Some Non-Abelian Extensions over Zp

Some Non-Abelian Extensions over Z

-Extensions and

p-Class Field Towers

Z

p

Satoshi FUJII

A dissertation submitted for the degree of Doctor of Science at Waseda University

2005

6

-1

Acknowledgements

I would like to express my sincere gratitude to Professor Norio Adachi. During my activities as a graduate student of doctorial course, he unceasingly gave very warm en- couragements and helpful advice.

I would like to express my thanks to Professor Manabu Ozaki for many valuable discus- sions and advice. His great works motivated me to study the subjects of this thesis.

I would also like to express my thanks to Doctor Tsuyoshi Itoh, Doctor Yasushi Mizusawa and Doctor Gen Yamamoto for many helpful suggestions and a lot of advice.

I would also like to express my thanks to Keiji Okano for collaboration with me on the work in Chapter 2. A part of contents in Chapter 2 is work in his master thesis.

I would also like to express my thanks to Professor Ki-ichiro Hashimoto, Professor Keiichi Komatsu and all other members of the number theory seminar held every week at Waseda University for giving me favorable environment.

I would like to mention that the research of this thesis is partially supported by Waseda University Grant for Special Research Projects. No: 2004B-895.

Finally, I am grateful to my parents Makoto and Takiko, to my brothers Takashi and Shinpei, and to all of my friends.

Satoshi FUJII

Department of Mathematical Sciences School of Science and Engineering Waseda University

3-4-1, Ohkubo, Shinjuku-ku Tokyo, 169-8555

JAPAN

Introduction

The theory of Z_p-extensions was initiated by K. Iwasawa in the middle of the last century. He studied aZ_p-extension, a certain infinite extension of the rational number field Q, to obtain arithmetic properties about the ideal class groups and the unit groups of finite extensions of Q. Let k/Q be a finite extension. To specific an algebraic extensionK/k is called “Z_p-extension” ifK/k is a Galois extension and its Galois group Γ = Gal(K/k) is isomorphic to the additive group of p-adic integers Z_p as topological groups. Then there is a unique intermediate field kn of K/k such that [kn : k] = pⁿ for each non-negative integer n, and we callk_n the n-th layer ofK/k. Then we obtain a sequence of fields

k=k0 ⊆k1 ⊆k2 ⊆ · · · ⊆kn ⊆ · · · ⊆K =∪n≥0kn

with

Gal(k_n/k)'Z/pⁿZ.

The field k(µ_p^∞) obtained by adjoining all p-power-th roots of unity contains the unique Z_p-extension k_∞ of k, which is called the cyclotomic Z_p-extension.

The Galois group Γ acts on various arithmetic objects ofK, for example, the ideal class group, the unit group and several abelian Galois groups over K, which is also a Galois extension ofk, via the inner automorphism. In particular, the action of Γ onthe Iwasawa module X = lim←−Akn, the projective limit of the p-primary part Akn of the ideal class group of k_n by norm map provided the following great formula.

Theorem (Iwasawa [13], 1959). Let K/k be a Z_p-extension of a finite extension k/Q and k_n its n-th layer. Then there are non-negative integers λ(K/k), µ(K/k) and an integer ν(K/k) such that

#A_kn =pλ(K/k)n+µ(K/k)pⁿ+ν(K/k)

for all sufficiently large n.

The above integers λ(K/k), µ(K/k) and ν(K/k) are called the Iwasawa λ-, µ- and ν-invariant of K/k. By class field theory the Iwasawa module X is also defined as the

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Galois group of the maximal unramified pro-p abelian extension LK of K. Then X is a Z_p[[Γ]]-module, the complete group ring of Γ with coefficients in Z_p, and Z_p[[Γ]] is isomorphic to Λ = Z_p[[T]], the formal power series ring with coefficients in Z_p, since X is compact. Hence X is regarded as a Λ-module. The Iwasawa module X is a finitely generated Λ-torsion Λ-module, and the kernel of the natural restriction maps X → A_kn can be written by the action of Γ on X. Also, the integers λ(K/k) and µ(K/k) are the invariants of X as Λ-modules. Iwasawa’s formula above is essentially deduced from these properties. Once we take the limit, to regard as Λ-modules, we can deal more easily the arithmetic objects inK than in finite number fields.

On the other hand, there is a strong theory in the cyclotomic Z_p-extensions. The class number of abelian fields is closely related to special values of Dirichlet’sL-functions.

