining our earlier considerations with the known validity of the Main Conjecture of Iwasawa theory we prove, modulo the conjectural vanishing of certain μ-invariants, a (corrected version of a) conjecture of Snaith and the ‘rank zero component’ of Kato’s Generalized Iwasawa Main Conjecture for Tate motives of strictly positive weight. We next use the validity of this case of Kato’s conjecture to prove a conjecture of Chinburg, Kolster, Pappas and Snaith and also to compute explicitly the Fitting ideals of certain natural étale cohomology groups in terms of the values of Dirichlet L-functions at negative integers. This computation improves upon results of Cornacchia and Østvær, of Kurihara and of Snaith, and, modulo the validity of a certain aspect of the Quillen-Lichtenbaum Conjecture, also verifies a finer and more general version of a well known conjecture of Coates and Sinnott.

2000 Mathematics Subject Classification: 11R42 11R33 11R70 Keywords and Phrases: Iwasawa theory, values of L-functions, Euler characteristics, Fitting ideals

1. Introduction

In a beautiful series of papers in 1993 Kato formulated and studied a ‘Generalized Iwasawa Main Conjecture’ for motives over number fields with respect to certain commutative coefficient rings [24, 25, 26]. This conjecture refined the ‘Tamagawa number conjectures’ previously formulated by Bloch and Kato in [3] and by Fontaine and Perrin-Riou in [20], and led naturally to the subsequent formulation by Flach and the first named author in [9] of a Tamagawa number conjecture for motives over number fields with respect to more general coefficient rings which, in particular, need not be commutative. The above approach has already led to some remarkable new insights and results in a number of rather different contexts. We recall, for example, that it has led to a universal approach to and refinement of the ‘refined Birch and Swinnerton-Dyer Conjectures’ for abelian varieties with complex multiplication which were formulated by Gross (cf. [9, Rem. 10 and §3.3, Examples c), d) e)]) and of all of the ‘refined abelian Stark conjectures’ which were formulated by Gross, by Tate, by Rubin and by Darmon (cf. [6, 7]). At the same time, the approach has led to a natural re-interpretation and refinement of all of the central conjectures of classical Galois module theory (cf. [8, 5, 2]) and has also more recently become the focus of attempts to formulate a natural ‘Main Conjecture of non-abelian Iwasawa theory’ (see, for example, the forthcoming articles of Huber and Kings and of Weiss and the first named author in this regard). It is certainly a great pleasure, on the occasion of Kato’s fiftieth birthday, to offer in this manuscript some additional explicit evidence in support of his Generalized Iwasawa Main Conjecture and, more generally, to demonstrate yet further the enormous depth and significance of the approach that he introduced in [24, 25, 26]. To describe the main results of the present manuscript in some detail we now fix a totally real number field k and a finite Galois extension K of k which is either totally real or a CM field, and we set G := Gal(K/k). We also fix a rational prime number p and an algebraic closure ℚ_{p}^{c} of the field of p-adic rationals ℚ_{p}. We recall that if G is abelian, then a key ingredient of Wiles’ proof [39] of the Main Conjecture of Iwasawa theory is the construction by Deligne and Ribet of an element of the power series ring ℤ_{p}[G][[T]] which is uniquely characterized by its relation to the p-adic L-series associated to the extension K/k. In this manuscript we first relax the restriction that G is abelian and discuss the possible existence of elements of the (non-commutative) power series ring ℤ_{p}[G][[T]] which are related in a precise manner to the p-adic Artin L-functions associated to irreducible ℚ_{p}^{c}-valued characters of G. In particular, under a certain natural hypothesis on G (which does not require G to be abelian), and assuming the vanishing of certain μ-invariants, we apply an appropriate version of the Weierstrass Preparation Theorem for the ring ℤ_{p}[G][[T]] to derive relations between two hypothetical generalizations of the power series of Deligne and Ribet.

Values of L-Functions at Negative Integers 159

In the remainder of the manuscript our main aim is to show that, if G is abelian, then the above considerations can be combined with the constructions of Deligne and Ribet and the theorem of Wiles to shed light on a number of interesting questions. To describe these applications we assume for the rest of this introduction that G is abelian. Our first application is to the ‘Wiles Unit Conjecture’ which is formulated by Snaith in [34, Conj. 6.3.4]. Indeed, by using the above approach we are able to show that the validity of a slightly amended version of Snaith’s conjecture follows directly from the (in general conjectural) vanishing of certain natural μ-invariants, and also to show that the original version of Snaith’s conjecture does not hold in general (cf. Remark 5). To describe the next application we fix an integer r with r > 1. Under the aforementioned hypothesis concerning μ-invariants we shall prove the Generalized Iwasawa Main Conjecture of [25, Conj. 3.2.2] for the pair (h^{0}(Spec (K))(1 − r), 𝔄_{r}), where 𝔄_{r} is a natural ring which annihilates the space ℚ ⊗_{ℤ} K_{2r−1}(K). If k = ℚ, then (by a result of Ferrero and Washington in [18]) the appropriate μ-invariants are known to vanish and hence we obtain in this way a much more direct proof of the relevant parts of the main result (Cor. 8.1) of our earlier paper [11]. We remark however that the proofs of all of our results in this area involve a systematic use of the equivariant Iwasawa theory of complexes which was initiated by Kato in [25] and subsequently extended by Nekovář in [30]. As a further application, we combine our result on the Generalized Iwasawa Main Conjecture with certain explicit cohomological computations of Flach and the first named author in [8] to prove (modulo the aforementioned hypothesis on μ-invariants) that the element Ω_{r−1}(K/k) of Pic(ℤ[G]) which is defined by Chinburg, Kolster, Pappas and Snaith in [13] belongs to the kernel of the natural scalar extension morphism Pic(ℤ[G]) → Pic(𝔄_{r}). As a final application we then combine our approach with a development of a purely algebraic observation of Cornacchia and the second named author in [15] to compute explicitly certain Fitting ideals which are of arithmetical interest. To be more precise in this regard we assume that p is odd, we fix a finite set of places S of K which contains all archimedean places and all places which either ramify in K/k or are of residue characteristic p and we write O_{K,S} for the ring of S-integers of K. Writing τ for the complex conjugation in G we let e_{r} denote the idempotent \frac{1}{2}(1 + (−1)^{r}τ) of ℤ_{p}[G]. Then, under the aforementioned hypothesis on μ-invariants, we prove that the Fitting ideal of the étale cohomology module e_{r}·H^{2}(Spec(O_{K,S})_{ét}, ℤ_{p}(r)) over the ring ℤ_{p}[G]e_{r} can be completely described in terms of the values at 1 − r of the S-truncated Dirichlet L-functions which are associated to K/k. This result improves upon previous results of Cornacchia and Østvær [16, Thm. 1.2], of Kurihara [28, Cor. 12.5] and of Snaith [34, Thm. 1.6, Thm. 2.4, Thm. 5.2] and also implies a natural analogue of the main result of Solomon in [36] concerning relations between Bernoulli numbers and the structure of certain ideal class groups (cf. Remark 8). We finally remark that, under the assumed validity of a particular case of the Quillen-Lichtenbaum Conjecture, our result verifies a finer and more general version of the well known conjecture formulated by Coates and Sinnott in [14, Conj. 1]. Acknowledgements. The authors are very grateful to Masato Kurihara and Otmar Venjakob for illuminating discussions relating to the results of this manuscript. In addition, they are grateful to the referee for several very helpful remarks.

2. Equivariant Weierstrass Preparation

In this section we discuss a natural generalization of the classical p-adic Weierstrass Preparation Theorem. We let 𝔄 be a ring, write rad(𝔄) for its Jacobson radical and set \overline{𝔄} := 𝔄/ rad(𝔄). In the sequel we shall say that 𝔄 is strictly admissible if it is both separated and complete in the rad(𝔄)-adic topology and is also such that \overline{𝔄} is a skew field. More generally, we shall say that 𝔄 is admissible if it is a finite product of strictly admissible rings.

Remark 1. Let G be a finite group and p any prime number. It can be shown that the group ring 𝔄 = ℤ_{p}[G] is admissible if G is the direct product of a pgroup and an abelian group (and, in particular therefore, if G is itself abelian). Note also that in any such case the ring \overline{𝔄} is a product of finite skew fields and is therefore commutative.

In this manuscript we define the power series ring 𝔄[[T]] over 𝔄 just as for commutative base rings; in particular, we require that the variable T commutes with all elements of the coefficient ring 𝔄. We observe that, with this definition, an element f of 𝔄[[T]] is invertible if and only if its constant term f(0) is invertible in 𝔄. If f is any element of 𝔄[[T]], then we write \overline{f} for its image under the obvious reduction map 𝔄[[T]] → \overline{𝔄}[[T]]. Assume for the moment that 𝔄 is admissible, with a decomposition 𝔄 = ∏_{i∈I} 𝔄_{i} for strictly admissible rings 𝔄_{i}. If f = (f_{i})_{i∈I} is any element of 𝔄[[T]], then we define the degree deg(f), respectively reduced degree rdeg(f), of f to be the vector (deg(f_{i}))_{i∈I}, respectively (deg(\overline{f}))_{i∈I}, where by convention we regard the zero element of each ring 𝔄_{i}[[T]] and \overline{𝔄_{i}}[[T]] to be of degree +∞. We observe that if 𝔄 = ℤ_{p}, then rdeg(f) is finite if and only if the μ-invariant of the ℤ_{p}[[T]]-module ℤ_{p}[[T]]/(f) is zero. By analogy, if 𝔄 is any admissible ring, then we shall write ‘μ_{𝔄}(f) = 0’ to express the fact that (each component of) rdeg(f) is finite. If 𝔄 is strictly admissible, respectively admissible, then we shall say that an element of 𝔄[[T]] is a distinguished polynomial if it is a monic polynomial all of whose non-leading coefficients are in rad(𝔄), respectively if all of its components are distinguished polynomials (of possibly varying degrees).

Proposition 2.1. (‘Equivariant Weierstrass Preparation’) Let 𝔄 be an admissible ring. If f is any element of 𝔄[[T]] for which μ_{𝔄}(f) = 0, then there exists a unique distinguished polynomial f^{∗} and a unique unit element u_{f} of 𝔄[[T]] such that f = f^{∗} · u_{f}.

Proof. By direct verification one finds that the argument of [4, Chap.VII, §3, no. 8] extends to the present (non-commutative) context to prove the following ‘Generalized Division Lemma’: if f is any element of 𝔄[[T]] for which μ_{𝔄}(f) = 0, then each element g of 𝔄[[T]] can be written uniquely in the form g = fq + r where q and r are elements of 𝔄[[T]] and each component of deg(r) is strictly less than the corresponding component of rdeg(f) . The deduction of the claimed result from this Generalized Division Lemma now proceeds exactly as in [31, V.5.3.3-V.5.3.4]. □

Remark 2. i) If 𝔄 is a discrete valuation ring, then it is (strictly) admissible and Proposition 2.1 is equivalent to the classical Weierstrass Preparation Theorem (cf. [38, Thm. 7.1]). ii) Shortly after the first version of this manuscript was circulated (in December 2001) we learnt of a recent preprint [37] of Venjakob in which a Weierstrass Preparation Theorem is proved under conditions which are considerably more general than those of Proposition 2.1. We remark that if 𝔄 is strictly admissible, then it can be shown that the result of Proposition 2.1 is indeed equivalent to a special case of the main result of loc. cit. iii) For any prime p, any ℤ_{p}-order 𝔄 which is not admissible and any element f of 𝔄[[T]] a natural interpretation of the equality ‘μ_{𝔄}(f) = 0’ would be that, for each primitive central idempotent ϵ of 𝔄, the element ϵ · f is not divisible by p (indeed, this interpretation recovers that given above in the case that 𝔄 is admissible). However, under this interpretation the product decomposition of Proposition 2.1 is not always possible. For example, if 𝔄 = M_{2}(ℤ_{p}), then

the constant series f := (\begin{matrix}1 & 0 \\ 0 & p \end{matrix}) satisfies μ_{𝔄}(f) = 0 (in the above sense) and

yet cannot be written in the stated form f^{∗} · u_{f}. Indeed, if it did admit such a decomposition, then the ℤ_{p}-module 𝔄[[T]]/f ·𝔄[[T]] would be finitely generated and this is not true since f·𝔄[[T]] is equal to the subset of M_{2}(ℤ_{p}[[T]]) consisting of those matrices which have both second row entries divisible by p. iv) In just the same way as Proposition 2.1, one can prove that if 𝔄 is admissible, then every element f of 𝔄[[T]] for which μ_{𝔄}(f) = 0 can be written uniquely in the form u^{f} · f_{∗} with f_{∗} a distinguished polynomial and u^{f} a unit of 𝔄[[T]]. However, as the following example shows, the relation between the elements f^{∗} and f_{∗} (and u^{f} and u_{f}) is in general far from clear.

