On stronger versions of Brumer s conjecture

Transcription

1 On stronger versions of Brumer s conjecture Masato Kurihara Abstract. Let k be a totay rea number fied and L a CM-fied such that L/k is finite and abeian. In this paper, we study a stronger version of Brumer s conjecture that the Stickeberger eement times the annihiator of the group of roots of unity in L is in the Fitting idea of the idea cass group of L, and aso study the dua version. We mainy study the Teichmüer character component, and determine the Fitting idea in a certain case. We wi see that these stronger versions hod in a certain case. It is known that the stronger version (SB) does not hod in genera. We wi prove in this paper that the dua version (DSB) does not hod in genera, either. 0 Introduction 0.1. In number theory, it is very important to know the Gaois action on the idea cass group of a number fied. Concerning the Gaois action, an interesting phenomenon is the annihiation of the cass group by some anaytic eement whose origin is the zeta functions in some Gaois group ring. Let k be a totay rea number fied, and L be a CM-fied such that L/k is finite and abeian. We fix an odd prime number p and consider the p-component A L of the idea cass group C L of L. Put R L = Z p [Ga(L/k)], and regard A L as an R L -modue. Let μ p (L) be the group of roots of unity with order a power of p in L, and I L = Ann RL (μ p (L)) the annihiator idea of μ p (L) inr L. We denote by θ L/k Q[Ga(L/k)] the Stickeberger eement, which is defined by the vaues of partia zeta functions (see 2.1). Then by Deigne and Ribet [3] we know I L θ L/k R L. Brumer s conjecture caims that I L θ L/k Ann RL (A L ). We consider the foowing property (SB) which is stronger than this; (SB) I L θ L/k Fitt RL (A L ) 1

2 where Fitt RL (A L ) is the Fitting idea of A L (see 2.2). Since Fitt RL (A L ) Ann RL (A L ), (SB) is certainy stronger than Brumer s conjecture. The main resut in the paper [12] by Miura and the author impies that (SB) hods if k = Q. In the paper [7] by Greither and the author, we showed that (SB) does not hod in genera, and the dua version is more natura and ikey to hod. Let (A L ) be the Pontryagin dua of A L with cogredient Gaois action, namey (σf)(x) =f(σx) for σ Ga(L/k), f (A L ) and x A L. Then the dua version means (DSB) I L θ L/k Fitt RL ((A L ) ). Our expicit exampe in [7] 3.2 does not satisfy (SB), but satisfies (DSB). In [6], Greither proved a beautifu theorem that the equivariant Tamagawa number conjecture impies (DSB) if μ p (L) is cohomoogicay trivia. Concerning the exposition of Brumer s conjecture and the Fitting ideas, see Greither [4]. In this paper, we prove the existence of number fieds for which neither (DSB) nor (SB) hods (see Coroary 0.5). (Under the assumptions of Coroary 0.5, μ p (L) is not cohomoogicay trivia.) We study the Iwasawa theoretic version of (DSB). Let L be as above and L /L be the cycotomic Z p -extension. We define A L = im A Ln where L n is the intermediate fied such that [L n : L] =p n for n>0. Put Λ L = Z p [[Ga(L /k)]]. We study the Pontryagin dua (A L ) which is a finitey generated torsion Λ L -modue. We define I L = Ann ΛL (μ p (L )) Λ L. By Deigne and Ribet [3], there is a unique eement θ L /k in the tota quotient ring of Λ L, which is the projective imit of θ Ln/k (more precisey, see 2.1). We study the Iwasawa theoretic version (IDSB) I L θ L /k Fitt ΛL ((A L ) ) of (DSB). Theorem A.5 in [11] impies that (IDSB) hods outside the Teichmüer character component if we assume the Leopodt conjecture and the vanishing of the Iwasawa μ-invariant of L. In particuar, if μ p L where μ p is the group of p-th roots of unity, (IDSB) is true under the above assumptions. In this paper, we mainy study the case μ p L. We assume that L k = k where k is the cycotomic Z p -extension of k. We denote by Γ(L/k) the p-component of Ga(L/k), so Ga(L/k) = Γ(L/k) Δ for some abeian group Δ with p #Δ. Suppose at first that Γ(L/k) Z/pZ. If Γ(L/k) is cycic, by Theorem 3, Proposition 4 in Greither [5] and Coroary A. 13 in [7], we know 2

3 that (IDSB) hods, assuming the vanishing of the μ-invariant. Moreover, Fitt ΛL ((A L ) ) is determined outside the component of the Teichmüer character by Theorem 3 in Greither [5] and Coroary A. 13 in [7]. In this paper, we determine the Teichmüer character component of the Fitting idea in the case that Γ(L/k) Z/pZ. Suppose that μ p L. We denote by K the subfied of L such that Γ(L/k) = Ga(L/K), so [K : k] is prime to p. Let ω be the Teichmüer character giving the action of Ga(K/k) onμ p. Since [K : k] is prime to p, A L is decomposed into A L = χ Aχ L where χ runs over a equivaence casses of Q p -vaued characters of Ga(K/k) (see 1.2). In particuar, we know that determining Fitt ΛL ((A L ) ) is equivaent to determining Fitt Λ χ ((A χ L L ) ) for a characters χ of Ga(K/k) (for the definition of the χ-components A χ L,Λ χ L, see 1.2). For any odd character χ such that ((A χ L ) ) is determined (see (2.3.2)). We study the χ ω, we know Fitt Λ χ L ω-component. To do this, we may assume K = k(μ p ) (see 2.2). We take a generator γ of Ga(L /L) = Ga(K /K). We denote the cycotomic character by κ : Ga(L /k) Z p. The cycotomic character of Ga(K /k) is aso denoted by κ. Suppose that Ga(L /K ) is generated by σ. Then I L is generated by γ κ(γ) and σ 1. In the foowing, we suppose that L 0 /k is a finite abeian p-extension (so L 0 is aso totay rea) and L = L 0 (μ p ). We wi state our theorems without expaining a notations. Theorem 0.1 Suppose that L 0 /k is a cycic extension of degree p with L 0 k = k. We put L = L 0 (μ p ), and assume that the Iwasawa μ-invariant of L /L vanishes, namey μ((a L ) )=0. We denote by S the set of primes of k which are prime to p and which are ramified in L. (1) Suppose at first that S is not empty. Then we have Fitt Λ ω L ((A ω L ) )=( ( 1 γ κ(γ) (1,ν( 1 κ(ϕ S )ϕ 1 )))(1,ν( 1 κ(ϕ )ϕ 1 ))ϑ L /k) ω. S Here, the right hand side is defined in (2.3.8); ν is the map defined in 2.1, ϕ is the Frobenius of in Ga(K /k), and ϑ L /k is a modified Stickeberger eement of Greither, which is defined in (2.3.4), and which is described by using θ L /k and θ K /k (see (2.3.7)). In this case, we have I L θ L /k Fitt ΛL ((A L ) ). (2) Suppose that S = φ, namey L/k is unramified outside p. Then we have Fitt Λ ω L ((A ω L ) )=(I L θ L /k) ω. 3

