On the

Equivariant Tamagawa Number Conjecture

for Abelian Extensions

of a Quadratic Imaginary Field

W. Bley

Received: October 17, 2005

Communicated by Peter Schneider

Abstract. Let k be a quadratic imaginary field, p a prime which splits in k/ℚ and does not divide the class number h_{k} of k. Let L denote a finite abelian extension of k and let K be a subextension of L/k. In this article we prove the p-part of the Equivariant Tamagawa Number Conjecture for the pair (h^{0}(Spec(L)), ℤ[Gal(L/K)]).

2000 Mathematics Subject Classification: 11G40, 11R23, 11R33, 11R65 Keywords and Phrases: L-functions, Iwasawa theory, Euler systems

1 Introduction

The aim of this paper is to provide new evidence for the validity of the Equivariant Tamagawa Number Conjectures (for short ETNC) as formulated by Burns and Flach in [4]. We recall that these conjectures generalize and refine the Tamagawa Number Conjectures of Bloch, Kato, Fontaine, Perrin-Riou et al. In the special case of the untwisted Tate motive the conjecture also refines and generalizes the central conjectures of classical Galois module theory as developed by Fröhlich, Chinburg, Taylor et al (see [2]). Moreover, in many cases it implies refinements of Stark-type conjectures formulated by Rubin and Popescu and the ‘refined class number formulas’ of Gross. For more details in this direction see [3]. Let k denote a quadratic imaginary field. Let L be a finite abelian extension of k and let K be any subfield of L/k. Let p be a prime number which does not divide the class number h_{k} of k and which splits in k/ℚ. Then we prove the ’p-part’ of the ETNC for the pair (h^{0}(Spec(L), ℤ[Gal(L/K)])) (see Theorem 4.2).

To help put the main result of this article into context we recall that so far the ETNC for Tate motives has only been verified for abelian extensions of the rational numbers ℚ and certain quaternion extensions of ℚ. The most important result in this context is due to Burns and Greither [5] and establishes the validity of the ETNC for the pair (h^{0}(Spec(L)(r), ℤ[\frac{1}{2}][Gal(L/K)])), where L/ℚ is abelian, ℚ ⊆ K ⊆ L and r ≤ 0. The 2-part was subsequently dealt with by Flach [8], who also gives a nice survey on the general theory of the ETNC, including a detailed outline of the proof of Burns and Greither. Shortly after Burns and Greither, the special case r = 0 was independently shown (up to the 2-part) by Ritter and Weiss [22] using different methods. In order to prove our main result we follow very closely the strategy of Burns and Greither, which was inspired by previous work of Bloch, Kato, Fontaine and Perrin-Riou. In particular, in [13] Kato formulates a conjecture whose proof is one of the main achievements in the work of Burns and Greither. Roughly speaking, we will replace cyclotomic units by elliptic units. More concretely, the ETNC for the pair (h^{0}(Spec(L), ℤ[Gal(L/K)])) conjecturally describes the leading coefficient in the Laurant series of the equivariant Dirichlet L-function at s = 0 as the determinant of a canonical complex. By Kronecker’s limit formula we replace L-values by sums of logarithms of elliptic units. In this formulation we may pass to the limit along a ℤ_{p}-extension and recover (an analogue) of a conjecture which was formulated by Kato in [13]. As in [5] we will deduce this limit conjecture from the Main Conjecture of Iwasawa Theory and the triviality of certain Iwasawa μ-invariants (see Theorem 5.1). Combining the validity of the limit theorem with Iwasawa-theoretic descent considerations we then achieve the proof of our main result. The Main Conjecture in the elliptic setting was proved by Rubin in [24], but only in semi-simple case (i.e. p ∤ [L : k]). Following Greither’s exposition [10] we adapt Rubin’s proof and obtain the full Main Conjecture (see Theorem 3.1) for ray class fields L and primes p which split in k/ℚ and do not divide the class number h_{k} of k. The triviality of μ-invariants in the elliptic setting is known from work of Gillard [9], but again only in the ordinary case when p is split in k/ℚ. The descent considerations are particularly involved in the presence of ’trivial zeros’ of the associated p-adic L-functions. In this case we make crucial use of a recently published result of the author [1] concerning valuative properties of certain elliptic p-units. As in the cyclotomic case it is possible to use the Iwasawa-theoretic result of Theorem 5.1 and Iwasawa descent to obtain the p-part of the ETNC for (h^{0}(Spec(L)(r), ℤ[Gal(L/K)])), r < 0. We refer to thesis of Johnson [12] who deals with this case. We conclude this introduction with some remarks on the non-split situation. Generically this case is more complicated because the corresponding Iwasawa extension is of type ℤ_{p}^{2}. The main issue, if one tries to apply the above described strategy in the non-split case, is to prove μ = 0. Note that we already use the triviality of μ in our proof of the Iwasawa Main Conjecture (see Remark 3.9).

During the preparation of this manuscript I had the pleasure to spend three months at the department of mathematics in Besançon and three weeks at the department of mathematics at Caltech, Pasadena. My thanks go to the algebra and number theory teams at both places for their hospitality and the many interesting mathematical discussions.

2 Elliptic units

The aim of this section is to define the elliptic units that we will use in this paper. Our main references are [20], [21] and [1]. We let L ⊆ ℂ denote a ℤ-lattice of rank 2 with complex multiplication by the ring of integers of a quadratic imaginary field k. We write N = N_{k/ℚ} for the norm map from k to ℚ. For any O_{k}-ideal 𝔞 satisfying (N(𝔞), 6) = 1 we define a meromorphic function

ψ(z; L, 𝔞) := F̃(z; L, 𝔞^{−1}L), z ∈ ℂ,

where F̃ is defined in [20, Théorème principale, (15)]. This function ψ coincides with the function θ(z; 𝔞) used by Rubin in [23, Appendix] and it is a canonical 12th root of the function θ(z; L, 𝔞) defined in [6, II.2]. The basic arithmetical properties of special values of ψ are summarized in [1, §2]. We choose a ℤ-basis w_{1}, w_{2} of the complex lattice L such that Im(w_{1}/w_{2}) > 0 and write η(τ), Im(τ) > 0, for the Dedekind η-function. Let η_{1}, η_{2} denote the quasi-periods of the Weierstrass ζ-function and for any z = a_{1}w_{1} + a_{2}w_{2} ∈ ℂ, a_{1}, a_{2} ∈ ℝ, put z^{∗} = a_{1}η_{1} + a_{2}η_{2}. Writing σ(z; L) for the Weierstrass σfunction attached to L we define

φ(z; w_{1}, w_{2}) := 2πie^{−zz^{∗}/2}σ(z; L)η^{2} (\frac{w_{1}}{w_{2}} ) w_{2}^{−1}.  (1)

Note that φ is exactly the function defined in [20, (4)]. The function φ is not a function of lattices but depends on the choice of a basis w_{1}, w_{2}. Its 12th power does not depend on this choice and we will also write φ^{12}(z; L). We easily deduce from [20, Sec. 3, Lemme] and its proof that the relation between φ and ψ is given by

ψ^{12}(z; L, 𝔞) = \frac{φ^{12N(𝔞)}(z; L)}{φ^{12}(z; 𝔞^{−1}L)}.  (2)

3 The Iwasawa main conjecture

For any O_{k}-ideal 𝔟 we write k(𝔟) for the ray class field of conductor 𝔟. In this notation k(1) denotes the Hilbert class field. We let w(𝔟) denote the number of roots of unity in k which are congruent to 1 modulo 𝔟. Hence w(1) is the number of roots of unity in k. This number will also be denoted by w_{k}.

Let p denote an odd rational prime which splits in k/ℚ, and let 𝔭 be a prime ideal of k lying over p. We assume p ∤ h_{k}. For each n ≥ 0 we write

Gal(k(𝔭^{n+1})/k) = Gal(k(𝔭^{n+1})/k(𝔭)) × H,

where H is isomorphic to Gal(k(𝔭)/k) by restriction. We set

k_{n} := k(𝔭^{n+1})^{H}, k_{∞} := ⋃_{n≥0} k_{n},

and note that k_{∞}/k is a ℤ_{p}-extension. More precisely, k_{∞}/k is the unique ℤ_{p}- extension of k which is unramified outside 𝔭. The prime 𝔭 is totally ramified in k_{∞}/k. Let now 𝔣 be any integral ideal of k such that (𝔣, 𝔭) = 1. Let F = k(𝔣𝔭) denote the ray class field of conductor 𝔣𝔭. We set K_{n} := F k_{n} = k(𝔣𝔭^{n+1}) and K_{∞} := ∪_{n≥0}K_{n}. Then K_{∞}/K_{0} is a ℤ_{p}-extension in which each prime divisor of 𝔭 is totally ramified. For any number field L we denote the p-part of the ideal class group of L by A(L). Set A_{∞} := lim_{n^{←}} A(K_{n}), the inverse limit formed with respect to the norm

maps. We write E_{n} for the group of global units of K_{n}. For a divisor 𝔤 of 𝔣 we let C_{n,𝔤} denote the subgroup of primitive Robert units of conductor 𝔣𝔭^{n+1}, n ≥ 0. If 𝔤 ≠ (1), then C_{n,𝔤} is generated by all ψ(1; 𝔤𝔭^{n+1}, 𝔞) with (𝔞, 𝔤𝔭) = 1 and the roots of unity in K_{n}. If 𝔤 = (1), then the elements ψ(1; 𝔭^{n+1}, 𝔞) are no longer units. By [1, Th. 2.4] a product of the form ∏ ψ(1; 𝔭^{n+1}, 𝔞)^{m(𝔞)} is a unit, if and only if ∑ m(𝔞)(N(𝔞) − 1) = 0. We let C_{n,𝔤} denote the group generated by all such products and the roots of unity in K_{n}. We let C_{n} be the group of units generated by the subgroups C_{n,𝔤} with 𝔤 running over the divisors of 𝔣. We let U_{n} denote the semi-local units of K_{n}⊗_{k}k_{𝔭} which are congruent to 1 modulo all primes above 𝔭, and let Ē_{n} and C̄_{n} denote the closures of E_{n} ∩ U_{n} and C_{n} ∩ U_{n}, respectively, in U_{n}. Finally we define

Ē_{∞} := lim_{n^{←}} Ē_{n}, C̄_{∞} := lim_{n^{←}} C̄_{n},

both inverse limits formed with respect to the norm maps. We let

Λ = lim_{n^{←}} ℤ_{p}[Gal(K_{n}/k)]

denote the completed group ring and for a finitely generated Λ-module and any abelian character χ of Δ := Gal(K_{0}/k) we define the χ-quotient of M by

M_{χ} := M⊗_{ℤ_{p}[Δ]}ℤ_{p}(χ),

where ℤ_{p}(χ) denotes the ring extension of ℤ_{p} generated by the values of χ. For the basic properties of the functor M ↦ M_{χ} the reader is referred to [30, §2]. The ring Λ_{χ} is (non-canonically) isomorphic to the power series ring ℤ_{p}(χ)[[T]]. If M_{χ} is a finitely generated torsion Λ_{χ}-module, then we write char(M_{χ}) for the characteristic ideal.

Theorem 3.1 Let p be an odd rational prime which splits into two distinct primes in k/ℚ. Then

char(A_{∞,χ}) = char((Ē_{∞}/C̄_{∞})_{χ}).

Remarks 3.2 a) If p ∤ [F : k] and p does not divide the number of roots of unity in k(1), then the result of Theorem 3.1 is already proved by Rubin, see [24, Th. 4.1(i)].

b) The Main Conjecture of Iwasawa theory for abelian extensions of ℚ was first proved by Mazur and Wiles [16] using deep methods from algebraic geometry. They proved the version which identifies the characteristic power series of the projective limit over the p-class groups with a p-adic L-function. These methods were further developed by Wiles [32] who in 1990 established the Main Conjecture for p ≠ 2 and Galois extensions L/K of a totally real base field K. Under the condition that p ∤ |Gal(L/ℚ)| the result of Mazur and Wiles implies a second version of the Main Conjecture where the p-adic L-function is replaced by the characteristic power series of “units modulo cyclotomic units”. It is this version which is needed in the context of this manuscript. Due to work of Kolyvagin and Rubin there is a much more elementary proof of the Main Conjecture for abelian extension L/ℚ with p ∤ |Gal(L/ℚ)|. This approach uses the Euler system of cyclotomic units. Replacing cyclotomic units by elliptic units (amongst many other things) Rubin achieves the result mentioned in part a) of this remark. In 1992 Greither [10] refined the method of Rubin and used the Euler system of cyclotomic units to give an elementary (but technical) proof of the second version of the Main Conjecture for L/ℚ abelian and all primes p. Our proof of Theorem 3.1 will closely follow Greither’s exposition. Finally we mention recent work of Huber and Kings [11]. They apply the machinery of Euler systems and simultaneously prove the Main Conjecture and the Bloch-Kato conjecture for all primes p ≠ 2 and all abelian extensions L/ℚ.

The rest of this section is devoted to the proof of Theorem 3.1. Let C(𝔣) denote the Iwasawa module of elliptic units as defined in [6, III.1.6]. Then C(𝔣) ⊆ C̄_{∞}, so that char((Ē_{∞}/C̄_{∞})_{χ}) divides char((Ē_{∞}/C(𝔣))_{χ}). By [6, III.2.1, Theorem] it suffices to show that char(A_{∞,χ}) divides char((Ē_{∞}/C(𝔣))_{χ}) for all characters χ of Δ = Gal(K_{0}/k) in order to prove the equality char(A_{∞,χ}) = char((Ē_{∞}/C(𝔣))_{χ}). Hence it is enough for us to prove

char(A_{∞,χ}) divides char((Ē_{∞}/C̄_{∞})_{χ})  (3)

for all characters χ of Δ. For an abelian character χ of Δ we write

e_{χ} := \frac{1}{|Δ|} ∑_{δ∈Δ} Tr(χ(δ))δ^{−1}

for the idempotent of ℚ_{p}[Δ] corresponding to χ with Tr denoting the trace map from ℤ_{p}(χ) to ℤ_{p}. We also set Tr_{Δ} = ∑_{δ∈Δ} δ. For any ℤ_{p}[Δ]-module M we have an epimorphism

M_{χ} = M⊗_{ℤ_{p}[Δ]}ℤ_{p}(χ) ⟶ |Δ|e_{χ}M, m⊗α ↦ |Δ|λ_{α}e_{χ}m,

where λ_{α} ∈ ℤ_{p}[Δ] is an element which maps to α under ℤ_{p}[Δ] → ℤ_{p}(χ). If Z denotes the kernel, then it is easily seen that |Δ|Z = 0. Let now M = A_{∞} and χ = 1. Then

Z ⟶ A_{∞,χ} ⟶ Tr_{Δ}A_{∞} ⟶ 0

is exact. Since Tr_{Δ}A_{n} is contained in the p-Sylow subgroup of the ideal class group of k_{n}, which is trivial by our assumption p ∤ h_{k} and [31, Th. 10.4], we see that A_{∞,χ} is annihilated by |Δ|. By the main result of [9] the Iwasawa μ-invariant of A_{∞,χ} is trivial. From this we deduce char(A_{∞,χ}) = (1), thus establishing (3) for the trivial character. The rest of this section is devoted to the proof of the divisibility relation (3) for non-trivial characters χ. As already mentioned we will closely follow Greither’s exposition [10]. Whenever there are only minor changes we shall be very brief, but emphasize those arguments which differ from the cyclotomic situation. To see the Euler system method applied in an easy setting the reader is advised to have a look at [26]. The strategy of the proof of our Theorem 3.1 is essentially the same, but there are additional difficulties because we allow p to divide |Δ|. If p ∤ |Δ|, the functor M ↦ M_{χ} is exact and the Euler system machinery directly produces a divisibility result of the form char(A_{∞,χ}) | char((Ē_{∞}/C̄_{∞})_{χ}). If p | |Δ|, the functor M ↦ M_{χ} is no longer exact, but Greither’s paper [10] shows how to adapt the Euler system method to produce a weaker divisibility relation of the form char(A_{∞,χ}) | ηchar((Ē_{∞}/C̄_{∞})_{χ}) with an additional factor η ∈ Λ_{χ}which is essentially a product of powers of p and γ−1. Because of Lemma 3.7 and the triviality of the μ-invariant of A_{∞,χ}, the factor η is coprime with char(A_{∞,χ}), so that we again derive a clean divisibility result as in the case p ∤ |Δ|. We will need some notation from Kolyvagin’s theory. Let M be a large power of p and define L = L_{F,M} to be the set of all primes 𝔩 of k satisfying

(i)  𝔩 splits completely in F/k,
(ii)  N_{k/ℚ}(𝔩) ≡ 1(mod M).

By [24, Lem. 1.1] there exists a unique extension F (𝔩) of F of degree M in F k(𝔩). Further F (𝔩)/F is cyclic, totally ramified at all primes above 𝔩 and unramified at all other primes. We write J = ⊕_{λ}ℤλ for the group of fractional ideals of F and for every prime 𝔩 of k we let J_{𝔩} = ⊕_{λ|𝔩}ℤλ denote the subgroup of J generated by the prime divisors of 𝔩. If y ∈ F^{×} we let (y)_{𝔩} ∈ J_{𝔩} denote the support of the principal ideal (y) = yO_{F} above 𝔩. Analogously we write [y] ∈ J/MJ and [y]_{𝔩} ∈ J_{𝔩}/MJ_{𝔩}.

For 𝔩 ∈ L we let

φ_{𝔩} : \begin{matrix}(O_{F} /𝔩O_{F}) & × \\ ((O_{F} /𝔩O_{F})^{×}) & M \end{matrix} ⟶ J_{𝔩}/MJ_{𝔩}

denote the Gal(F/k)-equivariant isomorphism defined by [24, Prop. 2.3]. For every 𝔩 ∈ L we also write φ_{𝔩} for the induced map

φ_{𝔩} : {y ∈ F^{×}/ (F^{×})^{M} : [y]_{𝔩} = 0} ⟶ J_{𝔩}/MJ_{𝔩}, y ↦ φ_{𝔩}(u),

where y = z^{M}u, z ∈ F^{×}, u a unit at all places above 𝔩. We write S = S_{F,M} for the set of squarefree integral ideals of k which are only divisible by primes 𝔩 ∈ L. If 𝔞 ∈ S, 𝔞 = ∏_{i=1}^{k} 𝔩_{i}, we write F (𝔞) for the compositum F (𝔩_{1}) ⋯ F (𝔩_{k}) and F (O_{k}) = F. For every ideal 𝔤 of O_{k} let S(𝔤) ⊆ S be the subset {𝔞 ∈ S : (𝔞, 𝔤) = 1}. We write F̄ for the algebraic closure of F and let U(𝔤) denote the set of all functions

α : S(𝔤) ⟶ F̄^{×}

satisfying the properties (1a)-(1d) of [24]. Any such function will be called an Euler system. Define U_{F} = U_{F,M} = ∐ U(𝔤). For α ∈ U_{F} we write S(α) for the domain of α, i.e. S(α) = S(𝔤) if α ∈ U(𝔤). Given any Euler system α ∈ U_{F}, we let κ = κ_{α} : S(α) ⟶ F^{×}/ (F^{×})^{M} be the map defined in [24, Prop. 2.2]. Then we have:

Proposition 3.3 Let α ∈ U_{F}, κ = κ_{α}, 𝔞 ∈ S(α), 𝔞 ≠ 1, and 𝔩 a prime of k. If 𝔞 = 𝔩 we also assume that α(1) satisfies v_{λ}(α(1)) ≡ 0(mod M) for all λ | 𝔩 in F/k. Then:

If 𝔩 ∤ 𝔞, then [κ(𝔞)]_{𝔩} = 0.
If 𝔩 | 𝔞, then [κ(𝔞)]_{𝔩} = φ_{𝔩}(κ(𝔞/𝔩)).

