Global Units modulo Circular Units : descent without Iwasawa's Main Conjecture.

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Iwasawa’s Main Conjecture.

Jean-Robert Belliard

To cite this version:

Jean-Robert Belliard. Global Units modulo Circular Units : descent without Iwasawa’s Main Conjec- ture.. Canadian Journal of Mathematics = Journal Canadien de Mathématiques, Louigi Addario-Berry McGill University, 2009, 61 (3), pp.518–533. �hal-00440905�

without Iwasawa’s Main Conjecture.

Jean-Robert Belliard

Universit´e de Franche-Comt´e, Laboratoire de math´ematiques UMR 6623, 16 route de Gray, 25030 Besan¸con cedex, FRANCE.

belliard@math.univ-fcomte.fr

September 28 2006

Abstract

Iwasawa’s classical asymptotical formula relates the orders of the p-parts X_n of the ideal class groups along a Z_p-extension F_∞/F of a number field F, to Iwasawa structural invariants λ and µ attached to the inverse limit X_∞ = lim

← X_n. It relies on ”good” descent properties satisfied byX_n. IfF is abelian andF_∞ is cyclotomic it is known that thep-parts of the orders of the global units modulo circular unitsU_n/C_nare asymptotically equivalent to the p-parts of the ideal class numbers. This suggests that these quotientsU_n/C_n, so to speak unit class groups, satisfy also good descent properties. We show this directly, i.e. without using Iwasawa’s Main Conjecture.

0 Introduction

LetKbe a number field andpan odd prime (p6= 2) and letK∞/K be aZ_p-extension (quite soonK_∞/K will be the cyclotomicZ_p-extension). Recall the usual notations : Γ = Gal(K∞/K) is the Galois group of K∞/K, Kn is the n^th-layer of K∞ (so that [Kn:K] =pⁿ), Γn = Gal(K∞/Kn), andGn = Gal(Kn/K)∼= Γ/Γn. Let us consider a sequence (M_n)_n∈N of Z_p[G_n]-modules equipped with norm maps M_n −→ M_n−1 and the inverse limit of this sequence M∞ = lim

← Mn seen as a Λ = Z_p[[Γ]]-module.

The general philosophy of Iwasawa theory is to study the simpler Λ-structure of M∞, then to try and recollect information about the Mn’s themselves from that structure. For instance, if M∞ is Λ-torsion, one can attach two invariants λ and µ toM∞. If we assume further that

(i) the Γn coinvariants (M∞)Γn are finite,

∗2000Mathematics Subject Classification. Primary 11R23.

1

(ii) the sequence (Mn)n∈N behaves ”nicely” viz descent;

then one can prove that the orders of the Mn are asymptotically equivalent to p^λn+µpn. These two assumptions are expected to occur whenever one chooses for (Mn) significant modules (from the number theoretic point of view). However prov- ing them may require some effort : actually, (i) for the canonical Iwasawa module tor_Λ(X_∞) is (one of the many equivalent formulations of) Leopoldt’s conjecture.

The historical example occurs when we specializeMn =Xn, the p-part of ideal class group of Kn. Then the asymptotic formulas are a theorem of Iwasawa. The proof of this theorem uses an auxiliary moduleY∞ which is pseudo-isomorphic to X∞ but with better descent properties.

In the present paper we are interested in similar statements for unit class groups, that is for Mn =Un/Cn, the p-part of the quotient of units modulo the subgroup of circular units of Kn. In order that the latter merely exist we need to assume that all Kn are abelian over Q : hence K is abelian and K∞/K is cyclotomic. Let us assume further that K is totally real, which is not a loss of generality as long as we are only interested in p 6= 2. Now by Sinnott index formulas we know that the orders of Un/Cn are asymptotically equivalent to the orders of Xn. And as a consequence of Iwasawa’s Main Conjecture, the structural invariantsλandµof both inverse limitsU∞/C∞= lim