Example 1. Let a and b be elements of rad(𝔄), and set f := (T − a)(1 + bT). Then it is clear that μ_{𝔄}(f) = 0, f^{∗} = T − a and u_{f} = 1 + bT. On the other hand it may be shown that f_{∗} = T −c where c is the unique element of 𝔄 which satisfies T − c ∈ 𝔄 · f, and that u^{f} = (1 − ab + bc) + bT. Upon calculating c as a power series in the noncommuting variables a and b, one finds that

c = a − baa + aba + bbaaa − baaba − abbaa + ababa + ….

However, describing c completely is tricky. For example, a convenient option is to use the context-free formal language with four productions S → T a, S → T bSS, T → ϵ, T → T ab. Indeed, it may be shown that the degree n part of the series c is equal to the weighted sum of all words w of length n in this language, counted with weight (−1)^{e(w)} where e(w) denotes the number of times that b occurs in w not immediately preceded by a. For background and a similar example, we refer the reader to [32], in particular Chapter VI.

3. p-adic L-functions

In this section we apply Proposition 2.1 in the context of Iwasawa theory. The main result of this section (Theorem 3.1) was first motivated by the observation that the constructions of Deligne and Ribet which are used by Wiles in [39, p. 501f] can be combined with Proposition 2.1 to shed light upon the ‘Wiles Unit Conjecture’ formulated by Snaith in [34, Conj. 6.3.4]. In particular, by these means we shall prove that the validity in the relative abelian case of a corrected version of Snaith’s conjecture is a direct consequence of the (in general conjectural) vanishing of certain natural μ-invariants. We first introduce some necessary notation. Throughout this section we fix an odd prime p and a finite group G. We recall that ℚ_{p}^{c} is a fixed algebraic closure of ℚ_{p}, and we write Irr_{p}(G) for the set of irreducible ℚ_{p}^{c}-characters of G. For each ρ ∈ Irr_{p}(G) we write ℤ_{p}(ρ) for the extension of ℤ_{p} which is generated by the values of ρ. Following Fröhlich [21, Chap. II], we now define for each element f of ℤ_{p}[G][[T]] a canonical element Det(f) of Map(Irr_{p}(G), ℚ_{p}^{c}[[T]]). To do this we fix a subfield N of ℚ_{p}^{c} which is of finite degree over ℚ_{p} and over which all elements of Irr_{p}(G) can be realized, and we write O_{N} for the valuation ring of N. For each character ρ ∈ Irr_{p}(G) we choose a finitely generated O_{N}[G]-module L_{ρ} which is free over O_{N} (of rank n say) and is such that the space L_{ρ} ⊗_{O_{N}} N has character ρ, and we write r_{ρ} : G → GL_{n}(O_{N}) for the associated homomorphism. If now f = ∑_{i≥0} c_{i}T^{i}, then r_{ρ}(f) := ∑_{i≥0} r_{ρ}(c_{i})T^{i} belongs to M_{n}(O_{N}[[T]]) and we define Det(f)(ρ) := det(r_{ρ}(f)) ∈ O_{N}[[T]] (which is indeed independent of the choices of field N and lattice L_{ρ}). We observe in particular that if ρ is any element of Irr_{p}(G) which is of dimension 1, then one has Det(f)(ρ) = ρ(f). In the remainder of this manuscript we assume given a finite Galois extension of number fields K/k for which Gal(K/k) = G. We write k_{∞} (or k_{∞}^{p} if we need to be more precise) for the cyclotomic ℤ_{p}-extension of k, and K_{∞} (or K_{∞}^{p}) for the compositum of K and k_{∞}. In the rest of this section we assume that k is totally real and that K is either totally real or a CM field. We also fix a finite set S of non-archimedean places of k which contains all non-archimedean places which ramify in K/k. We write Irr_{p}^{+}(G) for the subset of Irr_{p}(G) consisting of those characters which are even (that is, factor through characters of the Galois group of the maximal totally real extension K^{+} of k in K). We fix a topological generator γ of Gal(k_{∞}/k) and, with χ_{cyclo} denoting the cyclotomic character, we set u := χ_{cyclo}(γ) ∈ ℤ_{p}^{×}.

We recall that for each ρ ∈ Irr_{p}^{+}(G) there exists a p-adic L-function L_{p,S}(−, ρ) and an associated element f_{S,ρ} of the quotient field of ℤ_{p}(ρ)[[T]] such that

(1)  L_{p,S}(1 − s, ρ) = f_{S,ρ}(u^{s} − 1)

for almost all s ∈ ℤ_{p}. To be more precise about the denominator of f_{S,ρ} we set H_{ρ} := 1 unless ρ is induced by a multiplicative character of Gal(k_{∞}/k) (that is, ρ is a character of ‘type W’ in the terminology of Wiles [39]) in which case we set H_{ρ} := ρ(γ)(1 + T) − 1 ∈ ℤ_{p}(ρ)[[T]]. Then there exists an element G_{S,ρ} of ℤ_{p}(ρ)[[T]][\frac{1}{p}] such that f_{S,ρ} = G_{S,ρ} · H_{ρ}^{−1} (cf. [39, Thm. 1.1]). We hope that the reader will not in the sequel be confused by our notation: whenever G occurs without a subscript it denotes a Galois group; whenever G is adorned with a subscript it denotes a power series.

By the classical Weierstrass Preparation Theorem, each series G_{S,ρ} can be decomposed as a product

(2)  G_{S,ρ} = π(ρ)^{μ(S,ρ)} · G_{S,ρ}^{∗} · U_{S,ρ}

where π(ρ) is a uniformising parameter of ℤ_{p}(ρ), μ(S, ρ) is an integer, G_{S,ρ}^{∗} is a distinguished polynomial and U_{S,ρ} is a unit of ℤ_{p}(ρ)[[T]]. We now proceed to describe four natural hypotheses relating to the Weierstrass decompositions (2). The main result of this section will then describe certain relations that exist between these hypotheses.

Hypothesis (μ_{p}): For each ρ ∈ Irr_{p}^{+}(G) one has μ(S, ρ) = 0.

Remark 3. i) It is a standard conjecture that Hypothesis (μ_{p}) is always valid (the first statement of this was due to Iwasawa [23]). However, at present the only general result one has in this direction is that Hypothesis (μ_{p}) is valid for K/k when k = ℚ and G is abelian. Indeed, this is proved by Ferrero and Washington in [18]. ii) In this remark we describe a natural Iwasawa-theoretical reinterpretation of Hypothesis (μ_{p}) in the case that K is totally real. To do this we write S_{p} for the union of S and the set of places of k which lie above p, and we let Y (S_{p}) denote the Galois group of the maximal abelian pro-p-extension of K_{∞} which is unramified outside the set of places which lie above any element of S_{p}. (We note that, since p is odd, any pro-p-extension of K_{∞} is automatically unramified at all archimedean places.)

Lemma 1. If K is totally real, then Hypothesis (μ_{p}) is valid for K/k if and only if the μ-invariant of the ℤ_{p}[Gal(K_{∞}/K)]-module Y (S_{p}) is 0.

Proof. We set L := K(ζ_{p}) and Δ := Gal(L/K) and let ω : Δ → ℤ_{p}^{×} denote the Teichmüller character. For each ℤ_{p}[Δ]-module M and integer i we write M^{(i)} for the submodule consisting of those elements m which satisfy δ(m) = ω^{i}(δ)·m for all δ ∈ Δ. (Since p ∤ |Δ|, each such functor M ↦ M^{(i)} is exact.) We write Y, respectively Y_{L}, for the Galois group of the maximal abelian prop-extension of K_{∞}, respectively of L_{∞}, which is unramified outside all places above p. For each ρ ∈ Irr_{p}(Gal(K/k)) we also write G_{ρ} for the element of ℤ_{p}(ρ)[[T]][\frac{1}{p}] which is defined just as G_{S,ρ} but with S taken to be the empty set. We first observe that μ(Y (S_{p})) = 0 if and only if μ(Y) = 0 and that for each ρ ∈ Irr_{p}(Gal(K/k)) one has μ(S, ρ) = 0 if and only if μ(G_{ρ}) = 0. We also note that μ(Y) = 0 if and only if μ(X_{L}^{(1)}) = 0, where X_{L} denotes the Galois group of the maximal unramified abelian pro-p-extension of L_{∞}. For the reader’s convenience, we briefly sketch the argument. By Kummer duality, for each even integer i the module Y_{L}^{(i)} is isomorphic to Hom(Cl(L_{∞})^{(1−i)}, ℚ_{p}/ℤ_{p})(1) and hence in turn pseudo-isomorphic to X_{L}^{(1−i),#}(1) where # indicates contragredient action of Gal(L_{∞}/k) [31, (11.1.8), (11.4.3)]. This implies, in particular, that Y = Y_{L}^{(0)} is pseudoisomorphic to X_{L}^{(1),#}(1), and this in turn implies the claimed result. We next recall that, as a consequence of [39, Thm. 1.4], one has μ(X_{L}^{(1)}) = 0 if and only if μ(G_{η_{0}}) = 0, where η_{0} denote the trivial character of Δ. To finish the proof of the lemma, we now need only invoke the inductive property of L-functions and Iwasawa series. Indeed, one has G_{η_{0}} = G_{χ_{reg}}, where χ_{reg} is the character of the regular representation of Gal(K/k) (note that this is equal to the induction of η_{0} from K to k), and by its very definition, one has G_{χ_{reg}} = ∏_{ρ} G_{ρ}^{deg(ρ)}, where ρ runs over all elements of Irr_{p}(Gal(K/k)). It is therefore clear that μ(G_{η_{0}}) = 0 if and only if for all ρ ∈ Irr_{p}(Gal(K/k)) one has μ(G_{ρ}) = 0 (or equivalently μ(S, ρ) = 0), as required. □

We continue to introduce further natural hypotheses relating to the decompositions (2). If f and f′ are elements of ℤ_{p}[G][[T]], then we say that f is right associated, respectively left associated, to f′ if there exists a unit element u of ℤ_{p}[G][[T]] such that f = f′ · u, respectively f = u · f′.

Hypothesis (EPS) (‘Equivariant Power Series’) Assume that Hypothesis (μ_{p}) is valid for K/k. Then there exist elements G_{S} = G_{S}(T) and H = H(T) of ℤ_{p}[G][[T]] which are each right associated to distinguished polynomials and are such that for all ρ ∈ Irr_{p}^{+}(G) the quotient Det(G_{S})(ρ)/ Det(H)(ρ) is defined and equal to G_{S,ρ}/H_{ρ} = f_{S,ρ}.

Remark 4. i) If ℤ_{p}[G] is admissible, then Proposition 2.1 (and Remark 2iv)) implies that an element f of ℤ_{p}[G][[T]] is right associated to a distinguished polynomial if and only if it is left associated to a distinguished polynomial and that these conditions are in turn equivalent to an equality μ_{ℤ_{p}[G]}(f) = 0. ii) The completed group ring ℤ_{p}[[Gal(K_{∞}/k)]] is naturally isomorphic to the power series ring ℤ_{p}[Gal(K_{∞}/k_{∞})][[T]]. If K ∩ k_{∞} = k, then this ring can be identified with ℤ_{p}[G][[T]] but in general no such identification is possible.

We write c_{ρ} for the leading coefficient of H_{ρ} (so, explicitly, one has c_{ρ} = 1 unless ρ is a non-trivial character of ‘type W’ in which case c_{ρ} = ρ(γ)). We observe that the polynomial H_{ρ}^{∗} := c_{ρ}^{−1} · H_{ρ} is distinguished.

The following hypothesis is directly motivated by the ‘Wiles Unit Conjecture’ which is formulated by Snaith in [34, Conj. 6.3.4]. (We shall explain the precise connection at the end of this section.)

Hypothesis (EUS) (‘Equivariant Unit Series’) There exists a unit element U_{S} of ℤ_{p}[G][[T]] which is such that for all ρ ∈ Irr_{p}^{+}(G) one has Det(U_{S})(ρ) = c_{ρ}^{−1}U_{S,ρ}.

