Fitting ideals of class groups for CM abelian extensions

Let $K/k$ be a finite abelian CM-extension and $T$ a suitable finite set of finite primes of $k$. In this paper, we determine the Fitting ideal of the minus component of the $T$-ray class group of $K$, except for the $2$-component, assuming the validity of the equivariant Tamagawa number conjecture. As an application, we give a necessary and sufficient condition for the Stickelberger element to lie in that Fitting ideal.


Introduction
In number theory, the relationship between class groups and special values of L-functions is of great importance.In this paper we discuss such a phenomenon for a finite abelian CM-extension K/k, that is, a finite abelian extension such that k is a totally real field and K is a CM-field.We focus on the minus components of the (ray) class groups of K, except for the 2-components, and study the Fitting ideals of them.
Let Cl K denote the ideal class group of K. Let (−) − denote the minus component after inverting the multiplication by 2. When k = Q, Kurihara and Miura [10] succeeded in proving a conjecture of Kurihara [7] on a description of the Fitting ideal of Cl − K using the Stickelberger elements.However, for a general totally real field k, the problem to determine the Fitting ideal of Cl − K is still open.There seems to be an agreement that the Pontryagin duals (denoted by (−) ∨ ) of the class groups are easier to deal with (see Greither-Kurihara [4]).In [3], Greither determined the Fitting ideal of Cl ∨,− K , assuming that the equivariant Tamagawa number conjecture (eTNC for short) holds and that the group of roots of unity in K is cohomologically trivial.Subsequently, Kurihara [9] generalized the results of Greither on Cl ∨,− K to results on Cl T,∨,− K , where Cl T K denotes the T -ray class group, for a finite set T of finite primes of k.This enables us, by taking suitably large T , to remove the assumption that the group of roots of unity is cohomologically trivial, though we still need to assume the validity of eTNC.In recent work [2], Dasgupta and Kakde succeeded in proving unconditionally the same formula as Kurihara on the Fitting ideal of Cl T,∨,− As an application of the description, we will obtain a necessary and sufficient condition for the Stickelberger element to be in the Fitting ideal of Cl T,− K (assuming eTNC).Note that the question for the dualized version Cl T,∨,− K is called the strong Brumer-Stark conjecture and is answered affirmatively by Dasgupta-Kakde [2] unconditionally.
Though we mainly assume the validity of eTNC in this paper, we also obtain interesting unconditional results.For instance, we will show that the Fitting ideal of Cl T,− K is always contained in that of Cl T,∨,− K , and that the inclusion is often proper.In the rest of this section, we give precise statements of the main results.
1.1.Description of the Fitting ideal.Let K/k be a finite abelian CM-extension and put G = Gal(K/k).Let S ∞ (k) be the set of archimedean places of k.Let S ram (K/k) be the set of places of k which are ramified in K/k, including S ∞ (k).For each finite prime v ∈ S ram (K/k), let I v ⊂ G denote the inertia group of v in G and ϕ v ∈ G/I v the arithmetic Frobenius of v.We then define elements g v and h v by where we put ν Iv = τ ∈Iv τ .These elements are introduced in [3, Lemmas 6.1 and 8.3] and [9, §2.2, equations ( 7) and (10)] (up to the involution).Moreover, we define a Z[G]-module , where j is the complex conjugation in G.For any Z[G]-module M, we also define the minus component by M − = M ⊗ Z[G] Z[G] − .Note that we are implicitly inverting the action of 2. For any x ∈ M, we write x − for the image of x under the natural map M → M − .
In general, for a set S of places of k, we write S K for the set of places of K which lie above places in S. We take and fix a finite set T of finite primes of k satisfying the following.
> 0 for all primes w ∈ T K } is torsion free.Here, ord w denotes the normalized additive valuation.Note that, if we fix an odd prime number p and are concerned with the p-components, the last condition can be weakened to that K × T is p-torsion-free.We consider the T -ray ideal class group of K defined by where w runs over the finite primes of K which are not in T K .For a character ψ of G, we write L(s, ψ) for the primitive L-function for ψ; this function omits exactly the Euler factors of primes dividing the conductor of ψ.For any finite prime v of k, we put N(v) = #F v , where F v is the residue field of v.We then define the T -modified L-function by where ψ runs over the characters of G and e ψ = 1 #G σ∈G ψ(σ)σ −1 is the idempotent of the ψ-component.
Now the first main theorem of this paper is the following, whose proof will be given in §3.
Theorem 1.1.Assume that eTNC for K/k holds.Then we have where Fitt In the second main result below, we will obtain a concrete description of , which completes the description of the Fitting ideal of Cl T,− K .We do not review the precise statement of eTNC (see e.g.[1,Conjecture 3.6]).
In order to compare with Theorem 1.1, we recall the result for the dualized version: As already mentioned, Kurihara [9,Corollary 3.7] showed this formula under the validity of the eTNC, and recently Dasgupta-Kakde [2, Theorem 1.4] removed the assumption.Here, for a general G-module M, we equip the Pontryagin dual M ∨ with the G-action by (σf )(x) = f (σx) for σ ∈ G, f ∈ M ∨ , and x ∈ M.This convention is the opposite of [9] and [2], so the right hand side of the formula (1.2) differs from those by the involution.We now briefly outline the proof of Theorem 1.1.An important ingredient is an exact sequence where A − is free of finite rank #S ′ .Here, S ′ is an auxiliary finite set of places of k.This sequence was constructed by Kurihara [9], based on preceding work such as Ritter-Weiss [12] and Greither [3], and played a key role in proving (1.2) under eTNC.Our novel idea is to construct an explicit injective homomorphism from Moreover, assuming eTNC, we will compute the determinant of the composite map A − ֒→ W − S∞ ֒→ (Z[G] − ) ⊕#S ′ .By using these observations, we obtain an exact sequence to which the theory of shifts of Fitting ideals can be applied, and then Theorem 1.1 follows.
