On the ideal class groups of the maximal real subfields of number ...

c Springer-Verlag & EMS 1999

J. Eur. Math. Soc. 1, 35–49

Masato Kurihara

On the ideal class groups of the maximal real subfields of number fields with all roots of unity Received January 15, 1998 / final version received July 31, 1998 Abstract. In this paper, for a totally real number field k we show the ideal class group of k (∪n>0 µn )+ is trivial. We also study the p-component of the ideal class group of the cyclotomic Z p -extension.

1. Introduction For a positive integer n > 0, µn denotes the group of n-th roots of unity. Iwasawa studied the ideal class groups of number fields containing µ p∞ = ∪n>0 µ pn with a prime number p, and established a theory which clarifies their deep arithmetical meaning. One of the motivation lay in an analogy between a number field containing µ p∞ and a function field over an algebraically closed field, so it is natural to ask how is the ideal class group of a field containing all roots of unity. For an algebraic extension K/Q, C K = Pic(O K ) denotes the ideal class group of the integer ring O K , namely the group of isomorphism classes of invertible O K -submodules of K. So we have C K = lim Ck where k ranges → over all intermediate fields with [k : Q] < ∞. For a totally real number field k, let k∞ = ∪n>0 k(µn ) be the field obtained from k by adjoining all the roots of unity. For example, Q∞ is the maximal abelian extension Qab of Q. The class group Ck∞ was studied by Brumer [1], and Horie [6] (cf. also [15]), but the following result does not seem to be known. Theorem 1.1. Let k be a totally real number field. We denote by (k∞ )+ the maximal real subfield of k∞ . Then, we have C(k∞ )+ = 0. In particular, C(Qab )+ = 0. M. Kurihara: Department of Mathematics, Tokyo Metropolitan University, Hachioji, Tokyo 192-03, Japan Mathematics Subject Classification (1991): 11R23, 11R29,11R18

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On the other hand, we know that the Pontrjagin dual of the minus part Ck−∞ = Coker(C(k∞ )+ −→ Ck∞ ) is generated by infinitely many elements ˆ even as a Z[[Gal(k ∞ /k)]]-module (for example, cf. [11]). After some preparation in Sect. 2, we will prove this theorem in Sect. 3. For the proof, we only need class field theory. Using the same method, in Sect. 4 we will study the p-primary component of the ideal class group of the cyclotomic Z p -extension of a real abelian field. For a number field F, A F denotes the p-primary component ( p-Sylow subgroup) of the ideal class group C F . Let ϕ be an even Dirichlet character of the first kind, and kϕ (resp. kϕ,∞, p ) be the real abelian field corresponding to the kernel of ϕ (resp. the cyclotomic Z p -extension of kϕ ). Let ki be the subfield of kϕ,∞, p such that [ki : kϕ ] = pi , and consider X kϕ,∞, p = lim Aki where the projective limit is ← taken with respect to the norm maps. We denote by X ϕ the ϕ-component of X kϕ,∞, p . We will see in Sect. 4 that there are many ϕ’s such that X ϕ is finite (in other words, such that Aϕkϕ,∞, p = 0) (cf. Propositions 4.3, 4.4, 4.6, 4.1). I would like to express my hearty thanks to Professor K. Iwasawa for his warm encouragement. I learned a lot from his papers [8, 9] on the relation between A K and A L for a p-extension L/K. Notation. For an abelian group A and an integer n > 0, the cokernel (resp. kernel) of the multiplication by n is denoted by A/n (resp. A[n]). (Even in the case A is multiplicative, we use A/n instead of A/An .) For a number field F, its integer ring is denoted by O F . For an integer n > 0, µn denotes the group of n-th roots of unity. 2. Some lemmas In this section, we assume that K is a totally real number field of finite degree over Q, and p is a prime number. Let ` be a prime number which is different from p, and n > 0 be a positive integer. In the following lemma, we consider a finite extension L/K of totally real fields such that L/K is cyclic of degree pn , and that L/K is unramified outside `, and totally ramified at all primes of K lying over `. For a place w of L, let L w be the completion of L at w. We denote by E L (resp. E L w ) the unit group of the integer ring of L (resp. L w ). (If w is an infinite place, we define E L w = L × w .) We have an exact sequence Y E L w −→ C L −→ C L −→ 0 0 −→ E L −→ w

where C L (resp. C L ) is the idele class group (resp. ideal class group) of L. We ˆ 0 (L/K, M) with H 2 (L/K, M) fix a generator of Gal(L/K) and identify H ˆ 0 is the Tate cohomology. for any Gal(L/K)-module M. Here, H