For each character ρ ∈ Irr_{p}^{+}(G) we next consider the vector space over N which is given by H^{0}(Gal(K_{∞}/k_{∞}), Hom_{N}(L_{ρ} ⊗_{O_{N}} N, Y (S_{p}) ⊗_{ℤ_{p}} N)). This space is finite-dimensional (over N) and also equipped with a canonical action of the quotient group Gal(K_{∞}/k)/ Gal(K_{∞}/k_{∞}) ≅ Gal(k_{∞}/k) and hence, in particular, of the automorphism γ. We write h_{S,ρ} for the characteristic polynomial of the endomorphism of the above space which is induced by the action of γ − 1.

Hypothesis (ECP) (‘Equivariant Characteristic Polynomials’) Assume that Hypothesis (μ_{p}) is valid for K/k. Then there exist distinguished polynomials G_{S}^{∗} = G_{S}^{∗}(T) and H^{∗} = H^{∗}(T) in ℤ_{p}[G][T] which are such that for all ρ ∈ Irr_{p}^{+}(G) the quotient Det(G_{S}^{∗})(ρ)/ Det(H^{∗})(ρ) is defined and equal to h_{S,ρ}/H_{ρ}^{∗}.

We can now state the main result of this section.

Theorem 3.1. Assume that Hypothesis (μ_{p}) is valid for K/k.

i) If Hypotheses (ECP) and (EUS) are both valid for K/k, then Hypothesis (EPS) is valid for K/k with G_{S} = G_{S}^{∗} · U_{S} and H = H^{∗}. ii) If ℤ_{p}[G] is admissible and Hypothesis (EPS) is valid for K/k, then

Hypotheses (EUS) and (ECP) are both valid for K/k. iii) If K/k is abelian, then Hypotheses (EPS), (ECP) and (EUS) are all

valid for K/k.

Proof. i) We suppose that Hypotheses (μ_{p}), (ECP) and (EUS) are all valid for K/k. Under these hypotheses we claim that the series G_{S} := G_{S}^{∗} · U_{S} ∈ ℤ_{p}[G][[T]] and H := H^{∗} ∈ ℤ_{p}[G][T] are as described in Hypothesis (EPS). To show this we first observe that for every character ρ ∈ Irr_{p}^{+}(G) one has

Det(G_{S})(ρ) · Det(H)(ρ)^{−1} = h_{S,ρ}c_{ρ}^{−1}U_{S,ρ} · (H_{ρ}^{∗})^{−1}
= h_{S,ρ} · U_{S,ρ} · H_{ρ}^{−1}
= (h_{S,ρ} · (G_{S,ρ}^{∗})^{−1}) · (G_{S,ρ} · H_{ρ}^{−1}),

where the last equality is a consequence of the decomposition (2) and our assumption that μ(S, ρ) = 0. It is therefore enough to show that for each ρ ∈ Irr_{p}^{+}(G) one has an equality

(3)  h_{S,ρ} = G_{S,ρ}^{∗}.

Now if ρ is a one-dimensional even character which is of ‘type S’, then this equality is equivalent to the Main Conjecture of Iwasawa theory as proved by Wiles [39, Thm. 1.3]. In the general case the equality has been verified by Snaith [34, Thm. 6.2.5] and, for the reader’s convenience, we briefly sketch the argument (for details see loc. cit.). One proceeds by reduction to the result of Wiles by means of Brauer’s Induction Theorem and the fact that characteristic polynomials and Iwasawa series enjoy the same inflation and induction properties. The only complication in this reduction is caused by the need to twist with characters which are of ‘type W ’ and by the denominator polynomials H_{ρ}, and this is resolved by using the fact, first observed by Greenberg [22], that the denominator of the Iwasawa series which is attached to each irreducible character ρ of dimension greater than 1 is trivial (this justifies our setting H_{ρ} = 1 for these ρ). We would like to point out that there is actually a very simple argument for this absence of denominator in the case that K ∩ k_{∞} = k. Indeed, in this case, if ρ is any irreducible character of G which has degree greater than 1, then Brauer induction implies that ρ = ∑_{i} n_{i} Ind_{k_{i}}^{k}(ρ_{i}) where the k_{i} are suitable intermediate fields and ρ_{i} is a one-dimensional character of Gal(K/k_{i}) which is of ‘type S’. It follows that the denominator of each series f_{ρ_{i}} is trivial unless ρ_{i} is itself the trivial character. Further, if χ_{0} denotes the one-dimensional trivial representation of G, then one has 0 = ⟨χ_{0}, ρ⟩ = ∑_{i} n_{i}⟨χ_{0}, Ind_{k_{i}}^{k}(ρ_{i})⟩. Since the latter sum is equal to the sum of the multiplicities n_{i} for which ρ_{i} is trivial, it follows that the denominator of f_{ρ} is indeed trivial. ii) We now suppose that ℤ_{p}[G] is admissible and that Hypotheses (μ_{p}) and (EPS) are both valid for K/k. Recalling Remark 4i) (and Proposition 2.1), we find that the series G_{S} and H (as given by Hypothesis (EPS)) admit canonical decompositions G_{S} = G_{S}^{∗} · U′_{S} and H = H^{∗} · V, where G_{S}^{∗} and H^{∗} are distinguished polynomials in ℤ_{p}[G][T] and U′_{S} and V are units of the ring ℤ_{p}[G][[T]]. It follows that for each character ρ ∈ Irr_{p}^{+}(G) one has an equality

\frac{G_{S,ρ}}{H_{ρ}} = \frac{Det(G_{S}^{∗})(ρ) Det(U′_{S})(ρ)}{Det(H^{∗})(ρ) Det(V)(ρ)}.

We now recall that G_{S,ρ} = G_{S,ρ}^{∗} · U_{S,ρ} = h_{S,ρ} · U_{S,ρ} (by (2) and (3)) and we set U_{S} := U′_{S} · V^{−1} ∈ ℤ_{p}[G][[T]]^{×}. Upon clearing denominators in the last displayed formula, we therefore obtain equalities

h_{S,ρ} · Det(H^{∗})(ρ) · U_{S,ρ} = Det(G_{S}^{∗})(ρ) · H_{ρ} · Det(U_{S})(ρ),

or equivalently

(4)  h_{S,ρ} · Det(H^{∗})(ρ) · c_{ρ}^{−1}U_{S,ρ} = Det(G_{S}^{∗})(ρ) · H_{ρ}^{∗} · Det(U_{S})(ρ).

Lemma 2. Let f be a distinguished polynomial in ℤ_{p}[G][[T]]. Then, for each ρ ∈ Irr_{p}(G), the series Det(f)(ρ) is a distinguished polynomial in O_{N}[[T]].

Proof. It is clear that the series Det(f)(ρ) is a polynomial in O_{N}[[T]] which is distinguished if and only if the polynomial Det(f)(ρ)^{p^{j}} is distinguished for any natural number j. In addition, since f is a distinguished polynomial, of degree d say, there exists a natural number j such that f^{p^{j}} ≡ T^{dp^{j}} (modulo p · ℤ_{p}[G][[T]]). Since Det(f^{p^{j}})(ρ) = Det(f)(ρ)^{p^{j}} we may therefore assume in the sequel that f is a monic polynomial which satisfies f ≡ T^{d} (modulo p · ℤ_{p}[G][[T]]).

We now use the notation introduced at the beginning of §3 (when defining the map Det(f)). We write f = T^{d} + ∑_{i=0}^{d−1} α_{i}T^{i} where α_{i} ∈ p · ℤ_{p}[G] for each integer i with 0 ≤ i < d, and for each such i we set R_{i} := r_{ρ}(α_{i}) ∈ M_{n}(O_{N}). Upon denoting the n × n identity matrix by R_{d}, we obtain an equality r_{ρ}(f) = ∑_{i=0}^{d} R_{i}T^{i} in M_{n}(O_{N}[T]). We observe that each off-diagonal entry of r_{ρ}(f) belongs to p·O_{N}[T] and is of degree strictly less than d, and that each diagonal entry of r_{ρ}(f) is a monic polynomial which is congruent to T^{d} modulo p·O_{N}[T]. From this description it is immediately clear that Det(f)(ρ) := det(r_{ρ}(f)) is a monic polynomial which is congruent to T^{nd} modulo p · O_{N}[T], and hence that it is distinguished, as claimed. □

Upon applying this lemma with f equal to G_{S}^{∗} and H^{∗} we deduce that the polynomials Det(G_{S}^{∗})(ρ) and Det(H^{∗})(ρ), and hence also Det(G_{S}^{∗})(ρ) · H_{ρ}^{∗} and h_{S,ρ} · Det(H^{∗})(ρ), are distinguished. When combined with the equality (4) and the uniqueness of Weierstrass product decompositions in the ring O_{N}[[T]], this observation implies that h_{S,ρ} · Det(H^{∗})(ρ) = Det(G_{S}^{∗})(ρ) · H_{ρ}^{∗}, as required by Hypothesis (ECP), and also that c_{ρ}^{−1}U_{S,ρ} = Det(U_{S})(ρ), as required by Hypothesis (EUS). iii) We now assume that G is abelian, so that ℤ_{p}[G] is admissible. Following claim ii), it is therefore enough for us to assume that Hypothesis (μ_{p}) is valid for K/k, and then to prove that Hypothesis (EPS) is valid for K/k. To verify Hypothesis (EPS) for K/k we first assume that K ∩ k_{∞} = k. In this case, we may use the constructions of Deligne and Ribet which are used by Wiles in [39, p.501f.]. To be explicit, we obtain the elements G_{S} and H as required by Hypothesis (EPS) by combining in the obvious way the elements G_{m,c,S} and H_{m,c} of loc. cit., where m runs over the ‘components’ of ℤ_{p}[G] (and for each non-trivial component m we set H_{m,c} := 1). The required equalities ρ(G_{S})/ρ(H) = f_{S,ρ} (for each ρ ∈ Irr_{p}^{+}(G)) and the fact that μ_{ℤ_{p}[G]}(H) = 0 then follow as direct consequences of the properties of the series G_{m,c,S} and H_{m,c} described by Wiles in loc. cit., and the fact that μ_{ℤ_{p}[G]}(G_{S}) = 0 follows from the assumed validity of Hypothesis (μ_{p}) for K/k. From Remark 4i) we therefore deduce that G_{S} and H are both right associated to distinguished polynomials, as required. In general one has K ∩ k_{∞} ≠ k, and in this case we proceed as follows. There exists an extension K′ of k such that K′ ∩ k_{∞} = k and K_{∞} is the compositum of K′ and k_{∞}. We observe that K′/k is a finite abelian extension and we set G′ := Gal(K′/k) and Γ := Gal(k_{∞}/k). Each character ρ ∈ Irr_{p}(G) can be lifted to a character of Gal(K_{∞}/k) ≅ G′ × Γ (which we again denote by ρ), and as such it has a unique factorisation ρ = ψκ where ψ ∈ Irr_{p}(G′) and κ is a character of Γ which has finite order. As a consequence of the formula [39, (1.4)] one has an equality G_{S,ψκ}(T) = G_{S,ψ}(κ(γ)(1 + T) − 1). By the very definition of the polynomials H_{ψκ} and H_{ψ} one also has an equality H_{ψκ}(T) = H_{ψ}(κ(γ)(1 + T) − 1). We now write G_{K′,S}(T) and H_{K′}(T) for the elements of ℤ_{p}[G′][[T]] which are afforded by Hypothesis (EPS) for the extension K′/k (which we know to be valid by the above argument since K′, which sits between k and some K_{n}, is again either totally real or a CM field). Let G̃_{S}(T) = G_{K′,S}(γ(1 + T) − 1) and H̃(T) = H_{K′}(γ(1 + T) − 1); these series lie in ℤ_{p}[[G′ × Γ]][[T]]. Then applying the character ρ of G (considered as a character of G′ × Γ by inflation) to G̃_{S}(T) yields ψ(G_{K′,S})(κ(γ)(1+T)−1), similarly for H̃(T), so applying ρ to the quotient G̃_{S}(T)/H̃(T) gives f_{ψ}(κ(γ)(1 + T) − 1) by Hypothesis (EPS) for K′/k. By the aforementioned formulas, we therefore have f_{ψ}(κ(γ)(1+T)−1) = f_{ρ}(T) and so we are almost done: indeed, it simply suffices to define G_{S}(T), respectively H(T), to be equal to the image of G̃_{S}(T), respectively H̃(T), under the map ℤ_{p}[[G′ × Γ]][[T]] → ℤ_{p}[G][[T]] which is induced by the epimorphism G′ × Γ → G. □

Remark 5. To end this section we now explain the precise connection between Hypothesis (EUS) and the ‘Wiles Unit Conjecture’ [34, Conj. 6.3.4] of Snaith. To do this we fix an integer n with n > 1 and, assuming Hypothesis (EUS) to be valid for K/k, we set α_{S,n} := U_{S}(u^{n} − 1) ∈ ℤ_{p}[G]^{×}. Then for each ρ ∈ Irr_{p}^{+}(G) one has an equality

Det(α_{S,n})(ρ) = c_{ρ}^{−1}U_{S,ρ}(u^{n} − 1).