1.2.Computation of the shift of Fitting ideal.In order to make the formula of Theorem 1.1 more explicit, in §4, we will compute Fitt [1] Z[G] (A v ).This will be accomplished by using a similar method as Greither-Kurihara [5, §1.2], which was actually a motivation for introducing the shifts of Fitting ideals in [6].
As the problem is purely algebraic, we deal with a general situation as follows (it should be clear from the notation how to apply the results below to the arithmetic situation; simply add subscripts v appropriately).Let G be a finite abelian group.Let I and D be subgroups of G such that I ⊂ D ⊂ G and that the quotient D/I is a cyclic group.We choose a generator ϕ of D/I and put which are non-zero-divisors.We define a finite Z[G]-module A by In order to state the result, we introduce some notations.We choose a decomposition (1.3) as an abelian group such that I l is a cyclic group for each 1 ≤ l ≤ s.Here, we do not assume any minimality on this decomposition, so we allow even the extreme case where I l is trivial for some l.
For each 1 ≤ l ≤ s, we put We also put (1).We then define an ideal J of Z[G] by Note that the definition of Z i does depend on the choice of the decomposition (1.3).On the other hand, it can be shown directly that the ideal J is independent from the choice.We omit the direct proof because, at any rate, the independency can be deduced from the discussion in §4.
When s = 2, we have In this setting, we can describe Fitt [1] Z[G] (A) as follows.It is convenient to state the result after multiplying by h.Theorem 1.4.We have 1.3.Stickelberger element and Fitting ideal.As an application of Theorems 1.1 and 1.4, we shall discuss the problem whether or not the Stickelberger element lies in the Fitting ideal of Cl T,− K .We return to the setup in §1.1.Let p be a fixed odd prime number and we shall work over Z p .Let G ′ denote the maximal subgroup of G of order prime to p.We put k p = K G ′ , which is the maximal p-extension of k contained in K.For each character χ of G ′ , we regard From now on, we fix an odd character χ of G ′ .We define K χ = K Ker(χ) .Then K χ is a CM-field, K χ ⊃ k p , and K χ /k p is a cyclic extension of order prime to p.
We put S χ = S ram (K χ /k) and consider the χ-component of the Stickelberger element defined by where ψ runs over characters of G whose restriction to G ′ coincides with χ and we write Note that, comparing (1.1) and (1.4), we have Concerning the dualized version, by Dasgupta-Kakde [2, Theorem 1.3], ) is always true.This is called the strong Brumer-Stark conjecture.More precisely, the displayed claim is a bit stronger than [2, Theorem 1.3] as we took S χ instead of S ram (K/k) in the definition of the Stickelberger element, but in any case it is an immediate consequence of the formula (1.2).
On the other hand, the corresponding claim without dual is known to be false in general (see [4]).However, we had only partial results and an exact condition was unknown.The following theorem is strong as it gives a necessary and sufficient condition.
Theorem 1.5.Assume that eTNC for K/k holds.Then, for each odd character χ of G ′ , the following are equivalent.
(i) We have This theorem will be proved in §5 as an application of Theorems 1.1 and 1.4.Note that there is an elementary equivalent condition for θ χ K/k,T = 0 as in Lemma 5.3.Theorem 1.5 indicates that the failure of the inertia groups to be cyclic is an obstruction for studying the Fitting ideal of the class group without dual.The same phenomenon will appear again in Theorem 1.6 below.We should say that this kind of phenomenon had been observed in preceding work, such as Greither-Kurihara [4].It is also remarkable that the obstruction does not occur in the absolutely abelian case (i.e. when k = Q), since in that case the inertia groups are automatically cyclic, apart from the 2-parts.This seems to fit the fact that Kurihara and Miura [7], [10] succeeded in studying the class groups without dual in the absolutely abelian case.
Let us outline the proof of Theorem 1.5.We assume that χ is a faithful character of G ′ (i.e.K χ = K); actually we can deduce the general case from this case.Since , where on both sides v runs over the elements of Obviously we may assume that θ χ K/k,T = 0.The proof of (ii) ⇒ (i) is the easier part.We will show that, under the assumption (ii), the inclusion of (1.6) holds even for every v before taking the product.On the other hand, the opposite direction (i) ⇒ (ii) is the harder part.That is because, roughly speaking, we have to work over the ring Z p [G] χ , whose ring theoretic properties are not very nice.A key idea to overcome this issue is to reduce to a computation in a discrete valuation ring.More concretely, we make use of a character ψ of G which satisfies ψ| G ′ = χ and a certain additional condition, whose existence is verified by Lemma 5.3, and we consider the . By investigating the ideals in (1.6) after base change from Z p [G] χ to O ψ , we will show (i) ⇒ (ii).
1.4.Unconditional consequences.Even if we do not assume the validity of eTNC, our argument shows the following.
Theorem 1.6.We have an inclusion This theorem follows immediately from Corollaries 3.6 and 4.2.Furthermore, by similar arguments as the proof of Theorem 1.5, we can observe that the inclusion is often proper.
As already remarked, Dasgupta-Kakde [2] proved the formula (1.2) unconditionally.Therefore, if I v is cyclic for every v ∈ S ram (K/k) \ S ∞ (k), we can also deduce from Theorem 1.6 that Fitt Z[G] − Cl T,− K also coincides with that ideal, and this removes the assumption on eTNC in Theorem 1.1.However, in Theorem 1.1 we still need to assume eTNC when I v is not cyclic for some v.