The values of Dirichlet’s L-functions satisfy some p-adic continuous conditions, which is called the Kummer congruence. From the Kummer congruence, Kubota and Leopoldt [23] constructed a p-adic analogue of Dirichlet’s L-functions, which is called the p-adic L-functions. Iwasawa [14] also constructed the p-adic L-functions by another method, and found that they are essentially considered as the limit of the Stickelberger elements in Λ. From this point of view, one could formulate the Main Conjecture, which asserts that the characteristic polynomial multipled by T on the Iwasawa module is essentially equal to the p-adic L-function. As a result, we have

Theorem. Let p be an odd prime number and ω the Teichm¨uller character of modulo p. Then

#A^ω_Q(µⁱ _p) = #Z_p/L(0, ω¹⁻ⁱ)Z_p

for all odd integer i with 3≤i≤p−2. Here A^ω_Q(µⁱ _p) stands for the ωⁱ-eigen component of A_Q(µp) and L(s, χ) the Dirichlet’sL-function of a character χ of Gal(Q(µp)/Q).

The Main Conjecture is proved by Mazur and Wiles in 1984 [26]. This is indicative of the advantage of studying the Iwasawa theory in order to know deep facts of the ideal class groups of finite extensions of Q. From this point of view, the Iwasawa theory occupies a highly crucial position not only in algebraic number theory, but also in arithmetic

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geometry.

The above theorems dealing withZ_p-extensions were deduced from the structure of the Iwasawa module X and observing the action of Γ onX, i.e. observing the splitting exact sequence

1−→X −→Gal(L_K/k_n)−→Γ^pn −→1.

LetM/K be a pro-pextension which is also a Galois extension overk. Then we also have the following splitting exact sequence of pro-p groups since Γ is a free pro-p group;

1−→Gal(M/K)−→Gal(M/k_n)−→Γ^pn −→1.

Then can we obtain some information concerning the arithmetic objects ink_n related to the extension M/K, if we know the structure of the Galois group Gal(M/K) and the action of Γ on Gal(M/K) by observing the above exact sequence? In particular, can we get results concerning non-abelian extensions, such as, the class field tower, the maximal pro-pextension unramified outside p? The studies of this thesis stands on this spirit. We will deal with three types of non-abelian extensions. This theory is said to non-abelian Iwasawa theory ofZ_p-extensions, which has been developed by Ozaki [30].

In chapter 1 we show an analogue of Iwasawa’s formula of unramified class two exten- sions ofk_n of certainZ_p-extensions. LetL_n the Hilbertp-class field and L⁽²⁾n the maximal class 2 unramified p-extension, of k_n. By class field theory, there is an isomorphism A_kn 'Gal(L_n/k_n). Thus Iwasawa’s formula describes of the behavior of [L_n :k_n]. Since [L⁽²⁾n :kn] is also finite, one expects that [L⁽²⁾n :kn] also has a formula like [Ln:kn]. In fact, the author showed an analogue of Iwasawa’s formula of [L⁽²⁾n :k_n] of certain Z_p-extension.

To prove this, we use results from the theory of central class fields and the structure of the unit groups inZp-extensions.

In chapter 2 we discuss two of unsolved problems in the theory of the p-class field towers by using the cyclotomic Z_p-extensions. The first one treats the existence of a number field k such that the maximal pro-p extension ˜L_k is infinite with the p-rank of the ideal class group of k is 2 for an odd prime number p. Schoof [33] showed that if k is a quadratic field and the p-rank of the ideal class group of k is greater than or equal

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to 3, then [ ˜L_k : k] = ∞. Also, if the p-rank of the ideal class group is 1, then ˜L_k/k is abelian, so that it is a finite extension by class field theory. The other one is concerning the structure of Gal( ˜L_k/k). The problem is whether Gal( ˜L_k/k) is torsion-free under the assumption [ ˜L_k : k] = ∞. Under the assumption that the Galois group Gal( ˜L_k∞/k_∞) of the maximal unramified pro-p extension of the cyclotomic Zp-extension k∞/k over a CM-field k is a free pro-p group, we will give a positive answer to the first problem, but give a negative answer to the other question.

In chapter 3 we give a relation of the Galois group Gal(M_k/k) of the maximal pro-pex- tensionM_kunramified outsidepof an imaginary quadratic fieldkby using the cyclotomic Zp-extension k∞/k. The structure of such extensions is related to the Iwasawa theory of Z^⊕d_p -extensions (see [24] and [25]). The author believes that to know the structure of Gal(M_k/k) is important for studying the Iwasawa theory. Also, observing the cyclotomic Zp-extensions was used to obtain some properties of Gal(Mk/k). Studying Gal(Mk/k) is also motivated by interests in the non-abelian Iwasawa theory of Z_p-extensions. In chapter 3 we also use the splitting exact sequence

1−→Gal(M_k/k_∞)−→Gal(M_k/k)−→Γ−→1.