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Lemma 2.1. The above exact sequence yields an exact sequence L ˆ 0 (L/K, C L ) −→ H ˆ 0 (L w /K v , E L w ))0 ˆ 0 (L/K, E L ) −→ ( v|` H −→ H L 1 1 1 −→ H (L/K, C L ) −→ H (L/K, E L ) −→ v|` H (L w /K v , E L w ) −→ ... . The notation is as follows. v ranges over all primes of K lying over `, and w is the unique prime of L lying over v. We define an isomorphism ˆ 0 (L w /K v , E L w ) ' Z/ pn by H 2 × n ˆ 0 (L w /K v , E L w ) ' H ˆ 0 (L w /K v , L × H w ) ' H (L w /K v , L w ) ' Z/ p

where the last map is the invariant map of local class field theory. (The first two are isomorphic because L w /K v is totally ramified.) The group L groups 0 ˆ ( v|` H (L w /K v , E L w ))0 denotes the kernel of M M 6 ˆ 0 (L w /K v , E L w ) ' H Z/ pn −→ Z/ pn v|`

v|`

where 6 is the map defined by the sum. Remark 2.2. It is well known that the kernel of the map M H 1 (L w /K v , E L w ) H 1 (L/K, E L ) −→ v|`

in Lemma 2.1 coincides with the kernel of the canonical map i L/K : C K −→ C L . In fact, let D L (resp. PL ) be the divisor group of L (resp. the group of principal divisors of L). From a commutative diagram of exact sequences 0 0 ↑ ↑ L n H 1 (L/K, E L ) −→ w|` Z/ p ↑ ↑ 0 −→ H 0 (L/K, PL ) −→ H 0 (L/K, D L ) −→ H 0 (L/K, C L ) ↑ ↑ ↑ 0 × DK −→ CK −→ 0 , H (L/K, L ) −→ ↑ 0 we have an exact sequence 0 −→ Ker(i L/K ) −→ H 1 (L/K, E L ) −→

M

Z/ pn

w|`

(where w ranges over all primes of L L lying over `). Here, the last map 1 coincides with H 1 (L/K, E L ) −→ v|` H (L w /K v , E L w ), so its kernel coincides with Ker(i L/K ).

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Proof of Lemma 2.1. Let A be the kernel of C L −→ C L . We will simply ˆ q (M). First of all, since L/K is unramified outˆ q (L/K, M) by H denote H ˆ q (5w|` E L w ). Consider a commutative ˆ q (5w E L w ) = H side `, we have H diagram of exact sequences 0 x  

' ˆ0 −→ H (C x L)   L 0 (L /K , E ) −→ H 0 (A) −→ H 1 (E ) −→ ... ˆ ˆ 0 (E L ) −→ ˆ H ... −→ H w v Lw L v|` x x     L ˆ 0 (L w /K v , E L ))0 −→ H 1 (C L ) ( v|` H w x x     0 0 .

Z/xpn  6

Here, we used H 1 (C L ) = 0, and the canonical map M ˆ 0 (L w /K v , E L w ) −→ H ˆ 0 (C L ) H v|`

coincides with the sum sequence

L

n v|` Z/ p

6

−→ Z/ pn . Hence, we have an exact

M

ˆ 0 (E L ) −→ ( ... −→ H

ˆ 0 (L w /K v , E L w ))0 H

v|`

−→ H 1 (C L ) −→ H 1 (E L ) −→ ... ˆ 0 (A) −→ On the other hand, from the above commutative diagram, H 0 1 ˆ H (C L ) is surjective. This fact together with H (C L ) = 0 implies that ˆ 0 (C L ) −→ H 1 (A) is bijective. Thus, H M ˆ 0 (C L ) −→ H ˆ 0 (E L ) H 1 (L w /K v , E L w ) −→ H H 1 (E L ) −→ v|`

is exact. This completes the proof of Lemma 2.1.

t u

We fix a prime number p, and denote by A F the p-primary component ( p-Sylow subgroup) of the ideal class group C F for a number field F. We also need the following lemma (cf. [1, Lemma 6]). Lemma 2.3. Let K be a totally real number field, and n > 0 a positive integer. We assume K contains Q(µ2 pn )+ . Let c ∈ A K be any element of A K . Then there exist infinitely many rational primes ` such that i(c) ∈ pn A K(µ` )+ where i : A K −→ A K(µ` )+ is the canonical homomorphism.

