On the Equivariant Tamagawa Number Conjecture for Abelian Extensions of a Quadratic Imaginary Field

Let k be a quadratic imaginary field, p a prime which splits in k/Q and does not divide the class number hk 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))).

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).

W. Bley
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[ 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 2 p .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.

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) := F (z; L, a −1 L), z ∈ C, where F 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].

W. Bley
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 where H is isomorphic to Gal(k(p)/k) by restriction.We set denote the ray class field of conductor fp.We set 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 Ēn and Cn denote the closures of E n ∩ U n and C n ∩ U n , respectively, in U n .Finally we define Cn , both inverse limits formed with respect to the norm maps.We let Λ = lim denote the completed group ring and for a finitely generated Λ-module and any abelian character χ of ∆ := Gal(K 0 /k) we define the χ-quotient of M by 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 If M χ is a finitely generated torsion Λ χ -module, then we write char(M χ ) for the characteristic ideal.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 for all characters χ of ∆.
For an abelian character χ of ∆ we write where 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 |∆|.If p ∤ |∆|, 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∞ / C∞ ) χ ).If p | |∆|, 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∞ / C∞ ) 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 ∤ |∆|.
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 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/M J and [y] l ∈ J l /M J l .

Documenta Mathematica 11 (2006) 73-118
For l ∈ L we let 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 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 = k i=1 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 F for the algebraic closure of F and let U(g) denote the set of all functions α : S(g) −→ F × 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 : Then we have: , 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: 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), Let pc be the precise power of p which divides the conductor f of K. Then there are infinitely many primes λ of K such that (2) If l = k ∩ λ, then N l ≡ 1(mod M ), and l splits completely in K.
(3) For all w ∈ W one has [w] l = 0 in J l /M J l and there exists a unit u ∈ (Z/M Z) × 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 The situation is clarified by the following diagram w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w Obviously p is totally ramified in k(µ M )/k.Hence p ramifies in K ′ /k of exponent at least ϕ Z (M ).On the other hand, p is ramified in K/k of exponent at most ϕ O k (p c ). Therefore any prime divisor of p ramifies in K ′ /K of degree at least 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.
We write M = p m .Since divisors of p are totally ramified in k(µ M )/k of degree ϕ Z (M ) and at most ramified in 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 M denote the image of W under the homomorphism 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 Documenta Mathematica 11 (2006) 73-118

W. Bley
The extension K ′ /K is cyclic and a Herbrand quotient argument shows .
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: Recall the notation introduced at the beginning of this section.In addition, we let ∆ . 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 ∆ we write ], so that our notation is consistent.We choose a generator h χ ∈ Λ χ of char ( Ē∞ / C∞ ) χ .By the general theory of finitely generated Λ χ -modules there is a quasi-isomorphism

and by definition, char(A
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 ∆.Then there exist constants where here im( Cn,χ ) denotes the image of Cn,χ in Ēn,χ .
Proof We mainly follow Greither's proof of [10,Lem. 3.9].We let U n and proceed to prove ].This can be proved similarly as [18,Th. 11

It follows that
If we apply the snake lemma to the diagram 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 Let W n denote the image of π n and set Then we have a commutative diagram (with exact lines) We write π n,χ for the composite map and obtain the exact sequence We claim that ker(π n,χ ) is annihilated by (γ − 1) 2 p 2κ : Let e ∈ ker(π n,χ ).Then 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 Ē∞,χ where we define ϑ n in the following manner: for e ∈ Ēn,χ there exists z ∈ Ē∞,χ such that π n,χ (z) = (γ − 1) 2 p 2κ e.We then set On the other hand, we have the exact sequence The structure theorem of Λ χ -torsion modules implies that h χ Ē∞ / C∞ χ is finite.Since α( Ē∞,χ )/α(im C∞,χ ) is a quotient of Ē∞,χ /im( C∞,χ ), the module h χ α( Ē∞,χ )/α(im C∞,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( C∞,χ ))).Therefore p 2s h χ ∈ α(im( C∞,χ )) and we conclude further: 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 n0 − 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 ∆.Then there exists a constant such that p c3 cokτ n = 0 for all n ≥ 0.Here ḡi denotes the image of 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.
Lemma 3.8 Let K/k be an abelian extension, G = Gal(K/k) and ∆ a subgroup of G. Let χ denote a character of ∆, 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 , where [W ] l denotes the subgroup of J l /M J l generated by elements Then there exists a G-homomorphism 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 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 with p c3 at the ith position.Choose c k+1 arbitrary.By Lemma 3.5 there exists an element It is now easy to show that there exists an actual elliptic unit ξ ∈ C n such that 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 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 where v ∈ (Z/M Z) × 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 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 Without loss of generality we may assume that c 1 ≥ 2. Then one has c 1 is a cyclic as a Λ n,χ -module and annihilated by

Note that by the definition of h
We now apply Lemma 3.8 with a = a i−1 , g = g i−1 , x = κ(a i−1 ), E = p c3 and η = (γ n0 − 1) ti .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 M and obtain a homomorphism We let c denote a preimage of c i and consider the homomorphism We again apply Theorem 3.4 and obtain λ i satifying (a), (b) and also 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/M Z)  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 n0 − 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.