This implies that Gal(M_k/k) ' Gal(K_k/k_∞)o Γ, hence to knowing the structure of Gal(M_k/k) is equivalent to knowing the structure of Gal(M_k/k_∞) as a pro-p Γ-operator group. Let 1→R →F →Gal(M_k/k)→1 be a minimal presentation of G_k(p) by a free pro-p group F. The Kummer theory over k_∞ tells us that the explicit structure of the maximal pro-p abelian quotient of Gal(Mk/k∞) as Λ-modules for some cases. Then we obtain a system of relations of Gal(M_k/k) modulo certain closed normal subgroup of F.

Here we set the notation in this thesis. Let Z be the ring of rational integers and Q the field of rational numbers . Throughout this thesis, we fix an algebraic closureQofQ.

A number field means a subfield of Q. For a number field F, which can have an infinite degree over Q, let F^× = F\{0} be the multiplicative group, O_F the ring of integers, E_F =O^×_F the unit group and Cl_F the ideal class group, of F, respectively. Fix a prime numberp, and letZ_p andQ_p be the ring ofp-adic integers and the field ofp-adic numbers respectively. Denote by AF =ClF[p^∞] the p-primary part of the ideal class group ofF.

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LetGbe a topological group. In this thesis, a subgroupHofGmeans a closed subgroup of G. A G-module M means a topological abelian group with a continuous G-action, namely the map G×M →M, (g, m)7→gm(g ∈G, m∈M) is continuous. Let M^G and MG be theG-invariant submodule andG-coinvariant module ofM, respectively. WhenG is a profinite group and M is a discrete G-module, let Hⁱ(G, M) be the i-th cohomology group for each i∈Z_≥0. Furthermore, if G is finite, then we denote by ˆHⁱ(G, M) the i-th Tate cohomology group for each i∈Z.

LetR be a commutative ring with the unity and N anR-module. Denote by Tor_R(N) the R-torsion submodule of N. For an elementr∈R, put N[r] ={n ∈N|rn= 0}.

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Contents

Introduction 1

1. On a Second Iwasawa’s Formula of Certain Z_p-Extensions 7

1.1. Notation and Result 7

1.2. Lemmas 8

1.3. Proof of Theorem 1.1 12

2. Some Problems on p-Class Field Towers 14

2.1. Preliminaries 15

2.2. Proof of Theorem 2.1 18

2.3. Proof of Theorem 2.2 and Theorem 2.3 20

3. On the Maximal Pro-p Extension Unramified Outside p of an Imaginary

Quadratic Field 24

3.1. Results 24

3.2. The Galois Module Structure of thep-Unit Group. 26 3.3. Iwasawa’s Theorem on the Kummer Pairing For Abelian Fields. 29

3.4. Some Properties of Pro-pGroups. 36

3.5. Proof of Theorem 3.1 and Proposition 3.1 37

3.6. Proof of Theorem 3.2. 39

3.7. Examples 44

References 46

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1. On a Second Iwasawa’s Formula of Certain Zp-Extensions

1.1. Notation and Result. Let p be a prime number and k a number field of finite degree over Q. Let K/k be a Z_p-extension over k. For an integer n ≥ 0, we denote by k_n the n-th layer of the extension K/k, namely k_n is the unique intermediate field of K/k such that [kn : k] = pⁿ. Let ˜LK/K and ˜Lkn/kn be the maximal unramified pro-p extensions and put ˜G_K = Gal( ˜L/K) and ˜G_kn = Gal( ˜L_n/k_n) for all non-negative n. We define the subgroups C_i( ˜G_K) of ˜G_K by the descending central series

G˜_K =C₁( ˜G_K)⊇C₂( ˜G_K)⊇ · · · ⊇C_i( ˜G_K)⊇ · · · , C_i+1( ˜G_K) = [C_i( ˜G_K),G˜_K].

Then we consider the modules X_K⁽ⁱ⁾ =Ci( ˜GK)/Ci+1( ˜GK), and call X_K⁽ⁱ⁾ the i-th Iwasawa module. We define the subgroupsC_i( ˜G_kn)⊆G˜_kn and the modulesX_k⁽ⁱ⁾_n similar toC_i( ˜G_K) and X_K⁽ⁱ⁾, respectively. Note that X_k⁽¹⁾_n is isomorphic to the Sylow p-subgroup of the ideal class groupA_kn ofk_nand thatX_K⁽¹⁾ is the Iwasawa moduleX_K ofK/k which is defined to be the projective limit lim←−A_kn with respect to the norm maps. By definition, the complete group ring ΛK/k =Zp[[Gal(K/k)]] acts on X_K⁽ⁱ⁾ in the natural way, namely Gal(K/k) acts via the inner automorphism. For i = 1, Iwasawa studied the Λ_K/k-module structure of X_K and deduced the following celebrated formula.

Theorem A (Iwasawa [13], 1959). There exist non-negative integers λ(K/k),µ(K/k) and an integer ν(K/k) such that

#A_kn =pλ(K/k)n+µ(K/k)pⁿ+ν(K/k)

for all sufficiently large n.