Definition of Fitting ideals and their shifts
In this section, we fix our notations concerning Fitting ideals.
(i) Let A be a matrix over R with m rows and n columns.For each integer 0 ≤ i ≤ n, we define Fitt i,R (A) as the ideal of R generated by the (n − i) × (n − i) minors of A.
For each integer i > n, we also define Fitt i,R (A) = ( 1).(ii) Let X be a finitely generated R-module.We choose a finite presentation A of X with m rows and n columns, that is, an exact sequence Here and henceforth, as a convention, we deal with row vectors, so we multiply matrices from the right.Then, for each i ≥ 0, we define the i-th Fitting ideal of X by Fitt i,R (X) = Fitt i,R (A).It is known that this ideal does not depend on the choice of A. When i = 0, we also write Fitt R (X) = Fitt 0,R (X) and call it the initial Fitting ideal.
We will later make use of the following elementary lemma.We omit the proof (cf.Kurihara [8,Lemma 3.3]).Lemma 2.2.Let X be a finitely generated R-module and I be an ideal of R. If X is generated by n elements over R, then 2.2.Shifts of Fitting ideals.In this subsection, we review the definition of shifts of Fitting ideals introduced by the second author [6].
Although we can deal with a more general situation, for simplicity we consider the following.Let Λ be a Dedekind domain (e.g.Λ = Z, Z[1/2], or Z p ).Let ∆ be a finite abelian group and consider the ring We define C as the category of R-modules of finite length.We also define a subcategory P of C by P = {P ∈ C | pd R (P ) ≤ 1}, where pd R denotes the projective dimension over R. Note that any module M in C satisfies pd Λ (M) ≤ 1.
Definition 2.3.Let X be an R-module in C and d ≥ 0 an integer.We take an exact sequence The well-definedness (i.e. the independence from the choice of the n-step resolution) is proved in [6, Theorem 2.6 and Proposition 2.7].
We also introduce a variant for the case where d is negative.Definition 2.4.Let X be an R-module in C and d ≤ 0 an integer.We take an exact sequence The well-definedness is proved in [6, Theorem 3.19 and Propositions 2.7 and 3.17].