After taking account of the equalities (2) and (3) one finds that this property of α_{S,n} is closely related to that which should be satisfied by the element α_{n,K^{+}/k} of ℤ_{p}[Gal(K^{+}/k)]^{×} whose existence is predicted by [34, Conj. 6.3.4]. However, there are two important differences: in loc. cit. the p-adic L-functions are untruncated and the factors c_{ρ} are omitted. Whilst, a priori, Hypothesis (EUS) and [34, Conj. 6.3.4] could be simultaneously valid, we now present an explicit example which shows that [34, Conj. 6.3.4] is not valid (because the relevant p-adic L-functions are untruncated). We remark that similarly explicit examples exist to show that [34, Conj. 6.3.4] must also be corrected by the introduction of the factors c_{ρ}.

Example 2. We set p := 3 and k := ℚ and we let K denote the composite of the cyclic cubic extension K_{1} of ℚ which has conductor 7 and the field K_{2} := ℚ(\sqrt{5}). We set G := Gal(K/ℚ) (which is cyclic), we write ρ_{0} for the nontrivial character of Gal(K_{2}/ℚ) (considered as a character of G) and ρ_{1} for any faithful character of G, and we set S := {5, 7}. Then ρ_{0} and ρ_{1} have conductors 5 and 35 respectively and Theorem 3.1iii) implies that there exists a unit element U_{S} of ℤ_{p}[G][[T]] such that, for i ∈ {1, 2}, the unit part U_{ρ_{i}} of the Iwasawa series f_{S,ρ_{i}} which is associated to L_{p,S}(−, ρ_{i}) is equal to ρ_{i}(U_{S}). We now let w(T) be the power series such that w(u^{n} − 1)L_{p,{5}}(1 − n, ρ_{0}) = L_{p,S}(−, ρ_{0}) for all natural numbers n. Then w(u^{n} − 1) = 1 − ρ_{0}(7)ω^{−n}(7)7^{n−1} where ω is the 3-adic Teichmüller character. Since ω(7) = 1 it follows that w(T) = 1 − ρ_{0}(7) · 7^{−1}(T + 1)^{a} with u^{a} = 7. Now ρ_{0}(7) = −1 and so w(0) = \frac{8}{7} ≡ −1 (mod 3); in particular w(T) ∈ ℤ_{p}[[T]]^{×} and so the unit part U′_{ρ_{0}} of the Iwasawa series f_{{5},ρ_{0}} is equal to w(T)^{−1}U_{ρ_{0}}. In this setting [34, Conj. 6.3.4] predicts the existence (for any given n) of an element α′_{n} of ℤ_{3}[G]^{×} which satisfies both ρ_{0}(α′_{n}) = U′_{ρ_{0}}(u^{n} − 1) and ρ_{1}(α′_{n}) = U_{ρ_{1}}(u^{n} − 1). If such an element existed, then the element q := U_{S}(u^{n} − 1)(α′_{n})^{−1} of ℤ_{3}[G]^{×} would satisfy both ρ_{0}(q) = w(u^{n} − 1) and ρ_{1}(q) = 1. However, the above calculation shows that w(u^{n} − 1) ≡ −1 (mod 3) and so no such element q can exist (indeed, ρ_{0} and ρ_{1} differ by a character of order 3 and so, for any q′ ∈ ℤ_{3}[G], the elements ρ_{0}(q′) and ρ_{1}(q′) must be congruent modulo the maximal ideal of ℤ_{3}[ζ_{3}]).

4. Algebraic Preliminaries

In the remainder of this manuscript our aim is to describe certain explicit consequences of Theorem 3.1iii) concerning the values of Dirichlet L-functions at strictly negative integers. However, before doing so, in this section we describe some necessary algebraic preliminaries. We now let K/k be any finite Galois extension of number fields of group G (which is not necessarily abelian). We fix any rational prime p and a finite set of places T of k which contains all archimedean places, all places which ramify in K/k and all places of residue characteristic p. For any extension E of k we write O_{E,T} for the ring of T_{E}-integers in E, where T_{E} denotes the set of places of E which lie above those in T. We set U := Spec(O_{k,T}) and we write G_{k,T} for the Galois group of the maximal algebraic extension of k which is unramified outside T. For each non-negative integer n we write K_{n} for the subextension of K_{∞}^{p} which is of degree p^{n} over K, and π_{n} : Spec(O_{K_{n},T}) → U for the morphism of schemes which is induced by the inclusion O_{k,T} ⊆ O_{K_{n},T}. If F is any finite G_{k,T} -module, then we use the same symbol to denote the associated locally-constant sheaf on the étale site U_{ét}. If F is any continuous G_{k,T} -module which is finitely generated over ℤ_{p}, then we let F_{∞} denote the associated pro-sheaf (F_{n}, t_{n})_{n≥0} on U_{ét} where, for each non-negative integer n, we set F_{n} := π_{n,∗} ∘ π_{n}^{∗}(F/p^{n+1}) and the transition morphism t_{n} is induced by the composite of the trace map π_{n+1,∗} ∘ π_{n+1}^{∗}(F/p^{n+2}) → π_{n,∗} ∘ π_{n}^{∗}(F/p^{n+2}) and the natural projection F/p^{n+2} → F/p^{n+1}. Let Λ be a pro-p ring. (Thus we depart here from the usual convention that Λ has the fixed meaning ℤ_{p}[[T]].) We write D(Λ) for the derived category of bounded complexes of Λ-modules and D^{p}(Λ), respectively D^{p,f}(Λ), for the full triangulated subcategory of D(Λ) consisting of those complexes which are perfect, respectively are perfect and have finite cohomology groups. If F is any (p-adic) étale sheaf of Λ-modules on U, then we follow the approach of [9, §3.2] to define the complex of compactly supported cohomology RΓ_{c}(U_{ét}, F) so as to lie in a canonical distinguished triangle in D(Λ)

(5)  RΓ_{c}(U_{ét}, F) ⟶ RΓ(U_{ét}, F) ⟶ ⨁_{v∈T} RΓ(Spec(k_{v})_{ét}, F).

We recall that the approach developed by Kato in [25, §3.1] and by Nekovář in [30] (cf. also [11, Rem. 4.1] in this regard) allows one to extend the definitions of each of these complexes in a natural manner to the case of pro-sheaves of Λ-modules of the form F_{∞} discussed above, and that in this case there is again a canonical distinguished triangle of the form (5). If p = 2, then we set RΓ_{∗}(U_{ét}, −) := RΓ_{c}(U_{ét}, −). If p is odd, then we let RΓ_{∗}(U_{ét}, −) denote either RΓ_{c}(U_{ét}, −) or RΓ(U_{ét}, −). In each degree i we then set H_{∗}^{i}(U_{ét}, −) := H^{i}RΓ_{∗}(U_{ét}, −). We next recall that if Λ is any ℤ_{p}-order which spans a finite dimensional semisimple ℚ_{p}-algebra Λ_{ℚ_{p}}, then to each object C of D^{p,f}(Λ) one can associate a canonical element χ_{Λ}^{rel}C of the relative algebraic K-group K_{0}(Λ, ℚ_{p}) (cf. [5, Prop. 1.2.1] or [9, §2.8, Rem. 4]). We recall further that the Whitehead group K_{1}(Λ_{ℚ_{p}}) of Λ_{ℚ_{p}} is generated by elements of the form [α] where α is an automorphism of a finitely generated Λ_{ℚ_{p}}-module, and we write δ_{Λ} : K_{1}(Λ_{ℚ_{p}}) → K_{0}(Λ, ℚ_{p}) for the homomorphism which occurs in the long exact sequence of relative K-theory (as described explicitly in, for example, [5, §1.1]).

Proposition 4.1. Let F be a continuous ℤ_{p}[G_{k,T}]-module which is both finitely generated and free over ℤ_{p}. Then RΓ_{∗}(U_{ét}, F_{∞}) is an object of D^{p}(ℤ_{p}[Gal(K_{∞}^{p}/k)]). Assume now that K ∩ k_{∞}^{p} = k. Let ϵ be a central idempotent of ℤ_{p}[G], set Λ := ℤ_{p}[G]ϵ, and let θ_{∞} be an injective ℤ_{p}[Gal(K_{∞}^{p}/k)]-equivariant endomorphism of ϵ · F_{∞}. If both

ci) in each degree i the ℤ_{p}-module H_{∗}^{i}(U_{ét}, ϵ·F_{∞}) is finitely generated, and cii) in each degree i the endomorphism H_{∗}^{i}(U_{ét}, θ_{∞}) ⊗_{ℤ_{p}} ℚ_{p} is bijective,

then RΓ_{∗}(U_{ét}, coker(θ_{∞})) is an object of D^{p,f}(Λ), and in K_{0}(Λ, ℚ_{p}) one has an equality

χ_{Λ}^{rel}RΓ_{∗}(U_{ét}, coker(θ_{∞})) = ∑_{i∈ℤ}(−1)^{i}δ_{Λ}([H_{∗}^{i}(U_{ét}, θ_{∞}) ⊗_{ℤ_{p}} ℚ_{p}]).

Proof. For each non-negative integer n we set Λ_{n} := (ℤ/p^{n+1})[Gal(K_{n}/k)]. We also set Λ_{∞} := \underleftarrow{lim}_{n} Λ_{n} where the limit is taken with respect to the natural projection morphisms ρ_{n} : Λ_{n+1} → Λ_{n}. In the sequel we identify Λ_{∞} with ℤ_{p}[Gal(K_{∞}^{p}/k)] in the natural way. We first note that, for each non-negative integer n, F_{n} is the sheaf which is associated to the free Λ_{n}-module Λ_{n} ⊗_{ℤ_{p}} F and that t_{n} is the morphism which is associated to the natural morphism of Λ_{n+1}-modules

Λ_{n+1} ⊗_{ℤ_{p}} F → Λ_{n} ⊗_{Λ_{n+1},ρ_{n}} (Λ_{n+1} ⊗_{ℤ_{p}} F) ≅ Λ_{n} ⊗_{ℤ_{p}} F.

By using results of Flach [19, Thm. 5.1, Prop. 4.2] we may therefore deduce that, for each such n, RΓ_{∗}(U_{ét}, F_{n}) is an object of D^{p}(Λ_{n}) which is acyclic outside degrees 0, 1, 2, 3 and is also such that there exists an isomorphism ψ_{n} in D^{p}(Λ_{n}) between Λ_{n} ⊗_{Λ_{n+1},ρ_{n}}^{𝕃} RΓ_{∗}(U_{ét}, F_{n+1}) and RΓ_{∗}(U_{ét}, F_{n}). We observe next that Λ_{n+1} is Artinian and that ker(ρ_{n}) is a two sided nilpotent ideal. By using the structure theory of [17, Prop. (6.17)] we may thus deduce that for any morphism of finitely generated projective Λ_{n}-modules ϕ_{n} : M_{n} → N_{n} there exists a morphism of finitely generated projective Λ_{n+1}- modules ϕ_{n+1} : M_{n+1} → N_{n+1} for which one has M_{n} = Λ_{n} ⊗_{Λ_{n+1},ρ_{n}} M_{n+1}, N_{n} = Λ_{n} ⊗_{Λ_{n+1},ρ_{n}} N_{n+1} and ϕ_{n} = Λ_{n} ⊗_{Λ_{n+1},ρ_{n}} ϕ_{n+1}. This fact allows one to adapt the constructions of Milne in [29, p.264-265] and hence to prove that, for each non-negative integer n, there exists a complex of finitely generated projective Λ_{n}-modules C_{n}^{·} with the following properties: C_{n}^{i} = 0 for i ∉ {0, 1, 2, 3}; C_{n}^{·} is isomorphic in D^{p}(Λ_{n}) to RΓ_{∗}(U_{ét}, F_{n}); there exists a Λ_{n+1}-equivariant homomorphism of complexes ψ′_{n} : C_{n+1}^{·} → C_{n}^{·} which is such that the morphism Λ_{n}⊗_{Λ_{n+1},ρ_{n}} ψ′_{n} : Λ_{n}⊗_{Λ_{n+1},ρ_{n}} C_{n+1}^{·} → C_{n}^{·} is bijective in each degree and induces ψ_{n}. In this way we obtain a bounded complex of finitely generated projective Λ_{∞}-modules C_{∞}^{·} := \underleftarrow{lim}_{ψ′_{n}} C_{n}^{·} which represents RΓ_{∗}(U_{ét}, F_{∞}). This proves the first claim of the proposition. We now assume that K ∩ k_{∞}^{p} = k and that θ_{∞} is an injective Λ_{∞}-equivariant endomorphism of the pro-sheaf ϵ · F_{∞}. By adapting the constructions of [29, Chap. VI, Lem. 8.17, Lem. 13.10] (but note that [loc. cit., Lem. 8.17] is incorrect as stated since the morphism ψ need not be a quasi-isomorphism) it may be shown that there exists a Λ_{∞}-equivariant endomorphism θ_{∞}^{·} of the complex D_{∞}^{·} := ϵ · C_{∞}^{·} which induces the morphism RΓ_{∗}(U_{ét}, θ_{∞}). In this way one obtains a canonical short exact sequence of complexes