These integers λ(K/k), µ(K/k) and ν(K/k) are called the Iwasawa invariants of K/k.

We remark that λ(K/k) and µ(K/k) are the invariants of the Λ_K/k-module X_K, namely the invariants of finitely generated Λ_K/k-torsion Λ_K/k-module XK. In the theory XK, these invariants play impotent roles, so we want to define such invariants in the theory of X_K⁽ⁱ⁾. But Ozaki showed the following proposition.

Proposition A (Ozaki[30]). (1) rank_ZpX_K⁽ⁱ⁾ = dim_QpX_K⁽ⁱ⁾⊗_ZpQ_p <∞.

(2) If µ(K/k) = 0, then X_K⁽ⁱ⁾ is a finitely generated Zp-module for each i≥1. Otherwise,

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X_K⁽ⁱ⁾ is not finitely generated even if over Λ_K/k for each i≥2.

By Proposition A, since λ(K/k) = dimQpXK ⊗ZpQp, we can define the i-th Iwasawa λ-invariant λ⁽ⁱ⁾(K/k) by λ⁽ⁱ⁾(K/k) = dim_QpX_K ⊗_ZpQ_p. Also, if µ(K/k) = 0, then X_K is a finitely generated Z_p-module with rank_ZpX_K = λ(K/k), and X⁽ⁱ⁾ is also a finitely generated Zp-module. In this case, the i-th Iwasawa λ-invariant λ⁽ⁱ⁾(K/k) and the λ- invariant of X_K⁽ⁱ⁾ as Λ_K/k-module coincide. When µ(K/k) > 0, we can not define the invariant likeµ(K/k) well. However, under the assumptionµ(K/k) = 0, we can raise the following question on the higher Iwasawa modules.

Question. Suppose µ(K/k) = 0. Then, for each i ≥ 2, does there exist an integer ν⁽ⁱ⁾(K/k) depending on only K/k and i such that #X_k⁽ⁱ⁾_n = p^{λ(i)(K/k)n+ν(i)(K/k)} for all sufficiently large n?

In this chapter, we will prove an asymptotic formula ofX_k⁽²⁾_n in terms of p^λ(2)(K/k)n. Theorem1.1. Suppose thatµ(K/k) = 0and that there is only one prime ofK ramified in K/k. Then #X_k⁽²⁾_n =p^λ(2)(K/k)n+O(1).

1.2. Lemmas. To prove Theorem 1.1, we use the following lemmas. Let L be a finite Galois extension ofF and G= Gal(L/F) its Galois group. We regard Z_p as a G-module with trivial action.

Lemma 1.1. Let F be a number field of finite degree and L/F an unramified finite p-extension of finite degree such that L contains the Hilbert p-class field of F and let G= Gal(L/F) (p-extension means a Galois extension withp-power degree). Put HL/F = E_F/E_F ∩N_L/FL^×. Then we have the exact sequence

0−→ H_L/F −→Hˆ⁻³(G,Z_p)−→(A_L)_G −→0.

Furthermore, for any subfield k of F such that L/k and F/k are Galois extensions, the above sequence is exact as Gal(F/k)-modules.

Proof. This lemma is well known as the central class field theory. For example, see

Fr¨ohlich [5] ¤

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LetL_kn =L⁽¹⁾_kn = ˜L_k^C_n^{2( ˜Gn)}andL⁽²⁾_kn = ˜L_k^C_n^{3( ˜Gkn)}. ThenL_kn is the Hilbertp-class field ofk_n and L⁽²⁾_kn is the central p-class field of L_kn/k_n, respectively. It follows from the definition of X_k⁽²⁾_n that

X_k⁽²⁾_n = Gal(L⁽²⁾_kn/Lkn)'(ALn)Gal(L_kn/kn).

Since L_kn/k_n is an abelian extension, we have ˆH⁻³(Gal(L_kn/k_n),Z_p)'A_kn∧A_kn, where

∧ means the exterior product. Form ≥n≥0, let ˜N_m,n :A_km∧A_km →A_kn∧A_kn be the homomorphisms induced by the norm maps. Note that the diagram

Hˆ⁻³(Gal(L_km/k_m),Z_p) −−−→^∼ A_km∧A_km

dm,n



y ^Nm,n0

 y Hˆ⁻³(Gal(L_kn/k_n),Z_p) −−−→^∼ A_kn∧A_kn

is commutative form≥n ≥0. Here, we denote byd_m,n the map induced by the restriction map Gal(L_km/k_m)→Gal(L_kn/k_n) (σ 7→σ|_Lkn). By Lemma 2.1, we have

Lemma 1.2. The following diagram is exact and commutative as Γ-modules 0 −−−→ H_Lm/km −−−→ A_km ∧A_km −−−→ X_k⁽²⁾_m −−−→ 0

norm

 y

 y^N˜m,n



yrestriction

0 −−−→ H_Ln/kn −−−→ A_kn ∧A_kn −−−→ X_k⁽²⁾_n −−−→ 0.

for m≥n ≥0. The action of σ∈Γ on Akn∧Akn is given by σ(x∧y) = (σx)∧(σy) for x, y ∈A_kn.