Fitting ideals of ideal class groups
In this section, we prove Theorem 1.1, which describes the Fitting ideal of Cl T,− K using shifts of Fitting ideals.We keep the notation in §1.1.

3.1.
Brief review of work of Kurihara.We first review necessary ingredients from Kurihara [9], which in turn relies on preceding work, in particular Ritter-Weiss [12] and Greither [3].
For each place w of K, let D w and I w denote the decomposition subgroup and the inertia subgroup of w in G, respectively.These subgroups depend only on the place of k which lies below w.
Let us introduce local modules W v .For any finite group H, we define ∆H as the augmentation ideal in Z[H].Definition 3.1.For each finite prime w of K, we define a Z[D w ]-module W Kw by where w runs over the finite primes of K which lie above v.Alternatively, W v can be regarded as the induced module of W Kw from D w to G, as long as we choose a place w of K above v.
We take an auxiliary finite set S ′ of places of k satisfying the following conditions.
By using local and global class field theory, Kurihara constructed an exact sequence of the following form.Proposition 3.2 (Kurihara [9, §2.2, sequence (5)]).We have an exact sequence In [9], the author took the linear dual of this sequence, and the resulting sequence played an important role to study Cl T,∨,− K .In this paper, we do not take the linear dual but instead study the sequence itself for the proof of Theorem 1.1.

3.2.
Definition of f v .Our key ingredient for the proof of Theorem 1.1 is the following homomorphism f v .Definition 3.3.For a finite prime w of K, we define a Z[D w ]-homomorphism where the last isomorphism depends on a choice of w.

In §1.1 we introduced a finite Z[G]-module
It is actually motivated by the following.Lemma 3.4.For any finite prime v of k, the map f v is injective and where the lower sequence is the trivial one, the commutativity of the left square is easy, and the right vertical arrow is the induced one.By the definition of W Kw , we have Then by the diagram f w also satisfies the desired properties.

For any
− by choosing w, so we fix this isomorphism and write f − v for it.Using these f − v , we consider the following commutative diagram where the upper sequence is that in Proposition 3.2 and the map ψ is defined by the commutativity.By Lemma 3.4 and the snake lemma, we get the following proposition.Proposition 3.5.We have an exact sequence

By Proposition 3.5, the Fitting ideal Fitt
− generated by a non-zero-divisor.Then we can describe the Fitting ideals of Cl T,− K and of Cl T,∨,− K as follows.
Corollary 3.6.We have Proof.The first formula follows directly from Proposition 3.5 and Definition 2.3.For the second formula, by [6, Proposition 4.7], we have By Proposition 3.5 and Definition 2.4, we also have This completes the proof.

3.3.
Fitting ideal of Cok ψ.Recall the definitions of ω T and of h v in §1.1.
Theorem 3.7.Assume that eTNC for K/k holds.Then we have ) as in [9, §2.2, equation ( 9)] (we do not recall the precise definition here).Then we can see that its dual basis e ′ v of W v ⊗ Q is given by where ϕ v is a lift of ϕ v .Then, by the definition of f v , this element satisfies f v (e ′ v ) = 1, where by abuse of notation f v denotes the homomorphism Therefore, the determinant of the composite map ψ of Ψ and v∈S ′ f − v , with respect to the basis of A − and the standard basis of (Z[G] − ) ⊕#S ′ , also coincides with T .This shows the theorem.
We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1.By Corollary 3.6 and Theorem 3.7, we have Then Theorem 1.1 follows.
Remark 3.8.Similarly, under the validity of eTNC, Corollary 3.6 and Theorem 3.7 also imply a formula Combining this with Proposition 4.1 below, we can recover the formula (1.2).This argument may be regarded as a reinterpretation of the work [9] by using the shifts of Fitting ideals.

Computation of shifts of Fitting ideals
In this section, we prove Theorem 1.4 on the description of Fitt [1] Z[G] (A).We keep the notations as in §1.2.

Computation of
We choose a lift ϕ ∈ D of ϕ and put which is again a non-zero-divisor.Obviously, g is then the natural image of g to Z[G/I].
Proposition 4.1.We have Therefore, we also have Proof.We have an exact sequence Since multiplication by g is injective on each of these modules, applying the snake lemma, we obtain an exact sequence By Definition 2.4, we then have This proves the former formula of the proposition.
Since we have ν I g = ν I h, the former formula implies h Fitt Then the latter formula follows from h ≡ 1 − ν I #I ϕ −1 (mod(ν I )).Before proving Theorem 1.4, we show a corollary.