Proof See [24, Prop. 2.4]. Note that the additional assumption in the case 𝔞 = 𝔩 is needed in (ii), both for its statement (φ_{𝔩}(κ(1)) may not be defined in general) and for its proof.

We now come to the technical heart of Kolyvagin’s induction procedure, the application of Chebotarev’s theorem.

Theorem 3.4 Let K/k be an abelian extension, G = Gal(K/k). Let M denote a (large enough) power of p. Assume that we are given an ideal class 𝔠 ∈ A(K), a finite ℤ[G]-module W ⊆ K^{×}/ (K^{×})^{M}, and a G-homomorphism

ψ : W ⟶ (ℤ/Mℤ) [G].

Let 𝔭̄^{c} be the precise power of 𝔭̄ which divides the conductor 𝔣 of K. Then there are infinitely many primes λ of K such that (1) [λ] = p^{3c+3}𝔠 in A(K).

(2) If 𝔩 = k ∩ λ, then N𝔩 ≡ 1(mod M), and 𝔩 splits completely in K.

(3) For all w ∈ W one has [w]_{𝔩} = 0 in J_{𝔩}/MJ_{𝔩} and there exists a unit u ∈ (ℤ/Mℤ)^{×} such that

φ_{𝔩}(w) = p^{3c+3}uψ(w)λ.

Proof We follow the strategy of Greither’s proof of [10, Th. 3.7], but have to change some technical details. Let H denote the Hilbert p-class field of K. For a natural number n we write μ_{n} for the nth roots of unity in an algebraic closure of K. We consider the following diagram of fields

[figure: K′′ = K(μ_{M}, W^{1/M}); K′ = K(μ_{M}); H; K]

Claim (a) [H ∩ K′ : K] ≤ p^{c} Proof: The situation is clarified by the following diagram

[figure: K′; K′ ∩ H; k(μ_{M}); K; ℚ(μ_{M}); •; k; ℚ]

We write φ_{ℤ} (resp. φ_{O_{k}}) for the Euler function in ℤ (resp. O_{k}). Obviously 𝔭̄ is totally ramified in k(μ_{M})/k. Hence 𝔭̄ ramifies in K′/k of exponent at least φ_{ℤ}(M). On the other hand, 𝔭̄ is ramified in K/k of exponent at most φ_{O_{k}}(𝔭̄^{c}). Therefore any prime divisor of 𝔭̄ ramifies in K′/K of degree at least φ_{ℤ}(M)/φ_{O_{k}}(𝔭̄^{c}). Since K′ ∩ H/K is unramified and [K′ : K] ≤ φ_{ℤ}(M), we derive [K′ ∩ H : K] ≤ φ_{O_{k}}(𝔭̄^{c}). Since p is split in k/ℚ we obtain φ_{O_{k}}(𝔭̄^{c}) = (p − 1)p^{c−1} < p^{c}, so that the claim is shown. In order to follow Greither’s core argument for the proof of Theorem 3.4 we establish the following two claims. Claim (b) Gal(H ∩ K′′/K) is annihilated by p^{2c+1}. Claim (c) The cokernel of the canonical map from Kummer theory

Gal(K′′/K′) ↪ Hom(W, μ_{M})

is annihilated by p^{c+2}.

We write M = p^{m}. Since divisors of 𝔭̄ are totally ramified in k(μ_{M})/k of degree φ_{ℤ}(M) and at most ramified in K/k of degree φ_{O_{k}}(𝔭̄^{c}), one has

[k(μ_{M}) : K ∩ k(μ_{M})] ≥ \frac{φ_{ℤ}(M)}{φ_{O_{k}}(𝔭̄^{c})} = \begin{cases}p^{m−c}, & if c ≥ 1, \\ (p − 1)p^{m−1}, & if c = 0. \end{cases}

Since k(μ_{M})/k is cyclic, there exists an element j ∈ Gal(k(μ_{M})/K ∩ k(μ_{M})) of exact order a = p^{m−c−1}. Let r ∈ ℤ such that j(ζ_{M}) = ζ_{M}^{r}. Then r^{a} ≡ 1(mod M) and r^{b} ≢ 1(mod M) for all 0 < b < a. We also write j ∈ Gal(K′/K) for the unique extension of j to K′ with j|_{K} = id. Let σ ∈ Gal(K′′/K′) and α ∈ K′′ such that α^{M} = w ∈ W. Then there exists an integer t_{w} such that σ(α) = ζ_{M}^{t_{w}}α. Since W ⊆ K^{×}/ (K^{×})^{M}, there is an extension of j to K′′/K such that j(α) = α for all α ∈ K′′ such that α^{M} ∈ W. Therefore, for any such α,

jσj^{−1}(α) = jσ(α) = j(ζ_{M}^{t_{w}}α) = ζ_{M}^{rt_{w}}α.

Hence j acts as σ ↦ σ^{r} on Gal(K′′/K′). Since Gal(K′/K) acts trivially on Gal(K′′ ∩ K′H/K′) this implies that r − 1 annihilates Gal(K′′ ∩ K′H/K′). On the other hand Gal(K′′ ∩ K′H/K′) is an abelian group of exponent M, so that also gcd(M, r − 1) annihilates. Suppose that p^{d} divides r − 1 with d ≥ 1. By induction one easily shows that r^{p^{m−d}} ≡ 1(mod p^{m}). Hence a = p^{m−c−1} divides p^{m−d}, which implies d ≤ c + 1. As a consequence, p^{c+1} annihilates Gal(K′′ ∩ K′H/K′) ≃ Gal(K′′ ∩ H/K′ ∩ H). Together with claim (a) this proves (b). We now proceed to demonstrate claim (c). Let W′ ⊆ K^{′×}/ (K^{′×})^{M} denote the image of W under the homomorphism

K^{×}/ (K^{×})^{M} ⟶ K^{′×}/ (K^{′×})^{M} .  (4)

Since Gal(K′′/K′) ≃ Hom(W′, μ_{M}), it suffices to show that the kernel U of the map in (4) is annihilated by p^{c+2}. By Kummer theory U is isomorphic to H^{1}(K′/K, μ_{M}).

The extension K′/K is cyclic and a Herbrand quotient argument shows

#H^{1}(K′/K, μ_{M}) = #H^{0}(K′/K, μ_{M}) = #\frac{μ_{M}(K)}{N_{K′/K}(μ_{M})}.

From [20, Lem. 7] we deduce that #μ_{M}(K) divides p^{c+2}. Hence U is annihilated by p^{c+2}. Now that claim (b) and (c) are proved, the core argument runs precisely as in [10, pg.473/474] (using Greiter’s notation the proof has to be adapted in the following way: p^{c+2}ιψ has preimage γ ∈ Gal(K′′/K′); γ_{1} = p^{c+2} (\frac{𝔠}{H/K} ) ∈

Gal(H/K); δ ∈ Gal(K′′H/K) with δ|_{H} = p^{2c+1}γ_{1}, δ|_{K′′} = p^{2c+1}γ.)

Recall the notation introduced at the beginning of this section. In addition, we let Δ = Gal(K_{0}/k), G_{n} = Gal(K_{n}/k), G_{∞} = Gal(K_{∞}/k) and Γ_{n} = Gal(K_{n}/K_{0}). We fix a topological generator γ of Γ = Gal(K_{∞}/K_{0}), and abbreviate the p^{n}th power of γ by γ_{n}. For any abelian character χ of Δ we write Λ_{χ} = ℤ_{p}(χ)[[T]] for the usual Iwasawa algebra. Note that Λ⊗_{ℤ_{p}[Δ]}ℤ_{p}(χ) ≃ ℤ_{p}(χ)[[T]], so that our notation is consistent. We choose a generator h_{χ} ∈ Λ_{χ} of char ((Ē_{∞}/C̄_{∞})_{χ}). By the general theory of finitely generated Λ_{χ}-modules there is a quasi-isomorphism

τ : A_{∞,χ} ⟶ ⨁_{i=1}^{k} Λ_{χ}/(g_{i})

with g_{i} ∈ Λ_{χ}, and by definition, char(A_{∞,χ}) = (g) with g := g_{1} ⋯ g_{k}. As in [10] we need the following lemmas providing the link to finite levels.

Lemma 3.5 Let χ ≠ 1 be an abelian character of Δ. Then there exist constants n_{0} = n_{0}(F), c_{i} = c_{i}(F), i = 1, 2, a divisor h′_{χ} of h_{χ} (all independent of n) and G_{n}-homomorphisms

ϑ_{n} : Ē_{n,χ} ⟶ Λ_{n,χ} := Λ_{χ}/(1 − γ_{n})Λ_{χ}

such that

(i) h′_{χ} is relatively prime to γ_{n} − 1 for all n

(ii) (γ_{n_{0}} − 1)^{c_{1}}p^{c_{2}}h′_{χ}Λ_{n,χ} ⊆ ϑ_{n}(im(C̄_{n,χ}))

where here im(C̄_{n,χ}) denotes the image of C̄_{n,χ} in Ē_{n,χ}.

Proof We mainly follow Greither’s proof of [10, Lem. 3.9]. We let

π_{n} : Ē_{∞}/(1 − γ_{n})Ē_{∞} ⟶ Ē_{n}

denote the canonical map and first prove Claim 1: There exists an integer κ (independent of n) such that

(γ − 1)p^{κ} ker(π_{n}) = 0 and (γ − 1)p^{κ}cok(π_{n}) = 0

This is shown as in Greither’s proof of [10, Lem. 3.9]. He uses [25, Lem. 1.2], which is stated under the additional assumption p ∤ |Δ|. As already remarked by Greither, this hypothesis is not necessary.

Next we define U_{∞} := lim_{n^{←}} U_{n} and proceed to prove

Claim 2 ℚ_{p}⊗_{ℤ_{p}}U_{∞} ≃ ℚ_{p}⊗_{ℤ_{p}}Λ = Λ[\frac{1}{p}]. This can be proved similarly as [18, Th. 11.2.5]. The assumption p ∤ |Δ| of loc.cit. is not needed, since we invert p. Alternatively, Claim 2 follows from [6, Prop. III.1.3], together with Exercise (iii) of [6, III.1.1].

It follows that ℚ_{p}⊗_{ℤ_{p}}U_{∞,χ} is free cyclic over ℚ_{p}⊗_{ℤ_{p}}Λ_{χ} = Λ_{χ}[\frac{1}{p}]. Since Λ_{χ}[\frac{1}{p}] is a principal ideal domain, the submodule ℚ_{p}⊗_{ℤ_{p}} Ē_{∞,χ} is also free cyclic over Λ_{χ}[\frac{1}{p}]. It follows that there exists a pseudo-isomorphism

f : Ē_{∞,χ} ⟶ C := ⨁_{i} Λ_{χ}/p^{n_{i}}Λ_{χ} ⊕ Λ_{χ}.

If we apply the snake lemma to the diagram

\xymatrix{ & 0 \ar[r] \ar[d] & Ē_{∞,χ} \ar[r]^{=} \ar[d]_{f} & Ē_{∞,χ} \ar[r] \ar[d]^{pr∘f=:α} & 0 \\ 0 \ar[r] & ⊕Λ_{χ}/p^{n_{i}}Λ_{χ} \ar[r] & C \ar[r]^{pr} & Λ_{χ} \ar[r] & 0}

we see that ker(α) is annihilated by some power of p and cok(α) is finite. We note that for any G_{∞}-module X one has

(X/(1 − γ_{n})X)_{χ} ≃ X_{χ}/(1 − γ_{n})X_{χ}.

Let W_{n} denote the image of π_{n} and set T := Tor_{ℤ_{p}[Δ]}(cok(π_{n}), ℤ_{p}(χ)). Then we have a commutative diagram (with exact lines)

[figure: T; φ; W_{n,χ}; =; Ē_{n,χ}; cok(π_{n})_{χ}; 0; ker(π_{n})_{χ}; τ; \frac{Ē_{∞,χ}}{(1−γ_{n})Ē_{∞,χ}}; π_{n}; W_{n,χ}; 0]

We write π_{n,χ} for the composite map and obtain the exact sequence

0 ⟶ ker(π_{n,χ}) ⟶ \frac{Ē_{∞,χ}}{(1 − γ_{n})Ē_{∞,χ}} ⟶^{π_{n,χ}} Ē_{n,χ} ⟶ cok(π_{n})_{χ} ⟶ 0

We claim that ker(π_{n,χ}) is annihilated by (γ − 1)^{2}p^{2κ}: Let e ∈ ker(π_{n,χ}). Then

π_{n}(e) = φ(t) for some t ∈ Tor_{ℤ_{p}[Δ]}(cok(π_{n}), ℤ_{p}(χ))
⟹ π_{n}((γ − 1)p^{κ}e) = φ((γ − 1)p^{κ}t) = 0
⟹ τ(c) = (γ − 1)p^{κ}e for some c ∈ ker(π_{n})_{χ}
⟹ 0 = τ((γ − 1)p^{κ}c) = (γ − 1)^{2}p^{2κ}e

So both ker(π_{n,χ}) and cok(π_{n,χ}) are annihilated by (γ − 1)^{2}p^{2κ}. Consider now the following commutative diagram

\xymatrix{Ē_{∞,χ} \ar[r]^{(γ−1)^{4}p^{4κ}α} \ar[d]^{π_{n,χ}} & Λ_{χ} \ar[d] \\ Ē_{n,χ} \ar[r]^{ϑ_{n}} & Λ_{n,χ} = Λ_{χ}/(1 − γ_{n})Λ_{χ}}

where we define ϑ_{n} in the following manner: for e ∈ Ē_{n,χ} there exists z ∈ Ē_{∞,χ} such that π_{n,χ}(z) = (γ − 1)^{2}p^{2κ}e. We then set

ϑ_{n}(e) := (γ − 1)^{2}p^{2κ}α(z)(mod (1 − γ_{n})Λ_{χ}).

On the other hand, we have the exact sequence

C̄_{∞,χ} ⟶ Ē_{∞,χ} ⟶ (Ē_{∞}/C̄_{∞})_{χ} ⟶ 0

so that

Ē_{∞,χ}/im(C̄_{∞,χ}) ↪ (Ē_{∞}/C̄_{∞})_{χ} .

The structure theorem of Λ_{χ}-torsion modules implies that h_{χ} (Ē_{∞}/C̄_{∞})_{χ} is finite. Since α(Ē_{∞,χ})/α(imC̄_{∞,χ}) is a quotient of Ē_{∞,χ}/im(C̄_{∞,χ}), the module h_{χ} (α(Ē_{∞,χ})/α(imC̄_{∞,χ})) is also finite. Since cok(α) is finite, there exists a power p^{s} such that p^{s} ∈ α(Ē_{∞,χ}) and p^{s}h_{χ}α(Ē_{∞,χ}) ⊆ α(im(C̄_{∞,χ}))). Therefore p^{2s}h_{χ} ∈ α(im(C̄_{∞,χ})) and we conclude further:

p^{2s+4κ}(γ − 1)^{4}h_{χ} = p^{4κ}(γ − 1)^{4}α(z) for some z ∈ im(C̄_{∞,χ})
⟹ ϑ_{n}(z_{n}) = p^{2s+4κ}(γ − 1)^{4}h_{χ} with z_{n} = π_{n,χ}(z) ∈ im(C̄_{n,χ})
⟹ p^{2s+4κ}(γ − 1)^{4}h_{χ}Λ_{n,χ} ⊆ ϑ_{n}(im(C̄_{n,χ}))  (5)

Since γ_{n} − 1 divides γ_{n+1} − 1 for all n there exists a positive integer n_{0} and a divisor h′_{χ} of h_{χ} such that h_{χ} divides (γ_{n_{0}} − 1)h′_{χ} and such that h′_{χ} is relatively prime with γ_{n} − 1 for all n. The assertions of the lemma are now immediate from (5).

Lemma 3.6 Let χ ≠ 1 be a character of Δ. Then there exists a constant c_{3} = c_{3}(F) (independent of n) and G_{n}-homomorphisms

τ_{n} : A_{n,χ} ⟶ ⨁_{i=1}^{k} Λ_{n,χ}/(ḡ_{i})

such that p^{c_{3}}cokτ_{n} = 0 for all n ≥ 0. Here ḡ_{i} denotes the image of g_{i} ∈ Λ_{χ} in Λ_{n,χ}.

Proof The proof is identical to Greither’s proof of [10, Lem. 3.10]. It is based on the following sublemma which will be used again at the end of the section.

Lemma 3.7 For n ≥ 0 the kernel and cokernel of multiplication with γ_{n} − 1 on A_{∞} are finite.

Proof See [25, pg. 705]. It is remarkable that one uses the known validity of Leopoldt’s conjecture in this proof.

The following technical lemma is the analogue of [10, Lem. 3.12].

Lemma 3.8 Let K/k be an abelian extension, G = Gal(K/k) and Δ a subgroup of G. Let χ denote a character of Δ, M a power of p, 𝔞 = 𝔩_{1} ⋯ 𝔩_{i} ∈ S_{M,K}. Let 𝔩 = 𝔩_{i} and let λ be a fixed prime divisor of 𝔩 in K. We write 𝔠 for the class of λ and assume that 𝔠 ∈ A = A(K), where as usual A(K) denotes the p-Sylow subgroup of the ideal class group of K. Let B ⊆ A denote the subgroup generated by classes of prime divisors of 𝔩_{1}, …, 𝔩_{i−1}. Let x ∈ K^{×}/ (K^{×})^{M} such that [x]_{𝔮} = 0 for all primes 𝔮 not dividing 𝔞, and let W ⊆ K^{×}/ (K^{×})^{M} denote the ℤ_{p}[G]-span of x. Assume that there exist elements

E, g, η ∈ ℤ_{p}[G]

satisfying

(i) E · ann_{(ℤ_{p}[G])_{χ}}(𝔠̄_{χ}) ⊆ g · (ℤ_{p}[G])_{χ}, where 𝔠̄_{χ} is the image of 𝔠 under A → A/B → (A/B)_{χ}.

(ii) # ((ℤ_{p}[G])_{χ} /g (ℤ_{p}[G])_{χ}) < ∞

(iii) M ≥ |A_{χ}| |η (\frac{J_{𝔩}/MJ_{𝔩}}{[W]_{𝔩}} )_{χ}|, where [W]_{𝔩} denotes the subgroup of J_{𝔩}/MJ_{𝔩}

generated by elements [w]_{𝔩}, w ∈ W.

Then there exists a G-homomorphism

ψ : W_{χ} ⟶ ((ℤ/Mℤ) [G])_{χ}

such that

gψ(x)λ_{χ} = (E · η[x]_{𝔩})_{χ}

in (J_{𝔩}/MJ_{𝔩})_{χ}.

Proof Completely analoguous to the proof of [10, Lem. 3.12].