(6)  0 → D_{∞}^{·} → Cone(θ_{∞}^{·}) → D_{∞}^{·}[1] → 0

and also an isomorphism in D^{p}(Λ_{∞}) between Cone(θ_{∞}^{·}) and RΓ_{∗}(U_{ét}, coker(θ_{∞})). Now Λ_{∞} is a free ℤ_{p}[G]-module and so D_{∞}^{·} is a bounded complex of projective Λ-modules. If also each ℤ_{p}-module H^{i}(D_{∞}^{·}) is finitely generated, as is implied by condition ci), then by a standard argument (see, for example, the proof of [12, Thm. 1.1, p.447]) it follows that D_{∞}^{·} belongs to D^{p}(Λ). The exact sequence (6) then implies that Cone(θ_{∞}^{·}) also belongs to D^{p}(Λ). Further, condition cii) now combines with the long exact sequence of cohomology which is associated to (6) to imply that each module H_{∗}^{i}(Cone(θ_{∞}^{·})) is finite and hence that Cone(θ_{∞}^{·}) belongs to D^{p,f}(Λ), as claimed. It only remains to prove the explicit formula for χ_{Λ}^{rel}RΓ_{∗}(U_{ét}, coker(θ_{∞})). To do this we let P^{·} be a bounded complex of finitely generated projective Λ-modules which is quasi-isomorphic to D_{∞}^{·} and θ̂^{·} : P^{·} → P^{·} a morphism of complexes which induces θ_{∞}^{·}. Condition cii) combines with the argument of [13, Lem. 7.10] to imply we may assume that in each degree i the map θ̂^{i} is injective (and so has finite cokernel). It follows that RΓ_{∗}(U_{ét}, coker(θ_{∞})) is isomorphic in D^{p,f}(Λ) to the complex coker(θ̂^{·}) which is equal to coker(θ̂^{i}) in each degree i and for which the differentials are induced by those of P^{·}, and hence that χ_{Λ}^{rel}RΓ_{∗}(U_{ét}, coker(θ_{∞})) = χ_{Λ}^{rel} coker(θ̂^{·}). We next recall that χ_{Λ}^{rel} is additive on exact triangles in D^{p,f}(Λ) [5, Prop. 1.2.2]. From the short exact sequences of complexes

0 → coker(θ̂^{i})[−i] → τ_{i} coker(θ̂^{·}) → τ_{i−1} coker(θ̂^{·}) → 0

(where, for each integer j, τ_{j} denotes the naive truncation in degree j) we may therefore deduce that χ_{Λ}^{rel} coker(θ̂^{·}) = ∑_{i∈ℤ} χ_{Λ}^{rel}(coker(θ̂^{i})[−i]). The claimed formula now follows directly from the fact that for each integer i one has χ_{Λ}^{rel}(coker(θ̂^{i})[−i]) = (−1)^{i}δ_{Λ}([θ̂^{i} ⊗_{ℤ_{p}} ℚ_{p}]) and in K_{1}(Λ_{ℚ_{p}}) there is an equality

[θ̂^{i} ⊗_{ℤ_{p}} ℚ_{p}] = [θ̂^{i+1} ⊗_{ℤ_{p}} ℚ_{p} |_{B^{i+1}}] + [H_{∗}^{i}(U_{ét}, θ_{∞}) ⊗_{ℤ_{p}} ℚ_{p}] + [θ̂^{i} ⊗_{ℤ_{p}} ℚ_{p} |_{B^{i}}].

Here we write B^{i} for the submodule of coboundaries of P^{·} ⊗_{ℤ_{p}} ℚ_{p} in degree i, and the displayed equality is a consequence of the natural filtration of P^{i}⊗_{ℤ_{p}} ℚ_{p} which has graded pieces isomorphic to B^{i+1}, H_{∗}^{i}(U_{ét}, ϵ·F_{∞})⊗_{ℤ_{p}} ℚ_{p} and B^{i}. □

Remark 6. In this remark we assume that G is abelian, but otherwise use the same notation and hypotheses as in the second part of Proposition 4.1. We write Det_{Λ} for the determinant functor introduced by Knudsen and Mumford in [27], and (both here and in the sequel) we identify any graded invertible Λ-module of the form (I, 0) with the underlying invertible Λ-module I. We recall that the assignment χ_{Λ}^{rel}C ↦ Det_{Λ} C (where C ranges over all objects of D^{p,f}(Λ)) induces a well-defined isomorphism between K_{0}(Λ, ℚ_{p}) and the multiplicative group of invertible Λ-lattices in Λ_{ℚ_{p}} (cf. [1, Lem. 2.6]). In particular, in this case the equality at the end of Proposition 4.1 is equivalent to the following equality in Λ_{ℚ_{p}}

Det_{Λ} RΓ_{∗}(U_{ét}, coker(θ_{∞})) = ∏_{i∈ℤ} det_{Λ_{ℚp}} (H_{∗}^{i}(U_{ét}, θ_{∞}) ⊗_{ℤ_{p}} ℚ_{p})^{(−1)^{i+1}} · Λ.

(We remark that the exponent (−1)^{i+1} on the right hand side of this formula is not a misprint!)

5. Values of Dirichlet L-functions

In this section we derive certain explicit consequences of Theorem 3.1iii) and Proposition 4.1 concerning the values of Dirichlet L-functions at strictly negative integers. To this end we continue to use the notation introduced in §3. In particular, we now assume that k is totally real and that K is a CM abelian extension of k and we set G := Gal(K/k). We also fix an odd prime p, algebraic closures ℚ^{c} of ℚ and ℚ_{p}^{c} of ℚ_{p}, and we set G^{∧} := Hom(G, ℚ^{c×}) and G^{∧,p} := Hom(G, ℚ_{p}^{c×}). We let τ denote the complex conjugation in G, and for each integer a we write e_{a} for the idempotent \frac{1}{2}(1 + (−1)^{a}τ) of ℤ[\frac{1}{2}][G], and G_{(a)}^{∧} and G_{(a)}^{∧,p} for the subsets of G^{∧} and G^{∧,p} respectively which consist of those characters ψ satisfying ψ(τ) = (−1)^{a}. For each element ψ of G^{∧}, respectively of G^{∧,p}, we write e_{ψ} for the associated idempotent \frac{1}{|G|} ∑_{g∈G} ψ(g)g^{−1} of ℚ^{c}[G], respectively of ℚ_{p}^{c}[G]. We fix a finite set S of non-archimedean places of k which contains all nonarchimedean places which ramify in K/k and, for each ψ ∈ G^{∧}, we write L_{S}(s, ψ) for the Dirichlet L-function of ψ which is truncated by removing the Euler factors at all places in S.

If r is any integer with r > 1, then each function L_{S}(s, ψ) is holomorphic at s = 1 − r and so we may set

L_{S}(1 − r) := ∑_{ψ∈G^{∧}} L_{S}(1 − r, ψ)e_{ψ} ∈ ℂ[G].

If k = ℚ, then this element can be interpreted in terms of higher Bernoulli numbers and is therefore a natural analogue of the classical Stickelberger element. In general, by a result of Siegel [33], one knows that L_{S}(1 − r) belongs to the unit group of the ring ℚ[G]e_{r}. In order to state our next result we assume that K ∩ k_{∞}^{p} = k. Under this hypothesis we set

h_{S} := ∑_{ρ∈G_{(0)}^{∧,p}} h_{S,ρ}e_{ρ} ∈ ℤ_{p}[G][[T]][\frac{1}{p}].

We also write e for the idempotent \frac{1}{|G|} ∑_{g∈G} g of ℚ_{p}[G] and then set

H′ := ∑_{ρ∈G_{(0)}^{∧,p}} H_{ρ}e_{ρ} = T e + (e_{0} − e) ∈ ℚ_{p}[G][T],

where the second equality is a consequence of our assumption that K ∩k_{∞}^{p} = k. For the purposes of the next result we also assume that K contains a primitive p-th root of unity, and we write ω for the Teichmüller character of G. For each integer b we then let tw_{b} denote the ℤ_{p}-linear automorphism of ℤ_{p}[G] which sends each element g of G to ω^{b}(g) · g.

Theorem 5.1. Assume that Hypothesis (μ_{p}) is valid for K/k, that K ∩k_{∞}^{p} = k and that K contains a primitive p-th root of unity. Then for each integer r > 1 one has an equality

L_{S}(1 − r) · ℤ_{p}[G] = tw_{r}(H′(u^{r} − 1)^{−1}h_{S}(u^{r} − 1)) · ℤ_{p}[G]e_{r}.

Proof. At the outset we fix an integer r > 1 and an embeddding j : ℚ^{c} → ℚ_{p}^{c} and, for each χ ∈ G^{∧}, we set ρ_{χ} := (j ∘ χ) · ω^{r} ∈ G^{∧,p}. We observe that ω belongs to G_{(1)}^{∧,p} and hence that χ belongs to G_{(r)}^{∧} if and only if ρ_{χ} belongs to G_{(0)}^{∧,p}. In addition, for all characters χ ∈ G_{(r)}^{∧} one has an equality

(7)  (j ∘ χ)(L_{S_{p}}(1 − r)) = j(L_{S_{p}}(1 − r, χ)) = L_{p,S}(1 − r, ρ_{χ}).

We now set Z_{S} := (H′)^{−1} · h_{S}. Upon comparing images under each character ρ ∈ G^{∧,p}, recalling the equality (3) and noting that in the present case c_{ρ} = 1 for all such ρ, we may deduce that Z_{S} is equal to the quotient G_{S}^{∗}/H^{∗} which occurs in Hypothesis (ECP). With U_{S} denoting the unit element which occurs in Hypothesis (EUS) for K/k, and setting G_{S} := G_{S}^{∗} · U_{S} and H := H^{∗} it therefore follows that

Z_{S} · U_{S} = (G_{S}^{∗} · U_{S}) · (H^{∗})^{−1}
= G_{S} · H^{−1}.

From Theorem 3.1i),iii) we know that the series G_{S} and H satisfy the conditions specified in Hypothesis (EPS). Hence the last displayed formula implies that, for each character χ ∈ G_{(r)}^{∧}, one has

ρ_{χ}(Z_{S} · U_{S}) = ρ_{χ}(G_{S})ρ_{χ}(H)^{−1}
= f_{S,ρ_{χ}}.

Upon combining this formula with the equalities (1) and (7) we deduce that

(j ∘ χ)(L_{S_{p}}(1 − r)) = ρ_{χ}(Z_{S}(u^{r} − 1) · U_{S}(u^{r} − 1))
= (j ∘ χ)(tw_{r}(Z_{S}(u^{r} − 1) · U_{S}(u^{r} − 1)))
= (j ∘ χ)(tw_{r}(Z_{S}(u^{r} − 1)) · tw_{r}(U_{S}(u^{r} − 1))).

