On the mu and lambda invariants of the logarithmic class group

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On the mu and lambda invariants of the logarithmic class group

Jose Ibrahim Villanueva Gutierrez

To cite this version:

Jose Ibrahim Villanueva Gutierrez. On the mu and lambda invariants of the logarithmic class group.

2019. �hal-01705140v4�

LOGARITHMIC CLASS GROUP

JOS´E-IBRAHIM VILLANUEVA-GUTI´ERREZ

Abstract. Letℓbe a rational prime number. Assuming the Gross-Kuz’min conjec- ture along aZℓ-extensionK∞of a number fieldK, we show that there exist integers e

µ,eλandνesuch that the exponent ˜enof the orderℓ^e˜nof the logarithmic class group Cℓfn for then-th layerKn ofK∞ is given by ˜en=µℓeⁿ+λne +ν, fore nbig enough.

We show some relations between the classical invariantsµandλ, and their logarith- mic counterparts µeand eλfor some class ofZℓ-extensions. Additionally, we provide numerical examples for the cyclotomic and the non-cyclotomic case.

1. Introduction

Let K be a number field and ℓ a prime integer. The logarithmic class group Cℓf_K of K is theZ_ℓ-module measuring the gap between the cyclotomic Z_ℓ-extension K^c of K and the maximal abelian pro-ℓ-extension K^lc of K which splits completely over K^c, i.e. Cℓf_K is isomorphic to the relative Galois group Gal(K^lc/K^c) (e.g. [9], [10, Thm. &

Conj. 2.3]). The logarithmic class group is conjectured to be finite for every number field, this is equivalent to the Gross-Kuz’min conjecture [13,§2]; its finiteness is verified in the abelian case and some other cases (e.g. [12, 11]). As an arithmetic invariant of a number field the logarithmic class group has importance in its own. The logarithmic class group is effective by computational methods [3, 2, 1] and it is related to the wild kernels inK-theory [14]. In spite of its tight relation to theℓ-class groupCℓ_K ofK, it behaves differently in several situations (see§5).

We are interested in the study of the logarithmic class group alongZ_ℓ-extensions, in the spirit of Iwasawa’s work for class groups [7, 8]. This has been done in the cyclotomic case by Jaulent in [9]; in this situation the logarithmic class group attached to the n-th layer of the cyclotomic Z_ℓ-extension has a nice interpretation as a quotient of Kuz’min-Tate modules [13]. This interpretation is no longer valid for non-cyclotomic Z_ℓ-extensions whenever the base field admits such an extension.

Assuming the Gross-Kuz’min conjecture, the main result of this work provides a method for the non-cyclotomic case, and therefore completes the study of the logarithmic class group alongZ_ℓ-extensions:

Theorem 1.1. Let K∞/K be a Z_ℓ-extension. Assume the Gross-Kuz’min conjecture is valid along the Z_ℓ-extension K_∞. Let Cℓf_n be the logarithmic class group of K_n and

Date: April 1, 2019.

This paper was written under the financial support of CONACYT (The Mexican Council of Science and Technology) as part of the author’s PhD project.

let ℓ^een be its order. There exist integers eλ,µe≥0 and νesuch that e

e_n=µℓe ⁿ+eλn+ν,e for n big enough.

This article is organized as follows. In Section 2 we recall briefly the basic definitions and properties of the logarithmic arithmetic that are helpful for our purposes, nonetheless, for more on logarithmic arithmetic we urge the reader to consult the seminal paper of Jaulent [10]. Then we show some results on logarithmic ramification in Z_ℓ-extensions.

In Section 3 we construct a Λ-module, which is the logarithmic analogue to the Λ- module X in [16,§1], to which we apply the codescent techniques. We show that this logarithmic Λ-module is noetherian and assuming the Gross-Kuz’min conjecture also torsion.

In Section 4 we prove Theorem 1.1 in the non-cyclotomic case as well as a set of results involving theµe and eλlogarithmic invariants for some class of Z_ℓ-extensions.

Finally, in section 5 we provide numerical examples, in both the cyclotomic and non- cyclotomic case, of logarithmic class groups in the first layers of Z_ℓ-extensions. Addi- tionally we compare this information with the results in [6] to explicitly compute the e

µ,eλand eν logarithmic invariants.

Acknowledgements: I am very grateful to professor Jean-Fran¸cois Jaulent for having introduced me to the field of logarithmic arithmetic and his exemplary guidance. The author sincerely thanks the anonymous referee for the remarks and careful reading of the text, aiming to make the paper more accessible.

2. Logarithmic arithmetic

From now on we fix a number field K and a prime numberℓ6= 2¹. Let Pl⁰_K be the set of finite places of K.

For any prime plet Qc^c_p be the cyclotomic Z-extension ofb Q_p. For a placep abovep we define the logarithmic ramification indexee_p and the logarithmic inertia degreefe_p as

e

ep= [Kp:Kp∩Qc^c_p] and fep= [Kp∩Qc^c_p :Qp]

respectively. Similarly, for a local extension LP/Kp we define the relative logarithmic ramification indexee_P|p and relative logarithmic inertia degree fe_P|p as

e

e_P|p = [L_P:L_P∩Qc^c_pK_p] and fe_P|p= [L_P∩Qc^c_pK_p :K_p].

Proposition 2.1. Let L_P/K_p be a local extension. The logarithmic ramification index and the logarithmic inertia degree satisfy the following properties:

(1) Multiplicative relation: n_p = [K_p :Q_p] =e_pf_p =ee_pfe_p, where e_p and f_p are the usual ramification index and inertia degree.

(2) For all q6=p: v_q(e_p) =v_q(ee_p).

(3) Multiplicative in towers: ee_P=ee_P|peep and fe_P=fe_P|pfep.

1The following results and statements have analogous formulations when ℓ= 2. In order to ease notations we will skip the treatment of such a prime.