We will now sketch the main argument of the proof of Theorem 3.1. We fix a natural number n ≥ 1 and let K = K_{n} = F k_{n}. We view Δ as a subgroup of G = Gal(K/k). We let M denote a large power of p which we will specify in course of the proof. By Lemma 3.6 there exists for each i = 1, …, k an ideal class 𝔠_{i} ∈ A_{χ} such that

τ_{n}(𝔠_{i}) = (0, …, 0, p^{c_{3}}, 0, …, 0)

in ⨁_{i=1}^{k} Λ_{n,χ}/(ḡ_{i}) with p^{c_{3}} at the ith position. Choose 𝔠_{k+1} arbitrary. By Lemma 3.5 there exists an element ξ′ ∈ im(C̄_{n,χ}) such that ϑ_{n}(ξ′) = (γ_{n_{0}} − 1)^{c_{1}}p^{c_{2}}h′_{χ} in Λ_{n,χ}. It is now easy to show that there exists an actual elliptic unit ξ ∈ C_{n} such that

ϑ_{n}(ξ) = (γ_{n_{0}} − 1)^{c_{1}}p^{c_{2}}h′_{χ}(mod MΛ_{n,χ}).  (6)

By [24, Prop. 1.2] there exists an Euler system α ∈ U_{K,M} such that α(1) = ξ. Set d := 3c+3, where c was defined in Theorem 3.4. Following Greither we will use Theorem 3.4 to construct inductively prime ideals λ_{i} of K, 1 ≤ i ≤ k + 1, such that

(a) [λ_{i}]_{χ} = p^{d}𝔠_{i}

(b) 𝔩_{i} = λ_{i} ∩ k ⊆ S_{M,K}

(c) one has the equalities

(v_{λ_{1}}(κ(𝔩_{1})))_{χ} = u_{1}|Δ|(γ_{n_{0}} − 1)^{c_{1}}p^{d+c_{2}}h′_{χ},
(g_{i−1}v_{λ_{i}}(κ(𝔩_{1} ⋯ 𝔩_{i})))_{χ} = u_{i}|Δ|(γ_{n_{0}} − 1)^{c_{1}^{i−1}}p^{d+c_{3}} (v_{λ_{i−1}}(κ(𝔩_{1} ⋯ 𝔩_{i−1})))_{χ}

for 2 ≤ i ≤ k + 1. These are equalities in Λ_{n,χ}/MΛ_{n,χ}. The elements u_{i} are units in ℤ/Mℤ and v_{λ}(x) ∈ (ℤ/Mℤ) [G] ≃ Λ_{n}/MΛ_{n} is defined by v_{λ}(x)λ = [x]_{𝔩} in J_{𝔩}/MJ_{𝔩}, if 𝔩 = λ ∩ k ∈ L_{M,K}.

We briefly descibe this induction process. For i = 1 we let 𝔠 ∈ A be a preimage of 𝔠_{1} under the canonical epimorphism A → A_{χ}. We apply Theorem 3.4 with the data 𝔠, W = E/E^{M} (with E := O_{K}^{×}) and

ψ : W ⟶^{v} Ē_{n,χ}/Ē_{n,χ}^{M} ⟶^{ϑ_{n}} Λ_{n,χ}/MΛ_{n,χ} ⟶^{ε_{χ}} (ℤ/Mℤ) [G]

where v ∈ (ℤ/Mℤ)^{×} is such that each unit x ∈ K⊗k_{𝔭} satisfies x^{v} ≡ 1 modulo all primes above 𝔭. The map ε_{χ} is defined in [10, Lemma 3.13]. Theorem 3.4 provides a prime ideal λ = λ_{1} which obviously satisfies (a) and (b) and, in addition,

φ_{𝔩}(w) = p^{d}uψ(w)λ for all w ∈ E/E^{M}.

From this equality we conclude further

v_{λ}(κ(𝔩))λ = [κ(𝔩)]_{𝔩} = φ_{𝔩}(κ(1)) = φ_{𝔩}(ξ)
= p^{d}uψ(ξ)λ = (p^{d}uv(ε_{χ} ∘ ϑ_{n})(ξ)) λ

in J_{𝔩}/MJ_{𝔩} = (ℤ/Mℤ) [G]λ. Projecting the equality v_{λ}(κ(𝔩)) = p^{d}uv(ε_{χ} ∘ϑ_{n})(ξ) to ((ℤ/Mℤ) [G])_{χ} = Λ_{n,χ}/MΛ_{n,χ} and using [10, Lemma 3.13] together with (6) we obtain equality (c) for i = 1. For the induction step i − 1 ↦ i we set 𝔞_{i−1} := 𝔩_{1} ⋯ 𝔩_{i−1}. Using (c) inductively we see that (v_{λ_{i−1}}(κ(𝔞_{i−1})))_{χ} divides

(\underbrace{|Δ|^{i−1}p^{(i−2)(d+c_{3})+(d+c_{2})}}_{=:D_{i}}(γ_{n_{0}} − 1)^{c_{1}+∑_{s=1}^{i−2} c_{1}^{s}}h′_{χ})_{χ} .

Without loss of generality we may assume that c_{1} ≥ 2. Then one has c_{1} + ∑_{i=1}^{i−2} c_{1}^{s} ≤ c_{1}^{i−1}, so that (v_{λ_{i−1}}(κ(𝔞_{i−1})))_{χ} also divides D_{i}(γ_{n_{0}} − 1)^{t_{i}}h′_{χ} with t_{i} := c_{1}^{i−1}. The module

N = (γ_{n_{0}} − 1)^{t_{i}} (J_{𝔩_{i−1}}/ (M, [κ(𝔞_{i−1})]_{𝔩_{i−1}}))_{χ}

is a cyclic as a Λ_{n,χ}-module and annihilated by D_{i}h′_{χ}. Consequently

|N| ≤ |Λ_{n,χ}/(D_{i})| · |Λ_{n,χ}/(h′_{χ})|.

Note that by the definition of h′_{χ} the quotient Λ_{n,χ}/(h′_{χ}) is finite. If we choose M such that

M ≥ max (|A_{χ}| · |Λ_{n,χ}/(D_{k+1})| · |Λ_{n,χ}/(h′_{χ})|, p^{n})

then one has |N| ≤ M|A_{χ}|^{−1}. We now apply Lemma 3.8 with 𝔞 = 𝔞_{i−1}, g = g_{i−1}, x = κ(𝔞_{i−1}), E = p^{c_{3}} and η = (γ_{n_{0}} − 1)^{t_{i}}. Following Greither it is straight forward to check the hypothesis (a), (b) and (c) of Lemma 3.8. Note that for (b) one has to use the fact that char(A_{∞,χ}) is relatively prime to γ_{n} − 1 for all n, which is an immediate consequence of Lemma 3.5. We let W denote the ℤ_{p}[G]-span of κ(𝔞_{i−1}) in K^{×}/ (K^{×})^{M} and obtain a homomorphism

ψ_{i} : W_{χ} ⟶ ((ℤ/Mℤ) [G])_{χ}

such that g_{i−1}ψ_{i}(κ(𝔞_{i−1})) = (p^{c_{3}}(γ_{n_{0}} − 1)^{t_{i}}v_{λ_{i−1}}(κ(𝔞_{i−1})))_{χ}. We let 𝔠 denote a preimage of 𝔠_{i} and consider the homomorphism

ψ : W ⟶ W_{χ} ⟶^{ψ_{i}} Λ_{n,χ}/MΛ_{n,χ} ⟶^{ε_{χ}} (ℤ/Mℤ) [G]

We again apply Theorem 3.4 and obtain λ_{i} satifying (a), (b) and also

φ_{𝔩_{i}}(κ(𝔞_{i−1})) = p^{d}uψ(κ(𝔞_{i−1}))λ_{i}.

As in the case i = 1 one now establishes equality (c). This concludes the inductive construction of λ_{1}, …, λ_{k+1}. Using (c) successively we obtain (suppressing units in ℤ/Mℤ)

(g_{1} ⋯ g_{k}v_{λ_{k+1}}(κ(𝔩_{1} ⋯ 𝔩_{k+1}))) = ηh′_{χ}

(as an equality in Λ_{n,χ}/MΛ_{n,χ}) with

η = (|Δ|^{k+1}p^{k(d+c_{3})+d+c_{2}}(γ_{n_{0}} − 1)^{c_{1}+∑_{s=1}^{k} c_{1}^{s}})_{χ} .

Therefore g = g_{1} ⋯ g_{k} divides ηh′_{χ} in Λ_{n,χ}/MΛ_{n,χ}, and since p^{n} | M we also see that g divides ηh′_{χ} in Λ_{n,χ}/p^{n}Λ_{n,χ}. As in [31, page 371, last but one paragraph] we deduce that g divides ηh′_{χ} in Λ_{χ}. By [6, III.2.1, Theorem] (together with [6, III.1.7, (13)]) we know that the μ-invariant of A_{∞,χ} is trivial. Hence g = char(A_{∞,χ}) is coprime with p. By Lemma 3.7 it is also coprime with γ_{n_{0}} − 1, and consequently |Λ_{χ}/(g, η)| < ∞. Therefore there exist α, β ∈ Λ_{χ} and N ∈ ℕ such that p^{N} = αg + βη and we see that g divides p^{N}h′_{χ}. Since g is prime to p we obtain g | h′_{χ}.

Remark 3.9 There are several steps in the proof where we use the assumption that p splits in k/ℚ. Among these the vanishing of μ(A_{∞,χ}) is most important. The proof of this uses an important result of Gillard [9]. If p is not split in k/ℚ our knowledge about μ(A_{∞,χ}) seems to be quite poor.

4 The conjecture

In this section we fix an integral O_{k}-ideal 𝔣 such that w(𝔣) = 1 and write

M = h^{0}(Spec(k(𝔣)), A = ℚ[G_{𝔣}], 𝔄 = ℤ[G_{𝔣}],

where for any O_{k}-ideal 𝔪 we let G_{𝔪} denote the Galois group Gal(k(𝔪)/k). For any commutative ring R we write D(R) for the derived category of the homotopy category of bounded complexes of R-modules and D^{p}(R) for the full triangulated subcategory of perfect complexes of R-modules. We write D^{pis}(R) for the subcategory of D^{p}(R) in which the objects are the same, but the morphisms are restricted to quasi-isomorphisms.

We let P(R) denote the category of graded invertible R-modules. If R is reduced, we write Det_{R} for the functor from D^{pis}(R) to P(R) introduced by Knudsen and Mumford [14]. To be more precise, we define

Det_{R}(P) := ( ⋀_{R}^{rk_{R}(P)} P, rk_{R}(P)) ∈ Ob(P(R))

for any finitely generated projective R-module P and for a bounded complex P^{•} of such modules we set

Det_{R}(P^{•}) := ⨂_{i∈ℤ} Det_{R}^{(−1)^{i}}(P^{i}).

If R is reduced, then this functor extends to a functor from D^{pis}(R) to P(R). For more information and relevant properties the reader is refered to [5, §2], or the original papers [14] and [15]. For any finite set S of places of k we define Y_{S} = Y_{S}(k(𝔣)) = ⊕_{w∈S(k(𝔣))}ℤw. Here S(k(𝔣)) denotes the set of places of k(𝔣) lying above places in S. We let X_{S} = X_{S}(k(𝔣)) denote the kernel of the augmentation map Y_{S} → ℤ, w ↦ 1. The fundamental line Ξ(_{A}M) is given by

Ξ(_{A}M)^{#} = Det_{A}^{−1} (O_{k(𝔣)}^{×}⊗_{ℤ}ℚ) ⊗_{A}Det_{A} (X_{{v|∞}}⊗_{ℤ}ℚ) ,

where the superscript # means twisting the action of G_{𝔣} by g ↦ g^{−1}. We let

R = R_{k(𝔣)} : O_{k(𝔣)}^{×}⊗_{ℤ}ℝ ⟶ X_{{v|∞}}⊗_{ℤ}ℝ,
u ↦ − ∑_{v|∞} log |u|_{v} · v

denote the Dirichlet regulator map. Let

Aϑ_{∞} : ℝ[G_{𝔣}] ⟶ Ξ(_{A}M)^{#}⊗_{ℚ}ℝ

be the inverse of the canonical isomorphism

Det_{ℝ[G_{𝔣}]}^{−1} (O_{k(𝔣)}^{×}⊗_{ℤ}ℝ) ⊗_{ℝ[G_{𝔣}]}Det_{ℝ[G_{𝔣}]} (X_{{v|∞}}⊗_{ℤ}ℝ)
⟶^{det(R)⊗1} Det_{ℝ[G_{𝔣}]}^{−1} (X_{{v|∞}}⊗_{ℤ}ℝ) ⊗_{ℝ[G_{𝔣}]}Det_{ℝ[G_{𝔣}]} (X_{{v|∞}}⊗_{ℤ}ℝ)
⟶^{eval} (ℝ[G_{𝔣}], 0) .

Following [19] we define for integral O_{k}-ideals 𝔤, 𝔤_{1} with 𝔤 | 𝔤_{1} and each abelian character η of G_{𝔤} ≃ cl(𝔤) (cl(𝔤) denoting the ray class group modulo 𝔤)

S_{𝔤}(η, 𝔤_{1}) = ∑_{c∈cl(𝔤_{1})} η(c^{−1}) log |φ_{𝔤}(c)|,

where η is regarded as a character of cl(𝔤_{1}) via inflation. For the definition of the ray class invariants φ_{𝔤}(c) we choose an integral ideal 𝔠 in the class c and set

φ_{𝔤}(c) = φ_{𝔤}(𝔠) = \begin{cases}φ^{12N(𝔤)}(1; 𝔤𝔠^{−1}), & if 𝔤 ≠ 1, \\ |\frac{N(𝔠^{−1})^{6}Δ(𝔠^{−1})}{(2π)^{12}} | , & if 𝔤 = 1, \end{cases}

where φ was defined in (1). Note that this definition does not depend on the choice of the ideal 𝔠 (see [20, pp. 15/16]). For an abelian character η of cl(𝔤) we write 𝔣_{η} for its conductor. We write L^{∗}(η) for the leading term of the Taylor expansion of the Dirichlet L-function L(s, η) at s = 0. From [20, Th. 3] and the functional equation satisfied by Dirichlet L-functions we deduce

L^{∗}(η^{−1}) = −\frac{S_{𝔣_{η}}(η, 𝔣_{η})}{6N(𝔣_{η})w(𝔣_{η})}.  (7)

We denote by Ĝ_{𝔣}^{ℚ} the set of ℚ-rational characters associated with the ℚirreducible representations of G_{𝔣}. For χ ∈ Ĝ_{𝔣}^{ℚ} we set e_{χ} = ∑_{η∈χ} e_{η} ∈ A, where we view χ as an Gal(ℚ^{c}/ℚ)-orbit of absolutely irreducible characters of G_{𝔣}. Then the Wedderburn decompostion of A is given by

A ≃ ∏_{χ∈Ĝ_{𝔣}^{ℚ}} ℚ(χ).  (8)

Here, by a slight abuse of notation, ℚ(χ) denotes the extension generated by the values of η for any η ∈ χ. For any character χ ∈ Ĝ_{𝔣}^{ℚ} the conductor 𝔣_{χ}, defined by 𝔣_{χ} := 𝔣_{η} for any η ∈ χ, is well defined. We put L^{∗}(χ) := ∑_{η∈χ} L^{∗}(η)e_{η} and note that L^{∗}(χ)^{#} := ∑_{η∈χ} L^{∗}(η^{−1})e_{η}. The statement L^{∗}(χ)^{#} ∈ Ae_{χ} (compare to [8, page 8]) is not obvious, but needs to be proved. This is essentially Stark’s conjecture. We fix a prime ideal 𝔭 of O_{k} and also choose an auxiliary ideal 𝔞 of O_{k} such that (𝔞, 6𝔣𝔭) = 1. For each η ≠ 1 we define elements

ξ_{η} := \begin{cases}ψ(1; 𝔣_{η}, 𝔞), & if 𝔣_{η} ≠ 1, \\ \frac{δ(O_{k},𝔞^{−1})}{δ(𝔭,𝔭𝔞^{−1})}, & if 𝔣_{η} = 1, η ≠ 1, \end{cases}  (9)

where δ denotes the function of lattices defined in [21, Th. 1]. We set ξ_{χ} := ξ_{η} for any η ∈ χ. We fix an embedding σ : ℚ^{c} ↪ ℂ and write w_{∞} = σ|_{k(𝔣)}. A standard computation leads to

R(e_{η}ξ_{η})
= \begin{cases}(N𝔞 − η(𝔞))w(𝔣_{η})[k(𝔣) : k(𝔣_{η})]L^{∗}(η^{−1})e_{η}w_{∞}, & 𝔣_{η} ≠ 1, \\ (1 − η(𝔭)^{−1})(N𝔞 − η(𝔞))w(1)[k(𝔣) : k(1)]L^{∗}(η^{−1})e_{η}w_{∞}, & 𝔣_{η} = 1, η ≠ 1. \end{cases} (10)

For the reader’s convenience we briefly sketch the computation for characters η ≠ 1 with 𝔣_{η} = 1. By definition of the Dirichlet regulator map and [21, Cor. 2] we obtain

R(e_{η}ξ_{η}) = −\frac{1}{6}[k(𝔣) : k(1)] ∑_{𝔠∈cl(1)} log |\frac{Δ(𝔠)^{N𝔞}Δ(𝔞^{−1}𝔠𝔭)}{Δ(𝔞^{−1}𝔠)Δ(𝔠𝔭)^{N𝔞}} | η(𝔠)e_{η}w_{∞}.  (11)

Since ∑_{𝔠∈cl(1)} Cη(𝔠) = 0 for any constant C we compute further

∑_{𝔠∈cl(1)} log |\frac{Δ(𝔠)^{N𝔞}Δ(𝔞^{−1}𝔠𝔭)}{Δ(𝔞^{−1}𝔠)Δ(𝔠𝔭)^{N𝔞}} | η(𝔠)

= ∑_{𝔠∈cl(1)} log |(\frac{(N𝔠)^{6}Δ(𝔠)}{(2π)^{12}})^{N𝔞}| η(𝔠) + ∑_{𝔠∈cl(1)} log |\frac{(N𝔞^{−1}𝔠𝔭)^{6}Δ(𝔞^{−1}𝔠𝔭)}{(2π)^{12}} | η(𝔠) −

− ∑_{𝔠∈cl(1)} log |(\frac{(N𝔠𝔭)^{6}Δ(𝔠𝔭)}{(2π)^{12}})^{N𝔞}| η(𝔠) − ∑_{𝔠∈cl(1)} log |\frac{(N𝔞^{−1}𝔠)^{6}Δ(𝔞^{−1}𝔠)}{(2π)^{12}} | η(𝔠)

= N𝔞 ∑_{𝔠∈cl(1)} log |φ_{1}(𝔠^{−1})| η(𝔠) + ∑_{𝔠∈cl(1)} log |φ_{1}(𝔞𝔠^{−1}𝔭^{−1})| η(𝔠) −

−N𝔞 ∑_{𝔠∈cl(1)} log |φ_{1}(𝔭^{−1}𝔠^{−1})| η(𝔠) − ∑_{𝔠∈cl(1)} log |φ_{1}(𝔞𝔠^{−1})| η(𝔠).

Recalling that φ_{𝔤}(c) is a class invariant we obtain

∑_{𝔠∈cl(1)} log |\frac{Δ(𝔠)^{N𝔞}Δ(𝔞^{−1}𝔠𝔭)}{Δ(𝔞^{−1}𝔠)Δ(𝔠𝔭)^{N𝔞}} | η(𝔠) = (N𝔞 − η(𝔞))(1 − η(𝔭)^{−1})S_{1}(η, 1)e_{η}w_{∞},

so that (10) is an immediate consequence of (7) and (11). According to the decomposition (8) we decompose Ξ(_{A}M)^{#} character by character and obtain a canonical isomorphism

Ξ(_{A}M)^{#} ⟶( ∏_{χ∈Ĝ_{𝔣}^{ℚ}} (Det_{ℚ(χ)}^{−1}(O_{k(𝔣)}^{×}⊗_{𝔄}ℚ(χ))⊗_{ℚ(χ)}Det_{ℚ(χ)}(X_{{v|∞}}⊗_{𝔄}ℚ(χ)))) .