4 We define the standard Iwasawa modue X L by X L = im A Ln where the imit is taken with respect to the norm maps. We consider the foowing property (ISB) I L θ L /k Fitt ΛL (X L ). In genera, (ISB) does not hod (see [7] Theorem 1.1 and aso 3.3). Suppose that L/k is as in Theorem 0.1. Then we encounter a phenomenon that (IDSB) hods, but (ISB) does not. We wi expain in 3.3 that X L is not isomorphic to (A L ) as a Z p [Γ(L/k)]-modue, in genera (see (3.3.1) and (3.3.2)) We summarize here some affirmative resuts for (SB) and (DSB). As we mentioned above, if we assume the μ-invariant of L /L vanishes and Γ(L/k) is cycic, (IDSB) hods by Theorem 3, Proposition 4 in Greither [5] and Coroary A. 13 in [7]. By the standard descent technique, we obtain the foowing resuts. Coroary 0.2 Suppose that L/k is a finite abeian extension such that Γ(L/k) is cycic and L K = K. We assume that the Iwasawa μ-invariant of L /L is zero. (1) Suppose that n is sufficienty arge such that a the primes above p of k are ramified in the n-th ayer L n of the cycotomic Z p -extension L /L. Then (DSB) hods for L n /k. (2) For any prime p of k above p, we assume at east one of the foowing; (i) no prime above p spits in L/L +,or (ii) p is ramified in L. Then both (DB) and (DSB) hod for L/k. This coroary wi be proved in Next, we consider the case that Γ(L/k) is not cycic, and wi obtain negative resuts for (IDSB) and (DSB). As in Theorem 0.1, we study the ω-component. Theorem 0.3 Suppose that L 0 /k is a finite abeian p-extension such that L 0 k = k and Ga(L 0 /k) is not cycic. We put L = L 0 (μ p ), and assume that L/k is unramified outside p. Then we have (γ κ(γ))θ L /k Fitt ΛL ((A L ) ). In particuar, (IDSB) does not hod, namey I L θ L /k Fitt ΛL ((A L ) ). 4

5 Remark 0.4 (1) There are many exampes of (k,l) satisfying the conditions of Theorem 0.3. For exampe, if dim Fp A k /pa k 2, there is an unramified extension L 0 /k with non-cycic Gaois group of order a power of p, sol = L 0 (μ p ) satisfies the condition. (We give an expicit exampe in Exampe 0.6 such that L 0 /k is ramified.) (2) In the setting of Theorems 0.1 and 0.3, μ p (L) is not cohomoogicay trivia because Γ(L/k) = Ga(L 0 /k) acts on μ p (L) triviay, which impies Ĥ 0 (Γ(L/k),μ p (L)) 0. (3) If k = Q, there is no L 0 as in Theorem 0.1 (2) or Theorem 0.3. In fact, if L 0 Q = Q and L 0 /Q is a finite abeian p-extension which is unramified outside p, we have L 0 = Q. Coroary 0.5 (1) Let L 0 /k be a finite abeian p-extension which is unramified outside p such that L 0 k = k and Ga(L 0 /k) is not cycic. Then, for sufficienty arge n, (DSB) does not hod for L 0 (μ p n)/k. (2) Suppose that L 0 is as above. We denote by P p,k the set of primes of k above p, and assume that #{P P p,k P spits competey in k(μ p ) and ramified in L 0 (μ p )} 2. Then, for sufficienty arge n, neither (SB) nor (DSB) hods for L 0 (μ p n)/k. We wi prove Theorems 0.1, 0.3 and Coroary 0.5 in 3. Exampe 0.6 Let F 1 be the minima spitting fied of x 3 39x 16 = 0. We know 79 F 1 and F 1 /Q( 79) is unramified everywhere. Let F 2 be the minima spitting fied of x 3 6x 3 = 0. We know 69 F 2 and F 2 /Q( 69) is unramified outside 3 and ramified at the prime above 3. Let F 2, /F 2 be the cycotomic Z 3 -extension. Since the cass number of Q( 69) is 1, the prime above 3 is totay ramified in F 2, /Q( 69). We put k = Q( 69, 79) and take p = 3. There are two primes p 1, p 2 of k above 3, and both of them are totay ramified in k. We denote by P 1 (resp. P 2 ) the prime of k above p 1 (resp. p 2 ). Since p 1 and p 2 spit in k(μ 3 )=k( 3) = k( 23), P 1 and P 2 spit in k (μ 3 ). Put L 0 = F 1 F 2. So Ga(L 0 /k) Z/3Z Z/3Z. Since p 1 and p 2 are totay ramified in F 2, k/k, P 1 and P 2 are totay ramified in F 2, k/k. Thus, L 0 /k satisfies a the conditions of Coroary 0.5. Therefore, neither (SB) nor (DSB) hods for L = L 0 (μ p n) with n 0. We can construct many exampes in this way. The author woud ike to thank heartiy C. Greither for the discussion on the subjects in this paper, and for his usefu comments by which the author 5

6 coud improve Theorem 0.3. (In the first draft, ony the case Γ(L/k) = Z/pZ Z/pZ was studied in Theorem 0.3.) Notation Throughout this paper, we fix an odd prime number p. For any number fied F, we denote by A F the p-component of the idea cass group of F. The cycotomic Z p -extension of F is denoted by F, and we define A F = im A Fn where F n is the n-th ayer of F /F for n>0. We denote by P F the set of a finite primes of F, and by P p,f the subset of P F consisting of primes above p. We define P F = P F \ P p,f. For a group G and a G-modue M, we denote by M G the G-invariant part of M (the maxima subgroup of M on which G acts triviay), and by M G the G-coinvariant of M (the maxima quotient of M on which G acts triviay). 1 Computation of some Tate cohomoogy groups 1.1. In this section we suppose that k is a totay rea base fied, and K and L are CM-fieds such that k K L and that L/k is a finite abeian extension. We assume that L/K is a p-extension. Put G = Ga(L/K). In this section we study the p-component A L of the idea cass group of L, especiay the minus part A L on which the compex conjugation acts as 1. Throughout this paper, for any modue M on which the compex conjugation ρ acts, we define the minus part M by M = {x M ρ(x) = x}. We compute the Tate cohomoogy groups Ĥ q (G, A L )=Ĥq (G, A L ) (cf. Serre [15] Chap. 8). By Lemma 5.1 (2) in [10], we have an exact sequence Ĥ 0 (G, E L ) Ĥ0 (G, E Lw ) Ĥ 1 (G, A L ) w P L H 1 (G, E L ) H 1 (G, E Lw ) Ĥ0 (G, A L ) w P L H 2 (G, E L ) H 2 (G, w P L E Lw ) where E L (resp. E Lw ) is the unit group of L (resp. of the oca fied L w ), and P L is the set of a finite primes of L as in Notation. For each prime v P K, we denote by G v the decomposition subgroup of G at v. We know Ĥ q (G, w P L E Lw ) = v P K Ĥ q (G v,e Lw ) where in the right hand side we fix a prime w of L above v for v P K (note that Ĥq (G v,e Lw )=0 if v is unramified in L). More concretey, we have Ĥ0 (G v,e Lw )=I v by oca cass fied theory where I v is the inertia subgroup of G at v. Since 6