← Un/Cn and ofX∞= lim

← Xnare equal. Using these two theorems we have immediately a somewhat indirect proof of an analogue of Iwasawa theorem for unit class groups. Clearly a direct proof of this fact must exist and the first goal of the present paper is to write it down. The theorem 5.3 shows that the orders of the unit class groupsUn/Cn along the finite steps of the Z_p-extension are those prescribed by the structural Iwasawa invariants of the inverse limit U_∞/C_∞. It makes no use of any precise link between ideal and unit class groups. E.g. it does not use the Main Conjecture, neither even Sinnott’s index formula. On the other hand if one does use some classical results together with the theorem 5.3, one gets easier proofs of beautiful and well known theorems (see §6). For instance Sinnott’s index formulas together with theorem 5.3 imply the equality of the λ and µ invariants of the two class groups without using Iwasawa Main Conjecture.

The equality between these two λ invariants is actually an important step (some times called ”class number trick”) in the proof of Iwasawa’s Main Conjecture. If we use further Ferrero-Washington’s theorem, we prove that all the µ-invariants involved here are trivial. Maybe some ideas in the present approach could be used in a different framework where the equality of the two characteristic ideals (that is Iwasawa’s Main Conjecture) is still open. In the paper [N2], §5, T. Nguyen Quang Do has also proven that unit classes have asymptotically good descent properties, by using Iwasawa’s Main Conjecture in its strongest form : that is, following the path outlined just above.

We conclude this introduction by recalling the (now) traditional notations of cyclotomic Iwasawa theory. For any number field F we put S =S(F) for the set of places of F dividing p. We will adopt the following notations :

For any abelian group A, we’ll denote by A the p-completion of A, i.e. the inverse limitA= lim

← A/A^pm. IfAis finitely generated overZ, then ¯A∼=A⊗Z_p.

UF is the group of units of F.

U_F^′ is the group ofS-units ofF : that is, elements ofF^× whose valuations are trivial for all finite places v of F such that v 6∈S.

XF is the p-part of cl(F) which in turn is the ideal class group of F.

X_F^′ is the p-part of cl^′(F) which in turn is the quotient of cl(F) modulo the subgroup generated by classes of primes inS.

X_F is the Galois group over F of the maximal S-ramified (i.e. unramified outsideS) abelian pro-p-extension ofF.

N_F is the multiplicative group of the semi-local numbers. As a mereZ_p-module NF =Q

v∈SF_v^× where Fv is the completion of F at the placev ∈S.

U_F is the group of semi-local units ofF. As a mere Z_p-moduleU_F =Q

v∈SU_v¹ where U_v¹ is the set of principal units of Fv, that is units ≡ 1 modulo the maximal ideal ofFv.

So for instance the Un’s above could have been understood as UKn⊗Z_p and XF = cl(F).

1 Consequences of Leopoldt’s Conjecture

IfF is Galois over Q, the groups U_F and N_F come equipped with the induced action of Gal(F/Q). In other words if we fix one placevinSand if we putGv = Gal(Fv/Qp) then we have (asZ_p-modules) :

U_F ∼=U_v¹⊗Zp[Gv]Z_p[Gal(F/Q)] and N_F ∼=F_v^×⊗Zp[Gv]Z_p[Gal(F/Q)].

These isomorphisms define the action of Gal(F/Q) on U_F and on N_F. Let us now consider the cyclotomic Z_p-extension K∞/K of our totally real abelian over Qnum- ber field K and the n^th-finite layers Kn/K, i.e. Kn is the unique subfield of K∞

with [Kn:K] =pⁿ. We will denote by Cn (and of course we’ll be more interested in Cn) the group of circular units of Kn as defined by Sinnott ([Si]). We will indicate consistently by the subscript narithmeticall object related to Kn. So Un,Un and so on make sense. For any extension L/F of global fields the formula

NL/F((xw)w|p) = (Y

w|v

NLw/Kv(xw))v|p

defines a Galois equivariant morphism NL/F: NL−→ NF which is compatible with the usual norms on global units for instance. As ever U∞,C∞ N∞, U∞,U^′_∞ and so on denotes the inverse limit (related to norm maps) of Un, Cn, N_n, U_n, U^′_n and so on. Before considering more precise properties of descent kernels (and cokernels) of unit classes, we need to first establish their finiteness. This can surely be extracted from various and older references as a consequence of Leopoldt’s conjecture (which

is true here since all Kn are abelian over Q). However a more precise and general result on quotients of semi-local units modulo circular units can be found in [T].