Proof: This is [10, Thm. 1.4] and subsequent remarks.

We endow K with a family ofℓ-adic logarithmic valuations (ev_p)_p∈Pl⁰

K taking values in Z_ℓ and which are defined as follows

e vp(x) =







v_p(x) ifp∤ℓ,

Log_ℓ(N_Kp/Q

ℓ(x))

degp ifp|ℓ,

forx∈K_p^×.

Here vp denotes the classical valuation, N_Kp/Qℓ is the local norm, Log_ℓ is Iwasawa’s logarithm and degpis a normalizing term defined²as follows

degp=fe_pdegp where degp=

Log_ℓ(p) if p6=ℓ, Log_ℓ(1 +ℓ) if p=ℓ.

Remark 2.2. It is clear that evp(x) takes values inZ_ℓ for p∤ℓ. This is in fact the case also for places p over ℓ. Consider the function

hp(x) = Log_ℓ(N_Kp/Qℓ(x)) npdegp .

If Kp =Q_ℓ we have im(h_ℓ) = Z_ℓ, since Q^×_ℓ ≃ℓ^Z·µ_ℓ−1·U_ℓ⁽¹⁾ and deg(ℓ) = ℓ·η where η ∈ Z^×_ℓ. In particular h_ℓ(1 +ℓ) = 1. For general K_p we have that im(h_p) is the Z_ℓ- module [Kp : Kp∩Qc^c_ℓ]⁻¹Z_ℓ [10, Prop. 1.2], moreover evp = eephp [10, Thm. 1.4]. In particular this makes the logarithmic divisor map below well defined.

We define the logarithmic divisorsDℓ_K as the freeZ_ℓ-module Dℓ_K=L

p∈Pl⁰_KZ_ℓp and we extend additively the mapping deg to all of Dℓ_K, that is, for a logarithmic divisor d:

deg(d) = deg



 X

p∈Pl⁰_K

app



= X

p∈Pl⁰_K

apdegp, withap∈Z_ℓ. Let us denoteRK=Z_ℓ⊗ZK^×. The logarithmic divisor map

divf :RK −→ Dℓ_K, x7→ X

p∈Pl⁰_K

e vp(x)p,

associates to each element in the tensor product a logarithmic divisor by means of the logarithmic valuation. It turns out that the image of an element inRK has degree 0 [10, Prop.& Def. 2.2]. The imagePℓ_K of all these elements is the subgroup of principal logarithmic divisors. Therefore it is natural to consider the following exact sequence:

0−→Ee^K −→ R^K −→ D^divf ℓK deg

−→Cℓf^∗_K −→0,

we call Ee^K the group of logarithmic units (defined in the obvious way) and Cℓf^∗_K ≃ Dℓ_K/Pℓ_K the group of logarithmic classes of arbitrary degree.

Definition 2.3. The logarithmic class group Cℓf_K ⊂ Cℓf^∗_K consists of the classes of degree 0.

2The definition of deg(ℓ) given here differs to the one given in [10, Def. 1.1]. Both definitions differ by a unit inZℓand this does not affect subsequent results. With this definition it is clearer that there is an element with logarithmic valuation 1 as shown in Remark 2.2.

The group Cℓf^∗_K of logarithmic classes of arbitrary degree is isomorphic to the Galois group Gal(K^lc/K) of the maximal abelianℓ-extension ofK which splits completely over the cyclotomic Z_ℓ-extension K^c ofK. The logarithmic class group is isomorphic to the relative Galois group Gal(K^lc/K^c) [10, Thm. & Conj. 2.3]. The Gross-Kuz’min conjec- ture asserts that the Galois group Gal(K^lc/K) is aZ_ℓ-module of rank 1, equivalently, asserts the finiteness of the logarithmic class group.

Logarithmic ramification. Letp be a finite place of K. LetL/K be a finite extension, we say that p is logarithmically unramified if ˜e_P|p = 1 for every place P above p.

Hence if L/K is infinite, we say that pis logarithmically unramified if and only if it is logarithmically unramified for every finite extensionK^′/Kcontained inL. In the Galois case it might be more convenient to work with inertia groups to study ramification, so we give the corresponding description in the logarithmic setting in the very special case when Lis an abelian ℓ-extension of K.

The so called ℓ-adic class field theory [9, I.1.2] establishes a correspondence between the decomposition group Dp of the maximal abelian ℓ-extension K^ab of K with the pro-ℓ-completion Rp of the groups K_p^×, i.e. D_p ≃ Rp = lim

←−K_p^×/K_p^×ℓk. We can ex- tend the logarithmic valuations ˜vp to the R^p; in the sense that we have an analogous decomposition³ Rp = ˜π_p^ZℓUep to the classical decompositionRp =π^Z_p^ℓUp, where ˜π_p is a logarithmic uniformizer andUep is the subgroup of logarithmic units, i.e. the kernel un- der the corresponding logarithmic valuation. The image ˜I_p⊂Gal(K^ab/K) ofUep is said to be the subgroup of logarithmic inertia. Hence, we define the group of logarithmic inertia ˜I_p(L/K) of the abelianℓ-extension L/K as the image ofUep in the Galois group Gal(L/K). We sayp is logarithmically unramified inL/K if ˜Ip⊆Gal(K^ab/L). Notice that all the extensions we are considering are abelian, so we do not specify the placeP above psince the logarithmic inertia subgroups are isomorphic.

In general, ℓ-adic class field theory (e.g. [9, Thm. 1.1.13.], [10, §2]) establishes a correspondence between the Galois group of the maximal abelianℓ-extension ofK and the following quotient:

Gal(K^ab/K)≃ JK/RK, whereJK is the restricted productQ

Rp of the ℓ-adic completionsRp defined above.