7 H 1 (G v,l w) = 0, from the exact sequence 0 E Lw L w Z 0 we have H 1 (G v,e Lw )=Z/e v Z where e v is the ramification index of v in L/K, and H 2 (G v,e Lw ) Br(K v )[p ] where Br(K v )[p ] is the subgroup of the Brauer group of K v consisting of eements with order a power of p. Consider the cycotomic Z p -extensions K /K, L /L, and the n-th ayers K n, L n. We assume that L K = K. By the above descriptions, we have im Ĥ 0 (G v,e Ln,w ) = im H 2 (G v,e Ln,w )=0. v P Kn v P Kn Suppose that w n+1 is a prime of L n+1 and w n, v n+1, v n are the primes of L n, K n+1, K n beow w n+1, respectivey. Then the natura map H 1 (G vn,e Lwn )= Z/e vn Z H 1 (G vn+1,e Lwn+1 )=Z/e vn+1 Z is the mutipication by e wn+1,n which is the ramification index of w n in L n+1 /L n. Each prime above p is totay ramified in L /L n for sufficienty arge n, and a prime v which is not above p is unramified in L /L. It foows that im H 1 (G v,e Ln,w )= Z/e v Z v P Kn v P K where P K denotes the set of a finite primes of K which are prime to p (as in Notation), and e v is the ramification index of v in L /K. We define A L = im A Ln as in Notation, and E L = im E Ln. Taking the direct imit of the above exact sequence, we obtain the foowing emma. Lemma 1.1 We have an exact sequence 0 Ĥ 1 (G, A L ) H 1 (G, E L ) ( Z/e v Z) v P K Ĥ0 (G, A L ) H 2 (G, E L ) Now, we assume that [K : k] is prime to p. Hence Γ(L/k) = G in the terminoogy of 0. The group ring Z p [Ga(K/k)] is a product of discrete vauation rings. More expicity, it is described as foows. For two Q p -vaued characters χ 1 and χ 2 of Ga(K/k), we say χ 1 and χ 2 are Q p -conjugate if σχ 1 = χ 2 for some σ Ga(Q p /Q p ). For a Q p -vaued character χ of Ga(K/k), we put O χ = Z p [Image χ]. We regard O χ as a Z p [Ga(K/k)] modue by σ x = χ(σ)x for any σ Ga(K/k). We have (1.2.1) Z p [Ga(K/k)] = χ O χ where the sum is taken over the equivaence casses of Q p -vaued characters of Ga(K/k) (we choose a character χ from each equivaence cass). 7

8 For any Z p [Ga(K/k)]-modue M, we define M χ = M Zp[Ga(K/k)] O χ. Note that M χ is a direct summand of M because (1.2.1) impies M = χ M χ. We put Λ K = Z p [[Ga(K /k)]] and Λ L = Z p [[Ga(L /k)]]. Since Ga(K/k) is a direct summand of Ga(L /k), any Λ L -modue M is naturay a Z p [Ga(K/k)]-modue and M χ is defined. For such M, M χ can be written as M χ = M Zp[Ga(K/k)] O χ = M ΛL O χ [[Ga(L /K)]]. In the same way, for a Λ K -modue M, M χ = M Zp[Ga(K/k)] O χ = M ΛK O χ [[Ga(K /K)]]. When K contains a primitive p-th root of unity, we denote by ω the Teichmüer character which gives the action on μ p. Proposition 1.2 (1) Suppose that χ is an odd character such that χ ω. Then we have Ĥ 1 (G, A χ L )=0 and Ĥ 0 (G, A χ L )=( Z/e v Z) χ. v P K (2) Suppose that K = k(μ p ) and [L : K] =p. LetP k be the set of a finite primes of k which are prime to p as in Notation, and put If S is not empty, we have S = { P k is ramified in L/k}. Ĥ 1 (G, A ω L )=0 and Ĥ 0 (G, A ω L ) = Coker(μ p L S k μ p ) where μ p is the group of p-th roots of unity, the map is the diagona map, and S k is the set of primes of k which are above S. (3) Next, we assume K = k(μ p ) and L/K is a (genera abeian) p-extension which is unramified outside p (namey, the set S defined above is empty). Then we have 2 Ĥ 1 (G, A ω L ) G (1) and Ĥ 0 (G, A ω L ) ( G) (1) where G, ( 2 G) are the Pontryagin duas of G, 2 G (the second exterior power of G), and (1) is the Tate twist. In particuar, Ĥ 0 (G, A ω L )=0if and ony if G is cycic. Proof. First of a, since Ga(K/k) acts on G triviay, we have Ĥq (G, M) χ = Ĥ q (G, M χ ) for any χ and any Λ L -modue M. (1) Since χ ω, we know (E L ) χ = 0. This impies that H q (G, E L ) χ = H q (G, (E L ) χ ) = 0. Therefore, Lemma 1.1 impies the concusion of (1). 8

9 (2) Since μ p K, we know (E L ) ω = μ p = Q p /Z p (1). Therefore, using G = Z/pZ, we have H 1 (G, E L ) ω = H 1 (G, Q p /Z p (1)) = μ p and H 2 (G, E L ) ω = H 2 (G, Q p /Z p )(1) = Ĥ0 (G, Q p /Z p )(1) = 0. Suppose that is in S. Since is unramified in K = k(μ p ), S impies that the inertia group of in Ga(L/k) is of order divisibe by p. Hence N() 1 (mod p) where N() is the norm of. This impies that spits competey in K = k(μ p ). Let L be a prime of k above, and et v be a prime of K above L. Then v is ramified in L and [L : K ]=p, sov is totay ramified in L. Hence G = G v (where G v is the decomposition group of G at v). Since spits competey in K, L spits competey in K and we have ( H 1 (G v, Q p /Z p (1))) ω =( μ p ) ω = μ p. v L v L It foows that ( v P K H 1 (G v,e L,w )) ω = L S k μ p. Therefore, the natura map H 1 (G, E L ) ω ( v P K H 1 (G v,e L,w )) ω is the diagona map μ p L S k μ p. In particuar, it is injective because S φ. Thus, by Lemma 1.1 we obtain Ĥ 1 (G, A ω L ) = 0, and Ĥ 0 (G, A ω L ) = Coker(μ p L S k μ p ). (3) In this case, since v P K Z/e v Z = 0, using Lemma 1.1, we have and Ĥ 1 (G, A ω L )=H 1 (G, E L ) ω = H 1 (G, Q p /Z p (1)) = G (1) Ĥ 0 (G, A ω L )=H 2 (G, E L ) ω = H 2 (G, Q p /Z p )(1). Therefore, the next emma impies the concusion. This competes the proof of Proposition 1.2. Lemma 1.3 Suppose that G is an abeian p-group. Then H 2 (G, Q p /Z p ) is isomorphic to Hom( 2 G, Q p /Z p ). In particuar, if G is not cycic, we have H 2 (G, Q p /Z p ) 0. Proof. In fact, by the duaity Proposition 7.1 in Chap. VI in [1] (or the universa coefficient sequence (cf. page 60 in Chap. III in [1])), we have H 2 (G, Q p /Z p ) Hom(H 2 (G, Z), Q p /Z p ). Since G is abeian, we know H 2 (G, Z) = 2 G (Theorem 6.4 (iii) in Chap. V in [1]). This competes the proof of Lemma