As in the cyclotomic case one has

dim_{ℚ(χ)} e_{χ} (O_{k(𝔣)}^{×}⊗_{𝔄}ℚ(χ)) = dim_{ℚ(χ)} e_{χ} (X_{\{v|∞\}}⊗_{𝔄}ℚ(χ)) = \begin{cases}1, & χ ≠ 1, \\ 0, & χ = 1. \end{cases}

(12)

Upon recalling that Det_{ℚ}(0) = (ℚ, 0) in P(ℚ) we get a canonical isomorphism

Ξ(_{A}M)^{#} ⟶ ℚ × (∏_{χ≠1} ((O_{k(𝔣)}^{×}⊗_{𝔄}ℚ(χ))^{(−1)}⊗_{ℚ(χ)}(X_{{v|∞}}⊗_{𝔄}ℚ(χ)))) .

From (10) we deduce

(_{A}ϑ_{∞}(L^{∗}(_{A}M, 0)^{−1}))_{χ}

= \begin{cases}w(𝔣_{χ})[k(𝔣) : k(𝔣_{χ})](N𝔞 − σ(𝔞))e_{χ}ξ_{χ}^{−1}⊗w_{∞}, & 𝔣_{χ} ≠ 1, \\ w(1)[k(𝔣) : k(1)](1 − σ(𝔭)^{−1})(N𝔞 − σ(𝔞))e_{χ}ξ_{χ}^{−1}⊗w_{∞}, & 𝔣_{χ} = 1, χ ≠ 1 \\ L(χ, 0)^{−1}, & χ = 1. \end{cases}

In particular, this proves the equivariant version of [8, Conjecture 2]. We fix a prime p and put A_{p} := A⊗_{ℚ}ℚ_{p} = ℚ_{p}[G_{𝔣}], 𝔄_{p} := 𝔄⊗_{ℤ}ℤ_{p} = ℤ_{p}[G_{𝔣}]. Let S = S_{ram} ∪ S_{∞} be the union of the set of ramified places and the set of archimedian places of k. Let S_{p} = S ∪ {𝔭 | p} and put

Δ(k(𝔣)) := RHom_{ℤ_{p}}(RΓ_{c}(O_{k(𝔣),S_{p}}, ℤ_{p}), ℤ_{p})[−3]

Then Δ(k(𝔣)) can be represented by a perfect complex of 𝔄_{p}-modules whose cohomology groups H^{i}(Δ(k(𝔣)) are trivial for i ≠ 1, 2. For i = 1 one finds

H^{1}(Δ(k(𝔣)) ≃ O_{k(𝔣),S_{p}}^{×}⊗_{ℤ}ℤ_{p},

and H^{2} fits into an short exact sequence

0 ⟶ Pic(O_{k(𝔣),S_{p}})⊗_{ℤ}ℤ_{p} ⟶ H^{2}(Δ(k(𝔣))) ⟶ X_{{w|𝔣p∞}}⊗_{ℤ}ℤ_{p} ⟶ 0

We have an isomorphism

Aϑ_{p} : Ξ(_{A}M)^{#}⊗_{ℚ}ℚ_{p} ⟶ Det_{A_{p}} (Δ(k(𝔣))⊗_{ℤ_{p}}ℚ_{p})

given by the composite

Det_{A_{p}}^{−1}(O_{k(𝔣)}^{×}⊗_{ℤ}ℚ_{p})⊗_{A_{p}}Det_{A_{p}}(X_{{v|∞}}⊗_{ℤ}ℚ_{p})
⟶^{φ_{1}} Det_{A_{p}}^{−1}(O_{k(𝔣),S_{p}}^{×}⊗_{ℤ}ℚ_{p})⊗_{A_{p}}Det_{A_{p}}(X_{{v|𝔣p∞}}⊗_{ℤ}ℚ_{p})
⟶^{φ_{2}} Det_{A_{p}}^{−1}(O_{k(𝔣),S_{p}}^{×}⊗_{ℤ}ℚ_{p})⊗_{A_{p}}Det_{A_{p}}(X_{{v|𝔣p∞}}⊗_{ℤ}ℚ_{p})
⟶^{φ_{3}} Det_{A_{p}} (Δ(k(𝔣))⊗_{ℤ_{p}}ℚ_{p}) .

Here φ_{1} is induced by the split short exact sequences

0 ⟶ O_{k(𝔣)}^{×}⊗_{ℤ}ℚ_{p} ⟶ O_{k(𝔣),S_{p}}^{×}⊗_{ℤ}ℚ_{p} ⟶ Y_{{w|𝔣p}}⊗_{ℤ}ℚ_{p} ⟶ 0  (13)
0 ⟶ X_{{w|∞}}⊗_{ℤ}ℚ_{p} ⟶ X_{{w|𝔣p∞}}⊗_{ℤ}ℚ_{p} ⟶ Y_{{w|𝔣p}}⊗_{ℤ}ℚ_{p} ⟶ 0 (14)

The isomorphism φ_{2} is multiplication with the Euler factor ∏_{v∈S_{p}} E_{v}^{#} ∈ A^{×} where E_{v} is defined by

E_{v} = ∑_{η|_{Dv} =1} |D_{v}/I_{v}| e_{η} + ∑_{η|_{Dv} ≠1} (1 − η(f_{v}))^{−1} e_{η},  (15)

where f_{v} ∈ D_{v} denotes a lift of the Frobenius element in D_{v}/I_{v} and I_{v} ⊆ D_{v} ⊆ G_{𝔣} are the inertia and decomposition subgroups for a place w | v in k(𝔣)/k. Finally φ_{3} arises from the explicit description of the cohomology groups H^{i}(Δ(k(𝔣))), i = 1, 2, and the canonical isomorphism

Det_{A_{p}}(Δ(k(𝔣))⊗_{𝔄_{p}}ℚ_{p}) ≃ ⨂_{i∈ℤ} Det_{A_{p}}^{(−1)^{i}} (H^{i}(Δ(k(𝔣))⊗_{𝔄_{p}}ℚ_{p}))  (16)

([14, Rem. b) following Th. 2]). We are now in position to give a very explicit description of the equivariant version of [8, Conjecture 3].

Conjecture 4.1 _{A}ϑ_{p} (_{A}ϑ_{∞}(L^{∗}(_{A}M, 0)^{−1})) 𝔄_{p} = Det_{𝔄_{p}}(Δ(k(𝔣))).

The main result of this article reads:

Theorem 4.2 Let k denote a quadratic imaginary field and let p be an odd prime which splits in k/ℚ and which does not divide the class number h_{k} of k. Then Conjecture 4.1 holds.

Corollary 4.3 Let k denote a quadratic imaginary field and let p be an odd prime which splits in k/ℚ and which does not divide the class number h_{k} of k. Let L be a finite abelian extension of k and k ⊆ K ⊆ L. Then the p-part of the ETNC holds for the pair (h^{0}(Spec(L), ℤ[Gal(L/K)])).

Proof This is implied by well known functorial properties of the ETNC.

5 The limit theorem

Following [8] or [5] we will deduce Theorem 4.2 from an Iwasawa theoretic result which we will describe next. Let now p = 𝔭𝔭̄ denote a split rational prime and 𝔣 an integral O_{k}-ideal such that w(𝔣) = 1. In addition, we assume that 𝔭̄ divides 𝔣 whenever 𝔭 divides 𝔣. We write 𝔣 = 𝔣_{0}𝔭^{ν}, 𝔭 ∤ 𝔣_{0}. We put Δ := Gal(k(𝔣_{0}𝔭)/k) = G_{𝔣_{0}𝔭} and let

Λ = lim_{n^{←}} ℤ_{p}[G_{𝔣𝔭^{n}}] ≃ ℤ_{p}[Δ][[T]]

denote the completed group ring. The element T = γ −1 depends on the choice of a topological generator γ of Γ := Gal(k(𝔣_{0}𝔭^{∞})/k(𝔣_{0}𝔭)) ≃ ℤ_{p}. We will work in the derived category D^{p}(Λ) and define

Δ^{∞} := lim_{n^{←}} Δ(k(𝔣_{0}𝔭^{n})).

Then Δ_{∞} can be represented by a perfect complex of Λ-modules. For its cohomology groups one obtains H^{i}(Δ^{∞}) = 0 for i ≠ 1, 2,

H^{1}(Δ^{∞}) ≃ U_{S_{p}}^{∞} := lim_{n^{←}} (O_{k(𝔣_{0}𝔭^{n}),S_{p}}^{×}⊗_{ℤ}ℤ_{p})

and H^{2}(Δ^{∞}) fits into the short exact sequence

0 ⟶ P_{S_{p}}^{∞} ⟶ H^{2}(Δ^{∞}) ⟶ X_{{w|𝔣_{0}𝔭∞}}^{∞} ⟶ 0,

where

P_{S_{p}}^{∞} := lim_{n^{←}} (Pic(O_{k(𝔣_{0}𝔭^{n}),S_{p}})⊗_{ℤ}ℤ_{p}) ,

X_{{w|𝔣_{0}𝔭∞}}^{∞} := lim_{n^{←}} (X_{{w|𝔣_{0}𝔭∞}}(k(𝔣_{0}𝔭^{n}))⊗_{ℤ}ℤ_{p}) .

The limits over the unit and Picard groups are taken with respect to the norm maps; the transition maps for the definition of X_{{w|𝔣_{0}𝔭∞}}^{∞} are defined by sending each place to its restriction. For 𝔤 | 𝔣_{0} we put

η_{𝔤} := {ψ(1; 𝔤𝔭^{n+1}, 𝔞)}_{n≥0} ∈ U_{S_{p}}^{∞},
σ_{∞} := {σ|_{k(𝔣_{0}𝔭^{n+1})}}_{n≥0} ∈ Y_{{w|𝔣p∞}}^{∞},

where σ is our fixed embedding ℚ^{c} ↪ ℂ. For any commutative ring R we write Q(R) for its total ring of fractions. Then Q(Λ) is a finite product of fields,

Q(Λ) ≃ ∏_{ψ∈Δ̂^{ℚp}} Q(ψ),  (17)

where Δ̂^{ℚ_{p}} denotes the set of ℚ_{p}-rational characters of Δ which are associated with the set of ℚ_{p}-irreducible representations of Δ. For each ψ ∈ Δ̂^{ℚ_{p}} one has

Q(ψ) = Q (ℤ_{l}(ψ)[[T]][\frac{1}{p}]) .

As in [8] one shows that for each ψ ∈ Δ̂^{ℚ_{p}} one has

dim_{Q(ψ)} (U_{S_{p}}^{∞}⊗_{Λ}Q(ψ)) = dim_{Q(ψ)} (Y_{{w|𝔣p∞}}^{∞}⊗_{Λ}Q(ψ)) = 1

It follows that the element e_{ψ}(η_{𝔣_{0}}^{−1}⊗σ_{∞}) is a Q(ψ)-basis of

Det_{Q(ψ)}(Δ^{∞}⊗_{Λ}Q(ψ)) ≃ Det_{Q(ψ)}^{−1}(U_{S_{p}}^{∞}⊗_{Λ}Q(ψ))⊗Det_{Q(ψ)}(X_{{w|𝔣p∞}}^{∞}⊗_{Λ}Q(ψ)).

Theorem 5.1 Λ · L = Det_{Λ}(Δ^{∞}) with L = (N𝔞 − σ(𝔞)) (η_{𝔣_{0}}^{−1}⊗σ_{∞}).

Proof By [8, Lem. 5.3] it suffices to show that the equality

Λ_{𝔮} · L = Det_{Λ_{𝔮}}(Δ^{∞}⊗_{Λ}Λ_{𝔮})  (18)

holds for all height 1 prime ideals of Λ. Such a height 1 prime is called regular (resp. singular) if p ∉ 𝔮 (resp. p ∈ 𝔮). We first assume that 𝔮 is a regular prime. Then Λ_{𝔮} is a discrete valuation ring, in particular, a regular ring. Hence we can work with the cohomology groups of Δ^{∞}, and in this way, the equality Λ_{𝔮} · L = Det_{Λ_{𝔮}}(Δ^{∞}⊗_{Λ}Λ_{𝔮}) is equivalent to

(N𝔞 − σ(𝔞))Fitt_{Λ_{𝔮}} (ℤ_{p,𝔮}) Fitt_{Λ_{𝔮}} (U_{S_{p},𝔮}^{∞}/η_{𝔣_{0}}Λ_{𝔮})
= Fitt_{Λ_{𝔮}} (P_{S_{p},𝔮}^{∞}) Fitt_{Λ_{𝔮}} (Y_{{w|𝔣_{0}𝔭∞},𝔮}^{∞}/Λ_{𝔮}σ_{∞}) .  (19)

Attached to each regular prime 𝔮 there is a unique character ψ = ψ_{𝔮} ∈ Δ̂^{ℚ_{p}}. To understand this notion we recall that

Λ[\frac{1}{p}] ≃ ∏_{ψ∈Δ̂^{ℚp}} (ℤ_{p}(ψ)[[T]]) [\frac{1}{p}].

If p ∉ 𝔮, then Λ_{𝔮} is just a further localisation of Λ[\frac{1}{p}], so that exactly one of the above components survives the localization process. We set

U^{∞} := lim_{n^{←}} (O_{k(𝔣_{0}𝔭^{n})}^{×}⊗_{ℤ}ℤ_{p}) ,
P^{∞} := lim_{n^{←}} (Pic(O_{k(𝔣_{0}𝔭^{n})})⊗_{ℤ}ℤ_{p}) .

Remark 5.2 Note that, using the notation of Section 3, one has P^{∞} = A_{∞}. We put K_{n} := k(𝔣_{0}𝔭^{n+1}). Mimicking the proof of Leopoldt’s conjecture, one can show that for each n ≥ 0 the natural map O_{K_{n}}^{×}⊗_{ℤ}ℤ_{p} → U_{n} (semi-local units in K_{n}⊗_{k}k_{𝔭} which are congruent to 1 mod 𝔭) is an injection. It follows that U^{∞} = Ē_{∞}, where Ē_{∞} is, as in Section 3, the projective limit over the closures of the global units.

There is an exact sequence of Λ-modules

0 ⟶ U^{∞} ⟶ U_{S_{p}}^{∞} ⟶ Y_{{w|𝔣_{0}𝔭},β}^{∞} ⟶ P^{∞} ⟶ P_{S_{p}}^{∞} ⟶ 0,  (20)

where

Y_{{w|𝔣_{0}𝔭},β}^{∞} = lim_{n^{←}} (Y_{{w|𝔣_{0}𝔭}}(k(𝔣_{0}𝔭^{n}))⊗ℤ_{p})

with respect to the transition maps

Y_{{w|𝔣_{0}𝔭}}(k(𝔣_{0}𝔭^{n+1}) ⟶^{β_{n+1/n}} Y_{{w|𝔣_{0}𝔭}}(k(𝔣_{0}𝔭^{n}))

induced by w ↦ f_{w|v}v, if v denotes the restriction of w and f_{w|v} the residue degree. If now 𝔟 is a prime divisor of 𝔣_{0} and n_{0} ∈ ℕ such that there is no further splitting of primes above 𝔟 in k(𝔣_{0}𝔭^{∞})/k(𝔣_{0}𝔭^{n_{0}}), then β_{m|n}(w) = p^{m−n}w|_{k(𝔣_{0}𝔭^{n+1})} for all m ≥ n ≥ n_{0}. Letting m tend to infinity this shows that Y_{{w|𝔟},β}^{∞} = 0. Hence we have an exact sequence of Λ-modules

0 ⟶ U^{∞} ⟶ U_{S_{p}}^{∞} ⟶ Y_{{w|𝔭},β}^{∞} ⟶ P^{∞} ⟶ P_{S_{p}}^{∞} ⟶ 0.  (21)

In addition, one has the exact sequence

0 ⟶ X_{{w|𝔣_{0}}}^{∞} ⟶ X_{{w|𝔣_{0}𝔭∞}}^{∞} ⟶ Y_{{w|𝔭}}^{∞} ⊕ Y_{{w|∞}}^{∞} ⟶ 0.  (22)

Remark 5.3 Note that the transition maps in the first two limits are induced by restriction, which coincides with β_{n+1|n} for the places above 𝔭 and ∞. Hence Y_{{w|∞}} = Y_{{w|∞},β} and Y_{{w|𝔭}} = Y_{{w|𝔭},β}.

We observe that Y_{{w|∞},𝔮}^{∞} = Λ_{𝔮} · σ_{∞}. Putting together (21) and (22) we therefore deduce that (19) is equivalent to

(N𝔞 − σ(𝔞))Fitt_{Λ_{𝔮}} (U_{𝔮}^{∞}/η_{𝔣_{0}}Λ_{𝔮}) = Fitt_{Λ_{𝔮}} (P_{𝔮}^{∞}) Fitt_{Λ_{𝔮}} (X_{{w|𝔣_{0}},𝔮}) . (23)

Let 𝔡 be a divisor of 𝔣_{0} such that ψ_{𝔮} has conductor 𝔡 or 𝔡𝔭. For any prime divisor 𝔩 | 𝔣_{0} we write I_{𝔩} ⊆ D_{𝔩} ⊆ G_{𝔣_{0}𝔭^{∞}} for the inertia and decomposition subgroups at 𝔩. Let Fr_{𝔩} denote a lift of the Frobenius element in D_{𝔩}/I_{𝔩}. We view ψ as a character of G_{𝔣_{0}𝔭^{∞}} via inflation and note that if 𝔩 ∤ 𝔡 (i.e. ψ|_{I_{𝔩}} = 1), then Fr_{𝔩} is a well defined element in Λ_{𝔮}.

Lemma 5.4 Let

ε = \begin{cases}0, & ψ ≠ 1, \\ 1, & ψ = 1. \end{cases}

Then:

Fitt_{Λ_{𝔮}}(Λ_{𝔮}T^{ε}η_{𝔡}/Λ_{𝔮}η_{𝔣_{0}}) = T^{−ε} ∏_{𝔩|𝔣_{0},𝔩∤𝔡}(1 − Fr_{𝔩}^{−1})Λ_{𝔮} = Fitt_{Λ_{𝔮}}(X_{{w|𝔣_{0}},𝔮}^{∞}).

Lemma 5.5

Fitt_{Λ_{𝔮}}(U_{𝔮}^{∞}/Λ_{𝔮}T^{ε}η_{𝔡}) = (N𝔞 − σ(𝔞))Fitt_{Λ_{𝔮}}(P_{𝔮}^{∞})

Proof of Lemma 5.5: Let ψ = ψ_{𝔮}. By the Iwasawa main conjecture (Theorem 3.1) and Remark 5.2 we have

char(P_{ψ}^{∞}) = char ((U^{∞}/C̄_{∞})_{ψ}) ,

where (again by a slight abuse of notation) for a Λ-module M we set M_{ψ} := M_{η} for any η ∈ ψ. The corollary to [16, App. Prop. 2] implies that

Fitt_{Λ_{𝔮}}(P_{𝔮}^{∞}) = Fitt_{Λ_{𝔮}} ((U^{∞}/C̄_{∞})_{𝔮}) .