Remark 2.4. Notice that in the finite case this characterisation of logarithmic ramification for abelianℓ-extensions in terms of inertia subgroups, coincides with the definition given at the beginning of the section.

In fact, as we have seen, if L/K is a finite abelian ℓ-extension and P is a place of L above p ∈Pl⁰_K, the logarithmic inertia subgroup I˜_p(L/K) of the extension L/K is the image of Uep inside the Galois group Gal(L/K)≃ JK/N_L/K(JL)RK, hence:

I˜p(L/K)≃Ue^p/Ue^p∩N_L/K(JL)RK;

3Consider the short exact sequence ofZℓ-modules 1→Ue_p → R_p →^evp Zℓ→0, where the logarithmic valuation vep is a Zℓ-epimorphism by [10, Thm 1.4]. Then R_p is the direct sum of Ue_p and a free Zℓ-module of rank 1.

on the other hand, the logarithmic ramification index is defined as the degree ee_P|p = [L_P :L_P∩K_p^c], which is equal to the order of the Galois group

Gal(L_P/L_P∩K_p^c) =UepN_LP/Kp(RP)/N_LP/Kp(RP).

SinceL/K is abelian ([9, Ch. 1§2.a]) we have

N_LP/Kp(RP) =R^p∩N_L/K(JL)RK

and soI˜_p(L/K)≃Gal(L_P/L_P∩K_p^c).

Let K_∞ be a Z_ℓ-extension of K. We denote K_n the n-th layer of K_∞, i.e. K_n is the unique subfield of K_∞ with Gal(K_n/K)≃Z/ℓⁿZ.

As a consequence of the classic case, logarithmic ramification behaves well at the tame places inZ_ℓ-extensions.

Proposition 2.5. Z_ℓ-extensions are logarithmically unramified at the places p ∤ ℓ of K.

Proof: Let K_∞/K be aZ_ℓ-extension. Let p be a place of K over p withp6=ℓ. Let P be a place inK_n lying over p.

For everynwe have thatKn is unramified atp[18, Prop. 13.2], i.e. e_P|p= 1. We have thatvq(e_P|p) =vq(ee_P|p) = 0 for everyq 6=p, for the completions K_n,Pand Kp are both extensions of Q_p, hence theq-parts of the relative indices coincide due to (2) and (3) in Proposition 2.1. Moreover ee_P|p has no p-part since p ∤|Gal(K_n/K)|. We conclude

thatee_P|p= 1 as desired.

In terms of ramification, the locally cyclotomic extension K^lc of K corresponds to the maximal abelianℓ-extension of K which is logarithmically unramified. Unlike the classical case, this extensionK^lc is infinite since it contains the cyclotomicZ_ℓ-extension K^c of K. Since K^lc is a subextension of the maximal abelian ℓ-ramified ℓ-extension of K, the group Cℓf^∗_K ≃ Gal(K^lc/K) is a noetherian Z_ℓ-module, and assuming the Gross-Kuz’min Conjecture it has rank 1. This yields the following result.

Lemma 2.6. Let K be a number field verifying the Gross-Kuz’min conjecture for the prime ℓ, let K_∞/K be a non-cyclotomic Z_ℓ-extension. Then at least one place p above ℓis logarithmically ramified in K_∞/K.

Proof: Let us suppose that K_∞ is logarithmically unramified, hence it is contained in K^lc, but then Gal(K^lc/K) would have Z_ℓ-rank two which contradicts the Gross-

Kuz’min conjecture.

Remark 2.7. The cyclotomic Z_ℓ-extension K^c of K is logarithmically unramified.

Under the same assumptions and as a consequence of the preceding lemma we have the following corollary.

Corollary 2.8. There is an integer dsuch that K_∞/K_d is logarithmically totally ram- ified for a non-empty subset of places above ℓ and logarithmically unramified (possibly inert) for the rest of the places.

Proof: Let p₁, . . . ,p_s be the places in K above ℓ which are logarithmically ramified.

The logarithmic inertia subgroups ˜Ii are closed and non zero in Γ = Gal(K∞/K) (i= 1, . . . , s), therefore we have

I˜_i≃Γ^ℓdi,

for somedi∈N(i= 1, . . . , s), takingd= maxdi, it is clear that the extensionK∞/K_d is logarithmically totally ramified at every logarithmically ramified place.

Lemma 2.9. For aZ_ℓ-extension K∞ of K the following conditions are equivalent:

(i) There exists a place p of K such that ee_P|p > 1 for some place P|p of the subextensionK₁ of K.

(ii) K₁/K is logarithmically ramified.

(iii) K_∞ is globally disjoint from the locally cyclotomic extension K^lc of K.

Proof: (i)⇔ (ii) is clear. (ii) ⇔ (iii): If K_∞∩K^lc =K_n for some n >0, then K_n/K is logarithmically unramified. In particular K1/K is logarithmically unramified.

Definition 2.10. AZ_ℓ-extensionK_∞ofKis called locally disjoint from the cyclotomic Z_ℓ-extension K^c/K if it satisfies the above conditions.

LetK_∞be a non-cyclotomicZ_ℓ-extension ofK andp₁, . . . ,p_sbe the places inK above ℓwhich ramify logarithmically. Definen₀ = min{d_i |I˜_i ≃Γ^ℓdi}. Then theZ_ℓ-extension K_∞/K_n0 ofK_n0 is locally disjoint from the cyclotomic Z_ℓ-extension K_n^c₀/K_n0.

3. The logarithmic class group along non-cyclotomic Z_ℓ-extensions Let K be a number field admitting a non-cyclotomic Z_ℓ-extension K_∞ with Galois group Γ = Gal(K_∞/K), and assume the Gross-Kuz’min conjecture is verified for K.

We denote K_n the unique subextension ofK_∞ such that Gal(K_n/K)≃Z/ℓⁿZ.