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In fact, let H be the Hilbert class field of K, namely H/K is the unramified extension such that Gal(H/K) ' C K . We consider an abelian extension H(µ2 pn )/K whose Galois group is Gal(H(µ2 pn )/K) = C K × {±1} where {±1} = Gal(K(µ2 pn )/K). By Tschebotareff density, there exist infinitely many primes λ of K, of degree 1, whose Frobenius coincides with (c, 1) ∈ C K × {±1} = Gal(H(µ2 pn )/K). We denote by ` the prime number which is below λ. We may suppose ` is unramified in K. Our assumption implies that ` splits completely in Q(µ2 pn ), and that the class of λ in C K is c. Hence, ` ≡ 1 (mod 2 pn ), and K(µ` )/K is totally ramified at λ. Let w be the prime of K(µ` )+ lying over λ. Then, we have λ = w(`−1)/2 in K(µ` )+ , hence the image i(c) of the class of λ in A K(µ` )+ belongs to pn A K(µ` )+ . 3. Proof of Theorem 1.1 In this section, we prove Theorem 1.1. We will use the same notation as in Sect. 2. In particular, p denotes a prime number, and C F (resp. A F ) denotes the ideal class group of O F (resp. the p-Sylow subgroup of C F ) for a number field F. Let k be a totally real number field. In this section, we call an extension K/k an rcf extension (a real cyclotomic finite extension) if K ⊂ (k∞ )+ and [K : k] < ∞. We will show that for any rcf extension K/k, and any element c ∈ C K , there is an rcf extension L/k such that K ⊂ L and that the image of c vanishes in C L . 3.1. It is clear that we may assume [k : Q] is finite. We will first see that ˜ we may assume k/Q is Galois. Let k/Q be the Galois closure of k/Q, and ˜ put m = [k : k]. In order to show the above statement, we may assume the order of c is a power of p for some prime number p. By Lemma 2.3 there is an rcf extension K 0 /k such that c0 ∈ m A K 0 where c0 is the image of c in A K 0 . So we can take c˜ ∈ A K 0 k˜ such that N(˜c) = c0 where N : A K 0 k˜ −→ A K 0 is ˜ k˜ such that L˜ ⊃ K 0 k˜ the norm map. Assume that there is an rcf extension L/ and the image of c˜ is zero in A L˜ . We may suppose L˜ = L k˜ where L/k is an rcf extension. Then, the image of c in A L is zero. In the following, we assume k/Q is a finite Galois extension. 3.2. In this subsection, we will define two homomorphisms 8 and φ` . Let p be a prime number, and let K/k be an rcf extension. Note that K/Q is a finite Galois extension by our assumption. For a ring R and a group G, we define R[G]0 = Ker(R[G] −→ R) (6aσ σ 7 → 6aσ ) the augmentation ideal. Let E K be the unit group of O K . By Dirichlet’s unit theorem E K ⊗R ' R[Gal(K/Q)]0 , E K ⊗Q is isomorphic to Q[Gal(K/Q)]0 (cf. for example [14, Cor to Th 30 in Sect. 13]), so we

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can take a Gal(K/Q)-homomorphism 8 : E K ⊗ Z p −→ Z p [Gal(K/Q)]0 with finite cokernel. We take such a 8 and fix it. For a finite place v of K, E K v denotes the unit group of the integer ring O K v of the completion K v at v, and κ(v) denotes the residue field of v. For a prime number `, we have a canonical homomorphism E K −→

M

E K v −→

v|`

M

κ(v)× .

v|`

We assume that ` splits completelyL in K, and that ` ≡ 1 (mod 2 pn ). Fixing a prime v0 of K over `, we identify v|` κ(v)× / pn with Z/ pn [Gal(K/Q)]⊗ × × n n n (F× ` / p ). Fixing also a generator of F` and an isomorphism F` / p ' Z/ p , we have a Gal(K/Q)-homomorphism φ` : E K −→ (

M

κ(v)× )/ pn ' Z/ pn [Gal(K/Q)].

v|` n Our assumption on ` implies that the canonical map EQ = {±1} −→ F× ` /p is zero, hence the image of φ` is in (Z/ pn [Gal(K/Q)])0 .

3.3. Let K/k be an rcf extension, and c be any element in C K . We will show that there is an rcf extension L/k such that K ⊂ L and that the image of c vanishes in C L . We may assume that the order of c is a power of p for some prime number p, namely c is in A K . We may assume K is big enough, so we may assume K(µ p∞ )+ /K is totally ramified at all primes of K lying over p. Further, by Lemma 2.3 we may assume c ∈ 2A K . We fix 8 : E K ⊗ Z p −→ Z p [Gal(K/Q)]0 as in 3.2. We take a positive integer n such that pn > (#A K )2 · #(Coker(48 : E K ⊗ Z p −→ Z p [Gal(K/Q)]0 )). We choose a prime number ` such that (i) ` splits completely in K (ii) ` ≡ 1 (mod 2 pn ) (iii) There is a prime v0 of K lying over ` such that the class of v0 in A K is c. L (iv) Image(φ` : E K −→ v|` κ(v)× / pn ' Z/ pn [Gal(K/Q)]) coincides with Image(48) mod pn in Z/ pn [Gal(K/Q)].