Corollary 4.2. We have an inclusion
Moreover, if I is a cyclic group, the inclusion is an equality.Proof.By Definition 1.2, the ideal J is contained in Z[G] and we have J = Z[G] if I is cyclic.Hence this corollary immediately follows from Theorem 1.4 and Proposition 4.1.

Computation of Fitt [1]
Z[G] (A).This subsection is devoted to the proof of Theorem 1.4.We fix the decomposition (1.3) of I.For each 1 ≤ l ≤ s, we choose a generator σ l of I l and put τ and τ l ν l = 0.As in §4.1, we put g = 1 − ϕ −1 + #I after choosing ϕ.
We recall ) and also put ).Then we have I I = (τ 1 , . . ., τ s ) and We begin with a proposition.
Proposition 4.3.We have Proof.We have the tautological exact sequence Since multiplication by g is injective on each of these modules, by applying snake lemma, we obtain an exact sequence Then Definition 2.3 implies (I I / gI I ).Since I I is generated by the s elements τ 1 , . . ., τ s , we have by Lemma 2.2.Thus we obtain the proposition.
Our next task is to determine Fitt i,Z[G] (I I ) for 0 ≤ i ≤ s.The result will be Proposition 4.9 below.For that purpose, we construct a concrete free resolution of Z over Z[I], using an idea of Greither-Kurihara [5, §1.2] (one may also refer to [6, §4.3]).
For each 1 ≤ l ≤ s, we have a homological complex Then the homology groups are H n (C l ) = 0 for n = 0 and H 0 (C l ) ≃ Z.
We define a complex C over Z[I] by which is the tensor product of complexes over Z (we do not specify the sign convention as it does not matter to us; we define it appropriately so that the descriptions of d 1 and d 2 below are valid).Explicitly, the degree n component C n of C is defined as Clearly the tensor product is zero unless n 1 , . . ., n s ≥ 0, and in that case . ., n s ≥ 0, following [5].Then, for each n ≥ 0, the module C n is a free module on the set of monomials of x 1 , . . ., x s of degree n.
A basic property of tensor products of complexes implies that H n (C) = 0 for n = 0 and It will be necessary to investigate some components of C of low degrees.Note that C 0 is free of rank one with a basis 1(= and C 2 is a free module on the set S 2 ∪ S ′ 2 where 2 are described as follows.We have Let M denote the presentation matrix of d 2 .For clarity, we define M formally as follows.
Definition 4.4.We define a matrix with the columns (resp.the rows) indexed by S 1 (resp.
for 1 ≤ l < l ′ ≤ s, and the other components are zero.
Here, we do not specify the orders of the sets S 1 and S 2 ∪ S ′ 2 .The ambiguity does not matter for our purpose.
For later use, we also define a matrix as the submatrix of M with the rows in S 2 removed.More precisely, we define the matrix N s (τ 1 , . . ., τ s ) with the columns (resp.rows) indexed by S 1 (resp.S ′ 2 ), by for 1 ≤ l < l ′ ≤ s, and the other components are zero.Therefore, by choosing appropriate orders of rows and columns, we have Example 4.5.When s = 3, we have Here, we use the order x 2 x 3 , x 1 x 3 , x 1 x 2 for the set S ′ (N s−j (τ a j+1 , . . ., τ as )).
Here, for each j, in the second summation a runs over subsets of {1, 2, . . ., s} of j elements, and for each a we define a 1 , . . ., a s by requiring The matrix N s−j (τ a j+1 , . . ., τ as ) is defined as in Definition 4.4 for s − j and τ a j+1 , . . ., τ as instead of s and τ 1 , . . ., τ s .
Proof.By the definition of higher Fitting ideals, Fitt For each H, we define j and a by (so clearly 0 ≤ j ≤ s − i) and A row H ∩ S 2 = {x 2 a 1 , . . ., x 2 a j }.Recall that the x 2 l row in the matrix M contains a unique non-zero component ν l in the x l column.Therefore, the assumption det(H) = 0 forces x a 1 , . . ., x a j ∈ A column where H ′ is the square submatrix of H of size (s − i) − j, with rows in A row Let N a denote the submatrix of N s (τ 1 , . . ., τ s ) obtained by removing the x a 1 , . . ., x a j columns.Then it is clear that det(H ′ )'s (for fixed j and a) as above generate Fitt i,Z[G] (N a ).The argument so far implies By the formula τ l ν l = 0, we may remove the components ±τ a 1 , . . ., ±τ a j from the matrix N a in the right hand side.It is easy to check that the resulting matrix is nothing but N s−j (τ a j+1 , . . ., τ as ) (with several zero rows added).This completes the proof.
We show the vanishing when s ≥ 1 and i = 0. Let R = Z[T 1 , . . ., T s ] be the polynomial ring over Z. Then we have a ring homomorphism f : R → Z[G] defined by sending T l to τ l .We define a matrix N s (T 1 , . . ., T s ) over R in the same way as in Definition 4.4, with τ • replaced by T • .Then, by the base change via f , we have Hence the left hand side would vanish if we show that Fitt R (N s (T 1 , . . ., T s )) = 0.
For each 1 ≤ l ≤ s, we consider the complex which satisfies H n ( C l ) = 0 for n = 0 and H 0 ( C l ) ≃ Z. Similarly as previous, by taking the tensor product of the complexes C l over Z, we obtain an exact sequence (Alternatively, this exact sequence is obtained from the Koszul complex for the regular sequence T 1 , . . ., T s .)This implies that Fitt R (N s (T 1 , . . ., T s )) is the Fitting ideal of the augmentation ideal of R. Since s ≥ 1, the augmentation ideal of R is generically of rank one, so we obtain Fitt R (N s (T 1 , . . ., T s )) = 0, as desired.
Finally we show the case where 1 ≤ i < s.Since the components of the matrix N s (τ 1 , . . ., τ s ) are either 0 or one of τ 1 , . . ., τ s , the inclusion ⊂ is clear.In order to show the other inclusion, we use the induction on s.
For a while we fix an arbitrary 1 ≤ l ≤ s.Then, by permuting the rows and columns, the matrix N s (τ 1 , . . ., τ s ) can be transformed into .
(The symbol (−) means omitting that term.)Here, the x l column is placed in the right-most, and the x 1 x l , . . ., x l−1 x l , x l x l+1 , . . ., x l x s rows are placed in the lower.We also reversed the signs of some rows for readability as that does not matter at all.This expression implies Proposition 4.9.For 0 ≤ i ≤ s, we define an ideal J i of Z[G] by (1 − ϕ −1 ) i−1 J i .