Letp₁, . . . ,p_s be the places ofK aboveℓwhich ramify logarithmically. There exist two integers n₀ and dsuch that

(1) 0≤n₀≤d,

(2) for everyn≥n0theZ_ℓ-extensionK∞/Knis locally disjoint from the cyclotomic Z_ℓ-extensionK_n^c =K_nK^c ofK_n,

(3) K∞/K_d is completely logarithmically ramified at all the places that are loga- rithmically ramified in K_∞/K_d.

The inverse limit lim

←−Cℓf^∗_Kn of the logarithmic classes of arbitrary degree of all the K_n is isomorphic to the Galois group Ce^∗ = Gal(K_∞^lc/K_∞) of the maximal abelian logarithmically unramifiedℓ-extension ofK_∞. HenceK_n^lc is the maximal sub-extension of K_∞^lc which is abelian over Knand splits completely over K_n^c, equivalentlyK_n^lc is the maximal sub-extension ofK_∞^lc which is abelian and logarithmically unramified overK_n. Hence for n≥dwe have the following isomorphism of groups

Cℓf^∗_Kn ≃Gal(K_n^lc/K_n)≃Gal(K_∞K_n^lc/K_∞).

The profinite group Γ acts onCe^∗by inner automorphisms. Since Γ is procyclic the action is uniquely defined by a topological generator γ ∈Γ. The group G= Gal(K_∞^lc/K) is

then isomorphic to the semidirect product Γ⋉Ce^∗. We can extend the action of Γ to the Iwasawa algebra Λ =Z_ℓ[[Γ]] of Γ, making ofCe^∗ a Λ-module.

For everyn≥0, we set

(1) ωn=γ^ℓn−1.

We call cyclotomic factors the termω₀ and the quotients ω_n/ω_n−1 forn≥1.

Proposition 3.1. Ce^∗ is a noetherianΛ-module. Moreover, if the Gross-Kuz’min con- jecture holds forℓ and every K_n in K_∞, it is torsion.

Proof: The locally cyclotomic extensionK_∞^lc ofK_∞is contained in the maximal abelian ℓ-extension K_∞^lr of K_∞ which is ℓ-ramified. The Λ-module Ce^∗ can be written as a quotient of the noetherian Λ-module X = Gal(K_∞^lr/K_∞) [8, Thm. 4]. This yields the noetherian property.

For n ≥ 0 let us denote L_∞/n the maximal abelian ℓ-extension of K_n which is loga- rithmically unramified over K_∞, i.e. the maximal abelian extension of K_n contained inK_∞^lc. Then

K_∞⊆L_∞/0⊆L_∞/1⊆ · · · ⊆L_∞/n ⊆ · · · ⊆K_∞^lc, and we haveK_∞^lc =S

n≥0L_∞/n. Then we have the following group isomorphisms ω_nCe^∗ ≃Gal(K_∞^lc/L_∞/n) Ce^∗/ω_nCe^∗ ≃Gal(L_∞/n/K_∞), n≥0.

Let d be as in Corollary 2.8, for n ≥ d let p₁, . . . ,p_s be the primes of K_n which are logarithmically ramified inK_∞; the number of these primes is the same for all n≥d.

For each i = 1, . . . , s let ˜I_i be the logarithmic inertia subgroup of p_i in the exten- sion L_∞/n/K_n. Since p_i is totally logarithmically ramified in K_∞ and the extension L_∞/n/K_∞ is logarithmically unramified, the groups ˜I_i are pro-cyclic isomorphic to the free Z_ℓ-module Γ^ℓn ≃ Z_ℓ. So we restrict ˜I_i to Γ^ℓn. On the other hand, since none of the places apart ofp₁, . . . ,p_s are logarithmically ramified in L_∞/n, it follows that the product ˜I1I˜2· · ·I˜s fixes the maximal logarithmically unramified subextension of L_∞/n over Kn, which is nothing else than K_n^lc, i.e. Gal(L_∞/n/K_n^lc) = ˜I1I˜2· · ·I˜s. Assuming the Gross-Kuz’min conjecture,Cℓf^∗_Kn≃Gal(K_n^lc/K_n) is a noetherianZ_ℓ-module of rank 1. Then

rank_ZℓGal(L_∞/n/K_n) = rank_ZℓGal(K_n^lc/K_n) + rank_ZℓGal(L_∞/n/K_n^lc)

≤ 1 +s from which follows that

rank_ZℓGal(L_∞/n/K_∞) = rank_ZℓGal(L_∞/n/K_n)−1≤s,

so rank_ZℓCe^∗/ω_nCe^∗ ≤s, for all n≥d, therefore for alln≥0. HenceCe^∗ is Λ-torsion by

[8, Lemma 3 (ii)].

Remark 3.2. We have just mimicked the classical proof of Iwasawa as in[8, Thm. 5].

In our case due to the structure of Cℓf^∗_Kn we have that

rank_ZℓCe^∗/ω_nCe^∗ ≤s, for alln≥0.

Proposition 3.3. Assume the Gross-Kuz’min conjecture for all K_n in K_∞. Let Ce= Gal(K_∞^lc/K_∞^c ) be the subgroup of Ce^∗ that corresponds to the inverse limit lim

←−Cℓf_Kn of the logarithmic class groups Cℓf_Kn. There is an isomorphism of Z_ℓ-modules

(2) Ce^∗ ≃C ⊕e Z_ℓ.

Proof: Consider the following exact sequence

0→ Cℓf_Kn →Cℓf^∗_Kn →Z_ℓ→0.

We take the inverse limit lim

←− for the arithmetic norm maps in each of the terms. It is a well known fact that profinite groups satisfy the Mittag-Leffler property (e.g. [15, II.7]). So we obtain the following short exact sequence after taking inverse limits

0→C →e Ce^∗→Z_ℓ →0.

The claim follows sinceZ_ℓ is free and hence projective.