Since this equality is valid for every character χ in G_{(r)}^{∧} it implies that the elements L_{S_{p}}(1 − r) and tw_{r}(Z_{S}(u^{r} − 1)) of ℚ_{p}[G]e_{r} differ by the factor tw_{r}(U_{S}(u^{r} − 1)) which is a unit of the ring ℤ_{p}[G]e_{r}. It now only remains for us to show that the elements L_{S_{p}}(1 − r) and L_{S}(1 − r) differ by a unit of ℤ_{p}[G]e_{r}. But L_{S_{p}}(1 − r) = L_{S}(1 − r)x where x is a product of Euler factors of the form 1−Nv^{r−1} ·f_{v} where v is a place of k which divides p and does not belong to S, Nv is the absolute norm of v and f_{v} is the Frobenius automorphism of v in G. Further, since r > 1, it is clear that each such element 1 − Nv^{r−1} · f_{v} is a unit of ℤ_{p}[G]. □

Our next result concerns a special case of Kato’s Generalized Iwasawa Main Conjecture. However, before stating this result, it will be convenient to introduce some further notation. For the remainder of this section we let Σ denote the (finite) set of rational primes ℓ which satisfy either ℓ = 2 or K ∩ k_{∞}^{ℓ} ≠ k. We also write ℤ_{Σ} for the subring of ℚ which is generated by the inverses of each element of Σ. For any extension E of k and any finite set of places V of k we let O_{E,V} denote the ring of V_{E}-integers in E, where V_{E} denotes the set of all places of E which are either archimedean or lie above a place in V. We set U_{k} := Spec(O_{k,S_{p}}) and for each p-adic étale sheaf F on U_{k} and each finite Galois extension E/k which is unramified at all non-archimedean places outside S_{p} we write F_{E} for the étale sheaf of ℤ_{p}[Gal(E/k)]-modules π_{∗}π^{∗}F on U_{k} where π denotes the morphism Spec(O_{E,S_{p}}) → U_{k} which is induced by the inclusion O_{k,S_{p}} ⊆ O_{E,S_{p}}. We recall that, since π_{∗} is exact, the complexes RΓ(U_{k,ét}, F_{E}) and RΓ_{c}(U_{k,ét}, F_{E}) are canonically isomorphic in D(ℤ_{p}[Gal(E/k)]) to RΓ(Spec(O_{E,S_{p}})_{ét}, π^{∗}F) and RΓ_{c}(Spec(O_{E,S_{p}})_{ét}, π^{∗}F) respectively, and in the sequel we shall often use such identifications without explicit comment. For each integer r > 1 we set

C_{K,1−r} := RΓ_{c}(U_{k,ét}, e_{r}ℤ_{p}(1 − r)_{K})

and we recall that (since r > 1) this complex is an object of D^{p,f}(ℤ_{p}[G]e_{r}) (see the upcoming proof of Lemma 3 for further details in this regard). From the equalities of [8, (11),(12)] (with r replaced by 1 − r), it therefore follows that the ‘Equivariant Tamagawa Number Conjecture’ of [9, Conj. 4(iv)] is for the pair (h^{0}(Spec(K))(1 − r), ℤ_{Σ}[G]e_{r}) equivalent to asserting that if p does not belong to Σ, then in ℚ_{p}[G]e_{r} one has an equality

(8)  Det_{ℤ_{p}[G]e_{r}}^{−1} C_{K,1−r} = L_{S}(1 − r) · ℤ_{p}[G]

(cf. Remark 6). Before proceeding, we remark that the above equality is in general strictly finer than the corresponding case of the Generalized Iwasawa Main Conjecture which Kato formulates in [25, Conj. 3.2.2 and 3.4.14]. Indeed, since graded determinants are not used in [25] the central conjecture of loc. cit. is in this case only well defined to within multiplication by elements of (ℚ_{p}[G]e_{r})^{×} of square 1 which reflect possible re-ordering of the factors in tensor products. For more details in this regard we refer the reader to [loc cit., Rem. 3.2.3(3) and 3.2.6(3),(5)] and [9, Rem. 9]. We recall also that a direct comparison of [9, Conj. 4(iv)] with the central conjecture formulated by Kato in [24, Conj. (4.9)] can be found in [10, §2].

Theorem 5.2. Assume that p does not belong to Σ and that Hypothesis (μ_{p}) is valid for K/k. Then for each integer r > 1 the equality (8) is valid. In particular, the Generalized Iwasawa Main Conjecture of Kato is valid for each such pair (h^{0}(Spec(K))(1 − r), ℤ_{p}[G]e_{r}).

Remark 7. i) It is straightforward to describe explicit conditions on K/k which ensure that Σ = {2}. For example, if [K : k] is coprime to the class number of k, then Σ = {2} whenever the conductor of K/k is not divisible by the square of any prime ideal which divides [K : k]. ii) If K/ℚ is abelian, then Hypothesis (μ_{p}) is known to be valid for all p (Remark 3i)) and so Theorem 5.2 gives an alternative proof of parts of the main result (Cor. 8.1) of [11]. The reader will find that the approach of loc. cit. is considerably more involved than that used here. We remark that, nevertheless, the approach of loc. cit. can be extended to improve upon Theorem 5.2 by showing that [9, Conj. 4(iv)] is valid for the pair (h^{0}(Spec(K))(1−r), ℤ[\frac{1}{2}][G]e_{r}) under the assumption that Hypothesis (μ_{p}) is valid for K/k at all odd p.

Proof of Theorem 5.2. For the purposes of this argument we set 𝔄 := ℤ_{p}[G]e_{r} and A := ℚ_{p}[G]e_{r}. We first remark that, when verifying the equality (8), the functorial behaviour of compactly supported étale cohomology and of Dirichlet L-functions under Galois descent allows us to replace K by the extension of K which is generated by a primitive p-th root of unity (cf. [9, Prop. 4.1b)]). We may therefore henceforth assume that K contains a primitive p-th root of unity and is such that K ∩ k_{∞}^{p} = k. After taking into account the result of Theorem 5.1 it is therefore enough for us to prove that in ℚ_{p}[G]e_{r} one has an equality

(9)  Det_{𝔄}^{−1} C_{K,1−r} = tw_{r}(H′(u^{r} − 1)^{−1}h_{S}(u^{r} − 1)) · 𝔄.

We now use the notation of Proposition 4.1. We regard γ as a topological generator of Gal(K_{∞}^{p}/K) ≅ Gal(k_{∞}^{p}/k), we set γ̂ := 1−γ ∈ ℤ_{p}[Gal(K_{∞}^{p}/k)] and we observe that the action of γ̂ induces an injective ℤ_{p}[Gal(K_{∞}^{p}/k)]-equivariant endomorphism γ̂_{1−r} of the pro-sheaf e_{r} · ℤ_{p}(1 − r)_{∞} on U_{k,ét}.

Lemma 3. Let T denote the union of S_{p} and the set of archimedean places of k, and set U := U_{k}, F := ℤ_{p}(1 − r), ϵ := e_{r} and θ_{∞} := γ̂_{1−r}. If Hypothesis (μ_{p}) is valid for K/k, then this data satisfies the conditions ci) and cii) of Proposition 4.1, and in A one has an equality

∏_{i∈ℤ} det_{A}(H_{c}^{i}(U_{k,ét}, γ̂_{1−r}) ⊗_{ℤ_{p}} ℚ_{p})^{(−1)^{i}}

= det_{A}(γ̂ | e_{0}Y (S_{p}) ⊗_{ℤ_{p}} ℚ_{p}(−r)) · det_{A}(γ̂ | ℚ_{p}(−r))^{−1}

where γ̂ acts diagonally on e_{0}Y (S_{p}) ⊗_{ℤ_{p}} ℚ_{p}(−r).

Proof. We assume that Hypothesis (μ_{p}) is valid for K/k, and we recall (from Remark 3ii)) that this is equivalent to asserting that e_{0}Y (S_{p}) is a finitely generated ℤ_{p}-module. For each integer i we set H_{c}^{i}(1 − r) := H_{c}^{i}(U_{ét}, e_{r} · ℤ_{p}(1 − r)_{∞}). To verify that condition ci) of Proposition 4.1 is satisfied by the given data and also to prove the claimed equality, it is clearly enough to show that H_{c}^{i}(1 − r) vanishes if i ∉ {2, 3} and that H_{c}^{2}(1 − r) and H_{c}^{3}(1 − r) are canonically isomorphic to e_{0}Y (S_{p}) ⊗_{ℤ_{p}} ℤ_{p}(−r) (endowed with the natural diagonal action of γ) and ℤ_{p}(−r) respectively. To show this we first observe that, since p is odd and k is totally real, for each archimedean place v of k and any non-negative integer n the complex RΓ(Spec(k_{v})_{ét}, e_{r} · ℤ_{p}(1 − r)_{n}) is acyclic. This implies that our definition of compactly supported cohomology (as in (5)) coincides with that used by Nekovář in [30, (5.3)], and hence that the complex RΓ_{c}(U_{ét}, e_{r} · ℤ_{p}(1 − r)_{∞}) coincides with the complex RΓ_{c,Iw}(K_{∞}/k, ℤ_{p}(1 − r)) which is defined in [loc. cit., (8.5.4)]. To compute H_{c}^{i}(1 − r) we may therefore use the fact that there are natural isomorphisms of ℤ_{p}[Gal(K_{∞}/k)]-modules

(10)  H_{c}^{i}(1 − r) ≅ e_{r}(H_{c}^{i}(U_{ét}, ℤ_{p}(1)_{∞} ⊗_{ℤ_{p}} ℤ_{p}(−r)))

≅ (\underleftarrow{lim}_{n} H_{c,n}^{i,+}(1)) ⊗_{ℤ_{p}} ℤ_{p}(−r)

≅ (\underleftarrow{lim}_{n} H_{c,n,n}^{i,+}(1)) ⊗_{ℤ_{p}} ℤ_{p}(−r)

where, for each non-negative integer n, we set H_{c,n}^{i,+}(1) := e_{0}·H_{c}^{i}(U_{k,ét}, ℤ_{p}(1)_{K_{n}}) and H_{c,n,n}^{i,+}(1) := e_{0} · H_{c}^{i}(U_{k,ét}, (μ_{p^{n+1}})_{K_{n}}), each limit over the integers n ≥ 0 is taken with respect to the natural projection maps, Gal(K_{∞}/k) acts diagonally on each tensor product, and the second and third isomorphisms follow as a consequence of [loc. cit., Prop 8.5.5(ii), respectively Lem. (4.2.2)]. Now to compute explicitly each group H_{c,n}^{i,+}(1) for i ≠ 2 it is enough to combine the long exact sequence of cohomology of the triangle (5) (with U = U_{k} and F = ℤ_{p}(1)_{K_{n}}) together with certain standard results of Kummer theory and class field theory. To describe the result we write

λ_{n} : O_{K_{n},T}^{×} ⊗_{ℤ} ℤ_{p} → ∏_{w_{n}} \underleftarrow{lim}_{m≥1} K_{n,w_{n}}^{×}/(K_{n,w_{n}}^{×})^{p^{m}}

for the natural ‘diagonal’ morphism where on the right hand side w_{n} runs over all places of K_{n} which lie above places in T and the limit over m is taken with respect to the natural projection morphisms. Then one finds that H_{c,n}^{i,+}(1) vanishes if i ∉ {1, 2, 3}, that H_{c,n}^{3,+}(1) identifies with ℤ_{p} and that H_{c,n}^{1,+}(1) is isomorphic to e_{0} · ker(λ_{n}). Upon passing to the inverse limit over n (and by using (10)) one finds that H_{c}^{i}(1 − r) vanishes if i ∉ {2, 3} and that H_{c}^{3}(1 − r) is canonically isomorphic to ℤ_{p}(−r). To proceed we next recall that, for each pair of non-negative integers m and n, the Artin-Verdier Duality Theorem induces a canonical isomorphism in D(ℤ/p^{m}ℤ[Gal(K_{n}/k)])

RΓ_{c}(U_{ét}, e_{r}(μ_{p^{m}}^{⊗(1−r)})_{K_{n}}) ≅ Hom_{ℤ/p^{m}ℤ}(RΓ(U_{ét}, e_{r}(μ_{p^{m}}^{⊗r})_{K_{n}}), ℤ/p^{m}ℤ[−3])

where the linear dual is endowed with the contragredient action of Gal(K_{n}/k) (cf. [30, Prop. (5.4.3)(i), (2.11)] with R = ℤ/p^{m}ℤ[Gal(K_{n}/k)], J = R[0], K = k, S = T and X = e_{r}(μ_{p^{m}}^{⊗(1−r)})_{K_{n}}[0]). Now if K_{n,T}^{ab,n} denotes the maximal abelian extension of K_{n} which is unramified outside T and of exponent dividing p^{n+1}, then the above isomorphism (with r = 0 and m = n + 1) implies that H_{c,n,n}^{2,+}(1) is canonically isomorphic to e_{0} · Gal(K_{n,T}^{ab,n}/K_{n}). These isomorphisms are compatible with the natural transition morphisms as n varies and hence upon passing to the inverse limit (and using (10)) we obtain a canonical isomorphism between H_{c}^{2}(1 − r) and e_{0}Y (S_{p}) ⊗_{ℤ_{p}} ℤ_{p}(−r) (endowed with the diagonal action of Gal(K_{∞}/k)), as required. At this stage we need only verify that condition cii) of Proposition 4.1 is satisfied by the specified data. However, this is so because coker(γ̂_{1−r}) is isomorphic to the constant pro-sheaf e_{r}ℤ_{p}(1 − r)_{K} and all cohomology groups of the complex C_{K,1−r} are finite. To explain the latter fact we recall that, for any given n and r, the above displayed duality isomorphisms are compatible with the natural transition morphisms as m varies and hence (in the case n = 0) induce upon passing to the inverse limit a canonical isomorphism in D(𝔄)