The next lemma tells us that the knowledge ofAkn∧Akn gives information aboutHLn/kn

and X_k⁽²⁾_n.

Lemma 1.3. Let {An} and {Bn}(n ≥ 0) denote projective systems of finite abelian p-groups with the following exact commutative diagram:

0 −−−→ A_n+1 −−−→ B_n+1

surjective

 y



y^surjective 0 −−−→ A_n −−−→ B_n, and let A= lim←−A_n, B = lim←−B_n.

(1) Suppose that B is a finitely generated Zp-module. Then B= TorZpB is isomorphic to

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a subgroup of Bn for all sufficiently large n.

(2) Suppose that B is a finitely generated Z_p-module and

Ker(B_n+1/B→B_n/B) = (B_n+1/B)[p]

for all sufficiently large n (here we regard B a subgroup of Bn for all sufficiently large n by (1).) Then there exist integers a and b such that

#An =p^λ(A)n+a,

#B_n =p^λ(B)n+b

for all sufficiently large n. Here we denote by λ(M) the Zp-rank of a finitely generated Z_p-module M.

Proof. (1) Let b_n = Ker(B →B_n). Then {b_n} is a system of fundamental neighbor- hoods of B. Since Bis finite, there isn₀ such that b_n∩B= 0 forn≥n₀. It follows from the finiteness of B_n that b_n is a free Z_p-module of rank λ(B) for all sufficiently large n.

From the freeness of bn, we see thatB maps toBn injectively andBn is a product of the image of Band some subgroup of B_n.

(2) Let B⁰ =B/B. By (1), we have

0−→b_n −→B⁰ −→B_n/B−→0 (exact),

and dim_Z/pZ(B_n/B)/p(B_n/B) = λ(B) for all sufficiently large n. By the commutative diagram

0 −−−→ b_n+1 −−−→ B⁰ −−−→ B_n+1/B −−−→ 0

 y

°°

°

 y

0 −−−→ b_n −−−→ B⁰ −−−→ B_n/B −−−→ 0, and the snake lemma, we have

Ker(B_n+1/B→B_n/B) = (B_n+1/B)[p]'b_n/b_n+1. It follows from

(B_n+1/B)[p]'(B_n+1/B)/p(B_n+1/B)'(Z/pZ)^⊕λ(B) that b_n+1 =pb_n.

Fix an integer n0 ≥ 0 such that bn ∩B = 0 and bn+1 = pbn for all n ≥ n0. Let

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#(B_n0/B) = p^λ(B)n0+b0 for an integer b⁰. Since

#Ker(B_n+1/B→B_n/B) = #(B_n+1/B)[p] =p^λ(B),

we have #(Bn/B) = p^λ(B)n+b0 if n≥n0. Then #Bn= #(Bn/B)#B=p^λ(B)n+b0#B. Let p^b =p^b0#B. Then #B_n=p^λ(B)n+b for all sufficiently large n.

Leta_n = Ker(A→A_n) and A= Tor_ZpA. SinceA ⊆B we have A=A∩B. It follows from the exact commutative diagram

0 −−−→ A_n+1/A −−−→ B_n+1/B

 y

 y 0 −−−→ A_n/A −−−→ B_n/B

that Ker(An+1/A → An/A) = (An+1/A)[p]. Then an+1 = pan for all sufficiently large n.

The remaining part is proved as in the case of B_n. ¤

Lemma 1.4 (Grandet-Jaulent [7]). Let K/k be a Z_p-extension with a number field k.

Assume that the Iwasawa µ-invariant of K/k is 0. Then we have the following commu- tative diagram for all sufficiently large n:

Akn+1

−−−→∼ ³L_λ(K/k)

i=1 Z/p^ai+n+1Z

´

⊕TorZpXK norm

 y



ynatural surjection

A_kn −−−→^∼ ³L_λ(K/k)

i=1 Z/p^ai+nZ

´

⊕Tor_ZpX_K,

where a₁, . . . , a_λ(K/k) are integers independent of n and satisfy the inequalities a₁ ≤ a₂ ≤

· · · ≤a_λ(K/k).

The following is keystone of the proof of main theorem.

Lemma1.5. LetK/k be aZ_p-extension, and putΓ_n = Gal(K/k_n)andΓ_m,n = Γ_n/Γ_m = Gal(km/kn). For m ≥ n ≥ 0, let Bm⁽ⁿ⁾ = {c∈ Akm|^∃a ∈ c s.t. σa = a for all σ ∈ Γn} ⊆ A_k^Γ_mⁿ. If p is not decomposed in K/Q, then A_k^Γ_mⁿ/Bm⁽ⁿ⁾ 'Hˆ⁰(Γ_m,n, E_km).