Assumption on the logarithmic ramification. Before approaching our problem in great generality, let us assume first that the places of K which ramify logarithmically inK_∞ are totally logarithmically ramified. We can always do so by rewriting theZ_ℓ-extension in order to have K=K_d, withdas in Corollary 2.8.

For each place p_i ramifying logarithmically, let G = Gal(K_∞^lc/K) ≃ Ce^∗ ⋊Γ and let I˜i ⊂G its logarithmic inertia subgroup. Since K_∞^lc/K∞ is logarithmically unramified, we have that each logarithmic inertia subgroup has trivial intersection with the group Gal(K_∞^lc/K∞): ˜Ii∩Gal(K_∞^lc/K∞) = 1. SinceK∞/K is totally logarithmically ramified over p_i, the groups of logarithmic inertia ˜I_i restrict to Γ.

Since we are mainly interested in the study of the logarithmic class groups (of zero degree) Cℓf_Kn, we consider the group H = Gal(K_∞^lc/K^c). We take advantage of the fact that the cyclotomic extension is logarithmically unramified, so the logarithmic ramification groups ˜I_i ⊂Gremain unchanged inH, i.e. they are procyclic and restrict to Γ. The group H is the semi-direct product of Γ and the normal subgroup Ce, and can be written as

H= ˜IiCe=CeI˜i for i= 1, . . . , s.

Fixing topological generators ˜σ_i of ˜I_i, there exists ˜a₁, . . . ,˜a_s ∈Cesuch that

˜

σ_i = ˜a_iσ˜₁.

In particular we can take ˜σ₁ =γ for a topological generator γ of Γ.

LetH^′be the closure of the commutator subgroup ofH. We know thatH^′ ≃ω₀C ≃e TCe, the last is the isomorphism Z_ℓ[[Γ]≃Z_ℓ[[T]] given byγ 7→T+ 1 [18, Lemma 13.14].

Remark 3.4. After what we have been discussing we have the next result whose proof goes the same way as in the classical case of [18, Lemma 13.15]. Nonetheless we repro- duce the proof in our special setting.

Proposition 3.5. Let Y₀ be the Z_ℓ-submodule of Cegenerated by {˜a_i|2 ≤i ≤s} and ω₀Ce=TCe. Let Y_n= ^ω_ωⁿ

0Y₀, where ω_n

ω₀ = 1 +γ+γ²+· · ·+γ^ℓn−1 = (1 +T)^ℓn−1

T .

Then Cℓf_Kn ≃Ce/Y_n for n≥0.

Proof: First, we show the case Cℓf_K ≃Ce/Y₀. The locally cyclotomic extension K^lc is the maximal abelian extension logarithmically unramified ofK^c contained inK_∞^lc. Its Galois group Gal(K^lc/K^c) is hence equal to Gal(K_∞^lc/K^c)/Z₀, whereZ₀ is the smallest subgroup ofH= Gal(K_∞^lc/K^c) containing the commutator subgroupH^′ as well as the logarithmic inertia subgroups ˜I_i, 1≤i ≤s. The decomposition of H in a semi-direct product of ˜I₁ and Ceyields

Cℓf_K ≃ Gal(K^lc/K^c)

≃ H/Gal(K_∞^lc/K^lc)

≃ CeI˜₁/hI˜₁,a˜₂. . . ,˜a_s, ω₀Cie

≃ Ce/ha˜₂, . . . ,˜a_s, ω₀Cie

= Ce/Y₀.

Now let us assume n ≥ 1. If we replace K^c by K_n^c, the extension K_∞^c /K_n^c is also a Z_ℓ-extension. The above result can be applied by replacing the topological generatorγ byγ^ℓn and the ˜σ_i become ˜σ_i^ℓn. Notice that for every integer k, we have:

˜

σ_i^k = (˜a_iσ˜₁)^k

= ˜a_i(˜σ₁˜a_iσ˜⁻¹₁ )(˜σ₁²˜a_iσ˜₁⁻²)· · ·(˜σ₁^k−1˜a_iσ˜^−(k−1)₁ )˜σ^k₁

= ˜a^{1+˜σ1+...+˜σk}

−1 1

i σ˜₁^k

and particularly, for k = ℓⁿ, the isomorphism ˜I1 ≃ Γ gives ˜σ^ℓ_iⁿ = ˜a^ω_i^n/ω0(˜σ1)^ℓn. This implies that the groups of logarithmic inertia of H_n = Gal(K_∞^lc/K_n^c) are generated by the ^ω_ωⁿ₀˜a_i. Finally, the commutator subgroup H_n^′ of H_n is given by ω_nCe. From which

the result follows.

General case. In what follows we come back to the general situation exposed in the beginning of the section. LetK_∞be a non-cyclotomicZ_ℓ-extension ofK. Letp₁, . . . ,p_s be the places ofK which ramify logarithmically and letdthe smallest integer such that theZ_ℓ-extension K_∞/K_d is totally logarithmically ramified at all thep_i, 1≤i≤s.

Proposition 3.6. LetK_∞ be a non-cyclotomic Z_ℓ-extension of K. There exists d≥0 such that

Cℓf_Kn ≃Ce/Y_n for n≥d.

Proof: We apply the Proposition 3.5 to the extension K∞/K_d, the groupCebeing the same as the original extension. The elementγ^ℓd generates Gal(K∞/K_d^c), and naturally ω_dCegenerates the commutator subgroup of Gal(K_∞/K_d). For n≥dwe have

ω_n

ω_d = 1 +γ^ℓd+γ^2ℓd+· · ·+γ^ℓn−ℓd,

it comes that the commutator subgroup of Gal(K_∞/K_n) is given by ^ω_ωⁿ

d(ω_dCe) =ω_nCe; in general for theY_n we have Y_n= ^ω_ωⁿ

dY_d.