(11)  C_{K,1−r} ≅ R Hom_{ℤ_{p}}(RΓ(U_{ét}, e_{r}ℤ_{p}(r)_{K}), ℤ_{p}[−3]).

where the linear dual is endowed with the action of 𝔄 which is induced by the contragredient action of G. (The existence of such an isomorphism also follows from the exactness of the central column of [8, diagram (114)] (where L corresponds to our field K) and the fact that each complex RΓ_{Δ}(L_{w}, ℤ_{p}(1−r))^{∗} which occurs in that diagram becomes acyclic upon multiplication by e_{r}.) Now, after taking (11) into account, it is enough for us to prove that all of the groups H^{i}(U_{ét}, e_{r}ℤ_{p}(r)_{K}) are finite and this follows, for example, as a consequence of the description of [11, Lem. 3.2ii)] and the fact that e_{r}(K_{2r−i}(O_{K,T}) ⊗_{ℤ} ℤ_{p}) is finite for both i ∈ {1, 2}. □

We next observe that (as can be verified by explicit computation)

det_{A}(γ̂ | ℚ_{p}(−r)) = (1 − u^{−r})e_{ω^{−r}} + (e_{r} − e_{ω^{−r}})
= tw_{r}(v_{r} · H′(u^{r} − 1))

where v_{r} := u^{−r}e + (e_{0} − e), and also

det_{A}(γ̂ | e_{0}Y (S_{p}) ⊗_{ℤ_{p}} ℚ_{p}(−r))
= ∑_{ρ∈G_{(r)}^{∧,p}} det_{ℚ_{p}^{c}}(1 − γ | e_{ρ}(Y (S_{p}) ⊗_{ℤ_{p}} ℚ_{p}^{c}(−r)))e_{ρ}

= ∑_{ρ∈G_{(r)}^{∧,p}} det_{ℚ_{p}^{c}}(1 − u^{−r}γ | e_{ρω^{r}}(Y (S_{p}) ⊗_{ℤ_{p}} ℚ_{p}^{c}))e_{ρ}

= tw_{r}(v′_{r} · h_{S}(u^{r} − 1)),

where v′_{r} := ∑_{ρ∈G_{(0)}^{∧,p}} u^{−rd_{ρ}}e_{ρ} with d_{ρ} := dim_{ℚ_{p}^{c}}(e_{ρ}(Y (S_{p}) ⊗_{ℤ_{p}} ℚ_{p}^{c})) for each

ρ ∈ G_{(0)}^{∧,p}. We remark that in proving the last displayed equality one uses the fact that for each κ ∈ G^{∧,p} the ℚ_{p}^{c}[γ]-module e_{κ}(Y (S_{p}) ⊗_{ℤ_{p}} ℚ_{p}^{c}) is isomorphic to H^{0}(Gal(K_{∞}/k_{∞}), Hom_{ℚ_{p}^{c}}(ℚ_{p}^{c} · e_{κ}, Y (S_{p}) ⊗_{ℤ_{p}} ℚ_{p}^{c})). Upon combining the last two displayed formulas with the result of Lemma 3, the quasi-isomorphism C_{K,1−r} ≅ RΓ_{c}(U_{k,ét}, coker(γ̂_{1−r})) and the equality of Remark 6 we find that

Det_{𝔄}^{−1} C_{K,1−r} = tw_{r}(v_{r}^{−1}v′_{r}) tw_{r}(H′(u^{r} − 1)^{−1}h_{S}(u^{r} − 1)) · 𝔄.

The required equality (9) is thus a consequence of the following observation.

Lemma 4. v_{r}^{−1}v′_{r} is a unit of ℤ_{p}[G]e_{0}.

Proof. We start by making a general observation. For this we set 𝔅 := ℤ_{p}[G]e_{0}, and we let f_{1}(T) and f_{2}(T) denote any elements of 𝔅[[T]] which satisfy μ_{𝔅}(f_{1}(T)) = μ_{𝔅}(f_{2}(T)) = 0. For i = 1, 2 we write f_{i}^{∗}(T) and U_{i}(T) for the distinguished polynomial and unit series which occur in the product decomposition of f_{i}(T) afforded by Proposition 2.1 (with 𝔄 = 𝔅). We also set f_{i,r}(T) := f_{i}(u^{r}(1 + T) − 1) ∈ 𝔅[[T]] and, observing that μ_{𝔅}(f_{i,r}(T)) = 0, we write U_{i,r}(T) for the unit series which occurs in the product decomposition of f_{i,r}(T) afforded by Proposition 2.1. Then, by explicit computation, one verifies that the element (U_{1}(u^{r} − 1)U_{2,r}(0))(U_{2}(u^{r} − 1)U_{1,r}(0))^{−1} of 𝔅^{×} is equal to ∑_{ρ∈G_{(0)}^{∧,p}} u^{−rδ_{ρ}}e_{ρ} where δ_{ρ} := deg(ρ(f_{1}^{∗}(T))) − deg(ρ(f_{2}^{∗}(T))). We now apply this observation with f_{1}(T) and f_{2}(T) equal to the series G_{S} and H which occur in Hypothesis (EPS). We observe that, in this case, the equality (3) implies that for each ρ ∈ G_{(0)}^{∧,p} one has deg(ρ(G_{S}^{∗})) − deg(ρ(H)) = d′_{ρ} where here d′_{ρ} := d_{ρ} − 1 if ρ is trivial, and d′_{ρ} := d_{ρ} otherwise. From the general observation made above we may therefore deduce that the element v_{r}^{−1}v′_{r} = ∑_{ρ∈G_{(0)}^{∧,p}} u^{−rd′_{ρ}}e_{ρ} belongs to 𝔅^{×}, as claimed. □

This completes our proof of Theorem 5.2. □

We next use Theorem 5.2 to prove a result concerning the element Ω_{r−1}(K/k) of Pic(ℤ[G]) which is defined by Chinburg, Kolster, Pappas and Snaith in [13, §3]. We recall that it has been conjectured by the authors of loc. cit. that Ω_{r−1}(K/k) = 0. We write ρ_{Σ,r} for the natural scalar extension morphism Pic(ℤ[G]) → Pic(ℤ_{Σ}[G]e_{r}). With R denoting either ℤ or ℚ we also write ρ_{#} for the Rlinear involution of R[G] which is induced by sending each element of G to its inverse.

Corollary 1. Assume that (if k ≠ ℚ, then) Hypothesis (μ_{p}) is valid for K/k at each prime p ∉ Σ. Then for each integer r > 1 one has an equality ρ_{Σ,r}(Ω_{r−1}(K/k)) = 0.

Proof. The key point we use here is a result of Flach and the first named author. Indeed, the result of [8, Thm. 4.1] implies that ρ_{Σ,r}(Ω_{r−1}(K/k)) is equal to the class of the invertible ℤ_{Σ}[G]e_{r}-submodule of ℚ[G]e_{r} which is defined by means of the intersection

( ⋂_{p∉Σ} Det_{ℤ_{p}[G]e_{r}}^{−1} RΓ_{c}(U_{k,ét}, e_{r}ℤ_{p}(1 − r)_{K})) ⊗_{ℤ[G],ρ_{#}} ℤ[G].

(To see this one must recall that the normalisation of the determinant functor which is used in [8] is the inverse of that used here.) On the other hand, Theorem 5.2 implies that the above intersection is equal to the free ℤ_{Σ}[G]e_{r}-module which is generated by the element ρ_{#}(L_{S}(1 − r)). Hence one has ρ_{Σ,r}(Ω_{r−1}(K/k)) = 0, as required. □

Before stating our final result we introduce a little more notation. If V is any finite set of places of k, then for each rational prime ℓ we let V_{ℓ} denote the union of V and the set of places of k which are either archimedean or of residue characteristic ℓ. We set ℤ′ := ℤ[\frac{1}{2}] and we define a ℤ′[G]-module by setting

H^{2}(O_{K,V} , ℤ′(r)) := ⨁_{ℓ≠2} H^{2}(Spec(O_{K,V_{ℓ}})_{ét}, ℤ_{ℓ}(r)).

We let Σ′ denote the set {2}, respectively Σ, if k = ℚ, respectively k ≠ ℚ, and we write ℤ_{Σ′} for the subring of ℚ which is generated by the inverses of each element of Σ′. We also write μ_{ℚ^{c}} for the torsion subgroup of ℚ^{c×}.

Corollary 2. Assume that (if k ≠ ℚ, then) Hypothesis (μ_{p}) is valid for K/k at each prime p ∉ Σ. Then for each integer r > 1 one has an equality

ρ_{#}(L_{S}(1 − r)) · Ann_{ℤ[G]}(H^{0}(K, μ_{ℚ^{c}}^{⊗r})) ⊗_{ℤ} ℤ_{Σ′}

= e_{r} · Fit_{ℤ′[G]}(H^{2}(O_{K,S}, ℤ′(r))) ⊗_{ℤ′} ℤ_{Σ′}.

Remark 8. i) If T is any subset of S, then the localisation sequence of étale cohomology induces a natural inclusion of ℤ′[G]-modules H^{2}(O_{K,T} , ℤ′(r)) ⊆ H^{2}(O_{K,S}, ℤ′(r)). From this we may deduce that Ann_{ℤ′[G]}(H^{2}(O_{K,S}, ℤ′(r))) ⊆ Ann_{ℤ′[G]}(H^{2}(O_{K,T} , ℤ′(r))) and also if, for example, G is cyclic, that Fit_{ℤ′[G]}(H^{2}(O_{K,S}, ℤ′(r))) ⊆ Fit_{ℤ′[G]}(H^{2}(O_{K,T} , ℤ′(r))). In particular, if Σ′ = {2} (cf. Remark 7i)), and we write O_{K} in place of O_{K,∅} (which, in terms of our current notation, denotes the ring of algebraic integers in K), then the equality of Corollary 2 implies that

(12) ρ_{#}(L_{S}(1 − r)) · Ann_{ℤ[G]}(H^{0}(K, μ_{ℚ^{c}}^{⊗r})) ⊗_{ℤ} ℤ′ ⊆ Ann_{ℤ′[G]}(H^{2}(O_{K}, ℤ′(r))),

and also if, for example, G is cyclic, that

ρ_{#}(L_{S}(1 − r)) · Ann_{ℤ[G]}(H^{0}(K, μ_{ℚ^{c}}^{⊗r})) ⊗_{ℤ} ℤ′ ⊆ Fit_{ℤ′[G]}(H^{2}(O_{K}, ℤ′(r))).