10 2 Stickeberger eements and Fitting ideas 2.1. Let k be a totay rea number fied and L be a CM-fied such that L/k is finite and abeian. The Stickeberger eement θ L/k is defined by θ L/k = ζ(0,σ)σ 1 Q[Ga(L/k)] σ Ga(L/k) where ζ(s, σ) = ( L/k )=σ N(a) s is the partia zeta function (a runs over a integra ideas which are prime to the discriminant of L/k). Let L /L be the cycotomic Z p -extension and Λ L = Z p [[Ga(L /k)]]. We denote by κ : Ga(L /k) Z p the cycotomic character. By Deigne and Ribet ([3]), we know that there is a unique eement θ L /k in the tota quotient ring of Λ L satisfying the foowing property. For any σ Ga(L /k), (σ κ(σ))θ L /k is in Λ L and is a projective imit of (σ κ(σ))θ Ln/k Z p [Ga(L n /k)] for n 0. We denote by (2.1.1) κ :Λ L Λ L the ring homomorphism induced by κ(σ) =κ(σ)σ for a σ Ga(L /k). Ceary, κ is bijective. We extend κ to the tota quotient ring of Λ L. Then κ(θ L /k) is a pseudo-measure in the sense of Serre ([16]), and is the p-adic L-function of Deigne and Ribet. Suppose that K is the intermediate fied of L/k such that L/K is a p-extension and [K : k] is prime to p. Put Λ K = Z p [[Ga(K /k)]] and G = Ga(L/K). We assume that L k = k. Then we have Λ L =Λ K [G]. We regard Λ L as a Λ K -modue by this identification. We wi use two maps c and ν. The ring homomorphism c :Λ L Λ K is defined by the restriction σ σ K for σ Ga(L /k). The Λ K - homomorphism ν :Λ K Λ L is defined by σ Στ, where for σ Ga(K /k), τ runs over a eements in Ga(L /k) such that c(τ) =σ. We have (2.1.2) ν(x)y = ν(xc(y)) for a x Λ K and y Λ L. This impies (2.1.3) ν(x)ν(y) =ν(xc(ν(y))) = [L : K]ν(xy) for a x, y Λ K. Let Q(Λ L ) (resp. Q(Λ K )) be the tota quotient ring of Λ L (resp. Λ K ). We naturay extend c to the ring homomorphism c : Q(Λ L ) Q(Λ K ). 10

11 We can extend ν to the Λ K -homomorphism ν : Q(Λ K ) Q(Λ L ) such that ν(c(x))=σ σ G σx for x Q(Λ L ). We set We have S = { P k is unramified in K, and is ramified in L}. (2.1.4) c(θ L /k) =( S (1 ϕ 1 ))θ K /k where ϕ =( K ) is the Frobenius of /k in Ga(K /k) (Lemma 2.1 in [10]) For a commutative ring R and a finitey presented R-modue M such that R m f R n M 0 is exact, the Fitting idea of M is defined to be the idea of R generated by a n n minors of the matrix A f which corresponds to f. This idea does not depend on the choice of the exact sequence. We denote it by Fitt R (M). We obtain Fitt R (M) Ann R (M) from the definition. We consider X L = (A L ) and the minus part X L. As we mentioned in 1.2, we have decomposition X L = χ X χ L where χ runs over the equivaence casses of Q p -vaued characters of Ga(K/k). From X L = χ:odd X χ L, knowing Fitt ΛL (X L ) is equivaent to knowing Fitt Λ χ (X χ L L ) for a odd characters χ. We regard Ker χ Ga(K/k) as a subgroup of Ga(L/k) and denote by L χ (resp. K χ ) the subfied of L (resp. K) such that Ga(L/L χ ) = Ker χ (resp. Ga(K/K χ ) = Ker χ). Since [L : L χ, ] Ga(L is prime to p, A Lχ, A /L χ, ) L is an isomorphism. Therefore, X χ L X χ L χ, is bijective. Since we ceary have Λ χ L =Λ χ L χ,, we obtain Fitt Λ χ (X χ L L ) = Fitt Λ χ (X χ L Lχ, χ, ). So we may assume K = K χ when we study the χ-component. In particuar, we may assume that the conductor of K/k is the same as that of χ for the computation of the Fitting idea of X χ L In this subsection, we further assume that [L : K] =p. Wefixanodd character χ of Ga(K/k), and assume that the conductor of χ is equa to the conductor of K/k. 1) Suppose that χ ω. We extend χ to the ring homomorphism Q(Λ L ) Q(Λ χ L ) and the image of x Q(Λ L ) is denoted by x χ Q(Λ χ L ). We know θ χ K /k Λχ K and θ χ L /k Λχ L by Deigne and Ribet. Let S be as in 11

12 2.1. Foowing the idea of Greither [5] (cf. Theorem 7 in [5]), we consider 1 a fractiona idea (1,ν( )) of Λ L for S, and define 1 ϕ 1 (2.3.1) Θ=( 1 1 ϕ S(1,ν( 1 )))θ L /k which is a fractiona idea of Λ L. Consider the χ-component Θ χ. By (2.1.2), (2.1.3), and (2.1.4), we obtain Θ χ Λ χ L,soΘ χ is an idea of Λ χ L. By Theorem 3 in Greither [5] and Coroary A. 13 in [7], we have (2.3.2) Fitt Λ χ (X χ L L ) = Fitt Λ χ ((A χ L L ) )=Θ χ. We wi give another proof of (2.3.2) by the same method as the proof of Theorem 0.1 in Remark ) Next, we suppose that χ = ω and there is a prime P k which is ramified in L/K. We assume K = k(μ p ) (we may assume this as we expained in 2.2). Let S be as in 2.1. Note that S is not empty by our assumption. Foowing Greither [5] (cf. page 753 in [5]), we introduce a modified Stickeberger eement ϑ L /k (which corresponds to Ψ S in [5] though our eement is sighty modified). We put (2.3.3) ξ = ν( 1 1 κ(ϕ )ϕ 1 p 1 ϕ 1 )+(1 ν( 1 p )), S which is an eement of the tota quotient ring of Λ L where ϕ is the Frobenius of in Ga(K /k). We define (2.3.4) ϑ L /k = ξθ L /k. Using the definition of ϑ L /k and (2.1.4), we have (2.3.5) c(ϑ L /k) =( S (1 κ(ϕ )ϕ 1 ))θ K /k. Lemma 2.1 (C. Greither) If S is not empty, we have (2.3.6) ϑ L /k Λ L. Proof. This corresponds to Proposition 9 in Greither [5]. We give here a proof by computing ϑ L /k directy. Using (2.1.2) and (2.1.4), we compute (2.3.7) ϑ L /k = ν( S (1 κ(ϕ )ϕ 1 ) S (1 ϕ 1 p 12 ) θ K /k)+θ L /k.