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We can find infinitely many such `’s by Tschebotareff density theorem, by the same method as in [13, Theorem 3.1] (p odd), and [5, Theorem 3.7] 1/ pn ( p = 2). In fact, put F = K(µ2 pn ), and F 0 = F(E K ). Let H be the Hilbert p-class field of K. By our assumption that F + /K is totally ramified at every prime lying over p, we have F ∩ H = K , and we identify A K with Gal(HF/F). Since 2 Gal(HF ∩ F 0 /F) = 0 and c ∈ 2A K , c can be regarded as an element of Gal(HF/HF ∩ F 0 ). We choose a primitive pn -th root ζ pn of unity, and define ι : Z/ pn [Gal(K/Q)] −→ µ pn by ι(σ) = 1 if σ 6 = 1Gal(K/Q) (for σ ∈ Gal(K/Q)), and ι(1Gal(K/Q) ) = ζ pn . Then, 4ι ◦ 8 : E K −→ µ pn corresponds to an element γ of Gal(F 0 /HF ∩ F 0 ) by Kummer theory. We choose a prime λ of F, of degree 1, such that HF 0 /Q is unramified at ` which is the prime of Q below λ, and that the Frobenius of λ in Gal(HF/F) coincides with c, and that the Frobenius of λ in Gal(F 0 /F) coincides with γ . If we define ` (resp. v0 ) to be the prime of Q (resp. K ) below λ, the properties (i)-(iv) are satisfied. Let L be the totally real subfield of K(µ` ) such that L/K is cyclic of degree pn . We denote by i L/K : C K −→ C L the canonical homomorphism. We will show that i L/K (c) = 0. Suppose that i L/K (c) 6 = 0. L/K is totally ramified at v0 , and we denote by w0 the prime of L lying over v0 . Let [w0 ] (resp. [v0 ]) be the class of w0 pn Gal(L/K) in C L (resp. v0 in C K ). From w0 = v0 in L, we have [w0 ] ∈ A L . Let pr be the order of [w0 ] in ˆ 0 (L/K, C L ) = H ˆ 0 (L/K, A L ) = AGal(L/K) H /i L/K A K , L and α be a class in A K such that pr [w0 ] = i L/K (α). By our assumption, i L/K ( pn−r α) = pn [w0 ] = i L/K ([v0 ]) = i L/K (c) 6 = 0 in C L , hence pn−r α is not in the kernel of i L/K . In particular, the order of A K is greater than pn−r . This implies that pr > pn /#A K > #A K # Coker 48. Since ˆ 0 (L/K, C L ) ≥ pr , we have #H ˆ 0 (L/K, C L ) = # H ˆ 1 (L/K, C L ) > #A K # Coker 48. #H

(1)

Now we apply 2.1 to L/K. L L Lemma ˆ 0 (L w /K v , E L w ) = v|` κ(v)× / pn ' Z/ pn [Gal(K/Q)], First of all, v|` H L ˆ 0 (L/K, E L ) −→ v|` H ˆ 0 (L w /K v , E L w ) is induced from φ` . On the and H : C K −→ C L other hand, as we mentioned in Remark 2.2, the kernel L of i L/K 1 coincides with the kernel of H 1 (L/K, E L ) −→ H (L w /K v , E L w ). v|` Hence by Lemma 2.1, we have an exact sequence 0 −→ Coker(φ` : E K −→ Z/ pn [Gal(K/Q)]0 ) −→ → H 1 (L/K, C L ) −→ Ker(i L/K ) −→ 0.