We observe that if (as has been famously conjectured by Quillen and Lichtenbaum) the ℤ′[G]-module H^{2}(O_{K}, ℤ′(r)) is isomorphic to K_{2r−2}(O_{K}) ⊗_{ℤ} ℤ′, then the inclusion (12) is finer than (the image under − ⊗_{ℤ} ℤ′ of) the inclusion

#H^{0}(ℚ, μ_{ℚ^{c}}^{⊗r})ρ_{#}(L_{S}(1 − r)) · Ann_{ℤ[G]}(H^{0}(K, μ_{ℚ^{c}}^{⊗r})) ⊆ Ann_{ℤ[G]}(K_{2r−2}(O_{K}))

which was conjectured in the case k = ℚ by Coates and Sinnott in [14, Conj. 1]. We also recall that if k = ℚ, r is even and K is any abelian extension of ℚ, then Kurihara has recently used different methods to explicitly compute Fit_{ℤ′[G]}(H^{2}(O_{K}, ℤ′(r))) in terms of Stickelberger elements [28, Cor. 12.5 and Rem. 12.6]. ii) In the case that k = ℚ and the conductor of K/k is a prime power, the image under multiplication by e_{r} −e_{ω^{r}} of the equality of Corollary 2 has already been proved by Cornacchia and Østvær in [16, Thm. 1.2]. iii) If r is even, then the equality of Corollary 2 can be re-expressed as an equality in which K is replaced by K^{+} and the idempotent factor e_{r} is omitted. In a recent preprint [35], Snaith uses results from [11] to prove a weaker version of the equality of Corollary 2 in this context. More precisely, Snaith’s results [loc. cit., Th. 1.6, Th. 5.2] assume that K is totally real, that r is even and that Hypothesis (μ_{p}) is valid for K/k at all odd primes p, and involve chains of inclusions rather than a precise specification of Fitting ideals (see also Remark 9 in this regard). iv) In this remark we fix an odd prime p, an embedding j : ℚ^{c} → ℚ_{p}^{c} and a character χ ∈ G_{(r)}^{∧}. We set O := ℤ_{p}(j ∘χ) and we write lth_{O}(M) for the length of any finite O-module M. We let K_{χ} denote the (cyclic) extension of ℚ which corresponds to ker(χ). Then the image under the functor − ⊗_{ℤ′[G]} O of the equality of Corollary 2 with k = ℚ and K = K_{χ} is equivalent to an equality

val_{O}(j(L_{S}(1 − r, χ^{−1}))) =

lth_{O}(H^{2}(O_{K,S}, ℤ′(r)) ⊗_{ℤ′[G]} O) − lth_{O}(H^{0}(K, μ_{ℚ^{c}}^{⊗r}) ⊗_{ℤ[G]} O).

In addition, since in this case G is cyclic, for any finite ℤ′[G]-module N one has lth_{O}(N ⊗_{ℤ′[G]} O) = lth_{O}(Hom_{ℤ′[G]}(O, N)) and so the previous displayed equality provides a natural analogue of the main result (Thm. II.1) of Solomon in [36] concerning the relation between generalised Bernoulli numbers and the structure of certain ideal class groups. (The first named author is very grateful to Masato Kurihara for a most helpful conversation in this regard.)

Proof of Corollary 2. We now fix a prime p ∉ Σ′ and we assume that Hypothesis (μ_{p}) is valid for K/k (as is known in the case k = ℚ). We set 𝔄 := ℤ_{p}[G]e_{r}, μ(r) := H^{0}(K, μ_{ℚ^{c}}^{⊗r}) ⊗_{ℤ} ℤ_{p} and C := RΓ(U_{k,ét}, e_{r}ℤ_{p}(r)_{K}). In the sequel we shall say that a commutative ℤ_{p}-algebra Λ is ‘relatively Gorenstein over ℤ_{p}’ if Hom_{ℤ_{p}}(Λ, ℤ_{p}) (endowed with the natural action of Λ) is a free Λ-module of rank one. For each bounded object X of D(𝔄) we set X^{∗} := R Hom_{ℤ_{p}}(X, ℤ_{p}) which we endow with the action of 𝔄 which is induced by the contragredient action of G. We observe that 𝔄 is relatively Gorenstein over ℤ_{p} and hence that X^{∗} belongs to D^{p}(𝔄), respectively D^{p,f}(𝔄), if and only if X belongs to D^{p}(𝔄), respectively D^{p,f}(𝔄). Now for any 𝔄-module X there exists a canonical isomorphism between the 𝔄-modules Hom_{𝔄}(X, 𝔄) ⊗_{ℤ_{p}[G],ρ_{#}} ℤ_{p}[G] and Hom_{ℤ_{p}}(X, ℤ_{p}). This in turn implies that for any object X of D^{p}(𝔄) the lattice Det_{𝔄}^{−1} X^{∗}[−3] identifies canonically with (Det_{𝔄}^{−1} X) ⊗_{ℤ_{p}[G],ρ_{#}} ℤ_{p}[G]. Upon noting that (11) induces an isomorphism in D^{p,f}(𝔄) of the form C ≅ RΓ_{c}(U_{k,ét}, e_{r}ℤ_{p}(1 − r)_{K})^{∗}[−3], and recalling that the equality (8) is known to be valid as a consequence of Theorem 5.2 in the case k ≠ ℚ and as a consequence of [11, Cor. 8.1] in the case k = ℚ, we deduce that Det_{𝔄}^{−1} C = ρ_{#}(L_{S}(1 − r)) · 𝔄. The equality of Corollary 2 will therefore follow if we can show that

(13)  Det_{𝔄}^{−1} C · Ann_{𝔄}(μ(r)) = Fit_{𝔄}(H^{2}(C)).

We next observe that, since C belongs to D^{p,f}(𝔄) and is acyclic outside degrees 1 and 2, there exists an exact sequence of 𝔄-modules

(14)  0 → H^{1}(C) → Q ⟶^{d} Q′ → H^{2}(C) → 0

which is such that both Q and Q′ are finite and of projective dimension at most 1 and there exists an isomorphism ι in D^{p,f}(𝔄) between C and the complex Q ⟶^{d} Q′ (where the modules are placed in degrees 1 and 2, and the cohomology is identified with H^{1}(C) and H^{2}(C) by using the maps in (14)) for which H^{i}(ι) is the identity map in each degree i. This implies that Fit_{𝔄}(Q) and Fit_{𝔄}(Q′) are invertible ideals of 𝔄 and that Det_{𝔄}^{−1} C = Fit_{𝔄}(Q)^{−1} Fit_{𝔄}(Q′).

Lemma 5. Let R be any reduced commutative ℤ_{p}-algebra which is finitely generated, free and relatively Gorenstein over ℤ_{p}. If

0 → A ⟶ P ⟶ P′ ⟶ A′ → 0

is any exact sequence of finite R-modules in which P and P′ are both of projective dimension at most 1 over R, then Fit_{R}(P) and Fit_{R}(P′) are principal ideals of R and one has an equality

Fit_{R}(A^{∨}) Fit_{R}(P′) = Fit_{R}(P) Fit_{R}(A′),

where A^{∨} denotes the Pontryagin dual Hom_{ℤ_{p}}(A, ℚ_{p}/ℤ_{p}) (endowed with the natural action of R).

Proof. This is almost covered by the result of [15, Prop. 6]; however, the latter is stated only for rings R of the form O[G] with G a finite abelian p-group, and involves Fit_{R}(A) rather than Fit_{R}(A^{∨}). In addition, the second named author would like to take this opportunity to point out that the argument in loc. cit. makes the assumption that G (which is written as P in loc. cit.) is cyclic, which is unfortunately not stated explicitly at the appropriate place, and that the equality of [15, Prop. 6] does not appear to hold in general. We will therefore now quickly adapt the arguments of [15, Prop. 6] to better suit our present purpose. We first observe that, since R is semilocal, the Fitting ideal of any finite Rmodule P which is of projective dimension at most 1 is principal, being generated by the determinant of α in any presentation R^{n} ⟶^{α} R^{n} → P → 0. We may find the following data, proceeding exactly as in [15, p.456f.]: a nonzerodivisor f of R (it is in fact always possible to take f to be a large enough power of p); a natural number n; short exact sequences 0 → Q → Ã → A → 0 and 0 → A′ → Ã′ → Q′ → 0 in which Q and Q′ are both finite and of projective dimension at most 1, and a four term exact sequence

0 → Ã → (R/fR)^{n} → (R/fR)^{n} → Ã′ → 0.

In a similar way one obtains the equalities

Fit_{R}(P) Fit_{R}(Q) = f^{n}R = Fit_{R}(P′) Fit_{R}(Q′).

Now R/fR is Gorenstein of dimension zero, that is: (R/fR)^{∨} ≅ R/fR as Rmodules. Therefore the argument in loc. cit. starting with equation (4) applies to give an equality

Fit_{R}(Ã′) = Fit_{R}(Ã^{∨}).

(Note that we do not claim that Fit_{R}(Ã^{∨}) = Fit_{R}(Ã), as was done in loc. cit.) Applying the result of [15, Lem. 3] to the sequence 0 → A′ → Ã′ → Q′ → 0, respectively to the Pontryagin dual of the sequence 0 → Q → Ã → A → 0, we obtain an equality

Fit_{R}(Ã′) = Fit_{R}(A′) Fit_{R}(Q′),

respectively

Fit_{R}(Ã^{∨}) = Fit_{R}(A^{∨}) Fit_{R}(Q^{∨}).

Lemma 6. Fit_{R}(Q^{∨}) = Fit_{R}(Q).

Proof. Since Q is finite and of projective dimension at most 1 there exists an exact sequence of R-modules of the form

0 → R^{n} ⟶^{α} R^{n} → Q → 0.

Now Hom_{ℤ_{p}}(Q, ℤ_{p}) = 0 and Ext_{ℤ_{p}}^{1}(Q, ℤ_{p}) is isomorphic to Q^{∨} (as Q is finite), Ext_{ℤ_{p}}^{1}(R^{n}, ℤ_{p}) = 0 (as R is ℤ_{p}-free) and Hom_{ℤ_{p}}(R^{n}, ℤ_{p}) is isomorphic to R^{n} (as R is relatively Gorenstein over ℤ_{p}). From the long exact sequence of Ext_{ℤ_{p}}^{i}(−, ℤ_{p})-groups which is associated to the above sequence we therefore obtain a further exact sequence of R-modules

0 → R^{n} ⟶^{α^{t}} R^{n} → Q^{∨} → 0,

where α^{t} denotes the transpose of α. By using the two displayed sequences we now compute that Fit_{R}(Q) = det(α)R = det(α^{t})R = Fit_{R}(Q^{∨}), as claimed.

□

The equality of Lemma 5 now follows directly upon combining the equality of Lemma 6 with the four displayed equalities which immediately precede it. □

Upon applying Lemma 5 with R = 𝔄 (which is both reduced and relatively Gorenstein over ℤ_{p}) to the exact sequence (14) we obtain an equality

Det_{𝔄}^{−1} C · Fit_{𝔄}(H^{1}(C)^{∨}) = Fit_{𝔄}(Q)^{−1} Fit_{𝔄}(Q′) Fit_{𝔄}(H^{1}(C)^{∨})
= Fit_{𝔄}(H^{2}(C)).

To deduce the required equality (13) from this equality we now simply observe that the 𝔄-module H^{1}(C)^{∨} is isomorphic to the cyclic 𝔄-module μ(r)^{∨}, and hence that Fit_{𝔄}(H^{1}(C)^{∨}) = Fit_{𝔄}(μ(r)^{∨}) = Ann_{𝔄}(μ(r)^{∨}) = Ann_{𝔄}(μ(r)). This completes our proof of Corollary 2. □

Remark 9. Let Γ be any finite abelian group and ℓ any rational prime. If M is any finite ℤ_{ℓ}[Γ]-module, then Ann_{ℤ_{ℓ}[Γ]}(M) = Ann_{ℤ_{ℓ}[Γ]}(M^{∨}) and Ann_{ℤ_{ℓ}[Γ]}(M)^{n(M)} ⊆ Fit_{ℤ_{ℓ}[Γ]}(M) ⊆ Ann_{ℤ_{ℓ}[Γ]}(M) where n(M) denotes the minimal number of elements required to generate M over ℤ_{ℓ}[Γ]. Hence, if X is any object of D^{p,f}(ℤ_{ℓ}[Γ]) which is acyclic outside degrees 0 and 1 and t_{i} ∈ Ann_{ℤ_{ℓ}[Γ]}(H^{i}(X)) for i ∈ {0, 1}, then (since ℤ_{ℓ}[Γ] is both reduced and relatively Gorenstein over ℤ_{ℓ}) the equality of Lemma 5 implies that

t_{0}^{n(H^{0}(X)^{∨})} · Det_{ℤ_{ℓ}[Γ]} X ⊆ Fit_{ℤ_{ℓ}[Γ]}(H^{1}(X))

and also

t_{1}^{n(H^{1}(X))} · Det_{ℤ_{ℓ}[Γ]}^{−1} X ⊆ Fit_{ℤ_{ℓ}[Γ]}(H^{0}(X)^{∨}).

In particular, if n(H^{0}(X)^{∨}) = n(H^{0}(X)) (which, for example, is the case when H^{0}(X) = H^{0}(K, μ_{ℚ^{c}}^{⊗r}) ⊗_{ℤ} ℤ_{ℓ}), then Lemma 5 refines the main algebraic result (Thm. 2.4) of Snaith in [35].

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David Burns

Department of Mathematics King’s College London Strand London WC2R 2LS England david.burns@kcl.ac.uk Cornelius Greither Fakultät für Informatik Universität der Bundeswehr München, D-85577 Neubiberg Germany greither@informatik.unibwmuenchen.de 186

Documenta Mathematica · Extra Volume Kato (2003)