The point for our present approach is that the proofs of [T] don’t make any use of the Main Conjecture, and only need Coleman morphisms.

We will need and freely use the following consequence of theorem 3.1 in [T].

Theorem 1.1 Let G = Gal(K/Q). Recall that Γ = Gal(K_∞/K) and that Λ = Z_p[[Γ]]. Fix a generator γ of Γ. For all non-trivial Dirichlet characters of the first kindψ ofG, following [T] let us denote byb gψ(T)the Iwasawa power series associated to the Kubota-Leopoldt p-adic L-function Lp(ψ, s) (with our fixed choice of γ).

1. Assumep is tamely ramified inK so that every characters of Gare of the first kind. Then up to a power ofp, the characteristic ideal of the Λ-torsion module U_∞/C∞ is generated by the product Y

ψ∈G,ψ6=1b

gψ(T).

2. For all n∈N, the Γn-coinvariants (U_∞/C∞)Γn and (U∞/C∞)Γn are finite.

Proof. Let us consider ψ-partsM^ψ as defined in [T] for anyZ_p[G]-moduleM. These ψ parts are naturallyZ_p[ψ]-modules andZ_p[G]-submodules ofM. For technicalities about ψ-components which are often left to readers see Beliaeva’s thesis [Bt]. From theorem 3.1 of [T] we use that U_∞^ψ/C^ψ_∞ is Z_p-torsion free and that its characteristic series over Λ[ψ] isg_ψ(T). Letσ ∈Gal(Q_p[ψ]/Q_p); then (U_∞/C_∞)^ψσ is isomorphic to (U∞/C∞)^ψ andgψ^σ(T) =σ(gψ(T)). ForRequal to Λ or to Λ[ψ], let us abbreviate by CharR(M) the characteristic ideal of the R-module M. It is an easy linear algebra exercise to check that if a power series f generates Char_Λ[ψ](M) then NQp[ψ]/Qp(f) generates CharΛ(M). Hence we get

CharΛ((U_∞^ψ/C^ψ_∞)) = (NQp[ψ]/Qp(gψ(T))) =



 Y

σ∈Gal(Qp[ψ]/Qp)

gψ^σ(T)



.

Let us fix a set Ψ of representatives of Gb up to Q_p-conjugation classes. If a Z_p[G]- module M is Z_p-torsion free then the submodules of M, (M^ψ)ψ∈Ψ, are mutually direct summands and the quotientM/⊕_ψ∈ΨM^ψ is annihilated by #G. This applies toU_∞andC_∞, so that we may use the inclusionsL

ψ∈ΨU_∞^ψ ⊂ U_∞and L

ψ∈ΨC^ψ_∞ ⊂ C_∞ to form the snake diagram :

0 ^//L

ψ∈ΨC^ψ_∞ ^//

 _

L

ψ∈ΨU_∞^{ψ //}

 _

L

ψ∈ΨU_∞^ψ/C^ψ_∞ ^//

α

0

0 ^//C∞ //

U_∞ ^//

U_∞/C∞ //

0

Z_p−torsion ^//Z_p −torsion ^//Coker α ^//0

Now the kernel of α must be Z_p-torsion, as a submodule of the cokernel of the first inclusion, hence is trivial because all U_∞^ψ/C^ψ_∞ are Z_p-free. This proves that up to a

power ofpthe characteristic ideal CharΛ(U∞/C∞) is equal to CharΛ(L

ψ∈ΨU∞/C∞), which in turn is what we wanted.