The exact sequence

0→Y_d→C →e Cℓf_Kd→0

sets a pseudo-isomorphismY_d∼Ce, that is a morphismY_d→Ceof Λ-modules with finite kernel and cokernel.

4. Structural parameters Let us recall a fundamental result (e.g. [18, Thm 13.12]).

Theorem 4.1 (Structure theorem of noetherian Λ-modules). Let M be a noetherian Λ-module. There is a Λ-module morphism M → E to a unique elementary Λ-module E with finite kernel and finite cokernel:

M ∼E = Λ^ρ⊕ Ms

i=1

Λ/ℓ^miΛ

!

⊕



 Mt j=1

Λ/P_jΛ





with Pj distinguished polynomials, ordered by divisibility: P1|P2|. . .|Pt. The integer ρ is the rank of the Λ-module M, and the polynomial

χ= Ys i=1

ℓ^mi Yt j=1

Pj

is the characteristic polynomial of the torsion sub-module of M.

Definition 4.2. With the same notation of the above theorem let’s set ρ= rankΛM, µ=

Xs i=1

mi and λ= Xt j=1

deg(Pj);

we say that ρ,µ and λare the structural invariants of the Λ-moduleM.

We have that Ceis isomorphic to a noetherian Λ-module (Prop. 3.1). Let us write ˜ρ,

˜

µ and ˜λ for the invariants defined above forCe. Therefore assuming the Gross-Kuzmin conjecture we have ˜ρ= 0 and we obtain the following theorem.

Theorem 4.3. Let K_∞ be a non-cyclotomic Z_ℓ-extension of K. Let Cℓf_Kn be the logarithmic class group ofK_n. Assuming the Gross-Kuz’min conjecture forℓand every finite subextension K_n of K_∞/K there exist integers ˜λ,µ˜≥0 and ν, such that˜

|Cℓf_Kn|=ℓ^{µℓ˜ n+˜λn+˜ν}, for n big enough.

Remark 4.4. The above theorem together with Jaulent’s results in the cyclotomic case yield Theorem 1.

Let K_∞/K be a non-cyclotomic Z_ℓ-extension of K. The inverse limit lim

←−Cℓ of the ℓ-class groups of the K_n, corresponds to the Galois group C = Gal(K_∞^nr/K_∞) of the maximal abelian unramifiedℓ-extensionK_∞^nrofK_∞. It is well known thatCis a torsion Λ-module. Similarly the inverse limit lim

←−Cℓf^∗_Kn of the logarithmic classes of arbitrary

degree of theK_n, corresponds to the Galois groupCe^∗= Gal(K_∞^lc/K_∞) of the maximal abelian logarithmically unramified ℓ-extension K_∞^lc of K∞; after the propositions 3.1 and 3.3, assuming the Gross-Kuz’min conjectureCe^∗ is a noetherian torsion Λ-module, isomorphic as Z_ℓ-module to a direct sum Ce^∗ ≃C ⊕e Z_ℓ. As a consequence of theorem 4.1 we obtain the following proposition.

Proposition 4.5. Let Ce^∗ andCebe as before. The characteristic polynomials associated to these Λ-modules satisfy

χ_C∗e =χ_Ce·(γ−1), and we have the following relations for their parameters

(3) µe^∗=µe and eλ^∗=eλ+ 1,

where µe^∗ and eλ^∗ (respectively µe and eλ) are the parameters associated to the Λ-module Ce^∗ (respectively Ce).

Letp₁, . . . ,p_s be the places of K aboveℓ. We have the short exact sequence of pro-ℓ- groups (e.g. [1,§3])

(4) 0−→Cℓf^∗[ℓ]_K −→Cℓf^∗_K −→^Θ Cℓ^′_K −→0,

whereCℓf^∗[ℓ]_K is the subgroup ofCℓf^∗_K generated by the logarithmic classes of thep_i; the groupCℓ^′_K is the quotient of the ℓ-part of the ideal class group of K by the ℓ-part of the subgroup generated by the ideal classes of thep_i

(5) 0−→Cℓ^[ℓ]_K −→Cℓ_K−→Cℓ^′_K −→0;

and Θ is defined as

Θ :X

p

m_pp7→Y

p∤ℓ

p^mp.

Since the Mittag-Leffler property holds on profinite groups, taking the inverse limit in (4) and (5) gives short exact sequences. LetCe^∗[ℓ](respectivelyC^[ℓ]) be the subgroup ofCe^∗ (respectivelyC) that corresponds to lim

←−Cℓf^∗[ℓ]_n (respectively lim

←−Cℓ^[ℓ]n). The inverse limit lim←−Cℓ^′_Kn, corresponds to the Galois groupC^′ of the maximalℓ-extension ofK∞ which is unramified and splits completely over ℓ. Then we have the following isomorphisms (6) C^′ ≃Ce^∗/Ce^∗[ℓ]≃ C/C^[ℓ].

Le us denote byµandλ(respectivelyµ^′,λ^′) the parameters attached to the Λ-module C (respectivelyC^′). We have the following obvious relations

(7) µe^∗ =µe µe^∗ =µ^′+µe^∗[ℓ] λe^∗ =λ^′+eλ^∗[ℓ]

λe^∗=eλ+ 1 µ=µ^′+µ^[ℓ] λ=λ^′+λ^[ℓ]

with µ^[ℓ] and λ^[ℓ] (respectively µe^∗[ℓ] and eλ^∗[ℓ]) as the associated parameters to C^[ℓ]

(respectivelyCe^∗[ℓ]).

The following definition, if verified, allows us to have some control on the characteristic polynomials of the preceding modules. Specifically, we will show thatµ^[ℓ]=µe^∗[ℓ]= 0.

Definition 4.6. Let K_∞ be a Z_ℓ-extension of K. We say that a place p of K splits finitely in the towerK_∞/K, if its decomposition group in Gal(K_∞/K) is open, equiva- lently, if there is an integernpsuch that the layerKnp corresponds to the decomposition field ofp inK_∞/K.