(2)

Since the image of φ` coincides with the image of 48 mod pn , the inequality (1) implies that # Ker(i L/K ) = # Ker(A K −→ A L ) > #A K , which

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is a contradiction. Hence i L/K (c) = 0. This completes the proof of Theorem 1.1. 4. Remarks on cyclotomic Z p -extensions of real abelian fields In this section, we consider a simple situation, and assume p is an odd prime number. We also use the notation that A K is the p-primary component ( p-Sylow subgroup) of the ideal class group C K for a number field K . K ∞, p /K denotes the cyclotomic Z p -extension, and for i ≥ 0, K i denotes the intermediate field of degree pi . We define X K ∞, p = lim A K i ←

where the projective limit is taken with respect to the norm maps. Note that A K ∞, p = lim A K i . →

For a Dirichlet character ϕ of the first kind whose values are in an algebraic closure of Q p , kϕ denotes the abelian field corresponding to the kernel of ϕ, × and put G ϕ = Gal(kϕ /Q) (so ϕ : G ϕ ,→ Q p ). For a Z p [G ϕ ]-module M, we define M ϕ = M ⊗Z p [G ϕ ] Oϕ where we regard Oϕ = Z p [Image ϕ] as a Z p [G ϕ ]-module via ϕ, namely ϕ σx = ϕ(σ)x for any σ ∈ G ϕ and any x ∈ Oϕ . We simply denote Akϕ by Aϕ . From Gal(kϕ,∞, p /Q) = G ϕ ×Gal(kϕ,∞, p /kϕ ), we regard G ϕ as a subgroup of ϕ Gal(kϕ,∞, p /Q), and consider X kϕ,∞, p = X kϕ,∞, p ⊗Z p [G ϕ ] Oϕ which becomes an Oϕ [[Gal(kϕ,∞, p /kϕ )]]-module. We simply denote X ϕkϕ,∞, p by X ϕ . We will see in this section that there are many even ϕ’s with #X ϕ < ∞. We assume that χ is an even Dirichlet character of order prime to p such that χ( p) 6 = 1. 4.1. In this subsection, we assume that Aχ = 0. Then we have X χ = 0 since χ( p) 6 = 1 (cf. [9, Lemma 3]). We begin with the following simple observation. Proposition 4.1. Let χ be an even character of order prime to p, satisfying χ( p) 6 = 1 and Aχ = 0. For any n > 0, there exist infinitely many even Dirichlet characters ψ of order pn with conductor a prime number such that X χψ = 0. Put K = kχ . In fact, by Tschebotareff density, we can take infinitely many ` such that ` splits completely in K , ` ≡ 1 (mod pn ), and that

Ideal class groups

43

L (E K ⊗ Z/ pn )χ −→ ( v|` E K v ⊗ Z/ pn )χ is surjective. Let L be the subfield of K(µ` ) such that [L : K] = pn . For any Z p [Gal(L/Q)]-module M, we regard M as a Z p [G χ ]-module by regarding G χ as a subgroup of Gal(L/Q) by the decomposition Gal(L/Q) = G χ × Gal(L/K). We consider M χ = M ⊗Z p [G χ ] Oχ which is an Oχ [Gal(L/K)]-module. By Lemma 2.1, the above conditions on ` imply H 1 (L/K, A L )χ = χ χ χ 1 H (L/K, A L ) = 0 (cf. (2) in 3.3), hence A L = 0. For X L ∞, p = X L ∞, p ⊗Z p [G χ ] Oχ , by the standard argument (cf. [9, Lemma 3]), we have χ χ (X L ∞, p )Gal(L ∞, p /L) = A L . In fact, let γ be a generator of Gal(L ∞, p /L)
L and H = Ker X L ∞, p −→ A L . Since χ( p) 6 = 1, ( w| p Z p )χ = 0 where w ranges over all primes of L ∞, p over p, so (H/(γ − 1)X L ∞, p )χ = 0, χ χ χ which implies (X L ∞, p )Gal(L ∞, p /L) = A L . Hence X L ∞, p = 0 by Nakayama’s t u lemma, so X χψ = 0 for a character ψ of Gal(L/K). Concerning the finiteness of X χψ for a character ψ of conductor ` and of order p under the assumption Aχ = 0, by studying the group of cyclotomic units, we can easily see the following. Proposition 4.2. Let χ be an even character of order prime to p, satisfying χ( p) 6 = 1 and Aχ = 0. For a character ψ of order p and of conductor `, X χψ is finite if and only if ˆ 0 (L i /K i , E L i )χ = #(F p [[Gal(K ∞, p /Q)]]/(Frob` − 1))χ #H for some i ≥ 0 where Frob` is the Frobenius at ` in Gal(K ∞, p /Q). Concerning this kind of criterion, more general case is treated by Fukuda et al. [3]. L If (E K ⊗ Z p )χ −→ ( v|` E K v ⊗ Z p )χ is surjective as in the proof of ˆ 0 in Proposition 4.2 is satisfied for Proposition 4.1, then the condition on H √ all i ≥ 0. As an example, for p = 3, K = Q( 2), nontrivial character χ of K, and a prime number ` which splits completely L in K, and which satisfies ` ≡ 1 (mod p) with ` < 200, (E K ⊗ Z p )χ −→ ( v|` E K v ⊗ Z p )χ is surjective (so X χψ = 0 by Proposition 4.1) except ` = 79, 103 . For ˆ 0 (L/K, E L )χ = 3, hence ` = 79, and 103, by a table in [12] we know # H χψ by Proposition 4.2, X is finite (and nonzero). 4.2. Next we consider the case Aχ ' Oχ / p. Proposition 4.3. Let χ be an even character of order prime to p such that χ( p) 6 = 1, and Aχ ' Oχ / p. If pX χ 6 = 0, then for any n > 0 there exist infinitely many even Dirichlet characters ψ of order pn with a prime conductor such that X χψ is finite.