Hence it suffices to show that

C̄_{∞}(𝔞)_{𝔮} = Λ_{𝔮} · T^{ε}η_{𝔡},  (24)
Fitt_{Λ_{𝔮}}(C̄_{∞,𝔮}/C̄_{∞}(𝔞)_{𝔮}) = (N𝔞 − σ(𝔞))Λ_{𝔮}.  (25)

Here C̄_{∞}(𝔞) is the projective limit over

C̄_{n}(𝔞) = closure of ⟨ψ(1; 𝔤𝔭^{n+1}, 𝔞): 𝔤 | 𝔣_{0}⟩_{ℤ[Gal(k(𝔣_{0}𝔭^{n+1})/k)]} ∩ E_{n}.

(Note that Λ_{𝔮}η_{𝔡} is for ψ ≠ 1 a group of units. This is true even for 𝔡 = 1, because Λ_{𝔮}η_{1} = Λ_{𝔮}e_{ψ}η_{1} and e_{ψ} has augmentation 0.) In order to prove (24) we set

ψ_{n} := ψ(1; 𝔡𝔭^{n+1}, 𝔞), G_{n} := Gal(k(𝔣_{0}𝔭^{n+1})/k), Λ_{n} := ℤ_{p}[G_{n}].

If 𝔟_{n} denotes the annihilator of ψ_{n} in Λ_{n}, then we have the following exact sequence of inverse systems of finitely generated ℤ_{p}-modules

0 ⟶ (Λ_{n}/𝔟_{n})_{n} ⟶ (C̄_{n}(𝔞))_{n} ⟶ (C̄_{n}(𝔞)/Λ_{n}ψ_{n})_{n} ⟶ 0.

The topology of ℤ_{p} induces on each of these modules the structure of a compact topological group, so that [27, Prop. B.1.1] implies that lim_{n^{←}} is exact. Hence we

obtain the short exact sequence of Λ-modules

0 ⟶ lim_{n^{←}} (Λ_{n}/𝔟_{n}) ⟶ C̄_{∞}(𝔞) ⟶ lim_{n^{←}} (C̄_{n}(𝔞)/Λ_{n}ψ_{n}) ⟶ 0.

Again by [27, Prop. B.1.1] we obtain

lim_{n^{←}} (Λ_{n}/𝔟_{n}) ≃ Λ/ lim_{n^{←}} 𝔟_{n} ≃ Λη_{𝔡},

so that

C̄_{∞}(𝔞)/Λη_{𝔡} ≃ lim_{n^{←}}(C̄_{n}(𝔞)/Λψ_{n}).  (26)

For 𝔡 | 𝔣_{0} we identify Gal(k(𝔣_{0}𝔭^{n+1})/k(𝔡𝔭^{n+1})) and Gal(k(𝔣_{0}𝔭)/k(𝔡𝔭)). Then one has (in additive notation) for any 𝔤 with 𝔡 | 𝔤 | 𝔣_{0} the distribution relation

N_{k(𝔣_{0}𝔭)/k(𝔡𝔭)} (ψ(1; 𝔤𝔭^{n+1}, 𝔞))
= [k(𝔣_{0}𝔭) : k(𝔤𝔭)] ( ∏_{𝔩|𝔤,𝔩∤𝔡}(1 − Fr_{𝔩}^{−1})) ψ(1; 𝔡𝔭^{n+1}, 𝔞).  (27)

In addition, one obviously has

[k(𝔣_{0}𝔭) : k(𝔤𝔭)]ψ(1; 𝔤𝔭^{n+1}, 𝔞) = N_{k(𝔣_{0}𝔭)/k(𝔤𝔭)} (ψ(1; 𝔤𝔭^{n+1}, 𝔞)) .  (28)

Note that for ψ ≠ 1 and 𝔡 ∤ 𝔤 one has ψ(N_{k(𝔣_{0}𝔭)/k(𝔤𝔭)}) = 0. Hence, if ψ ≠ 1, then (27), (28) and (26) show that

A := ( ∏_{𝔤|𝔣_{0},𝔡∤𝔤}[k(𝔣_{0}𝔭) : k(𝔤𝔭)]) · N_{k(𝔣_{0}𝔭)/k(𝔡𝔭)}

annihilates C̄_{∞}(𝔞)/Λη_{𝔡}. Since ψ(A) ∈ ℤ_{p} is non-trivial and p is invertible in Λ_{𝔮}, the element A is a actually a unit in Λ_{𝔮}, which implies C̄_{∞}(𝔞)_{𝔮} = Λ_{𝔮}η_{𝔡}. If ψ = 1 we proceed in almost the same way, but now set ψ_{n} := ψ(1; 𝔭^{n+1}, 𝔞)^{γ−1}. In this case we have 𝔡 = 1.

Sublemma: Let {C_{n}, f_{n}}_{n≥0} be a projective system of finitely generated ℤ_{p}[G_{n}]-modules and set C_{∞} = lim_{n^{←}} C_{n}. Let 𝔮 denote a regular prime and let

ψ = ψ_{𝔮}. Then:

C_{∞,𝔮} ≃ (lim_{n^{←}} C_{n,ψ})_{𝔮}.

Proof of Sublemma: The natural map C_{n} ⟶ ⊕_{χ∈Δ̂^{ℚp}} C_{n,χ} has kernel and cokernel annihilated by |Δ|. Passing to the limit we obtain (again by [27, B.1.1]) an exact sequence of Λ-modules

0 ⟶ W_{∞} ⟶ C_{∞} ⟶ ⨁_{χ∈Δ̂^{ℚp}} lim_{n^{←}} C_{n,χ} ⟶ X_{∞} ⟶ 0,

where W_{∞} and X_{∞} are annihilated by |Δ|. Since |Δ| ∈ Λ_{𝔮}^{×} we obtain

C_{∞,𝔮} ≃ ( ⨁_{χ∈Δ̂^{ℚp}} lim_{n^{←}} C_{n,χ})_{𝔮} = (lim_{n^{←}} C_{n,ψ})_{𝔮} .

Arguing as in the case ψ ≠ 1 and applying the Sublemma we obtain

(C̄_{∞}(𝔞)/ΛT η_{𝔡})_{𝔮} ≃ (lim_{n^{←}}(C̄_{n}(𝔞)/Λ_{n}ψ_{n}))_{𝔮} ≃ (lim_{n^{←}}(C̄_{n}(𝔞)/Λ_{n}ψ_{n})_{ψ})_{𝔮} .

Hence it suffices to show that each of the modules (C̄_{n}(𝔞)/Λ_{n}ψ_{n})_{ψ} is annihilated by the unit N_{k(𝔣_{0}𝔭)/k(𝔭)}. If

∏_{𝔤≠1} ψ(1; 𝔤𝔭^{n+1}, 𝔞)^{α_{𝔤}} · ψ(1; 𝔭^{n+1}, 𝔞)^{α_{1}} with α_{1}, α_{𝔤} ∈ ℤ_{p}[G_{n}]

is a unit in K_{n} = k(𝔣_{0}𝔭^{n+1}), then the prime ideal factorization of the singular values ψ(1; 𝔤𝔭^{n+1}, 𝔞) (see [1, Th. 2.4]) implies that α_{1} has augmentation 0. It follows that ψ(α_{1}) ∈ ℤ_{p}[Gal(K_{n}/K_{0})] is divisible by γ −1. For any element σ ∈ G_{n} we write σ = γ(σ)δ(σ) according to the decomposition G_{n} = Gal(K_{n}/K_{0})× Δ. If 𝔤 ≠ 1 each of the factors ψ(1 − Fr_{𝔩}^{−1}) = 1 − γ(Fr_{𝔩})^{−1} in (27) is divisible by γ − 1. Altogether this implies that N_{k(𝔣_{0}𝔭)/k(𝔭)} annihilates (C̄_{∞}(𝔞)/ΛT η_{𝔡})_{𝔮}, hence C̄_{∞}(𝔞)_{𝔮} = Λ_{𝔮}T η_{𝔡}. It finally remains to prove (25). For any integral ideal 𝔪 and any two integral ideals 𝔞 and 𝔟 such that (𝔞𝔟, 6𝔪) = 1 one has the relation

ψ(1; 𝔪, 𝔞)^{N𝔟−σ(𝔟)} = ψ(1; 𝔪, 𝔟)^{N𝔞−σ(𝔞)}.  (29)

This is a straightforward consequence of [1, Prop. 2.2] and the definition of ψ, see in particular [20, Théorème principal (b) and Remarque 1 (g)]. Equality (29) shows that N𝔞−σ(𝔞) annihilates C̄_{∞,𝔮}/C̄_{∞}(𝔞)_{𝔮}. Using the same arguments as in the proof of Lemma 3.5 (see that paragraph following Claim 2), one shows that this module is generated by one element. By [16, App. 3 and 8 ] it therefore suffices to show that (N𝔞 − σ(𝔞))Λ_{𝔮} is the exact annihilator. From Lemma 5.6 below we obtain finitely many ideals 𝔞_{1}, …, 𝔞_{s} and n_{1}, …, n_{s} ∈ Λ_{𝔮} such that

1 = ∑_{i=1}^{s} n_{i}(N𝔞_{i} − σ(𝔞_{i})).

Consider the element η := T^{ε} ∏_{i=1}^{s} η_{𝔡}(𝔞_{i})^{n_{i}}, where η_{𝔡}(𝔞_{i}) := {ψ(1; 𝔡𝔭^{n+1}, 𝔞_{i})}_{n=0}^{∞}. One has

η^{N𝔞−σ(𝔞)} = T^{ε}η_{𝔡}.

As a consequence of Lemma 3.5, Claim 2, the module C̄_{∞}(𝔞)_{𝔮} = Λ_{𝔮}T^{ε}η_{𝔡} is Λ_{𝔮}-free. It follows that no divisor of N𝔞 − σ(𝔞) annihilates the quotient C̄_{∞,𝔮}/C̄_{∞}(𝔞)_{𝔮}. To complete the proof for the localization at regular primes 𝔮 we add the following

Lemma 5.6 Let ψ ∈ Δ̂^{ℚ_{p}}, η ∈ ψ and write R = ℤ_{l}(ψ) = ℤ_{l}(η). Let I denote the ideal of Λ_{ψ} = R[[Γ]] generated by the elements N𝔞−σ(𝔞) = N𝔞−η(𝔞)γ(𝔞), where 𝔞 runs through the integral ideals of O_{k} such that (𝔞, 6𝔣𝔭) = 1. Then IΛ_{ψ}[\frac{1}{p}] = Λ_{ψ}[\frac{1}{p}].

Proof As usual we identify R[[Γ]] with R[[T]] by identifying γ with 1 + T. We note that Λ_{ψ}[\frac{1}{p}] is a principal ideal domain whose irreducible elements are given by the irreducible distinguished polynomials f ∈ R[T]. We fix such f and write

f(T) = γ^{s} + a_{s−1}γ^{s−1} + … + a_{1}γ + a_{0}, a_{i} ∈ R.

For any n there exist ideals 𝔞_{0}, …, 𝔞_{s} (depending on n) such that (𝔞_{i}, 6𝔣𝔭) = 1 and σ(𝔞_{i})|_{K_{n}} = γ^{i}|_{K_{n}}. In particular, this implies η(𝔞_{i}) = γ^{i} and

∑_{i=0}^{s} a_{i}(N𝔞_{i} − σ(𝔞_{i})) ≡ ∑_{i=0}^{s} a_{i}N𝔞_{i} − f(T)(mod (γ^{p^{n}} − 1)Λ_{ψ}).

Inverting p we derive

∑_{i=0}^{s} a′_{i}(N𝔞_{i} − σ(𝔞_{i})) ≡ 1 − cf(T)(mod (γ^{p^{n}} − 1)Λ_{ψ}[\frac{1}{p}])

with a′_{1}, …, a′_{s}, c ∈ ℚ_{p}(ψ) = ℚ_{p}(η). Therefore

1 ∈ IΛ_{ψ}[\frac{1}{p}] + fΛ_{ψ}[\frac{1}{p}] + ⋂_{n}(γ^{p^{n}} − 1)Λ_{ψ}[\frac{1}{p}].

Since (γ^{p^{n}} − 1)Λ_{ψ}[\frac{1}{p}] is a strictly decreasing sequence of ideals in a principal ideal domain we obtain ⋂_{n}(γ^{p^{n}} − 1)Λ_{ψ}[\frac{1}{p}] = (0). Consequently, IΛ_{ψ}[\frac{1}{p}] + fΛ_{ψ}[\frac{1}{p}] = Λ_{ψ}[\frac{1}{p}] for every irreducible distinguished polynomial f and the lemma is proved.

We now assume that 𝔮 is a singular prime. We write Δ = Δ′ × P with p ∤ |Δ′| and note that the singular primes 𝔮 are in one-to-one correspondence with the ℚ_{p}-rational irreducible characters of Δ′ ([5, Lem. 6.2(i)]). Assume that in this way 𝔮 is associated with ψ ∈ Δ̂^{′ℚ_{p}} and set χ = ψ ×η, where η ∈ P̂ is arbitrarily chosen. From [6, III,2.1 Theorem] and [6, III,1.7 (13)] we know that the μinvariant of P_{χ}^{∞} := P^{∞}⊗_{ℤ_{p}[Δ]}ℤ_{p}(χ) vanishes. By [8, Lem. 5.6] it follows that P_{𝔮}^{∞} = 0. The module X_{{w|𝔣_{0}𝔭}}^{∞} is ℤ_{p}[[T]]-torsion and free over ℤ_{p}, hence has vanishing μ-invariant (as ℤ_{p}[[T]]-module). Again by [8, Lem. 5.6] we derive X_{{w|𝔣_{0}𝔭},𝔮}^{∞} = 0. Since P_{S_{p}}^{∞} is an epimorphic image of P^{∞} and because of the exactness of

0 ⟶ X_{{w|𝔣_{0}𝔭}}^{∞} ⟶ X_{{w|𝔣_{0}𝔭∞}}^{∞} ⟶ Y_{{w|∞}}^{∞} ⟶ 0

we derive

H^{2}(Δ^{∞})_{𝔮} = Y_{{w|∞},𝔮}^{∞} ≃ Λ_{𝔮}σ_{∞}.

We now compute H^{1}(Δ^{∞})_{𝔮}. Consider the filtration

Λ · η_{𝔣_{0}} ⊆ C̄_{∞}(𝔞) ⊆ C̄_{∞} ⊆ U^{∞} ⊆ U_{S_{p}}^{∞} = H^{1}(Δ^{∞}).

By (21) the quotient U_{S_{p}}^{∞}/U^{∞} injects into Y_{{w|𝔭}}^{∞}. This module is a finite free ℤ_{p}-module and hence has vanishing μ-invariant. The module U^{∞}/C̄_{∞} (or rather any of its χ-components) also has vanishing μ-invariant by [6, III, 2.1 Theorem and 1.7 (13)]. As shown above, the graded piece C̄_{∞}/C̄_{∞}(𝔞) is annihilated by N𝔞 − σ(𝔞). We claim that N𝔞 − σ(𝔞) ∈ Λ_{𝔮}^{×}. In order to prove the claim we note that N𝔞 − σ(𝔞) = N𝔞 − δ(𝔞)(1 + T)^{w} with w ∈ ℤ_{p} and w ≠ 0 (since σ(𝔞) has infinite order in G_{𝔣_{0}𝔭^{∞}}). Let π denote a prime element in ℤ_{p}(ψ). Then the explicit description of 𝔮 given in [5, Lem. 6.2] easily implies 𝔮 = (π, ΔP)[[T]], where ΔP is the kernel of the augmentation map ℤ_{p}(ψ)[P] → ℤ_{p}(ψ). Therefore Λ/𝔮 ≃ (ℤ_{p}(ψ)/π))[[T]]. Hence it suffices to show that the image of N𝔞 − σ(𝔞) under

Λ ⟶ ℤ_{p}(ψ)[[T]] ⟶ (ℤ_{p}(ψ)/π) [[T]] = Λ/𝔮 given by N𝔞 − ψ(𝔞)(1 + T)^{w} is non-trivial. This, in turn, is an easy exercise. Finally we will use the distribution relation

N_{k(𝔣_{0}𝔭^{n+1})/k(frg𝔭^{n+1})}ψ(1; 𝔣_{0}𝔭^{n+1}, 𝔞) = ( ∏_{𝔩|𝔣_{0},𝔩∤𝔤}(1 − Fr_{𝔩}^{−1})) ψ(1; 𝔤𝔭^{n+1}, 𝔞) (30)

to show that C̄_{∞}(𝔞)_{𝔮}/Λ_{𝔮}η_{𝔣_{0}} is trivial. Indeed, a statement similar to (26) shows that this quotient is annihilated by ∏_{𝔩|𝔣_{0}}(1−Fr_{𝔩}^{−1}), which is a unit in Λ_{𝔮} (same argument as with N𝔞 − σ(𝔞) as above). In conclusion, we have now shown that Δ_{𝔮}^{∞} has perfect cohomology, so that again (18) is equivalent to (19), which is trivially valid because all modules involved have trivial μ-invariants.

In the following we want to deduce Conjecture 4.1 from Theorem 5.1. Again we can almost word by word rely on Flach’s exposition [8]. We have a ring homomorphism

Λ ⟶ ℤ_{p}[G_{𝔣}] = 𝔄_{p} ⊆ A_{p} = ∏_{χ∈Ĝ_{𝔣}^{ℚp}} ℚ_{p}(χ),

a canonical isomorphism of complexes

Δ^{∞}⊗_{Λ}^{L}𝔄_{p} ≃ Δ(k(𝔣)),  (31)

and a canonical isomorphism of determinants

(Det_{Λ}Δ^{∞}) ⊗_{Λ}𝔄_{p} ≃ Det_{𝔄_{p}} (Δ(k(𝔣)))

It remains to verify that the image of the element L⊗1 in Det_{𝔄_{p}} (Δ(k(𝔣))) ⊆ Det_{A_{p}} (Δ(k(𝔣))⊗_{ℤ_{p}}ℚ_{p}) agrees with _{A}ϑ_{p} (_{A}ϑ_{∞}(L^{∗}(_{A}M, 0)^{−1})). Let δ denote the morphism such that the following diagram commutes

\xymatrix{Det_{Q(Λ)}(Δ^{∞}⊗_{Λ}Q(Λ))⊗_{Q(Λ)}A_{p} \ar[r]^{≃} \ar[d]_{≃} & Det_{Q(Λ)}(H^{•}(Δ^{∞}⊗_{Λ}Q(Λ)))⊗_{Q(Λ)}A_{p} \ar[d]^{δ} \\ Det_{A_{p}}(Δ(k(𝔣))⊗_{ℤ_{p}}ℚ_{p}) \ar[r]^{≃} & Det_{A_{p}}(H^{•}(Δ(k(𝔣))⊗_{ℤ_{p}}ℚ_{p}))}

We let

ϕ : Det_{ℚ_{p}(χ)} (Δ(k(𝔣))⊗_{𝔄_{p}}ℚ_{p}(χ))
≃ \begin{cases}Det_{ℚ_{p}(χ)}^{−1}(O_{k(𝔣)}^{×}⊗_{𝔄_{p}}ℚ_{p}(χ))⊗_{ℚ_{p}(χ)}Det_{ℚ_{p}(χ)}(X_{\{v|∞\}}⊗_{𝔄_{p}}ℚ_{p}(χ)), & χ ≠ 1, \\ ℚ, & χ = 1 \end{cases}

Documenta Mathematica 11 (2006) 73–118

denote the isomorphism induced by φ_{1}^{−1} and φ_{3}^{−1} (see (13), (14) and (16)). Note that ϕ is defined in terms of cohomology. Then we have to show that

( ∏_{v∈S_{p}} (E_{v}^{#})^{−1}) ϕ(δ(L⊗1))  (32)

= \begin{cases}w(𝔣_{χ})[k(𝔣) : k(𝔣_{χ})](N𝔞 − χ(𝔞))e_{χ}ξ_{χ}^{−1}⊗w_{∞}, & 𝔣_{χ} ≠ 1, \\ w(1)[k(𝔣) : k(1)](1 − χ(𝔭)^{−1})(N𝔞 − χ(𝔞))e_{χ}ξ_{χ}^{−1}⊗w_{∞}, & 𝔣_{χ} = 1, χ ≠ 1 \\ L(χ, 0)^{−1}, & χ = 1. \end{cases}

By abuse of notation we also write χ for the composite ring homomorphism Λ → ℚ_{p}(χ) and denote its kernel by 𝔮_{χ}. Then 𝔮_{χ} is a regular prime of Λ and Λ_{𝔮_{χ}} is a discrete valuation ring with residue field ℚ_{p}(χ). We consider χ as a character of Gal(k(𝔣_{0}𝔭^{∞})/k). If χ = ψ × η with ψ ∈ Δ̂ and η a character of Gal(k(𝔣_{0}𝔭^{∞})/k(𝔣_{0}𝔭)), then the quotient field of Λ_{𝔮_{χ}} is given by Q(ψ) (notation as in (17)). We set

𝔣_{1} = \begin{cases}𝔣, & if 𝔭 | 𝔣, \\ 𝔣𝔭, & if 𝔭 ∤ 𝔣. \end{cases}

Let p^{n} be the degree of k(𝔣_{1})/k(𝔣_{0}𝔭).