Proof. For the proof of this lemma, see Theorem 1 of Greenberg [8]. ¤ LetSbe a set of primes which is lying abovepandD_nbe the subgroup ofA_kn generated by the classes which contains a prime above S. Put A⁰_kn = A_kn/D_n and define A⁰_kn → A⁰km as the homomorphism induced by the natural inclusion kn→km.

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Lemma 1.6. The order of the kernel of the homomorphism A⁰kn →A⁰km is bounded for m≥n ≥0.

Proof. For the proof of this lemma, see Iwasawa [15] ¤ 1.3. Proof of Theorem 1.1. Let H = lim←− H^Ln/kn, where the projective limit is taken with respect to the norm maps, and I_n = Im(H → H_Ln/kn) the image of the projection map. Applying Lemma 1.3 toI_n and A_kn∧A_kn, we see that there exist integers a and b such that #Akn∧Akn =p^{λ(XK∧XK)n+a} and #In =p^λ(H)n+b for all sufficiently large n. By Lemma 1.2, we have

#X_k⁽²⁾_n = #A_kn ∧A_kn/#H_Ln/kn

=p^λ(2)(K/k)n+(a−b)/[H_Ln/kn :In]

Sinceλ(X_K∧X_K)−λ(H) =λ(X_K⁽²⁾) = λ⁽²⁾(K/k). Hence, we have to prove that [H_Ln/kn : I_n] is bounded for n ≥ 0. We can easily see that I_n = T

m≥nN_km/knH_Lm/km. Thus if

# ˆH⁰(Γ_m,n, E_km) is bounded form ≥n≥0, then [H_Ln/kn :I_n] is bounded forn according to the following commutative diagram:

E_km −−−→ H_Lm/km

norm

 y

 y^norm E_kn −−−→ H_Ln/kn.

Therefore, we have only to prove that # ˆH⁰(Γ_m,n, E_km) is bounded form≥n ≥0. Now we remark that we may assume that K/k is totally ramified at the prime Pwhich ramified inK/k.

Letpn be the unique prime of kn under Pand Dn the subgroup of Akn generated by a class which contains a power of p_n. Since p_n is Γ-invariant and #A^Γ_kn ≤ #A^Γ_knⁿ = #A_kn, we see that D_n⊆ A^Γ_kn and the order of D_n is bounded. Then there is a constant C₁ > 0 such that #Akn/#A⁰kn = #Dn ≤C1 for alln ≥0. Now we consider the homomorphism A⁰_kn → A⁰_km induced by the natural inclusion k_n → k_m. Clearly the image of the above map is contained inBm⁽ⁿ⁾/D_m. Conversely, let cmodD_m be an element ofBm⁽ⁿ⁾/D_m. Then there is an ideal a ∈ c of k_m such that σa = a for all σ ∈ Γ_n. Since every Γ_n-invariant ideal of k_m is a product of a power of the prime above p_m and an ideal of k_n, we may assume that the class c contains an ideal of kn. Therefore Im(A⁰kn →A⁰km) = Bm⁽ⁿ⁾/Dm.

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By Lemma 1.6, there is a constant C₂ >0 such that #A⁰_kn#D_m/#Bm⁽ⁿ⁾ ≤C₂. Hence we have

# ˆH⁰(Γm,n, Ekm) = #A^Γ_kmⁿ/#B_m⁽ⁿ⁾

≤ #A_knC₂

#A⁰_kn#D_m

≤C₁C₂/#D_m

≤C₁C₂,

because n≥n₀. This completes the proof of Theorem 1.1. ¤

We must mention that Ozaki recently showed that the Question of this chapter is always affirmatively answered for each i≥1.

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2. Some Problems on p-Class Field Towers

Letpbe a prime number andk a finite extension ofQ, the field of rational numbers. Let L_k be the maximal unramified abelian p-extension ofk and define Lⁱ_k by L⁰_k=k, Lⁱ⁺¹_k :=

L_Lⁱ

k (i≥0) inductively. Then we have a sequence of fields k =L⁰_k ⊆L¹_k ⊆L²_k ⊆ · · · ⊆Lⁱ_k ⊆ · · ·,

which is called the p-class field tower of k. Put ˜L_k=∪_i≥0Lⁱ_k and denote its Galois group Gal( ˜L_k/k) over k by ˜G_k. Then Gal(L_k/k) = ˜G^ab_k ' A_k, the p-primary part of the ideal class group of k, by the class field theory. The extension ˜Lk is the maximal unramified pro-p extension of k. For a number field k which can be an infinite extension of Q, we use the same notation, e.g. ˜L_k and ˜G_k, as in a finite number field. There are interesting problems concerning p-class field towers, one of which is about the existence of a finite extension k of Q such that # ˜G_k = ∞ with dim_Z/pZA_k/A^p_k = 2. In the case p= 2, Hajir showed in [9] that the 2-class field tower of k = Q(√

−5·11·461) is infinite and that A_k ' Z/2Z⊕Z/4Z. However, we have no such example of k for any odd prime number p. Another problem is whether ˜G_k is torsion-free under the condition that # ˜G_k = ∞.