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This proposition says that if Aχ ' Oχ / p, there is at least one character ψ (ψ is of order 1 or pn ) such that X χψ is finite. We will prove Proposition 4.3. We use the same notation as in the proof χ of Proposition 4.1. Let K = kχ , and c be a generator of A K . We take ` which satisfies the conditions that in K , ` ≡ 1 (mod pn ), and L` splits completely n χ n χ that (E K ⊗Z/ p ) −→ ( v|` E K v ⊗Z/ p ) is surjective, and that [v0 ] = c where v0 is a prime lying over `. Tschebotareff density asserts the existence of infinitely many ` satisfying these conditions. n Let L be the subfield of LK(µ` ) such thatn[Lχ : K ] = p . By the surjectivity n χ of (E K ⊗ Z/ p ) −→ ( v|` E K v ⊗ Z/ p ) , Lemma 2.1 implies that χ

χ

H 1 (L/K, A L )χ = Ker(i L/K : A K −→ A L )

(3)

(cf. (2) in 3.3). On the other hand, the prime w0 of L lying over ˆ 0 (L/K, A L )χ . So # H ˆ 0 (L/K, A L )χ = v0 yields a nontrivial class in H 1 χ #H (L/K, A L ) 6 = 1, hence by (3) we get i L/K (c) = 0. χ χ The isomorphism (3) also implies that (A L )Gal(L/K) ' A K . (Note that χ χ A L = A L ⊗Z p [G χ ] Oχ , and not A L ⊗Z p [Gal(L/Q)] Oχ .) So A L is generaχ ted by [w0 ]. We have A L ' Oχ / p by induction on n. In fact, let K 0 0 n−1 be the subfield of L such . Since the inclusion inLthat [K : K] =χ p L duces an isomorphism ( v|` E K v ⊗ Z/ p) ' ( v0 |` E K v0 0 ⊗ Z/ p)χ , the L map (E K 0 ⊗ Z/ p)χ −→ ( v0 |` E K 0 0 ⊗ Z/ p)χ is also surjective, hence v the same argument as above implies that i L/K 0 ([v00 ]) = 0 where v00 is the prime of K 0 lying above v0 . From p[w0 ] = i L/K 0 ([v00 ]) = 0, we have χ A L ' Oχ / p. Put G = Gal(L/K). Fixing a generator γ of Gal(K ∞, p /K) and identifying γ with 1 + T , we write 3 = Oχ [[Gal(K ∞, p /K)]] ' Oχ [[T ]], and 3 L = Oχ [[Gal(L ∞, p /K)]] ' Oχ [[T ]][G] = 3[G]. Let σ be a generator of G = Gal(L/K). We take a character ψ of G such that ψ(σ) = ζ pn where ζ pn is a primitive pn -th root of unity. We define ψ α : 3 L −→ 3 K = Oχ [[T ]] (resp. ψ : 3 L −→ 3 L = Oχψ [[T ]]) to be a ring homomorphism defined by σ 7 → 1 (resp. σ 7 → ψ(σ)). χ χ As in the proof of Proposition 4.1, we have (X L ∞, p )Gal(L ∞, p /L) ' A L χ from our assumption χ( p) 6 = 1. So X L ∞, p is cyclic as a 3 L -module by χ χ Nakayama’s lemma. We write X L ∞, p = 3 L /I . From (X L ∞, p )Gal(L ∞, p /L) ' 3 L /(I, T) = Oχ / p, I contains elements G(T) = σ −1 + Tg and H(T) = p + Th for some g, h ∈ 3 L . We assume X χψ is infinite. Since ψ(G(0)) = ζ pn − 1 is a prime element of Oχψ and µ = 0 [2], ψ(G(T))∗ is an Eisenstein polynomial, so irreducible where ψ(G(T))∗ is the polynomial obtained from ψ(G(T)) by Weierstrass