Lemma 5.7 The element ω̄ := 1 − γ^{p^{n}} is a uniformizing element for Λ_{𝔮_{χ}}.

Proof We have to show that after localisation at 𝔮_{χ} the kernel of χ is generated by ω̄. Since the idempotents e_{ψ} and e_{η} associated with ψ and η, respectively, are units in Λ_{𝔮_{χ}}, one has (Λ[\frac{1}{p}])_{𝔮_{χ}} = ((ℤ_{p}(ψ)[[T]][\frac{1}{p}])_{𝔮_{χ}} and

(ℚ_{p}(ψ)][Γ_{n}])_{𝔮_{χ}} = ℚ_{p}(χ). This immediately implies the result.

We apply [8, Lem. 5.7] to

R = Λ_{𝔮_{χ}}, Δ = Δ_{𝔮_{χ}}^{∞}, ω̄ = 1 − γ^{p^{n}}.

For a R-module M we put Mω̄ := {m ∈ M | ω̄m = 0} and M/ω̄ := M/ω̄M. As we already know, the cohomology of Δ is concentrated in degrees 1 and 2. We will see that the R-torsion subgroup of H^{i}(Δ), i = 1, 2, is annihilated by ω̄, hence H^{i}(Δ)_{tors} = H^{i}(Δ)ω̄. We define free R-modules M^{i}, i = 1, 2, by the short exact sequences

0 ⟶ H^{i}(Δ)ω̄ ⟶ H^{i}(Δ) ⟶ M^{i} ⟶ 0,

and consider the exact sequences of ℚ_{p}(χ)-vector spaces

0 ⟶ H^{i}(Δ)/ω̄ ⟶ H^{i}(Δ⊗_{R}^{L}ℚ_{p}(χ)) ⟶ H^{i+1}(Δ)ω̄ ⟶ 0

induced by the distinguished triangle

Δ ⟶^{ω̄} Δ ⟶ Δ⊗_{R}^{L}ℚ_{p}(χ) ⟶ Δ[1].

Then the map ϕω̄ of [8, Lem. 5.7] is induced by the exact sequence of ℚ_{p}(χ)- vector spaces

0 ⟶ M^{1}/ω̄ ⟶ H^{1}(Δ⊗_{R}^{L}ℚ_{p}(χ)) ⟶^{βω̄} H^{2}(Δ⊗_{R}^{L}ℚ_{p}(χ)) ⟶ M^{2}/ω̄ ⟶ 0,

(33)

where the Bockstein map βω̄ is given by the composite

H^{1}(Δ⊗_{R}^{L}ℚ_{p}(χ)) ⟶ H^{2}(Δ)ω̄ ⟶ H^{2}(Δ)/ω̄ ⟶ H^{2}(Δ⊗_{R}^{L}ℚ_{p}(χ)).

Note that for the exactness of (33) on the left we need to show that H^{1}(Δ) is torsion-free. We recall that Gal(k(𝔣_{0}𝔭^{n+1})/k(𝔣_{0}𝔭)) = Gal(K_{n}/K_{0}) is isomorphic to (1 + 𝔣_{0}𝔭)/(1 + 𝔣_{0}𝔭^{n+1}) ≃ (1 + pℤ_{p})/(1 + p^{n+1}ℤ_{p}) via the Artin map. As before we denote this isomorphism by σ : (1 + pℤ_{p})/(1 + p^{n+1}ℤ_{p}) → Gal(K_{n}/K_{0}) and also write σ : 1 + pℤ_{p} → Γ. Passing to the limit we obtain a character

χ_{ell} : Γ ⟶ 1 + pℤ_{p}

uniquely defined by σ(χ_{ell}(τ) mod (1 + p^{n+1}ℤ_{p})) = τ|_{K_{n}} for all τ ∈ Γ. Note that one has

ψ(1; 𝔣_{0}𝔭^{n+1}, 𝔞)^{τ} = ψ(χ_{ell}(τ); 𝔣_{0}𝔭^{n+1}, 𝔞)

for all n ≥ 0 and τ ∈ Γ. For a place w | 𝔭 in k(𝔣)/k and u ∈ k(𝔣) we write u_{w} = σ_{w}(u), where σ_{w} : ℚ^{c} → ℚ_{p}^{c} defines w.

Lemma 5.8 Define for 𝔩 | 𝔣_{0} the element c_{𝔩} ∈ ℤ_{p} by γ^{c_{𝔩}p^{n}} = Fr_{𝔩}^{−f_{𝔩}}, where f_{𝔩} ∈ ℤ is the residue degree at 𝔩 of k(𝔣)/k. Put c_{𝔭} = log_{p}(χ_{ell}(γ^{p^{n}}))^{−1} ∈ ℚ_{p}. Then βω̄ is induced by the map

H^{1}(Δ(k(𝔣)))⊗ℚ_{p} = O_{k(𝔣),S_{p}}^{×}⊗ℚ_{p} ⟶ X_{{w|𝔣𝔭∞}}⊗ℚ_{p} = H^{2}(Δ(k(𝔣)))⊗ℚ_{p}

given by

u ↦ ∑_{𝔩|𝔣_{0}} c_{𝔩} ∑_{w|𝔩} ord_{w}(u) · w + c_{𝔭} ∑_{w|𝔭} Tr_{k(𝔣)_{w}/ℚ_{p}} (log_{p}(u_{w})) · w.

Proof As in [8, Lem. 5.8].

Let 𝔞_{1}, 𝔞_{2} denote integral O_{k}-ideals and set 𝔟 = lcm(𝔞_{1}, 𝔞_{2}), 𝔠 = gcd(𝔞_{1}, 𝔞_{2}). In the following we will frequently apply the formulas

[k(𝔟) : k(𝔞_{1})k(𝔞_{2})] = \frac{w(𝔟)w(𝔠)}{w(𝔞_{1})w(𝔞_{2})}, k(𝔞_{1}) ∩ k(𝔞_{2}) = k(𝔠),

which follow easily from [28, (15)]. Without loss of generality we may assume that w(𝔣_{0}) = 1. We also note that w(𝔭) = 1, because 𝔭 ∤ 2 and 𝔭 ≠ 𝔭̄. This implies w(𝔤) = 1 for any multiple 𝔤 of 𝔣_{0} or 𝔭.

After these preparations we will now prove equality (32). Recall that this equality is equivalent to the statement that the image of L⊗1 in Det_{𝔄_{p}} (Δ(k(𝔣))) ⊆ Det_{A_{p}} (Δ(k(𝔣))⊗_{ℤ_{p}}ℚ_{p}) is equal to _{A}ϑ_{p} (_{A}ϑ_{∞}(L^{∗}(_{A}M, 0)^{−1})). This suggests to think of L as a p-adic L-function. The connection with the usual p-adic Lfunction is hidden in the fact that for each character χ the characteristic power series of the module of “semi-local units modulo elliptic units” is generated by the p-adic L-function (see [24, Remark after Theorem 4.1]). It is therefore intuitively clear that our descent considerations will be most difficult (and interesting) in the case of “trivial zeros” of the associated p-adic L-function.

The case of χ|_{D_{𝔭}} ≠ 1. We let χ ∈ Ĝ_{𝔣}^{ℚ_{p}} be a non-trivial character such that χ|_{D_{𝔭}} ≠ 1. This should be considered as the case of no trivial zeros. We first show that P_{𝔮_{χ}}^{∞} = 0. From Lemma 3.7 we know that multiplication by γ^{p^{n}} − 1 on P^{∞} has finite kernel and cokernel. It follows that the characteristic power series h ∈ ℤ_{p}[[Γ]] of P^{∞} (considered as a module over ℤ_{p}[[Γ]]) is coprime with γ^{p^{n}} − 1. Hence h is a unit in Λ_{𝔮_{χ}} which annihilates P_{𝔮_{χ}}^{∞}. From (21) and Remark 5.3 we obtain the short exact sequence

0 ⟶ U_{𝔮_{χ}}^{∞} ⟶ U_{S_{p},𝔮_{χ}}^{∞} ⟶ Y_{{w|𝔭},𝔮_{χ}}^{∞} ⟶ 0

Moreover, Y_{{w|𝔭}}^{∞} = ℤ_{p}[G_{∞}/D_{𝔭}], so that χ|_{D_{𝔭}} ≠ 1 implies Y_{{w|𝔭},𝔮_{χ}}^{∞} = 0. It follows that H^{1}(Δ) = U_{S_{p},𝔮_{χ}}^{∞} ≃ U_{𝔮_{χ}}^{∞} and Lemma 5.5 implies

U_{𝔮_{χ}}^{∞} = (N𝔞 − σ(𝔞))(1 − γ)^{ε}η_{𝔣_{χ,0}} · Λ_{𝔮_{χ}},

where 𝔣_{χ,0} is the divisor of 𝔣_{0} such that ψ has conductor 𝔣_{χ,0} or 𝔣_{χ,0}𝔭. Recall also that

ε = \begin{cases}0, & ψ ≠ 1, \\ 1, & ψ = 1. \end{cases}

If ψ = 1, then η ≠ 1 and 1 − χ(γ) = 1 − η(γ) ≠ 0, so that 1 − γ is a unit in Λ_{𝔮_{χ}}. Since also N𝔞 − σ(𝔞) ∈ Λ_{𝔮_{χ}}^{×}, we may choose β_{1} = η_{𝔣_{χ,0}} as Λ_{𝔮_{χ}}-basis of M^{1} = U_{𝔮_{χ}}^{∞}. Since P_{{w|𝔣𝔭}}^{∞} is a quotient of P^{∞} we obtain P_{{w|𝔣𝔭},𝔮_{χ}}^{∞} = 0. Therefore H^{2}(Δ) = X_{{w|𝔣_{0}𝔭∞},𝔮_{χ}}^{∞}. From the short exact sequence

0 ⟶ X_{{w|𝔣_{0}𝔭}}^{∞} ⟶ X_{{w|𝔣_{0}𝔭∞}}^{∞} ⟶ Y_{{w|∞}}^{∞} ⟶ 0

together with the fact that X_{{w|𝔣_{0}𝔭}}^{∞} is Λ-torsion, we derive

M^{2} = Y_{{w|∞}}^{∞} = Λ_{𝔮_{χ}} · β_{2} with β_{2} = σ_{∞}.

We now apply [8, Lem. 5.7] with ω̄ = 1 − γ^{p^{n}}. Recall that H^{2}(Δ)_{tors} = X_{{w|𝔣_{0}𝔭},𝔮_{χ}}^{∞} and this module is annihilated by ω̄. Indeed, ω̄ ∼ 1 − γ^{p^{m}} for m ≥ n. For large m one has γ^{p^{m}} ∈ D_{𝔩} for each 𝔩 | 𝔣_{0}𝔭. It follows that 1 − γ^{p^{m}} annihilates X_{{w|𝔣_{0}𝔭}}^{∞}, so that the assumptions of [8, Lem. 5.7] are satisfied. The element β̄_{1} ∈ M^{1}/ω̄ is the image of the norm-compatible system

η_{𝔣_{χ,0}} = {ψ(1; 𝔣_{χ,0}𝔭^{n+1}, 𝔞)}_{n≥0}

in M^{1}/ω̄ ⊆ O_{k(𝔣),S_{p}}^{×}⊗_{ℤ[G_{𝔣}]}ℚ_{p}(χ). We write

𝔣 = 𝔣_{0}𝔭^{ν}, 𝔣_{χ} = 𝔣_{χ,0}𝔭^{ν′}.

and recall the definition of ξ_{χ} in (9). We will show that

β̄_{1} = T_{χ}ξ_{χ}⊗[k(𝔣) : k(𝔣_{0}𝔭^{ν′})]^{−1}

with

T_{χ} = \begin{cases}(1 − χ^{−1}(𝔭)), & if 𝔣_{χ} ≠ 1, \\ 1, & if 𝔣_{χ} = 1. \end{cases}

If ν = 0, then 𝔣_{1} = 𝔣𝔭, 𝔣_{χ,0} = 𝔣_{χ} and we have the following diagram of fields

[figure: k(𝔣_{1}); w(𝔣_{χ}); k(𝔣)k(𝔣_{χ}𝔭); k(𝔣); k(𝔣_{χ}𝔭); k(𝔣_{χ})]

Hence we obtain from[1, Th. 2.3]

β̄_{1} = N_{k(𝔣_{1})/k(𝔣)}ψ(1; 𝔣_{χ}𝔭, 𝔞)⊗1 = T_{χ}ξ_{χ}⊗1.

Note that in this case [k(𝔣) : k(𝔣_{0}𝔭^{ν′})] = 1. If ν > 0 and ν′ = 0 we obtain the diagram

[figure: k(𝔣_{0}𝔭^{ν}) = k(𝔣); \frac{w(𝔣χ)}{w(𝔣χ𝔭^{ν})}; k(𝔣_{0})k(𝔣_{χ}𝔭^{ν}); k(𝔣_{0}); k(𝔣_{χ}𝔭^{ν}); k(𝔣_{χ,0}) = k(𝔣_{χ})]

Writing |G_{𝔣}|e_{χ} = t_{χ} and t̄_{χ} for the image of t_{χ} in ℤ[Gal(k(𝔣_{χ})/k)] we therefore have

β̄_{1} = ψ(1; 𝔣_{χ}𝔭^{ν}, 𝔞)⊗1
= t_{χ}ψ(1; 𝔣_{χ}𝔭^{ν}, 𝔞)⊗1/|G_{𝔣}|
= t̄_{χ}T_{χ}ξ_{χ}⊗\frac{w(𝔣_{χ}𝔭^{ν})}{w(𝔣_{χ})} \frac{1}{[k(𝔣_{χ}𝔭^{ν}) : k]}
= T_{χ}ξ_{χ}⊗\frac{w(𝔣_{χ}𝔭^{ν})}{w(𝔣_{χ})} \frac{1}{[k(𝔣_{χ}𝔭^{ν}) : k(𝔣_{χ})]}
= T_{χ}ξ_{χ}⊗[k(𝔣) : k(𝔣_{0})]^{−1}.

The case ν, ν′ > 0 is similar. Note that in this case χ(𝔭) = 0. For each 𝔩 | 𝔣_{0} we choose a place w_{𝔩} above 𝔩 in k(𝔣)/k. It is easy to see that

Y_{\{w|𝔩\}}⊗_{𝔄}ℚ_{p}(χ) = \begin{cases}0, & χ|_{D_{𝔩}} ≠ 1, \\ ℚ_{p}(χ) · w_{𝔩}, & χ|_{D_{𝔩}} = 1. \end{cases}

We choose for each 𝔩 | 𝔣_{0} with χ|_{D_{𝔩}} = 1 an element x_{𝔩} ∈ k(𝔣)^{×} such that

ord_{w_{𝔩}}(x_{𝔩}) ≠ 0
ord_{w}(x_{𝔩}) = 0 for all w ≠ w_{𝔩}.

Then ℚ_{p}(χ)x_{𝔩} ⟶^{val} Y_{{w|𝔩}}⊗_{ℤ_{p}[G_{𝔣}]}ℚ_{p}(χ) = ℚ_{p}(χ)w_{𝔩} is an isomorphism. We set

J = {𝔩 | 𝔣_{0} : χ|_{D_{𝔩}} = 1}, x_{J} := ⋀_{𝔩∈J} x_{𝔩}, w_{J} := ⋀_{𝔩∈J} w_{𝔩} and c_{χ} := ∏_{𝔩∈J} c_{𝔩}.

Since O_{k(𝔣)}^{×}⊗_{𝔄}ℚ_{p}(χ) is a ℚ_{p}(χ)-vector space of of dimension 1, the element β̄_{1} is necessarily a generator. Therefore {β̄_{1}} ∪ {x_{𝔩}: 𝔩 ∈ J} is a ℚ_{p}(χ)-basis of H^{1}(Δ⊗_{R}^{L}ℚ_{p}(χ)) = O_{k(𝔣),S_{p}}^{×}⊗_{𝔄}ℚ_{p}(χ). Moreover, {β̄_{2}} ∪ {w_{𝔩}: 𝔩 ∈ J} is a ℚ_{p}(χ)-basis of Y_{{w|𝔣_{0}𝔭∞}}⊗_{𝔄}ℚ_{p}(χ). Finally note that β̄_{2} = σ|_{k(𝔣)}. From (33) we deduce

(ϕ ∘ ϕ−1 ; ω̄)(β̄_{1}^{−1}⊗β̄_{2}) = ϕ(β̄_{1}^{−1} ∧ x_{J}^{−1}⊗βω̄(x_{J}) ∧ β̄_{2})

Applying Lemma 5.8 we obtain further

(ϕ ∘ ϕ−1 ; ω̄)(β̄_{1}^{−1}⊗β̄_{2}) = c_{χ}ϕ(β̄_{1}^{−1} ∧ x_{J}^{−1}⊗val(x_{J}) ∧ β̄_{2})
= c_{χ}(β̄_{1}^{−1}⊗β̄_{2})  (34)
= \underbrace{c_{χ}[k(𝔣) : k(𝔣_{0}𝔭^{ν′})]T_{χ}^{−1}ξ_{χ}^{−1}⊗σ|_{k(𝔣)}}_{=:A} .

In order to apply [8, Lem. 5.7] we compute the exponent e such that ω̄^{e}β_{1}^{−1}⊗β_{2} is a Λ_{𝔮_{χ}}-basis of Det_{Λ_{𝔮χ}} (Δ_{𝔮_{χ}}^{∞}). By the proof of [8, Lem. 5.7] one has

e = ∑_{i∈ℤ}(−1)^{i+1} dim_{ℚ_{p}(χ)} (H^{i}(Δ)ω̄)
= − dim_{ℚ_{p}(χ)} (X_{{w|𝔣_{0}𝔭}}^{∞}⊗_{𝔄}ℚ_{p}(χ))

=^{χ≠1} − dim_{ℚ_{p}(χ)} (⨁_{𝔩|𝔣_{0}𝔭} ℤ_{p}[G_{∞}/D_{𝔩}]⊗_{𝔄}ℚ_{p}(χ))
= −|J|.