Hajir presented this problem in [10], but we have no answer yet, even if any partial one.

In this chapter, we study the above two questions by using the theory of Z_p-extensions under the assumption that ˜Gk∞ is a free pro-pgroup, and deduce the following results.

Theorem 2.1. Let p be an odd prime number, k a CM-field with the maximal totally real field k⁺, k_∞ the cyclotomic Z_p-extension of k and k_n its n-th layer. Suppose that (1) ˜G_k∞ = Gal( ˜L_k∞/k_∞) is a free pro-p group with rank λ≥2,

(2) the prime p does not split in k_∞/Q, (3) the class number of k⁺ is prime to p.

If p≥5and dim_Z/pZAk0/A^p_k0 = 2, or p= 3 andAk0 'Z/3^aZ⊕Z/3^bZwith a, b≥2, then L˜_k1/k₁ is an infinite extension.

Theorem 2.2. Let p be an odd prime number, k a CM-field, ζ_p a primitive p-th root of unity, k⁺ the maximal totally real subfield of k and k_∞/k the cyclotomicZ_p-extension.

Suppose that

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(1) ˜G_k⁺_∞ 'Z/pZ, and G˜_k∞Lk∞⁺ is a free pro-p group, (2) k_∞/k is totally ramified at any prime lying above p, (3) λ(k_∞/k) ≥ 1 + 2√

1 +δ+s ; where λ(k_∞/k) is the Iwasawa λ-invariant of k_∞/k, s the number of primes of k_∞ lying above p, and δ= 1 or 0 according as ζ_p ∈k or not.

If dimZ/pZAkn/A^p_kn = dimZ/pZG˜^ab_k∞/( ˜G^ab_k∞)^p, then # ˜Gkn = ∞ and G˜kn has an element of order p.

Theorem 2.2 follows from the next theorem.

Theorem 2.3. Let p be an odd prime number, K/k an unramified Galois extension of degree p of CM-fields and denote by K⁺ and k⁺ the maximal totally real field of K and k respectively. Let K∞, k∞ the cyclotomic Zp-extension of K and k respectively.

Assume that Gal( ˜L_K∞/K_∞) is a free pro-p group and Gal( ˜L_K∞⁺/K_∞⁺) is trivial. Then G˜_k∞ = Gal( ˜L_k∞/k_∞) is expressed as

G˜_k∞ '





(Z/pZ×Z_p) `

F_{λ(k∞/k)−1} (if ζ_p ∈k) Z/pZ `

F_λ(k∞/k) (if ζ_p ∈/ k),

where ζ_p is a primitive p-th root of unity, F_d is a free pro-p group of rank d, λ(k_∞/k) is the Iwasawa λ-invariant of k_∞/k, and the symbol `

stands for the free pro-p product.

In each proof of Theorem 2.1 and Theorem 2.3, the assumption that the Galois groups G˜_k∞ and ˜G_K∞ are free plays a crucial role. But it is slightly difficult to check that these Galois groups are free. To find out such examples will be an important problem in the theory of p-class field towers.

In Theorem 2.1, let’s remark that if λ = 2 and n ≥ 0 then dim_Z/pZA_kn/A^p_kn = 2. It is known under some similar situations that # ˜G_kn = ∞ for all sufficiently large n. In Theorem 2.3, let’s remark that the assumption Gal( ˜L_K⁺_∞/K_∞⁺) = 0 is equivalent to the statement Gal( ˜L_k⁺_∞/k_∞⁺)'Z/pZ, and that the Iwasawa µ-invariants of K_∞/K and k_∞/k are zero, since Gal( ˜LK∞/K∞) is a free pro-pgroup.

2.1. Preliminaries. In the following, for any finite number field k, we use the nota- tion k_∞, λ(k_∞/k), µ(k_∞/k) for the cyclotomic Z_p-extension of k and the Iwasawa λ, µ-invariants of k∞/k respectively. For any odd prime p and any CM-field k, we also

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use the notation k⁺ for the maximal totally real subfield of k, and put λ(k∞/k)⁻ = λ(k_∞/k)−λ(k_∞⁺/k⁺),µ(k_∞/k)⁻=µ(k_∞/k)−µ(k⁺_∞/k⁺).