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45

preparation theorem. From the assumption #X χψ = ∞, this implies that ψ(G(T)) generates ψ(I). Hence, ψ(G(T)) divides ψ(H(T)). n n−1 Put 8 pn (S) = (S p − 1)/(S p − 1) the pn -th cyclotomic polynomial. Then, we can write H(T) = G(T)A(T) + 8 pn (σ)B(T) for some A(T), B(T) ∈ 3 L . Since p = α(H(0)) = pα(B(0)), α(B(0)) = 1 and B(T) is a unit in 3 L . So 8 pn (σ) is in I , which implies p ∈ α(I). Since we χ χ have a surjective homomorphism (X L ∞, p )G = 3/α(I) −→ X K ∞, p = X χ , t u this implies pX χ = 0, which contradicts our assumption. For example, by the same method, we have Corollary 4.4. Let K be a real quadratic field, and p be an odd prime number. We assume that p does not decompose in K, and #A K = p. If pX K ∞, p 6 = 0, for any n > 0, there exist infinitely many real cyclic fields L 0 of degree pn with a prime conductor such that X L ∞, p /i(X K ∞, p ) is finite where L = L 0 K and i : X K ∞, p −→ X L ∞, p is the canonical map. Proof. Let χ0 be the trivial character (χ0 = 1), and χ be the nontrivial character of K. Let K 1 be the subfield of K ∞, p such that [K 1 : K] = p. We take a prime number ` such that ` splits completely in K 1 , ` ≡ 1 L n χ0 n 0 χ0 ⊗ Z/ p ) −→ [( E (mod pn ), and that both (E K1 v|` K 1,v ⊗ Z/ p ) ] L and (E K ⊗ Z/ pn )χ −→ ( v|` E K v ⊗ Z/ pn )χ are surjective, and that [v0 ] is a generator of A K where v0 is a prime of K lying over `. Tschebotareff density asserts the existence of infinitely many such `. Then, by the method in the proof of Proposition 4.3 we get the conclusion. t u Remark 4.5. For p = 3, 5, 7, there are a lot of examples of quadratic fields K such that A K ' Z/ p, pX K ∞, p 6 = 0, and that X K ∞, p is finite (for √ √ example, K = Q( 254), Q( 473) for p = 3, cf. [7, 10]). For these K, by Corollary 4.4 we can find infinitely many real cyclic fields L (⊂ K(µ` ) for some `) of degree 2 pn such that A L 6= 0

and

#X L ∞, p < ∞.

We remark that in [9] Iwasawa constructed infinitely many K such that [K : Q] = p, A K 6 = 0, and #X K ∞, p < ∞ for arbitrary odd prime number p. 4.3. We will give one more Proposition which has the same nature as Proposition 4.3. Proposition 4.6. Let χ be an even character of order prime to p, such that χ( p) 6 = 1. We assume that X χ is isomorphic to a quotient of Oχ [[T ]]/( f(T)) as an Oχ [[Gal(kχ,∞, p /kχ )]] = Oχ [[T ]]- module where f(T) ∈ Oχ [[T ]] is an irreducible polynomial. Then at least either X χ is finite, or for any n > 0 there exist infinitely many even Dirichlet characters ψ of order pn with a prime conductor such that X χψ is finite.

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We use the same notation as in the proof of Proposition 4.3. Put K = kχ , and suppose AχK ' Oχ / pm 6 = 0. We take ` as in the proof of Proposition 4.3 and take L ⊂ K(µ` ) such that [L : K] = pn . By the same argument, we have AχL ' Oχ / pm . Let ψ be a faithful character of G = Gal(L/K). Assume both X χ and χψ X are infinite. Our assumption implies that X χ ' Oχ [[T ]]/( f(T)). We χ χ write X L ∞, p ' 3 L /I as in the proof of Proposition 4.3. Since A L ' Oχ / pm , I contains G(T) = σ − 1 + Tg and H(T) = pm + Th for some g, h ∈ 3 L . Let α and ψ be the homomorphisms defined in the proof of Proposition 4.3. Since α(H(0)) = f(0) = pm , α(H(T)) generates ( f(T)). So α(H(T)) divides α(G(T)). On the other hand, ψ(G(T)) divides ψ(H(T)) as in the proof of Proposition 4.3. In general, for a discrete valuation ring R with maximal ideal m R , and a power series f ∈ R[[T ]], we define λ( f) = ordT ( f mod m R ) where ordT is the normalized additive valuation of (R/m R )[[T ]] defined by T . The above divisibility and µ = 0 [2] imply that λ(α(H(T))) ≤ λ(α(G(T))) = λ(ψ(G(T))) ≤ λ(ψ(H(T))) = λ(α(H(T))) < ∞. This implies both α(H(T)) and α(G(T)) generate the same ideal in Oχ [[T ]]. This is a contradiction because α(H(0)) 6 = α(G(0)) = 0. t u Appendix After this paper was accepted to be published, I was suggested by John Coates to study about Ckab by the same method where k is an imaginary quadratic field and kab is the maximal abelian extension of k in an algebraic closure of k. Further I was informed by Nguyen Quang Do and R. Schoof of a paper by G. Gras [4]. I would like to express my hearty thanks to John Coates for his suggestion, and to Nguyen Quang Do and R. Schoof for telling me about [4]. By the same method as in Sect. 3, we can prove Theorem A.1. Let k be an imaginary quadratic field, and kab its maximal abelian extension in an algebraic closure. For any algebraic extension K/k, we have C Kkab = 0 where Kkab is the compositum of K and kab . For the maximal abelian extension K ab of a number field K, Gras [4, Conjecture 0.5] conjectures C K ab = 0 for K which is not totally real. From Theorem A.1, we obtain