13 As we expained in the proof of Proposition 1.2 (2), S satisfies N(ϕ ) 1 (mod p), so κ(ϕ ) 1 (mod p). Therefore, S (1 κ(ϕ )ϕ 1 ) S (1 ϕ 1) Λ K. p Let γ be a generator of Ga(L /L). Since (γ κ(γ))θ K /k Λ K and (γ κ(γ))θ L /k Λ L, we obtain (γ κ(γ))ϑ L /k Λ L. We have to show (γ κ(γ))ϑ L /k (γ κ(γ))λ L. Let κ be the automorphism of Q(Λ L ) defined in (2.1.1), and π L : Λ L Z p (resp. π K :Λ K Z p ) be the augmentation map. Since κ(ϑ L /k) is a pseudo-measure in the sense of Serre [16], it suffices to prove π L ((γ 1) κ(ϑ L /k)) = 0 (see [16] 1.14). Using (2.3.5), we compute π L ((γ 1) κ(ϑ L /k)) = π K ((γ 1) κ(( S (1 κ(ϕ )ϕ 1 ))θ K /k)) = π K (( (1 ϕ 1 ))(γ 1) κ(θ K /k)) S = 0. Note that we used S φ to obtain the fina equation. This competes the proof of Lemma 2.1. Note that 1 κ(ϕ )ϕ 1 is divisibe by γ κ(γ) inλ K. We consider a fractiona idea (1,ν((γ κ(γ))/(1 κ(ϕ )ϕ 1 ))). We define (2.3.8) S = ( 1 γ κ(γ) (1,ν( 1 κ(ϕ S )ϕ 1 )))(1,ν( 1 κ(ϕ )ϕ 1 S By (2.1.2), (2.1.3), (2.3.5) and (2.3.6), we obtain that (2.3.9) S Λ L, ))ϑ L /k. namey, S is an idea of Λ L. We study the ω-component S ω Λ ω L. Our S ω coincides with the idea in Greither [5] Proposition 10. Lemma 2.2 Suppose that I L is the idea of Λ L defined in 0. We have (I L θ L /k) ω S ω. Proof. We put Λ = Z p [[Ga(k /k)]]. Then Λ L = Λ[Ga(L/k)] and Λ ω L Λ[Ga(L/K)]. Let ψ : Ga(L/K) μ p be a faithfu character (namey, a bijective homomorphism). This ψ induces a ring homomorphism Λ ω L = 13

14 Λ[Ga(L/K)] Λ[μ p ], which we aso denote by ψ. By (2.3.7) and (2.3.8), we obtain ψ(θ ω L /k )=ψ(ϑω L /k ) ψ(sω ). On the other hand, concerning (I L θ L /k) ω,ifσ is a generator of Ga(L/K), we have ψ((i L θ L /k) ω )=(ψ(σ) 1,γ κ(γ))ψ(θ ω L /k ). Since (ψ(σ) 1,γ κ(γ)) Λ[μ p ], we have ψ(θ ω L /k ) ψ((i L θ L /k) ω ). Therefore, ψ((i L θ L /k) ω ) ψ(s ω ), and we obtain the concusion. We wi prove Fitt Λ ω L (X ω L ) = Fitt Λ ω L ((A ω L ) )=S ω (Theorem 0.1 (1)) in the next section. 3) Finay, we suppose that χ = ω and L/K is unramified outside p. In other words, we suppose S = φ. We cannot define a good eement ϑ L /k in Λ L in this case. We wi use θ L /k. Let I L be the idea of Λ L defined in 0. By Deigne and Ribet, we know I L θ L /k Λ L. We consider (I L θ L /k) ω which is an idea of Λ ω L. What we wi prove in the next section is Fitt Λ ω L (X ω L ) = Fitt Λ ω L ((A ω L ) )=(I L θ L /k) ω. 3 Proof of Theorems 3.1. We go back to the genera situation, and suppose that L 0 /k is a finite abeian p-extension such that L 0 k = k. We put K = k(μ p ) and L = L 0 (μ p ) (we do not assume [L : K] =p). We study XL ω =(A ω L ). Let L 0, be the cycotomic Z p -extension of L 0, and et M L0, /L 0, (resp. M k /k ) be the maxima abeian pro-p extension of L 0, (resp. k ) which is unramified outside p. By Washington [17] Proposition 13.32, we have canonica isomorphisms XL ω =(A ω L ) Ga(M L0, /L 0, )(1) and XK ω = (A ω K ) Ga(M k /k )(1) (note that our action is cogredient). Using these isomorphisms, we obtain Lemma 3.1 Let L 0 /k be the maxima subextension of L 0/k which is unramified outside p. Put G = Ga(L/K) = Ga(L 0 /k). Then we have an exact sequence 0 Ĥ0 (G, A ω L ) (XL ω f ) G XK ω Ga(L 0 /k)(1) 0. Proof. The cokerne of the natura map Ga(M L0, /L 0, ) Ga(M k /k ) is Ga((L 0 ) /k ) = Ga(L 0 /k). Therefore, the cokerne of f is Ga(L 0 /k)(1). For n>0, we regard A Ln as the Gaois group of the maxima unramified abeian p-extension of L n, and A Kn simiary. Then the norm map is the 14