As elements of (Det_{Λ}(Δ^{∞}))_{𝔮_{χ}} we have

L = (N𝔞 − σ(𝔞))η_{𝔣_{0}}^{−1}⊗σ_{∞}
= (N𝔞 − σ(𝔞))[k(𝔣_{0}𝔭) : k(𝔣_{χ,0}𝔭)][Tr_{k(𝔣_{0}𝔭)/k(𝔣_{χ,0}𝔭)}η_{𝔣_{0}}]^{−1}⊗σ_{∞},

because Tr_{k(𝔣_{0}𝔭)/k(𝔣_{χ,0}𝔭)} = [k(𝔣_{0}𝔭) : k(𝔣_{χ,0}𝔭)] as elements of Λ_{𝔮_{χ}} (multiply both sides with e_{χ}). From the distribution relation we derive further

L = (N𝔞 − σ(𝔞))[k(𝔣_{0}𝔭) : k(𝔣_{χ,0}𝔭)] ∏_{𝔩|𝔣_{0},𝔩∤𝔣_{χ,0}} \frac{1}{1 − Fr_{𝔩}^{−1}} η_{𝔣_{χ,0}}^{−1}⊗σ_{∞}

= \underbrace{(N𝔞 − σ(𝔞))[k(𝔣_{0}𝔭) : k(𝔣_{χ,0}𝔭)] ∏_{χ(𝔩)≠1^{𝔩|𝔣_{0},𝔩∤𝔣_{χ,0}}} \frac{1}{1 − Fr_{𝔩}^{−1}} ∏_{𝔩∈J} \frac{ω̄}{1 − Fr_{𝔩}^{−1}}}_{=:B} (ω̄^{e}β_{1}^{−1}⊗β_{2}) .

Now [8, Lem. 5.7] implies

ϕω̄(B^{−1}(L⊗1)) = β̄_{1}^{−1}⊗β̄_{2},

which in conjunction with (34) shows that ϕ(B^{−1}(L⊗1)) = A or ϕ(L⊗1) = AB. For 𝔩 ∈ J we have by definition of c_{𝔩} the equality Fr_{𝔩}^{−f_{𝔩}} = γ^{c_{𝔩}p^{n}} and therefore

χ (\frac{ω̄}{1 − Fr_{𝔩}^{−1}} ) = χ (\frac{(1 − γ^{p^{n}})(1 + Fr_{𝔩}^{−1} + … + Fr_{𝔩}^{−f_{𝔩}+1})}{1 − γ^{c_{𝔩}p^{n}}}) = \frac{f_{𝔩}}{c_{𝔩}} . (35)

Using [k(𝔣) : k(𝔣_{0}𝔭^{ν′})][k(𝔣_{0}𝔭) : k(𝔣_{χ,0}𝔭)] = w(𝔣_{χ})[k(𝔣) : k(𝔣_{χ})] it follows that

AB =

(N𝔞 − σ(𝔞))w(𝔣_{χ})[k(𝔣) : k(𝔣_{χ})] ( ∏_{χ(𝔩)≠1^{𝔩|𝔣_{0},𝔩∤𝔣_{χ,0}}} \frac{1}{1 − Fr_{𝔩}^{−1}} ) (∏_{𝔩∈J} f_{𝔩}) T_{χ}^{−1}ξ_{χ}^{−1}⊗σ|_{k(𝔣)}.

Documenta Mathematica 11 (2006) 73–118

Recalling the definition of the elements E_{v} from (15) we observe that this is exactly the equality (32).

The case of χ ≠ 1 and χ|_{D_{𝔭}} = 1. We let χ ∈ Ĝ_{𝔣}^{ℚ_{p}} be a non-trivial character such that χ|_{D_{𝔭}} = 1. This should be considered as the case of trivial zeros. Note that in this case 𝔭 ∤ 𝔣_{χ}, i.e. 𝔣_{χ,0} = 𝔣_{χ}. Before going into detail we brievely explain the strategy of the proof. We first point out that in one respect the elliptic case is easier than the cyclotomic case: there is no distinction between odd and even characters. Indeed, in the elliptic setting every non-trivial character behaves like an even character. Nevertheless, the strategy of the proof becomes most clear, if one recalls what happens in the cyclotomic case for odd characters. In order to avoid the trivial zero one divides the p-adic L-function by γ−1. As a consequence of a theorem of Ferrero and Greenberg [7] (which gives a formula for the first derivative of the p-adic L-function) one obtains that this quotient interpolates essentially the global L-function L(χ^{−1}, s) at s = 0 (for more details see [8, Lemma 5.11]). For even characters the strategy can be motivated by the fact that the p-adic Lfunction is closely related to norm-coherent sequences of cyclotomic (or in our case, of elliptic) units. In order to “avoid the trivial zero” we again divide by γ − 1, which means that we have to take the (γ − 1)-st root of a norm-coherent sequence of cyclotomic or elliptic units. In the cyclotomic case this is achieved by using a result of Solomon [29] which also provides enough information to work out the relation to the value of L(χ^{−1}, s) at s = 0. In the elliptic case we will use an analoguous result proved by the author in [1]. For any subgroup H of G_{∞} we define J_{H} to be the kernel of the canonical map ℤ_{p}[[G_{∞}]] → ℤ_{p}[[G_{∞}/H]]. As in the case of no trivial zeros we can show that P_{𝔮_{χ}}^{∞} = 0. From (21) we obtain the short exact sequence

0 ⟶ U_{𝔮_{χ}}^{∞} ⟶ U_{S_{p},𝔮_{χ}}^{∞} ⟶ Y_{{w|𝔭},𝔮_{χ}}^{∞} ⟶ 0  (36)

where now Y_{{w|𝔭},𝔮_{χ}}^{∞} ≃ ℤ_{p}[G_{∞}/D_{𝔭}]⊗_{Λ}Λ_{𝔮_{χ}} ≃ Λ/J_{D_{𝔭}}⊗_{Λ}Λ_{𝔮_{χ}} ≃ Λ_{𝔮_{χ}}/J_{D_{𝔭}}Λ_{𝔮_{χ}}. Since Γ ⊆ D_{𝔭} one has γ^{p^{n}} − 1 ∼ γ − 1. It follows that Y_{{w|𝔭},𝔮_{χ}}^{∞} ≃ ℚ_{p}(χ), and in addition, the structure theorem for modules over principal ideal rings implies (γ − 1)U_{S_{p},𝔮_{χ}}^{∞} = U_{𝔮_{χ}}^{∞}.

For a finite set S of places of k we set U_{k(𝔣_{χ}),S}^{∞} = lim_{n^{←}} (O_{k(𝔣_{χ}𝔭^{n+1}),S}^{×}⊗_{ℤ}ℤ_{p}).

Lemma 5.9 a) The sequence

0 ⟶ U_{k(𝔣_{χ}),S_{p}}^{∞} ⟶^{γ−1} U_{k(𝔣_{χ}),S_{p}}^{∞} ⟶ U_{k(𝔣_{χ}),S_{p},Γ}^{∞} ⟶ 0

is exact.

b) The canonical map U_{k(𝔣_{χ}),S_{p},Γ}^{∞} ⟶ O_{k(𝔣_{χ}𝔭),S_{p}}^{×}⊗_{ℤ}ℤ_{p} is injective.

Proof One has (U_{k(𝔣_{χ}),S_{p}}^{∞})^{Γ} = lim_{n^{←}} (O_{k(𝔣_{χ}𝔭),S_{p}}^{×}⊗_{ℤ}ℤ_{p}) = 0. Hence a) is imme-

diate. For b) one has to prove

(γ − 1)U_{k(𝔣_{χ}),S_{p}}^{∞} = {u ∈ U_{k(𝔣_{χ}),S_{p}}^{∞} | u_{0} = 1}.

The inclusion ”⊆” is obvious. Suppose that u_{0} = 1. Then for each n Hilbert’s Theorem 90 provides an element β_{n} ∈ k(𝔣_{χ}𝔭^{n+1})^{×}/k(𝔣_{χ}𝔭)^{×} such that

β_{n}^{γ−1} = u_{n} and N_{k(𝔣_{χ}𝔭^{n+2})/k(𝔣_{χ}𝔭^{n+1})}(β_{n+1}) ≡ β_{n}(mod k(𝔣_{χ}𝔭)^{×}).

Let S be a finite set of places of k containing S_{p} and such that Pic(O_{k(𝔣_{χ}𝔭),S}) = 0. Then we may assume that

β_{n} ∈ O_{k(𝔣_{χ}𝔭^{n+1}),S}^{×}/O_{k(𝔣_{χ}𝔭),S}^{×}.

In the following diagram all vertical maps are induced by the norm,

[figure: 0; O_{k(𝔣_{χ}𝔭),S}^{×}⊗ℤ_{p}; O_{k(𝔣_{χ}𝔭^{n+2}),S}^{×}⊗ℤ_{p}; \frac{O_{k(𝔣χ𝔭^{n+2}),S}^{×}⊗ℤ_{p}}{O_{k(𝔣χ𝔭),S}^{×}⊗ℤ_{p}}; 0; 0; O_{k(𝔣_{χ}𝔭),S}^{×}⊗ℤ_{p}; O_{k(𝔣_{χ}𝔭^{n+1}),S}^{×}⊗ℤ_{p}; \frac{O_{k(𝔣χ𝔭^{n+1}),S}^{×}⊗ℤ_{p}}{O_{k(𝔣χ𝔭),S}^{×}⊗ℤ_{p}}; 0]

The natural topology of ℤ_{p} induces on each finitely generated ℤ_{p}-module the structure of a compact topological group. By [17, Satz IV.2.7] the functor lim_{n^{←}} is

therefore exact on the above exact sequence of projective systems. In addition, the projective limit over the modules on the left hand side is obviously trivial and therefore

U_{k(𝔣_{χ}),S}^{∞} ≃ lim_{n^{←}} \frac{O_{k(𝔣_{χ}𝔭^{n+1}),S}^{×}⊗ℤ_{p}}{O_{k(𝔣_{χ}𝔭),S}⊗ℤ_{p}} .

Moreover, the argument used to prove (21) also shows that U_{k(𝔣_{χ}),S}^{∞} ≃ U_{k(𝔣_{χ}),S_{p}}^{∞} ≃ U_{k(𝔣_{χ}),{w|𝔭∞}}^{∞} for any set S ⊇ S_{p}, so that the inclusion ”⊇” follows.

We now choose an auxiliary prime ideal 𝔟 of O_{k} such that

(𝔟, 𝔣𝔭) = 1, w(𝔟) = 1, χ(𝔟) ≠ 1.

In order to be able to deal also with the case 𝔣_{χ} = 1 we introduce the element

η = {ψ(1; 𝔣_{χ}𝔟𝔭^{n+1}, 𝔞)}_{n=0}^{∞} ∈ lim_{n^{←}} O_{k(𝔣_{χ}𝔟𝔭^{n+1})}^{×}.

With respect to the injection U_{k(𝔣_{χ}),S_{p},Γ}^{∞} ⟶ O_{k(𝔣_{χ}𝔭),S_{p}}^{×}⊗ℤ_{p} the element N_{k(𝔣_{χ}𝔟𝔭)/F} (η) maps to N_{k(𝔣_{χ}𝔟𝔭)/F} (η^{0}), where here F denotes the decomposition subfield at 𝔭 in k(𝔣_{χ})/k. One has the following diagram of fields

[figure: k(𝔣); •; _{k(𝔣)_{D_{𝔭}}}; k(𝔣_{χ}); F; _{k(𝔣)_{ker(χ)}}]

Since by definition of F one has σ(𝔭)|_{F} = id, we derive from the distribution relation

N_{k(𝔣_{χ}𝔟𝔭)/F} (η^{0}) = (1 − σ(𝔭)^{−1})N_{k(𝔣_{χ}𝔟)/F} ψ(1; 𝔣_{χ}𝔟, 𝔞) = 1,

so that Lemma 5.9 yields a unique element z^{∞} ∈ U_{k(𝔣_{χ}),S_{p}}^{∞}⊗_{ℤ_{p}}ℚ_{p} such that

(γ − 1)z^{∞} = \frac{1}{[k(𝔣_{χ}𝔟𝔭) : F]}N_{k(𝔣_{χ}𝔟𝔭)/F} (η).  (37)

From Lemma 5.5 and N𝔞 − σ(𝔞) ∼ 1 we deduce U_{𝔮_{χ}}^{∞} = Λ_{𝔮_{χ}}η_{𝔣_{χ}}. Again from the distribution relations [1, Th. 2.3] we deduce

N_{k(𝔣_{χ}𝔟𝔭)/F} η = (1 − F r_{𝔟}^{−1})N_{k(𝔣_{χ}𝔭)/F} η_{𝔣_{χ}}.

Combining (36) and (37) we see that

H^{1}(Δ) = U_{S_{p},𝔮_{χ}}^{∞} = Λ_{𝔮_{χ}} · β_{1} with β_{1} = z^{∞}.

Note that

β̄_{1} = \begin{cases}z_{μ}, & if 𝔭 | 𝔣, 𝔣 = 𝔣_{0}𝔭^{μ+1}, μ ≥ 0, \\ N_{k(𝔣𝔭)/k(𝔣)}(z_{0}), & if 𝔭 ∤ 𝔣, \end{cases}

when we regard β̄_{1} as an element in O_{k(𝔣),S_{p}}^{×}⊗ℤ_{p}. Let v denote a place of k(𝔣) above w, where w | 𝔭 in F/k. Using the above diagram we compute

Tr_{k(𝔣)_{v}/ℚ_{p}} (log_{p} (N_{k(𝔣_{χ}𝔟)/k(𝔣_{χ})} (ψ(1; 𝔣_{χ}𝔟, 𝔞))))
= \frac{|D_{𝔭}|}{[k(𝔣_{χ}) : F]} log_{p} (N_{k(𝔣_{χ}𝔟)/F} (ψ(1; 𝔣_{χ}𝔟, 𝔞)))
= \frac{|D_{𝔭}|}{[k(𝔣_{χ}) : F]} log_{p}(χ_{ell}(γ)) \underbrace{\frac{1}{log_{p}(χ_{ell}(γ))} log_{p} (N_{k(𝔣_{χ}𝔟)/F} (ψ(1; 𝔣_{χ}𝔟, 𝔞)))}_{=:B}

By the main result of [1] the quantity B is well known. We briefly recall the construction of [1]. Let k_{∞} denote the unique ℤ_{p}-extension of k which is unramified outside 𝔭. Let k_{n} ⊆ k_{∞} denote the extension of degree p^{n} above k. We put F_{n} := F k_{n} and consider the diagram of fields

[figure: k(𝔣_{χ}𝔟𝔭^{n+1}); k(𝔣_{χ}𝔟𝔭); •; k(𝔣_{χ}𝔟); F_{n}; F = F_{0}]

For each n Hilbert’s Theorem 90 provides an element β_{n} ∈ F_{n}^{×}/F^{×} such that

β_{n}^{γ−1} = N_{k(𝔣_{χ}𝔟𝔭^{n+1})/F_{n}} (ψ(1; 𝔣_{χ}𝔟𝔭^{n+1}, 𝔞)) .

If we put κ_{n} := N_{F_{n}/F} (β_{n}) ∈ F^{×}/ (F^{×})^{p^{n}} and κ^{∞} := {κ_{n}}_{n=0}^{∞} ∈ lim F^{×}/ (F^{×})^{p^{n}}, then the main result of [1] says

B = ord_{w}(κ^{∞}).

From the construction of z^{∞} it is clear that one has

β_{n} = N_{k(𝔣_{χ}𝔟𝔭^{n+1})/F_{n}}(z_{n}) in F_{n}^{×}/F^{×},

and consequently,

κ^{∞} = {N_{k(𝔣_{χ}𝔟𝔭)/F} (z_{0})}_{n=0}^{∞}.

We let w′ | w denote the place in k(𝔣_{χ})/F defined by v and set c_{𝔭}(γ) := 112 W. Bley log_{p}(χ_{ell}(γ))^{−1}. Then

Tr_{k(𝔣)_{v}/ℚ_{p}} (log_{p} (N_{k(𝔣_{χ}𝔟)/k(𝔣_{χ})} (ψ(1; 𝔣_{χ}𝔟, 𝔞))))
= \frac{|D_{𝔭}|}{[k(𝔣_{χ}) : F]}c_{𝔭}(γ)^{−1}ord_{w}(N_{k(𝔣_{χ}𝔟𝔭)/F} z_{0})
= \frac{|D_{𝔭}|c_{𝔭}(γ)^{−1}}{[k(𝔣_{χ}) : F]} ord_{w}(N_{k(𝔣_{χ}𝔭)/F} N_{k(𝔣_{χ}𝔟𝔭)/k(𝔣_{χ}𝔭)}z_{0})
= \frac{|D_{𝔭}|c_{𝔭}(γ)^{−1}[k(𝔣_{χ}𝔟𝔭) : k(𝔣_{χ}𝔭)]}{[k(𝔣_{χ}) : F]}ord_{w}(N_{k(𝔣_{χ}𝔭)/F} z_{0})
= \frac{|D_{𝔭}|c_{𝔭}(γ)^{−1}w(1)[k(𝔟) : k(1)]}{[k(𝔣_{χ}) : F]}ord_{w}(N_{k(𝔣_{χ})/F} (N_{k(𝔣_{χ}𝔭)/k(𝔣_{χ})}z_{0}))
= \frac{|D_{𝔭}|c_{𝔭}(γ)^{−1}w(1)[k(𝔟) : k(1)]}{[k(𝔣_{χ}) : F]}f_{w′/w}ord_{w′}(N_{k(𝔣_{χ}𝔭)/k(𝔣_{χ})}z_{0})
= |D_{𝔭}|c_{𝔭}(γ)^{−1}w(1)[k(𝔟) : k(1)]e_{v/w′}^{−1}ord_{v}(N_{k(𝔣_{χ}𝔭)/k(𝔣_{χ})}z_{0})
= f_{𝔭}c_{𝔭}(γ)^{−1}w(1)[k(𝔟) : k(1)]ord_{v}(N_{k(𝔣_{χ}𝔭)/k(𝔣_{χ})}z_{0})

We now apply Lemma 5.8. The congruence in the following computation is modulo Y_{{w|𝔣_{0}}}⊗_{ℤ_{p}}ℚ_{p}.