Then we have
More generally we actually show (1 − ϕ −1 ) i−1 J i by induction on s ′ , for each 0 ≤ s ′ ≤ s.The case s ′ = 0 is trivial.For 1 ≤ s ′ ≤ s, we have Here, the second equality follows from the induction hypothesis and expanding the power g s ′ −1 .By (4.2), for 1 ≤ i ≤ s ′ − 1, we have (#I) s ′ −i J s ′ ⊂ J i .Therefore, we obtain This completes the proof of (4.3).The right hand side of (4.3) can be computed as Here, the first equality follows from the definition of J i , the second by putting i + j = k, the third by I D = (I I , 1 − ϕ −1 ), and the final by the definition of J .Then, combining this with (4.3), we obtain the formula (4.1).This completes the proof of Theorem 1.4.

Stickelberger element and Fitting ideal
In this section, we prove Theorem 1.5.As explained after the statement, we need to compare the ideals in the both sides of (1.6) for each v before taking the product.That task will be done in §5.1, and after that we complete the proof of Theorem 1.5 in §5.2.
In this section we fix an odd prime number p and always work over Z p .
We now prove that (i) implies (ii).Suppose that both (i) and the negation of (ii) hold.Since θ χ K/k,T = 0, we may take a character ψ as in Lemma 5.3.By applying ψ to (1.6), we obtain On the other hand, by Lemmas 5.1 and 5.2(2), for each v ∈ S fin , we have ψ(A v ) ⊂ ψ(B v ).Moreover, the inclusion is proper if and only if both the conditions (a) and (b) in (ii) are false.Therefore, by the hypothesis that (ii) fails, we obtain Thus we get a contradiction.This completes the proof of Theorem 1.5.

[ 1 ]
Z[G]− is the first shift of the Fitting ideal (see Definition 2.3).
where x denotes the image of x in Z[D w /I w ].For each finite prime v of k, we define the Z[G]-module W v by taking the direct sum as

T
the sequence in Proposition 3.2.Then, under eTNC, Kurihara[9, Theorem 3.with respect to a certain basis of A − as a Z[G] − -module and the basis (e ′,− v ) v∈S ′ of W − S∞ .Actually this is an easy reformulation of the result of Kurihara, which concerns the determinant of the linear dual of Ψ.