The conjecture
In this section we fix an integral O k -ideal f such that w(f) = 1 and write 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 for any finitely generated projective R-module P and for a bounded complex 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 Here S(k(f)) denotes the set of places of k(f) lying above places in S. We let where the superscript # means twisting the action of G f by g → g −1 .We let 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) 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 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 We denote by ĜQ f the set of Q-rational characters associated with the Qirreducible representations of G f .For χ ∈ ĜQ f 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 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 ∈ ĜQ f 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 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 : 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 Since c∈cl(1) Cη(c) = 0 for any constant C we compute further log ϕ 1 (ac −1 ) η(c).
According to the decomposition (8) we decompose Ξ( A M ) # character by character and obtain a canonical isomorphism As in the cyclotomic case one has Documenta Mathematica 11 (2006) 73-118

W. Bley
From (10) we deduce In particular, this proves the equivariant version of [8, Conjecture 2].We fix a prime p and put 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 Then ∆(k(f)) can be represented by a perfect complex of A p -modules whose cohomology groups H i (∆(k(f)) are trivial for i = 1, 2. For i = 1 one finds and H 2 fits into an short exact sequence We have an isomorphism given by the composite Here ϕ 1 is induced by the split short exact sequences The isomorphism ϕ 2 is multiplication with the Euler factor v∈Sp E # v ∈ A × where E v is defined by ([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].
The main result of this article reads: Proof This is implied by well known functorial properties of the ETNC.

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 = pp denote a split rational prime and f an integral O k -ideal such that w(f) = 1.In addition, we assume that p divides f whenever p divides f.We write f = f 0 p ν , p ∤ f 0 .We put ∆ := Gal(k(f 0 p)/k) = G f0p and let 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 Then ∆ ∞ can be represented by a perfect complex of Λ-modules.For its cohomology groups one obtains where 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|f0p∞} are defined by sending each place to its restriction.For g | f 0 we put 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, where ∆Qp denotes the set of Q p -rational characters of ∆ which are associated with the set of Q p -irreducible representations of ∆.For each ψ ∈ ∆Qp one has As in [8] one shows that for each ψ ∈ ∆Qp one has Proof By [8,Lem. 5.3] it suffices to show that the equality Documenta Mathematica 11 (2006) 73-118 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 ∆ ∞ , and in this way, the equality Attached to each regular prime q there is a unique character ψ = ψ q ∈ ∆Qp .
To understand this notion we recall that If p ∈ q, then Λ q is just a further localisation of Λ[ 1 p ], so that exactly one of the above components survives the localization process.We set 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 × Kn ⊗ 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 where with respect to the transition maps 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 n0 ), then β m|n (w) = p m−n w| k(f0p n+1 ) for all W. Bley 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 In addition, one has the exact sequence 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 We observe that Y ∞ {w|∞},q = Λ q • σ ∞ .Putting together ( 21) and ( 22) we therefore deduce that ( 19) is equivalent to 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 f0p ∞ 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 f0p ∞ 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 Then: Proof of Lemma 5.5: Let ψ = ψ q .By the Iwasawa main conjecture (Theorem 3.1) and Remark 5.2 we have where (again by a slight abuse of notation) for a Λ-module M we set M ψ := M η for any η ∈ ψ.