15 restriction map, so A ω L n A ω K n is surjective because Ga(K/k) acts on A ω L n via ω and acts on Ga(L n /K n ) triviay. This impies that the norm map A ω L A ω K is surjective. Therefore, N G A ω L coincides with the image of the natura map A ω K A ω L where N G =Σ σ G σ. This impies that A ω K (A ω L ) G Ĥ0 (G, A ω L ) 0 is exact. Taking the dua, we obtain the kerne of f We first prove Theorem 0.1. Suppose that [L : K] =[L 0 : k] =p, so G = Ga(L/K) is of order p. Put Λ = Z p [[Ga(k /k)]]. Then XL ω is a Λ ω L =Λ[G]-modue. Let ψ : G Q p be a faithfu character. We extend ψ to the ring homomorphism ψ :Λ ω L =Λ[G] Λ[μ p ] as in the proof of Lemma 2.2. For any Λ ω L -modue M, we define M ψ to be M Λ ω L Λ[μ p ] where Λ[μ p ] is regarded as a Λ ω L -modue via ψ. We have to prepare three more emmas. Lemma 3.2 Let M be a Λ[G]-modue such that M is a free Z p -modue of finite rank. We regard M ψ as a Z p [μ p ]-modue. If Ĥ 0 (G, M) =(Z/pZ) r, the maxima Z p [μ p ]-torsion submodue of M ψ is isomorphic to (Z/pZ) r. Proof. This is we-known. We know M is isomorphic to Z p [G] a Z p [μ p ] b Z c p as a Z p [G]-modue. Then Ĥ0 (G, M) =(Z/pZ) r impies c = r. We know M ψ = M Λ[G] Λ[μ p ]=M Zp[G] Z p [μ p ] Z p [μ p ] (a+b) (Z/pZ) c. Therefore, the Z p [μ p ]-torsion submodue is (Z/pZ) c. Suppose that G is generated by σ, and consider two homomorphisms c :Λ[G] Λ which is induced by σ 1, and ψ :Λ[G] Λ[μ p ] which is as above. Lemma 3.3 Let I and J be two ideas of Λ[G]. We assume that c(i) =c(j) and ψ(i) =ψ(j). Furthermore, we assume one of the foowing. i) c(i) is a principa idea generated by a non-zero eement g Λ, whose μ invariant is zero. ii) ψ(i) is a principa idea generated by a non-zero eement h Λ[μ p ], whose μ invariant is zero. Then we have I = J. Proof. We first assume i). Let x be an eement of I. We wi show x J. Put Φ = p 1 i=0 σi. The kerne of ψ :Λ[G] Λ[μ p ] is generated by Φ. Since ψ(i) =ψ(j), we can write x = y +Φz for some y J and z Λ[G]. We 15

16 have c(x) = c(y)+pc(z). Therefore, c(i) = c(j) =(g) impies that g divides pc(z). This shows that g divides c(z) because we assumed the μ-invariant of g is zero. Therefore, using c(i) =c(j), we can write z = u +(σ 1)v for some u J and v Λ[G]. We have x = y +Φu because Φ(σ 1) = 0. This shows that x J. Hence I J. The other incusion J I is obtained by the same method, so we have I = J. Suppose ii) is satisfied, and x I. Using c(i) =c(j), we can write x = y+(σ 1)z for some y J and z Λ[G]. Now, ψ(x) =ψ(y)+(ζ p 1)ψ(z) where ζ p = ψ(σ) is a primitive p-th root of unity. Therefore, h divides ψ(z). So we can write z = u +Φv for some u J and v Λ[G]. This impies that x = y +(σ 1)u J. Thus, I J. The other incusion is proved in the same way, and we have I = J. Lemma 3.4 Let R be the ring of integers of a finite extension of Q p, and A = R[[T ]]. Suppose that 0 M 1 M 2 M 3 0 is an exact sequence of finitey generated torsion A-modues, and that M 3 contains no nontrivia finite submodue. Then we have Fitt A (M 2 ) = Fitt A (M 1 ) char A (M 3 ) where char A (M 3 ) is the characteristic idea of M 3. Proof. (cf. aso [2] Lemma 3.) By [19] Proposition 2.1, M 3 has a free resoution of the form 0 A m A m M 3 0. Therefore, we have Fitt A (M 3 ) = char A (M 3 ), and we can appy Theorem 22 in Chapter 3 of Northcott [14] to obtain Fitt A (M 2 ) = Fitt A (M 1 ) Fitt A (M 3 ) = Fitt A (M 1 ) char A (M 3 ). Proof of Theorem 0.1 (1). Suppose that S φ. By Lemma 3.1, we have an exact sequence 0 Ĥ0 (G, A ω L ) (X ω L ) G X ω K 0 of Λ-modues because L 0 = k in this case. By Iwasawa [8] Theorem 18, X ω K contains no nontrivia finite submodue. Therefore, using Lemma 3.4 and the Iwasawa main conjecture proved by Wies [18], we have (3.2.1) Fitt Λ ((X ω L ) G ) = Fitt Λ (Ĥ0 (G, A ω L ) ) char Λ (X ω K ) = Fitt Λ (Ĥ0 (G, A ω L ) )((γ κ(γ))θ ω K /k ). 16

17 We wi compute Fitt Λ (Ĥ0 (G, A ω L ) ). Since Ĥ0 (G, A ω L ) is finite, we know Fitt Λ (Ĥ0 (G, A ω L ) ) = Fitt Λ (Ĥ0 (G, A ω L )) by [13] Appendix Proposition 3. Suppose that S = {1,..., r}. For S, we put α =(1 κ(ϕ )ϕ 1 ) ω, β = α /(γ κ(γ)) Λ ω K = Λ. By Proposition 1.2 (2), Ĥ 0 (G, A ω L ) is isomorphic to Coker(Z/pZ j r i=1 Λω K /(p, α i )) where the map j is defined by j(1) = (β 1,..., β r ). Therefore, a reation matrix of Ĥ0 (G, A ω L ) is p α p α p α r β 1 β 2... β r. This shows that Fitt Λ (Ĥ0 (G, A ω L )) is generated by the eements of the form i) p r #T T α (where T S and T φ), ii) p r 1 #T β T α (where S and T S \{}), and iii) p r. Since an eement of the form i) is a mutipe of some eement of the form ii), we ony need ii) and iii). We have Fitt Λ (Ĥ0 (G, A ω L )) = S Thus, it foows from (3.2.1) that c(fitt Λ[G] (X ω L )) = Fitt Λ ((X ω L ) G )= S ( S ( S (α,p))(β,p). (α,p))(β,p)(γ κ(γ))θ ω K /k. On the other hand, using the definition of S (see (2.3.8)), we have c(s ω ) = S = S ( S ( S (1, p α ))(1, p β )( S α )θ ω K /k (α,p))(β,p)(γ κ(γ))θ ω K /k. Therefore, we obtain c(fitt Λ[G] (X ω L )) = c(s ω ). 17