Remark 4.7. The preceding definition holds for every place p|ℓ in the cyclotomic Z_ℓ-extension of every number field.

With the notation of (7) and (1) we state the following theorem:

Theorem 4.8. Let K_∞/K be a non-cyclotomic Z_ℓ-extension. Assume that the places of K above ℓ split finitely in the extension K_∞/K. Then assuming the Gross-Kuz’min conjecture for ℓ and all layers ofK∞/K the structural invariants attached to C and Ce satisfy the following relations:

ρ=ρe= 0, µ=µe but λ might be different of ˜λ.

And the characteristic polynomials of the torsion Λ-modules C and Ce just differ by cyclotomic factors.

Proof: The first relation is trivial since C and Ce^∗ are Λ-torsion modules.

Let D_i be the decomposition subgroup for the place p_i in K_∞/K; since the D_i are open they are isomorphic to Γ^ℓmi for some m_i. Take m = max_1≤i≤s{m_i}, then Γ^ℓm acts trivially on C^[ℓ] and Ce^∗[ℓ], therefore ω_m = γ^ℓm −1 annihilates them, i.e. their characteristic polynomial has noℓ-power factors which implies µ^[ℓ]=µe^∗[ℓ]= 0.

We have the decomposition into cyclotomic factors ω_m=ω₀ω₁

ω0

ω₂

ω1· · · ω_m ωm−1

.

Since the minimal polynomial and the characteristic polynomials must have the same factors, the characteristic polynomials ofC^[ℓ]andCe^∗[ℓ]differ just by cyclotomic factors.

Hence the Λ-modulesCandCe^∗ have the sameµinvariants by (7) and theirλinvariants might be different. From (3) we recover the relations for the invariants ofCe. Corollary 4.9. With the same notation as above we have

µ=µe=µe^∗ =µ^′.

Remark 4.10. Assuming the Gross-Kuz’min conjecture in Proposition 3.3 we deduced that eλ^∗ =eλ+ 1. Observing that the Z_ℓ-submodule Ce^∗[ℓ] of Ce^∗ is infinite, for it contains the classes α·p for p a place over ℓ and α ∈Z_ℓ (see §2), we can apply an analogous argument to the Λ-modules Ce^[ℓ] and Ce^∗[ℓ], from which we deduce that eλ^∗[ℓ]=eλ^[ℓ]+ 1.

The following table summarizes our results.

Invariants attached to finiteZ_ℓ-modules.

Modules Definition Parameters

ρ λ µ

Cℓ^′_n Cℓ^′_n is the ℓ-group of ℓ-classes ofK_n 0 λ^′ µ^′ Cℓf_n Cℓf_n is the logarithmic class group of K_n 0 ˜λ=λ^′+eλ^[ℓ] µ^′ Cℓ_n Cℓ_nis the ℓ-class group of K_n 0 λ=λ^′+λ^[ℓ] µ^′ We give an example for which we can apply our results.

Example 4.11. Let K be a quadratic imaginary number field and ℓa prime number splitting in K, say (ℓ) = l¯l. The maximal abelian l-ramified ℓ-extension M_l of K contains a non-cyclotomicZ_ℓ-extension ofK, sayK_∞. For consider theℓ-Hilbert class fieldH of K, then

Gal(M_l/H) = Y

p

UpRK/Y

p6=l

UpRK

= U^l/U^l∩(Y

p6=l

U^pRK)

= Ul/s_l(µ_K)

= Z_ℓ,

wheres_l(µ_K) is the image of the generalized unitsµ_K of K under the localisation map s_l [9, Thm. & Def. 1.1.22]. Therefore Gal(M_l/K)≃Z_ℓ ×finite abelianℓ-group.

In particular, since K/Q is abelian, Lemma 2.6 states that at least one place l over ℓ ramifies logarithmically in K_∞. Finally, the image of the decomposition subgroupD¯l

inside Gal(M_l/K) is isomorphic toZ_ℓ×finite abelianℓ-group, hence it has finite index in Gal(M_l/K). For the image ofD¯l is

R^¯l/R^¯l∩(Y

p6=l

UpRK).

Therefore both places land ¯l split finitely.

5. Numerical examples

5.1. Quadratic fields. We use the functions and algorithms implemented by Belabas and Jaulent in [1] to compute the logarithmic class group of a number field. These algorithms are now available in PARI/GP [17]. We use PARI/GP to compute the extensions corresponding to the first layers of someZ_ℓ-extensions, afterwards we apply the logarithmic class group algorithms to each of these number fields.

LetK be a quadratic field of the formQ(√

d). For now we will restrict to the compu- tation of the 3-group of logarithmic classes in the layers K = K₀, K₁ and K₂ of the cyclotomic Z3-extension ofK.

In the next table we take−100< d <100. For sake of space, we just show all the cases for which the logarithmic class group is not trivial and for which 3 splits inQ(√

d).

Q(√

d) Q(√

d,cos(2π/9)) Q(√

d,cos(2π/27)) d Cℓf_K Cℓf^[ℓ]_K Cℓ^′_K Cℓf_K1 Cℓf^[ℓ]_K1 Cℓ^′_K1 Cℓf_K2 Cℓf^[ℓ]_K2 Cℓ^′_K2 -86 [3] [3] [] [9, 9] [3] [9, 3] [27, 27] [3] [27, 9]

-74 [9] [9] [] [27] [9] [3] [81] [9] [9]

-65 [3] [3] [] [9] [3] [3] [27] [3] [9]

-47 [9] [9] [] [27] [9] [3] [81] [9] [9]

-41 [27] [27] [] [81, 3] [27] [3, 3] [243, 9] [27] [9, 9]

-35 [3] [3] [] [9] [3] [3] [27] [3] [9]

-14 [3] [3] [] [9] [3] [3] [27] [3] [9]

67 [3] [3] [] [3] [] [3] [3] [] [3]

In the next table we present all the cases meeting with the following criteria:

−100< d <3 000, 3 splits inQ(√

d) and the groups Cℓf_K and Cℓ^′_K are not trivial.