Documenta Mathematica 11 (2006) 73-118
Hence it suffices to show that Here C∞ (a) is the projective limit over (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 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 The topology of Z p induces on each of these modules the structure of a compact topological group, so that 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 In addition, one obviously has Documenta Mathematica 11 (2006) 73-118
If ψ = 1 we proceed in almost the same way, but now set In this case we have d = 1.
Sublemma: Let {C n , f n } n≥0 be a projective system of finitely generated Let q denote a regular prime and let ψ = ψ q .Then: Proof of Sublemma: The natural map C n −→ ⊕ χ∈ ∆Qp C n,χ has kernel and cokernel annihilated by |∆|.Passing to the limit we obtain (again by [27, where W ∞ and X ∞ are annihilated by |∆|.Since |∆| ∈ Λ × q we obtain Arguing as in the case ψ = 1 and applying the Sublemma we obtain Hence it suffices to show that each of the modules ( Cn (a ), 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 Altogether this implies that N k(f0p)/k(p) annihilates C∞ (a)/ΛT η d q , hence C∞ (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 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 N a−σ(a) annihilates C∞,q / C∞ (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 (N a − 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 = Consider the element η As a consequence of Lemma 3.5, Claim 2, the module C∞ (a) q = Λ q T ε η d is Λ q -free.It follows that no divisor of N a − σ(a) annihilates the quotient C∞,q / C∞ (a) q .
To complete the proof for the localization at regular primes q we add the following , where a runs through the integral ideals of O k such that (a, 6fp) = 1.
] is a principal ideal domain whose irreducible elements are given by the irreducible distinguished polynomials f ∈ R[T ].We fix such f and write For any n there exist ideals a 0 , . . ., a s (depending on n) such that (a i , 6fp) = 1 and σ(a i )| Kn = γ i | Kn .In particular, this implies η(a i ) = γ i and given by N a − ψ(a)(1 + T ) w is non-trivial.This, in turn, is an easy exercise.Finally we will use the distribution relation to show that C∞ (a) q /Λ q η f0 is trivial.Indeed, a statement similar to (26) shows that this quotient is annihilated by l|f0 (1 − Fr −1 l ), which is a unit in Λ q (same argument as with N a − σ(a) as above).
In conclusion, we have now shown that ∆ ∞ 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 and a canonical isomorphism of determinants It remains to verify that the image of the element . Let δ denote the morphism such that the following diagram commutes (see ( 13), ( 14) and ( 16)).Note that φ is defined in terms of cohomology.Then we have to show that 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 qχ 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 ∈ ∆ and η a character of Gal(k(f 0 p ∞ )/k(f 0 p)), then the quotient field of Λ qχ is given by Q(ψ) (notation as in (17)).We set Let p n be the degree of k(f 1 )/k(f 0 p).Lemma 5.7 The element ω := 1 − γ p n is a uniformizing element for Λ qχ .
Proof We have to show that after localisation at q χ the kernel of χ is generated by ω.Since the idempotents e ψ and e η associated with ψ and η, respectively, are units in Λ qχ , one has Λ[ . This immediately implies the result.
We apply [8,Lem. 5.7] to For a R-module M we put M ω := {m ∈ M | ωm = 0} and M/ω := M/ωM .As we already know, the cohomology of ∆ is concentrated in degrees 1 and 2. We will see that the R-torsion subgroup of H i (∆), i = 1, 2, is annihilated by ω, hence H i (∆) tors = H i (∆) ω .We define free R-modules M i , i = 1, 2, by the short exact sequences and consider the exact sequences of Q p (χ)-vector spaces Documenta Mathematica 11 (2006) 73-118 Then the map φ ω of [8,Lem. 5.7] is induced by the exact sequence of Q p (χ)vector spaces where the Bockstein map β ω is given by the composite Note that for the exactness of (33) on the left we need to show that H 1 (∆) is torsion-free.We recall that Gal(k(f , 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 β ω is induced by the map given by u → 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 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 = p.This implies w(g) = 1 for any multiple g of f 0 or p. 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|f0p} , so that the assumptions of [8,Lem. 5.7] are satisfied.The element β1 ∈ M 1 /ω is the image of the norm-compatible system and recall the definition of ξ χ in (9).We will show that Note that in this case [k(f) : k(f 0 p ν ′ )] = 1.If ν > 0 and ν ′ = 0 we obtain the diagram Writing |G f |e χ = t χ and tχ for the image of t χ in Z[Gal(k(f χ )/k)] we therefore have 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 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 .
For l ∈ J we have by definition of c l the equality Fr −f l l = γ c l p n and therefore 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| Dp = 1.We let χ ∈ ĜQp f be a non-trivial character such that χ| Dp = 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 As in the case of no trivial zeros we can show that P ∞ qχ = 0. From ( 21) we obtain the short exact sequence , and in addition, the structure theorem for modules over principal ideal rings implies Lemma 5.9 a) The sequence The inclusion "⊆" is obvious.Suppose that u 0 = 1.Then for each n Hilbert's Theorem 90 provides an element Let S be a finite set of places of k containing S p and such that Pic(O k(fχp),S ) = 0. Then we may assume that In the following diagram all vertical maps are induced by the norm, Moreover, the argument used to prove (21) also shows that U ∞ k(fχ),S ≃ U ∞ k(fχ),Sp ≃ U ∞ k(fχ),{w|p∞} for any set S ⊇ S p , so that the inclusion "⊇" follows.
In order to be able to deal also with the case f χ = 1 we introduce the element With respect to the injection U ∞ k(fχ),Sp,Γ −→ O × k(fχp),Sp ⊗Z p the element N k(fχbp)/F (η) maps to N k(fχbp)/F (η 0 ), where here F denotes the decomposition subfield at p in k(f χ )/k.One has the following diagram of fields 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 k(f χ bp n+1 ) r r r r r r r r r r k(f χ bp) • q q q q q q q q q q q q k(f χ b) F n q q q q q q q q q q q q F = F 0 , then the main result of [1] says From the construction of z ∞ it is clear that one has and consequently, We

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 .

Documenta Mathematica 11 (
2006) 73-118 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 (∆(k(f))), i = 1, 2, and the canonical isomorphism

Theorem 4 . 2 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.Then Conjecture 4.1 holds.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)])).
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 thereforeU ∞ k(fχ),S ≃ lim ← n O × k(fχp n+1),S ⊗Z p O k(fχp),S ⊗Z p .
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∞ / C∞ ) χ ).Remarks 3.2 a) If p ∤ [F : k] 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)|.
[9]tained 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 |∆|.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.