18 By Proposition 1.2 (2), we have Ĥ0 (G, X ω L )) = Ĥ 1 (G, A ω L )) =0. Therefore, by Lemma 3.2, (X ω L ) ψ contains no nontrivia finite submodue. Hence we have Fitt Λ[μp]((X ω L ) ψ ) = char Λ[μp]((X ω L ) ψ )=(ψ(θ ω L /k )) by the main conjecture proved by Wies [18]. Since it is easy to see ψ(s ω )= (ψ(θ ω L /k )), we obtain ψ(fitt Λ[G](X ω L )) = ψ(s ω ). Therefore, the conditions of Lemma 3.3 are satisfied (the condition ii) is satisfied), and we obtain Fitt Λ[G] (X ω L )=S ω. Next, we wi prove I L θ L /k Fitt ΛL (X L ). We take a I L. It is easy to see ψ(aθ ω L /k ) ψ(sω ) and c(aθ ω L /k ) c(sω ) from the above descriptions of ψ(s ω ) and c(s ω ) (cf. aso (2.1.4)). By the same argument as the proof of Lemma 3.3, we have aθ ω L /k Sω. We saw in Lemma 2.2 that (I L θ L /k) ω S ω, so we obtain (I L θ L /k) ω Fitt ΛL (X ω L ). If χ is odd and χ ω, we have θ χ L /k Fitt Λ L (X χ L ) by (2.3.2). If χ is even, θ χ L /k = 0. Therefore, we obtain I L θ L /k Fitt ΛL (X L ). This competes the proof of Theorem 0.1 (1). Proof of Theorem 0.1 (2). We prove this statement by the same strategy as the proof of Theorem 0.1 (1). By Proposition 1.2 (3) and Lemma 3.1, we have an exact sequence 0 (X ω L ) G X ω K Z/pZ 0. Since X ω K contains no nontrivia finite submodue ([8] Theorem 18), (X ω L ) G aso has this property. Therefore, Fitt Λ ((X ω L ) G ) = char Λ ((X ω L ) G ) ([19] Proposition 2.1), and Fitt Λ ((X ω L ) G ) = char Λ ((X ω L ) G ) = char Λ (X ω K )=((γ κ(γ))θ ω K /k ) by the Iwasawa main conjecture [18]. Since c((i L θ L /k) ω )=((γ κ(γ))θ ω K /k ), we obtain c(fitt Λ[G] (X ω L )) = Fitt Λ ((X ω L ) G )=c((i L θ L /k) ω ). Next, we consider (X ω L ) ψ. It foows from Proposition 1.2 (3) that Ĥ 0 (G, X ω L )=Ĥ 1 (G, A ω L ) Z/pZ. Therefore, Lemma 3.2 impies that the maxima finite torsion submodue of (X ω L ) ψ is of order p. Thus, we have an exact sequence 0 Z/pZ (X ω L ) ψ M 0 18

19 of Λ[μ p ]-modues where M =(X ω L ) ψ /((X ω L ) ψ ) tors contains no nontrivia finite submodue. Using Lemma 3.4 and the main conjecture [18], we compute Fitt Λ[μp]((XL ω ) ψ ) = Fitt Λ[μp](Z/pZ) char Λ[μp](M) = Fitt Λ[μp](Z/pZ) char Λ[μp]((XL ω ) ψ ) = (ζ p 1,γ κ(γ))ψ(θl ω ) /k where ζ p = ψ(σ) which is a primitive p-th root of unity. On the other hand, it is cear that ψ((i L θ L /k) ω )=(ζ p 1,γ κ(γ))ψ(θl ω /k ). This shows that ψ(fitt Λ[G] (X ω L )) = Fitt Λ[μp]((X ω L ) ψ )=ψ((i L θ L /k) ω ). Therefore, Fitt Λ[G] (X ω L ))=(I L θ L /k) ω by Lemma 3.3 (now the condition i) is satisfied). This competes the proof of Theorem 0.1. Remark 3.5 We can prove (2.3.2) directy by the same method as above. In this Remark 3.5 we put Λ = Λ χ K and G = Ga(L/K). Then Λ χ L =Λ[G]. We use two maps c, ν as in 2.1. For χ ω, we use an exact sequence 0 ( Z/e v Z) χ (X χ L ) G X χ K 0, v P K which is obtained from Proposition 1.2 (1). Reca that S = { P k is unramified in K, and is ramified in L}. We put S χ = { S χ(( K/k ))=1} where ( K/k ) is the Frobenius of in Ga(K/k). We can compute Fitt Λ (( Z/e v Z) χ )= (p, α ) S χ v P K where α =(1 ϕ 1) χ. If S \ S χ, α is a unit of Λ. So S (p, α )= S χ (p, α ). Therefore, using Lemma 3.4 and the main conjecture [18], we have Fitt Λ ((XL ω ) G )=( α S(p, ))θχ K. /k On the other hand, by the definition (2.3.1) of Θ, c(θ χ )=( S (1, p α ))( α )θχ K =( /k (p, α ))θχ K. /k S S χ 19

20 Therefore, we have c(fitt Λ[G] (X χ L )) = c(θ χ ). Next, ψ(fitt Λ[G] (X χ L )) = (ψ(θ χ L /k )) = ψ(θχ ) can be easiy checked. Therefore, by Lemma 3.3 (the condition ii) is satisfied), we obtain Fitt Λ[G] (X χ L ))=Θ χ In this subsection, we compare X L with the standard Iwasawa modue. For a number fied L, the standard Iwasawa modue X L is defined by X L = im A Ln. In this subsection 3.3, we consider the case G = Ga(L/K) is cycic of order p. For simpicity, we ony consider the case that K = k(μ p ) and L/k is unramified outside p for the ω-component (the genera case can be treated by the same method). We assume μ(xk ω ) = 0. We use the same notation as 1.1. We put a = dim Fp Coker(μ p v P Kn I v ) ω where I v is the inertia group of v in G = Ga(L /K ) and the map is induced by the reciprocity map of oca cass fied theory. Using the argument in 1.1, we have Ĥ 1 (G, XL ω ) = Coker(μ p v P Kn I v ) ω (cf. aso Proposition 5.2 in [10]). Hence dim Fp Ĥ 1 (G, XL ω ) = a. We can aso get dim Fp Ĥ 0 (G, XL ω )=a +1. This together with Kida s formua impies that (3.3.1) X ω L Z p [G] λ a 1 (Z p [G]/N G ) a Z a+1 p (cf. Iwasawa [9] 9) as Z p [G]-modues where λ is the λ-invariant of X ω K. On the other hand, by Proposition 1.2 (3) we have This shows that Ĥ 1 (G, X ω L ) = 0 and Ĥ0 (G, X ω L ) Z/pZ. (3.3.2) X ω L Z p [G] λ 1 Z p. Therefore, if a>0, X ω L is not isomorphic to X ω L as a G-modue. We further remark that (ISB) does not hod if a>0 (note that (IDSB) aways hods by Theorem 0.1). This can be proved by the same method as Theorem 1.1 in [7]. Suppose that a>0. Then the natura map (X ω L ) G 20