Q(√

d) Q(√

d,cos(2π/9)) Q(√

d,cos(2π/27)) d Cℓf_K Cℓf^[ℓ]_K Cℓ^′_K Cℓf_K1 Cℓf^[ℓ]_K1 Cℓ^′_K1 Cℓf_K2 Cℓf^[ℓ]_K2 Cℓ^′_K2

1129 [3] [] [3] [3] [] [3] [3] [] [3]

1654 [3] [] [3] [3] [] [3] [3] [] [3]

1954 [3] [] [3] [3] [] [3] [3] [] [3]

2419 [3] [] [3] [3] [] [3] [3] [] [3]

2713 [3] [] [3] [9] [] [9] [9] [] [9]

2971 [3] [] [3] [3] [] [3] [3] [] [3]

In the next table we present all the cases issued with the following criteria:

−100< d <20 000, 3 splits inQ(√

d) and the groups Cℓf^[ℓ]_K and Cℓ^′_K are non trivial.

Q(√

d) Q(√

d,cos(2π/9)) Q(√

d,cos(2π/27)) d CℓfK Cℓf^[ℓ]_K Cℓ^′_K CℓfK1 Cℓf^[ℓ]_K1 Cℓ^′_K1 CℓfK2 Cℓf^[ℓ]_K2 Cℓ^′_K2

3739 [9] [3] [3] [27] [3] [9] [81] [3] [27]

7726 [3,3] [3] [3] [9,3,3] [] [9,3,3] [27,3,3] [] [27,3,3]

11545 [9] [3] [3] [9] [] [9] [9] [] [9]

17134 [9] [3] [3] [9] [] [9] [9] [] [9]

19330 [9] [3] [3] [9] [] [9] [9] [] [9]

5.2. Logarithmic invariants: Cyclotomic case. We are now interested in explicitly knowing the values of the logarithmic invariants µe eteλ. In [9] Jaulent showed (in the caseK∞/Ka cyclotomicZ_ℓ-extension) that the invariantµeattached to the logarithmic class group equals its classical counterpart µ. Hence we can state the following result.

Theorem 5.1 (Logarithmic Ferrero-Washington). Let K be an abelian number field over Q, and letK^c/K be its cyclotomic Z_ℓ-extension. Then

e µ= 0.

Remark 5.2. We have µe= 0 for the examples given in the preceding section 5.1.

In the cyclotomic case we have thatλ^′ =eλ, where λ^′ is the invariant attached to the ℓ-subgroup of ℓ-classes Cℓ^′_n. This follows from [9, IV.2.1] and the fact that in the cyclotomic Z_ℓ-extension we haveCℓf_n≃ C^′/ω_nC^′ (e.g. [13,§2]).

Lemma 5.3. Let K =Q(√

−d) be an imaginary quadratic number field and consider an odd prime ℓ such that ℓ ∤ d. For the cyclotomic Z_ℓ-extension of K the following relations hold

λ=

λ^′+ 1 if ^−d_ℓ

= 1, λ^′ if ^−d_ℓ

=−1;

where λand λ^′ are the invariants attached to theΛ-modules C and C^′.

Proof. Suppose that ℓ splits in K. Let p_n and ¯p_n be the ideals of K_n above ℓ. The productp_n¯p_n is an ideal in the n-th layer of the cyclotomic Z_ℓ-extension Q^c of Q. By a classical argument in Iwasawa theory (e.g. [16,§4]), the ℓ-part of the class groups of the finite layers of Q^c is trivial. This implies thatp_n and ¯p_n generate the same cyclic subgroup ofCℓ_n. Alsop^ℓ_nⁿ =p₀.

Let ℓ^a be the order of the subgroup generated by p₀ in Cℓ₀, then the order of p_n in Cℓ_n is p^ℓ_n^a+n. Hence the short exact sequence (5) becomes

0−→Z/ℓ^n+aZ−→Cℓ_n−→Cℓ^′_n−→0, for everyn≥0. Taking inverse limits we find thatλ=λ^′+ 1.

If ℓ is inert in K, then we have Cℓ^[ℓ] = 0. The submodule C^[ℓ] of C is a noetherian torsion Λ-module. Moreover, since there is only one place p_n in K_n above ℓ, by a classical argument in Iwasawa theory (e.g. [16,§4]) we have

Cℓ^[ℓ]_n ≃ C^[ℓ]/ω_nC^[ℓ] for n≥0.

From which we deduce C^[ℓ] = mC^[ℓ], where m is the maximal ideal of Λ. From the Nakayama’s lemma for compact Λ-modules [18, Lem. 13.16] we have C^[ℓ] = 0. Then

from a relation in (7) we haveλ=λ^′ as claimed.

Remark 5.4. Notice that if ^−d_ℓ

=−1 and Cℓ_K = 0 then λ=λ^′ =eλ= 0.

The above lemma together with the following theorem allow us to compute explicitly the value of eλfor the cyclotomicZ_ℓ-extension of some imaginary quadratic fields.

Theorem 5.5 (Gold [4, 5]). Let K =Q(√

−d) be an imaginary quadratic number field such that ℓ∤d. Let K_n be the n-th layer of the cyclotomic Z_ℓ-extension of K, and e_n be the exponent of the ℓ-part of the class group of K_n. If there is n ≥ 1, such that e_n−e_n−1 < ϕ(ℓⁿ), then

λ=e_n−e_n−1.

Moreover, ifℓ splits in K, it suffices that e_n−e_n−1 ≤ϕ(ℓⁿ).