To prove our results, we will use some lemmas in this section. First, we introduce pro-p group theoretical lemmas. The first one is a criterion for the infiniteness of a finitely generated pro-p group. The second one is a pro-p analogue of the theorem of Dyer and Scott in [3]. It has been proved in [32] by Scheiderer in the case where a given group is finitely generated, and extended by Herfort, Ribes and Zalesskii in [12] to the general case (but we need only the finitely generated case).

Lemma 2.1 (Theorem 7.21 of [20]). Let p be a prime number and G a finitely gen- erated pro-p group, i.e., G is generated by finite elements of G topologically. Put d :=

dim_Z/pZH¹(G,Z/pZ) and r := dim_Z/pZH²(G,Z/pZ). Let 1−→R−→F −→G−→1

be a minimal presentation of G by a free pro-p group F. Let I ⊆ Z/pZ[[F]] be the argumentation ideal of the complete group ring Z/pZ[[F]]of F with coefficients in Z/pZ, and put F(n) := {a ∈ F|a−1 ∈ Iⁿ} (The set {F(n)|n ≥ 1} is called the Zassenhaus filtration.)

(1) If a∈F and b ∈F(n), then [a, b] =aba⁻¹b⁻¹ ∈F(n+ 1) and b^p ∈F(pn).

(2) If G is finite and R ⊆F(n), then r >(dⁿ(n−1)ⁿ⁻¹)/nⁿ. ¤ Lemma2.2. (Scheiderer [32], Herfort-Ribes-Zalesskii [12])Letpbe a prime number and let G be a pro-p group that contains an open free pro-p subgroup H of index p. Then G is written as a free pro-p product:

G' Ãa

x∈X

(Z/pZ×F^(x))

!` F,

where F is a free pro-pgroup and F^(x)’s are free pro-p groups indexed by a profinite space X (the profinite space X is a quotient of the space of conjugacy classes of elements of

order p in G). ¤

The following two lemmas are basic facts in the theory of Z_p-extensions. Lemma 2.3 describes the relation rank of ˜G_kn by using ˜G_k∞. Note that Gal(k_∞/k)(' Z_p) acts on G˜k∞ for any Zp-extensionk∞ ofk via the inner automorphism, since thep-cohomological

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dimension of Gal(k∞/k) is 1. Lemma 2.4, called Kida’s formula, is an analogue of the formula of Riemann-Hurwitz.

Lemma 2.3. Let k be a finite extension of Q and k∞/k a Zp-extension which is to- tally ramified at all primes lying above p. Then we may assume that an extension

˜

γ ∈ Gal( ˜L_k∞/k) of a topological generator γ ∈ Γ = Gal(k_∞/k) generates the inertia subgroup of a prime lying above p. Let Rn be the kernel of the natural mapping from G˜_k∞ to G˜_kn and s the number of primes of k_∞ lying above p. Then there are elements g₂, . . . , g_s ∈G˜_k∞ such that R_n is generated by [Γ^pn,G˜_k∞] =hσ(g)g⁻¹|σ ∈Γ^pn, g ∈Gi and ν_n(g_i) = g_iγ(g_i)· · ·γ^pn−1(g_i) (2≤i≤s) as a closed normal subgroup of G˜_k∞.

Proof. For a proof, see [30] or Lemma 13.15 of [35]. ¤ Lemma 2.4. (Kida [18]) Letp be an odd prime, and letk andK be CM-fields such that K/k is a Galois p-extension. Let K_∞ (resp. k_∞) be the cyclotomic Z_p-extension of K (resp. k). Assumeµ(k_∞/k)⁻= 0. Then µ(K_∞/K)⁻= 0 and

λ(K∞/K)⁻ = [K∞ :k∞](λ(k∞/k)⁻−δ) +X

w+

(e(w⁺)−1) +δ,

where the sum on the right side is taken over all finite non-p-primes w⁺ on K_∞⁺ split in the extension K_∞/K_∞⁺, e(w⁺) is the ramification index in K_∞⁺/k_∞⁺ of w⁺, and δ = 1 or 0

according as ζ_p ∈k or not. ¤

As a corollary to Lemmas 2.1 and 2.3, we obtain an upper bound of the relation ranks of ˜G_kn and a sufficient condition for thep-class field tower to be infinite.

Corollary 2.1. Let k be a finite extension of Q and k_∞/k a Z_p-extension which is totally ramified at any prime lying above p. Put hⁱ_n = dim_Z/pZHⁱ( ˜G_kn,Z/pZ) and hⁱ = dim_Z/pZHⁱ( ˜G_k∞,Z/pZ). Suppose that h² <∞. Then h¹_n≤h²_n≤h¹_n+h²+s−1. In particular, if h¹_n≥2 + 2√

h²+s, then # ˜G_kn =∞.

Proof. Applying the five term sequence to the sequence 1−→Rn −→G˜k∞ −→G˜kn −→1,

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