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Corollary A.2. For a number field K which contains an imaginary quadratic field, we have C K ab = 0. Proof of Theorem A.1. In principle, instead of µn , we can use division points of an elliptic curve with complex multiplication. For an ideal a of Ok , k(a) denotes the ray class field modulo a of k. Note that [k(a) : k(1)] = #(Ok /a)× wa /wk where wk (resp. wa ) denotes the number of roots of unity (resp. the number of roots of unity congruent to 1 modulo a). We will apply the method in Sect. 3. Let K/k be a finite Galois extension. For any ideal class c ∈ A K , we will show the existence of an abelian extension k0 /k such that the image of c vanishes in A Kk0 . Gal(K(µ p∞ )/K) Suppose µ p∞ (K) = µ p∞ = µ pm . We may assume m is big enough. By the method of Lemma 2.3, for any n > 0 we can show that i(c) ∈ pn A Kk(λ1 ·...·λr ) for some primes λ1 ,...,λr of degree 1, so we may assume c ∈ pm A K . Further, we may assume that K contains the Hilbert class field of k. We take 8 : E K ⊗ Z p −→ Z p [Gal(K/k)]0 as in Sect. 3, and take n > 0 such that pn > (#A K )2 · #(Coker(2 pm 8)). As in 3.3, by Tschebotareff density theorem, we can choose a prime number ` such that ` splits completely in K , ` ≡ 1 (mod 12 pn ), c = [v0 ] where v0 is a prime of K lying over `, and that if λ is a prime of k below v0 , Image(φλ : E K −→

M

κ(λ)× / pn ' Z/ pn [Gal(K/k)])

v|λ

= Image(2 pm 8 mod pn ). In the notation of 3.3 we have pm Gal(HF ∩ F 0 /F) = 0, so this is possible. We define L to be the subfield of Kk(λ) such that [L : K ] = pn , and apply an exact sequence M

ˆ 0 (L/K, E L ) −→( . . . −→ H

ˆ 0 (L w /K v , E L w ))0 −→ H

v|λ

→H 1 (L/K, C L ) −→ . . . , which is obtained by the same method as Lemma 2.1. Then we have i L/K (c) = 0 by the same method as in 3.3.

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Remark A.3. In [4], Gras studied the capitulation of ideals of number fields, √ and gave some examples. For K = Q( 79), he showed the elements of C K (' Z/3) become trivial in C M where M is the compositum of K and the cubic field of conductor 97. Our method in this paper shows that we can take a field of smaller conductor for the capitulation. For example, the ideals of K become principal in the compositum of K and the cubic field of conductor 7, namely C K −→ C K(cos(2π/7)) is zero. (This can be also checked from a numerical calculation CK(cos(2π/7)) ' Z/3. Our method in this paper shows that the third and the fifth conditions in [4, page 421] can be replaced by a simpler condition that a prime over ` is not principal.) In general, we can show the following by the method in the proof of Proposition 4.3. We use the notation in Sect. 4. Let χ be a Dirichlet character with order prime to p, and K = kχ be the field corresponding to the kernel χ of χ. Assume that A K ' Oχ / pn . If ` is a prime number which splits χ completely in K, and satisfies ` ≡ 1 (mod 2 pn ), and L if A K ×is generated n χ by a prime above `, and if (E K ⊗ Z/ p ) −→ ( v|` κ(v) ⊗ Z/ pn )χ χ χ is surjective, then the canonical map A K −→ A L is zero where L is the + n subfield of K(µ` ) such that [L : K] = p .

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