21 XK ω has non-trivia kerne by Proposition 5.2 in [10]. This together with Lemma 3.4 impies that Fitt Λ ω K ((X ω L ) G ) Fitt Λ ω K (X ω K ). Since the main conjecture impies Fitt Λ ω K (XK ω ) = char Λ ω K (XK ω )=((γ κ(γ))θk ω /k ), we have (γ κ(γ))θ ω K /k Fitt Λ ω K ((Xω L ) G )=c(fitt Λ ω L (X ω L )) where c :Λ ω L Λ ω K is the natura map. Since L /K is unramified outside p, we know c((γ κ(γ))θl ω )=(γ /k κ(γ))θω K /k. It foows that (γ κ(γ))θl ω ) Fitt /k Λ ω (Xω L L ). Namey, (ISB) does not hod In this subsection, we prove Coroary 0.2. We wi first prove (1). By [17] Proposition 13.26, A L n A L is injective. This impies that Since (IDSB) hods, we have Fitt RLn ((X L ) Ga(L /L n)) Fitt RLn ((A L n ) ). c L /L n (I L θ L /k) Fitt RLn ((A L n ) ). On the other hand, by our assumption that a the primes of k above p are ramified in L n, we have c L /Ln (I L θ L /k) =I Ln θ Ln/k where c L /Ln is the natura restriction map. Since I Ln θ Ln/k is in the minus part of R Ln (namey, (I Ln θ Ln/k) + = 0), we get (DSB) for L n /k. Next, we wi prove (2). As we have seen above, hods. We have c L /L(I L θ L /k) Fitt RL ((A L ) ) c L /L(aθ L /k) = p T (1 ϕ 1 p )c L /L(a)θ L/k for a I L where T is the set of primes of k which are ramified in L and unramified in L. By our assumption (i) and (ii), if p is in T, the primes above p do not spit in L/L +. Therefore, (1 ϕ 1 p ) is a unit, and Π p T (1 ϕ 1 p ) is a unit. Since c L /L(I L )=I L, we obtain (DSB). On the other hand, since Γ(L/k) is cycic, we know This impies that (SB) is aso true. Fitt RL (A L ) = Fitt R L ((A L ) ). 21

22 3.5. In this subsection, we study the case that Γ(L/k) is not cycic, and wi prove Theorem 0.3 and Coroary 0.5. Proof of Theorem 0.3. Put K = k(μ p ), G = Ga(L/K) = Ga(L 0 /k), and Λ = Λ ω K. Then Λ ω L =Λ[G]. Let c be the restriction map in 2.1. Proposition 1.2 (3) impies that 2 Ĥ 0 (G, XL ω )=Ĥ 1 (G, A ω L ) =( G)(1). It foows from Lemma 3.1 that 0 ( 2 G)(1) (X ω f L ) G XK ω G(1) 0 is exact. By our assumption on G, we have 2 G 0. Since XK ω does not contain a nontrivia finite submodue, neither does Image f. Therefore, by Lemma 3.4 and the main conjecture [18], we obtain Fitt Λ ((XL ω ) G ) = 2 Fitt Λ (( G)(1)) charλ (Image f) (p, γ 1) char Λ (Image f) = (p, γ 1) char Λ (XK ω ) = (p, γ 1)(γ κ(γ))θk ω. /k If (γ κ(γ))θ ω L /k was in Fitt Λ L (X ω L ), c((γ κ(γ))θ ω L /k )=(γ κ(γ))θω K /k woud be in c(fitt ΛL (X ω L )) = Fitt Λ ((X ω L ) G ) (p, γ 1)(γ κ(γ))θ ω K /k. But this is impossibe. Therefore, (γ κ(γ))θ ω L /k is not in Fitt Λ L (X ω L ). Thus, we have obtained (γ κ(γ))θ L /k Fitt ΛL (X L ). Proof of Coroary 0.5. The statement (1) foows from Theorem 0.3 and Theorem 2.1 in [7]. Next, we wi prove (2). We consider X L = im A Ln and X K = im A Kn. By Coroary 5.3 in [10], we have an exact sequence Z p (1) ( v p I v (L /K )) ω (X ω L ) Ga(L /K ) X ω K 0 where v runs over a primes of K above p and I v (L /K ) is the inertia group of Ga(L /K )atv. Put P = {P P p,k P spits competey in k (μ p ) and ramified in L 0 (μ p )}. 22

23 We have ( v p I v (L /K )) ω P P I P (L 0, /k ) where I P (L 0, /k ) is the inertia group of Ga(L 0, /k )atp. We put N = Coker(Z p (1) P P I P(L 0, /k )). By our assumption #P 2, we have N 0. We appy Lemma 3.4 to the exact sequence to obtain 0 N (X ω L ) Ga(L /K ) X ω K 0 Fitt Λ ((X ω L ) G ) = Fitt Λ (N)(γ κ(γ))θ ω K /k (using the main conjecture). Since c(θ L /k) =θ K /k and Fitt Λ (N) Λ, we have (γ κ(γ))θ ω L /k Fitt Λ L (X ω L ) by the same argument as the proof of Theorem 0.3. By Theorem 2.1 in [7], for sufficienty arge n, we have (γ κ(γ))θ L0 (μ p n )/k Fitt RL0 (μ p n ) (A L 0 (μ p n )). This competes the proof of Coroary 0.5. References [1] Brown, K. S., Cohomoogy of groups, Graduate Texts in Math. 87, Springer-Verag [2] Cornacchia, P. and Greither, C., Fitting ideas of cass groups of rea fieds with prime power conductor, J. Number Theory 73 (1998) [3] Deigne, P. and Ribet, K., Vaues of abeian L-functions at negative integers over totay rea fieds, Invent. math. 59 (1980), [4] Greither, C., Arithmetic annihiators and Stark-type conjectures, Stark s Conjectures: Recent Work and New Directions, Contemporary Math. 358 (2004), [5] Greither, C., Computing Fitting ideas of Iwasawa modues, Math. Zeitschrift 246 (2004), [6] Greither, C., Determining Fitting ideas of minus cass groups via the equivariant Tamagawa number conjecture, Compositio Math. 143 (2007),

24 [7] Greither, C. and Kurihara, M., Stickeberger eements, Fitting ideas of cass groups of CM fieds, and duaisation, Math. Zeitschrift 260 (2008), [8] Iwasawa, K., On Z -extensions of agebraic number fieds, Ann. of Math. 98 (1973), [9] Iwasawa, K., Riemann-Hurwitz formua and p-adic Gaois representations for number fieds, Tôhoku Math. J. 33 (1981), [10] Kurihara, M., Iwasawa theory and Fitting ideas, J. reine angew. Math. 561 (2003), [11] Kurihara, M., On the structure of idea cass groups of CM-fieds, Documenta Mathematica, Extra Voume Kato (2003), [12] Kurihara, M. and Miura T., Stickeberger ideas and Fitting ideas of cass groups for abeian number fieds, Math. Annaen 350 (2011), [13] Mazur, B. and Wies, A., Cass fieds of abeian extensions of Q, Invent. math. 76 (1984), [14] Northcott, D. G., Finite free resoutions, Cambridge Univ. Press, Cambridge New York [15] Serre, J.-P., Corps Locaux, Hermann, Paris 1968 (troisième édition). [16] Serre, J.-P., Sur e résidu de a fonction zêta p-adique, Comptes Rendus Acad. Sc. Paris, 287 (1978), Série A, [17] Washington, L., Introduction to cycotomic fieds, Graduate Texts in Math. 83, Springer-Verag [18] Wies, A., The Iwasawa conjecture for totay rea fieds, Ann. of Math. 131 (1990), [19] Wingberg, K., Duaity theorems for Γ-extensions of agebraic number fieds, Compos. Math. 55 (1985), Masato Kurihara Department of Mathematics, Keio University, Hiyoshi, Kohoku-ku, Yokohama, , Japan kurihara@math.keio.ac.jp 24