βω̄ (N_{k(𝔣_{χ}𝔟)/k(𝔣_{χ})}(ψ(1; 𝔣_{χ}𝔟, 𝔞)))
≡ c_{𝔭} ∑_{v|𝔭} Tr_{k(𝔣)_{v}/ℚ_{p}} (log_{p}(N_{k(𝔣_{χ}𝔟)/k(𝔣_{χ})}(ψ(1; 𝔣_{χ}𝔟, 𝔞)))) · v
= \frac{c_{𝔭}}{c_{𝔭}(γ)}f_{𝔭}w(1)[k(𝔟) : k(1)] ∑_{v|𝔭} ord_{v} (N_{k(𝔣_{χ}𝔭)/k(𝔣_{χ})}(z_{0})) · v
= \begin{cases}\frac{f_{𝔭}}{p^{n}} [k(𝔟) : k(1)]\frac{w(1)}{w(𝔣_{χ})} ∑_{v|𝔭} ord_{v}(N_{k(𝔣_{0}𝔭)/k(𝔣_{0})}z_{0}) · v, & if 𝔭 ∤ 𝔣, \\ \frac{f_{𝔭}}{p^{n}} w(1)[k(𝔟) : k(1)][k(𝔣_{χ}𝔭) : k(𝔣_{χ})] ∑_{v|𝔭} ord_{v} (z_{0}) · v, & if 𝔭 | 𝔣 \end{cases}
= \begin{cases}\frac{f_{𝔭}}{p^{n}} [k(𝔟) : k(1)]\frac{w(1)}{w(𝔣_{χ})} ∑_{v|𝔭} ord_{v}(N_{k(𝔣_{0}𝔭)/k(𝔣_{0})}z_{0}) · v,  if 𝔭 ∤ 𝔣, \\ \frac{f_{𝔭}}{p^{n}} w(1)[k(𝔟) : k(1)][k(𝔣_{χ}𝔭) : k(𝔣_{χ})][k(𝔣_{χ}𝔭^{μ+1}) : k(𝔣_{χ}𝔭)] ∑_{v|𝔭} ord_{v} (z_{μ}) · v, \\ if 𝔭 | 𝔣 \end{cases}
= \begin{cases}\frac{f_{𝔭}}{p^{n}} [k(𝔟) : k(1)]\frac{w(1)}{w(𝔣_{χ})} ∑_{v|𝔭} ord_{v}(N_{k(𝔣_{0}𝔭)/k(𝔣_{0})}z_{0}) · v, & if 𝔭 ∤ 𝔣, \\ \frac{f_{𝔭}}{p^{n}} w(1)[k(𝔟) : k(1)][k(𝔣_{χ}𝔭^{μ+1}) : k(𝔣_{χ})] ∑_{v|𝔭} ord_{v} \binom{}{β̄_{1}} · v, & if 𝔭 | 𝔣 \end{cases}
= \begin{cases}\frac{f_{𝔭}}{p^{n}} \frac{w(1)}{w(𝔣_{χ})}[k(𝔟) : k(1)] ∑_{v|𝔭} ord_{v}(β̄_{1}) · v, & if 𝔭 ∤ 𝔣, \\ \frac{f_{𝔭}}{p^{n}} w(1)[k(𝔟) : k(1)][k(𝔣_{χ}𝔭^{ν}) : k(𝔣_{χ})] ∑_{v|𝔭} ord_{v} \binom{}{β̄_{1}} · v, & if 𝔭 | 𝔣 \end{cases}

Equivariant Tamagawa Number Conjecture 113

We first assume that 𝔭 | 𝔣 and use this data to compute

(ϕ ∘ ϕ−1 ; ω̄)(β̄_{1}^{−1}⊗β̄_{2})

= ϕ (β̄_{1}^{−1} ∧ [N_{k(𝔣_{χ}𝔟)/k(𝔣_{χ})}ψ(1; 𝔣_{χ}𝔟, 𝔞)]^{−1}
∧x_{J}^{−1}⊗βω̄(x_{J}) ∧ βω̄(N_{k(𝔣_{χ}𝔟)/k(𝔣_{χ})}(ψ(1; 𝔣_{χ}𝔟, 𝔞))) ∧ β̄_{2})
= c_{χ} \frac{f_{𝔭}}{p^{n}} w(1)[k(𝔟) : k(1)][k(𝔣_{χ}𝔭^{ν}) : k(𝔣_{χ})] ×

ϕ (β̄_{1}^{−1} ∧ [N_{k(𝔣_{χ}𝔟)/k(𝔣_{χ})}(ψ(1; 𝔣_{χ}𝔟, 𝔞))]^{−1} ∧ x_{J}^{−1}⊗val(x_{J}) ∧ val(β̄_{1}) ∧ β̄_{2})

= \underbrace{−c_{χ} \frac{f_{𝔭}}{p^{n}} w(1)[k(𝔟) : k(1)][k(𝔣_{χ}𝔭^{ν}) : k(𝔣_{χ})][N_{k(𝔣_{χ}𝔟)/k(𝔣_{χ})}(ψ(1; 𝔣_{χ}𝔟, 𝔞))]^{−1}⊗σ_{∞}|_{k(𝔣)}}_{=:A} .

On the other hand we note that 𝔣_{χ} = 𝔣_{χ,0} and compute

L = (N𝔞 − σ(𝔞))[k(𝔣_{0}𝔭) : k(𝔣_{χ}𝔭)] ∏_{𝔩|𝔣_{0},𝔩∤𝔣_{χ}} \frac{1}{1 − Fr_{𝔩}^{−1}} η_{𝔣_{χ}}^{−1}⊗σ_{∞}.

In addition, one has

(γ − 1)β_{1} = (γ − 1)z^{∞} = \frac{1}{[k(𝔣_{χ}𝔟𝔭) : F]}N_{k(𝔣_{χ}𝔟𝔭)/F} (η)

and

\frac{ω̄}{1 − γ} = T := 1 + γ + … + γ^{p^{n}−1}.

This implies the equality

ω̄β_{1} = (1 − γ)T β_{1} = −T \frac{1}{[k(𝔣_{χ}𝔟𝔭) : k(𝔣_{χ}𝔭)]}N_{k(𝔣_{χ}𝔟𝔭)/k(𝔣_{χ}𝔭)}(η)
= −T \frac{1}{[k(𝔣_{χ}𝔟𝔭) : k(𝔣_{χ}𝔭)]}(1 − σ(𝔟)^{−1})η_{𝔣_{χ}}
= −T \frac{1}{w(1)[k(𝔟) : k(1)]}(1 − σ(𝔟)^{−1})η_{𝔣_{χ}}

in U_{S_{p},𝔮_{χ}}^{∞}. Since e = −(|J| + 1) we obtain L = Bω̄^{e}(β_{1}^{−1}⊗β_{2}) with

B = −T (N𝔞 − σ(𝔞))[k(𝔣_{0}𝔭) : k(𝔣_{χ}𝔭)](w(1)[k(𝔟) : k(1)])^{−1} ×

(1 − σ(𝔟)^{−1}) ∏_{χ(𝔩)≠1^{𝔩|𝔣_{0},𝔩∤𝔣χ}} \frac{1}{1 − Fr_{𝔩}^{−1}} ∏_{𝔩∈J} \frac{ω̄}{1 − Fr_{𝔩}^{−1}} .

Again we deduce from [8, Lem. 5.7] that ϕ(L⊗1) = AB. From

[k(𝔣_{χ}𝔭^{ν}) : k(𝔣_{χ})][k(𝔣_{0}𝔭 : k(𝔣_{χ}𝔭)] = [k(𝔣) : k(𝔣_{χ})]

Documenta Mathematica 11 (2006) 73–118

(recall again that w(𝔭) = w(𝔣) = 1) and

N_{k(𝔣_{χ}𝔟)/k(𝔣_{χ})}(ψ(1; 𝔣_{χ}𝔟, 𝔞)^{w(𝔣_{χ})}) = \begin{cases}(1 − σ(𝔟)^{−1})ψ(1; 𝔣_{χ}, 𝔞), & if 𝔣_{χ} ≠ 1, \\ \frac{δ(O_{k},𝔞^{−1})}{δ(𝔟,𝔞^{−1}𝔟)}, & if 𝔣_{χ} = 1, \end{cases}

we compute

AB =
= w(𝔣_{χ})[k(𝔣) : k(𝔣_{χ})](N𝔞 − σ(𝔞)) ∏_{χ(𝔩)≠1^{𝔩|𝔣_{0}}} \frac{1}{1 − Fr_{𝔩}^{−1}} ∏_{𝔩∈J∪\{𝔭\}} f_{𝔩} ×

\begin{cases}ψ(1; 𝔣_{χ}, 𝔞)^{−1}⊗σ_{∞}|_{k(𝔣)}, & if 𝔣_{χ} ≠ 1, \\ (1 − σ(𝔟)^{−1})\frac{δ(O_{k},𝔞^{−1})}{δ(𝔟,𝔞^{−1}𝔟)}⊗σ_{∞}|_{k(𝔣)}, & if 𝔣_{χ} = 1. \end{cases}

Finally we use in the case 𝔣_{χ} = 1 the relation

(\frac{δ(O_{k}, 𝔞^{−1})}{δ(𝔟, 𝔞^{−1}𝔟)})^{1−σ(𝔭)^{−1}} = (\frac{δ(O_{k}, 𝔞^{−1})}{δ(𝔭, 𝔞^{−1}𝔭)})^{1−σ(𝔟)^{−1}}

and recover the equation (32). The case 𝔭 ∤ 𝔣 is completely analogous.

The case of the trivial character In this case β_{1} = η_{1} and we first have to compute β̄_{1}. If 𝔭 ∤ 𝔣, the β̄_{1} = N_{k(𝔣𝔭)/k(𝔣)}(ψ(1; 𝔮, 𝔞)) and the distribution relation [1, Th. 2.3 b)] implies

β̄_{1} = N_{k(𝔮)/k(1)}(ψ(1; 𝔮, 𝔞)^{w(1)}) = \frac{δ(O_{k}, 𝔞^{−1})}{δ(𝔭, 𝔭𝔞^{−1})},

where δ denotes the function of lattices defined in [21, Th. 1]. We recall that

δ(L, \underline{L})^{12} = \frac{Δ(L)^{[\underline{L}:L]}}{Δ(\underline{L})} .

If 𝔭 | 𝔣, then β̄_{1} = ψ(1; 𝔭^{ν}, 𝔞), where again 𝔣 = 𝔣_{0}𝔭^{ν}. We now want to compute (ϕ ∘ ϕ−1 ; ω̄)(β̄_{1}^{−1}⊗β̄_{2}). Since χ is now trivial we no longer have X_{{w|∞}}⊗_{𝔄}ℚ_{p}(χ) = Y_{{w|∞}}⊗_{𝔄}ℚ_{p}(χ), and therefore have to take into account the short exact sequence

0 ⟶ X_{{w|𝔣_{0}𝔭}}^{∞} ⟶ X_{{w|𝔣_{0}𝔭∞}}^{∞} ⟶ Y_{{w|∞}}^{∞} ⟶ 0  (38)

in the definition of ϕω̄. Recall here that H^{2}(Δ) = X_{{w|𝔣_{0}𝔭∞}}^{∞} and Y_{{w|∞}}^{∞} = M^{2}. A lift of σ|_{k(𝔣)} ∈ M^{2}/ω̄ = Y_{{w|∞}} is given by σ|_{k(𝔣)} − w_{𝔭}, where w_{𝔭} denotes a fixed place of k(𝔣) above 𝔭. We obtain

(ϕ ∘ ϕ−1 ; ω̄)(β̄_{1}^{−1}⊗β̄_{2}) = ϕ(β̄_{1}^{−1} ∧ x_{J}^{−1}⊗βω̄(x_{J}) ∧ (σ|_{k(𝔣)} − w_{𝔭}))
= c_{χ}ϕ(β̄_{1}^{−1} ∧ x_{J}^{−1}⊗val(x_{J}) ∧ (σ|_{k(𝔣)} − w_{𝔭}))

Next, we compute val(β̄_{1}) and express the result in terms of σ|_{k(𝔣)} −w_{𝔭}. If 𝔭 ∤ 𝔣, then

val(β̄_{1}) = \frac{1}{12}(N𝔞 − 1)val (\frac{Δ(O_{k})}{Δ(𝔭)}) .

We use Δ(O_{k})/Δ(𝔭) ∼ 𝔭^{12} and obtain in Y_{{w|𝔣_{0}𝔭}}⊗_{𝔄}ℚ_{p}(χ)

val(β̄_{1}) = (N𝔞 − 1) ∑_{w|𝔭} ord_{w}(𝔭) · w
= (N𝔞 − 1)|I_{𝔭}| ∑_{w|𝔭} w

= (N𝔞 − 1)|I_{𝔭}|\frac{|G_{𝔣}|}{|D_{𝔭}|}w_{𝔭}
= (N𝔞 − 1)\frac{[k(𝔣) : k]}{f_{𝔭}}w_{𝔭}

An explicit splitting of the short exact sequence (14) is given by

w ↦ w − \frac{1}{[k(𝔣) : k]}Tr_{k(𝔣)/k}σ|_{k(𝔣)}.

Under this map val(β̄_{1}) maps to −(N𝔞 − 1)\frac{[k(𝔣):k]}{f_{𝔭}}(σ|_{k(𝔣)} − w_{𝔭}) in X_{{w|𝔣𝔭∞}}⊗_{𝔄}ℚ_{p}(χ). Recall that φ_{O_{k}} denotes the Euler function attached to the ring O_{k}. In the case 𝔭 | 𝔣 we compute from [1, Th. 2.4]

val(β̄_{1}) = \frac{N𝔞 − 1}{φ_{O_{k}}(𝔭^{ν})} ∑_{w|𝔭} ord_{w}(𝔭) · w

= \frac{N𝔞 − 1}{φ_{O_{k}}(𝔭^{ν})} \frac{[k(𝔣) : k]}{f_{𝔭}}w_{𝔭}.

So we derive the closed formula

val(β̄_{1}) = −\frac{N𝔞 − 1}{φ_{O_{k}}(𝔭^{ν})} \frac{[k(𝔣) : k]}{f_{𝔭}}(σ|_{k(𝔣)} − w_{𝔭})

as elements of X_{{w|𝔣𝔭∞}}⊗_{𝔄}ℚ_{p}(χ). This implies

(ϕ ∘ ϕ−1 ; ω̄)(β̄_{1}^{−1}) = \underbrace{−c_{χ} \frac{φ_{O_{k}}(𝔭^{ν})}{N𝔞 − 1} \frac{f_{𝔭}}{[k(𝔣) : k]}}_{=:A}

On the other hand we compute for L⊗1

L⊗1 = (N𝔞 − σ(𝔞))η_{𝔣_{0}}^{−1}⊗σ
= (N𝔞 − σ(𝔞)) [k(𝔣_{0}) : k(1)] \frac{w(1)}{w(𝔣_{0})} [Tr_{k(𝔣_{0}𝔭^{n+1})/k(𝔭^{n+1})}η_{𝔣_{0}}^{−1}]^{−1} ⊗σ
= (N𝔞 − σ(𝔞)) [k(𝔣_{0}) : k(1)] \frac{w(1)}{w(𝔣_{0})} ∏_{𝔩|𝔣_{0}} \frac{1}{1 − Fr_{𝔩}^{−1}} η_{1}^{−1}⊗σ

= \underbrace{(N𝔞 − σ(𝔞)) [k(𝔣_{0}) : k(1)] \frac{w(1)}{w(𝔣_{0})} ∏_{𝔩|𝔣_{0}} \frac{ω̄}{1 − Fr_{𝔩}^{−1}}}_{=:B} ω̄^{e}(β_{1}^{−1}⊗β̄_{2}).

It follows from (35) together with

[k(𝔣) : k] = h_{k} \frac{w(𝔣)}{w(1)}φ_{O_{k}}(𝔣), [k(𝔣_{0}) : k(1)] = \frac{w(𝔣_{0})}{w(1)}φ_{O_{k}}(𝔣_{0})

that ϕ(L⊗1) = AB = −f_{𝔭} (∏_{𝔩|𝔣_{0}} f_{𝔩}) \frac{w(1)}{h_{k}}. Since ζ_{k}^{∗}(0) = −\frac{h_{k}}{w(1)} this concludes the proof.

References

[1] W. Bley, Wild Euler systems of elliptic units and the Equivariant Tamagawa Number Conjecture, J. Reine Angew. Math. 577 (2004) 117-146.

[2] M. Breuning, D. Burns, Leading terms of Artin L-functions at s = 0 and s = 1, preprint 2005, submitted for publication.

[3] D. Burns, Congruences between derivatives of abelian L-functions at s = 0, preprint 2005, submitted for publication.

[4] D. Burns, M. Flach, Tamagawa Numbers for motives with (noncommutative) coefficients, Doc. Math., J. DMV 6 (2001) 501-570.

[5] D. Burns, C. Greither, On the Equivariant Tamagawa Number Conjecture for Tate motives, Invent. math. 153 (2003) 303-359.

[6] E. deShalit, Iwasawa Theory of Elliptic Curves with Complex Multiplication , Perspectives in Math. Vol. 3, Academic Press 1987.

[7] B. Ferrero, R. Greenberg, On the behaviour of p-adic L-series at s = 0, Inv. Math. 50 (1978) 91-102.

[8] M. Flach, The Equivariant Tamagawa Number Conjecture: A survey, Stark’s conjecture: recent work and new directions, Contemp. Math. 358 (2004) 79-125.

[9] R. Gillard, Fonctions L p-adiques des corps quadratiques imaginaires et de leurs extensions abéliennes, J. Reine Angew. Math. 358 (1986) 76-91.

[10] C. Greither, Class groups of abelian fields and the main conjecture, Ann. Inst. Fourier, 42 (1992) 449-499.

[11] A. Huber, G. Kings, Bloch-Kato Conjecture and Main Conjecture of Iwasawa teory for Dirichlet characters, Duke Math. J. 119 (2003) 393-464.

[12] J.M. Johnson, Artin L-functions for abelian extensions of imaginary quadratic fields, Thesis, California Institute of Technology, Pasadena, California 2005

[13] K. Kato, Lectures on the approach to Iwasawa theory of Hasse-Weil Lfunctions via B_{dR}, Part I, In: Arithmetical algebraic Geometry, Ballico, E. (eds.) Lect.Notes Math. 1553, 50 - 163, New York, Springer 1993.

[14] 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.

[15] F. Knudsen, Determinant functors on exact categories and their extensions to categories of bounded complexes, Michigan Math.J. 50 (2002) 407-444.

[16] B. Mazur, A. Wiles, Class fields of abelian extensions of ℚ, Invent. Math. 76 (1984) 179-330.

[17] J. Neukirch, Algebraische Zahlentheorie, Springer, Heidelberg, 1992.

[18] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number fields, Springer, Berlin Heidelberg, 2000.

[19] G. Robert, Unités elliptiques, Bull.Soc.Math.France, Mémoire 36 (1973).

[20] G. Robert, Concernant la relation de distribution satisfaite par la fonction φ associée à un réseau complexe, Invent. math. 100 (1990) 231-257.

[21] G. Robert, La racine 12-ième canonique de Δ(L)^{[\underline{L}:L]}/Δ(\underline{L}), Séminaire de Théorie des Nombres Paris (1989-90) 209-232.

[22] J. Ritter, A. Weiss, Toward equivariant Iwasawa theory, munuscrpita mathematica 109 (2002) 131-146.

[23] K. Rubin, Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication, Invent.Math. 89 (1987), 527–560.

[24] K. Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent.Math. 103 (1991), 25–68.

[25] K. Rubin, On the main conjecture of Iwasawa theory for imaginary quadratic fields, Invent.Math. 93 (1988), 701–713.

[26] K. Rubin, The main conjecture, Appendix to the second edition of Serge Lang: Cyclotomic fields, Springer Verlag, 1990.

[27] K. Rubin, Euler Systems, Annals of mathematics studies 147, Princeton University Press, 2000.

[28] R. Schertz, Konstruktion von Potenzganzheitsbasen in Strahlklassenkörpern über imaginär-quadratischen Zahlkörpern, J. Reine Angew. Math. 398 (1989), 105–129.

[29] D. Solomon, On a construction of p-units in abelian fields, Invent. math. 109 (1992) 329-350.

[30] T. Tsuji, Semi-local Units modulo Cyclotomic Units , J.Number Theory. 78 (1999) 1-26.

[31] L. Washington, Introduction to cyclotomic fields, Springer Verlag 1998.

[32] A. Wiles, The Iwasawa conjecture for totally real fields , Ann. Math. 131 (1990) 493-540.

W. Bley

Universität Kassel Fachbereich 17 D-34109 Kassel Germany bley@mathematik.uni-kassel.de