The second claim is equivalent to the fact that both characteristic ideals are prime to all (T + 1)^pn −1 for all n ∈ N. From the inclusion U∞/C∞ ⊂ U∞/C∞

we see that the first characteristic ideal divides the second. Let ζ be a p-power order root of unity and let ρ be the unique character of the second kind such that ρ(γ) = ζ⁻¹. Then we have gψ(ζ −1) = Lp(1, ψρ). By Leopoldt’s conjecture (see [W] corollary 5.30) Lp(1, ψρ)6= 0. Hence in the special case where p is tame in K, 2 follows from 1. We now want to remove this hypothesis. Let Ip ⊂Gal(K∞/Q) be the inertia subgroup ofp, and let Ip ⊂Ip be its pro-p-part. Then the field L=K∞^Ip

is a finite abelian over Q number field and pis at most tamely ramified in L. Since all modulesM∞ depend on K∞ and not on K itself, to prove that the hypothesis is almost no loss of generality we just have to show the following lemma :

Lemma 1.2 L∞=K∞.

Proof. Let B_∞/Q be the Z_p-extension of Q. By construction we have L⊂K∞ and thereforeL∞⊂K∞. By definition ofL, the group Gal(K∞/L∞) is then a subgroup of Ip, hence K∞/L∞ is a p-extension totally ramified at p. But on the other hand B_∞⊂ L∞ and K∞ is abelian over Q. Let Q^ab be the maximal abelian extension of Q. Then Q^ab/B∞ is tamely ramified at (the only) prime of B_∞ abovep. This shows L∞=K∞.

Now, with that lemma we already have proven the finiteness assertion when we take coinvariants along Γen = Gal(K∞/Ln). We conclude the proof of the second part of theorem 1.1 by pointing that even if the first few Γn may differ from ˜Γn, we get Γn = ˜Γn as soon asK∞/Kn is totally ramified atp. Due to canonical surjections (M∞)Γn ։(M∞)Γn−1, this does not change anything to the finiteness of (U∞/C∞)Γn.

Remark : As long as one is only concerned with characteristic ideals up to a power of p and finiteness of coinvariants, theorem 1.1 was well known long before as part of the folklore, and can be tracked back to Iwasawa ([I1]; see also [G]) in some cases.

Now that the finiteness of (U_∞/C∞)Γn (resp. (U∞/C∞)Γn) has been proved, we want to relate these modules with their counterpartsU_n/C_n(resp. U_n/C_n) at finite levels. On the way we will study descent for various other multiplicative Galois modules.

2 Background: easy part of descent

Let M∞ = lim

←(Mn) be a Λ-module. Of course the projections M∞ −→ Mn factor through (M∞)Γn −→ Mn, but in general these maps have non trivial kernels and cokernels. Let us denote Kern(M∞) for the kernel of (M∞)Γn −→ Mn, Mfn ⊂ Mn

for its image, and Cokern(M∞) = Mn/Mfn for its cokernel. Letl be a prime number

(l =p is allowed) and E be a finite extension of Q_l. Put E∞/E for a Z_p-extension of E, and En for the n^th finite layers. Let us abbreviate LE = E^×, consistently Ln= (En)^×, and L∞= lim

← Ln. For alln ∈N∪ ∞, local class field theory identifies Ln with Gal(Mn/En), where Mn is the maximal abelianp-extension of En.

Lemma 2.1 The natural map (L∞)Γn −→Ln fits into an exact sequence 0 ^//(L∞)Γn

//L_n ^Artin//Gal(E∞/En) ^//0.

Proof. This is well known to experts. We follow the cohomological short cut of [N1].

The group Γn ≃ Z_p is pro-p-free hence we have H²(Γn,Q_p/Z_p) = 0. Consider the inflation-restriction sequence associated to the extension of groups

1 ^//H ^//G_n ^//Γ_n ^//1, where G_n is the absolute Galois group of En :

0 ^//H¹(Γ_n,Q_p/Z_p) ^//H¹(G_n,Q_p/Z_p) ^//H¹(H,Q_p/Z_p)^{Γn //}0.

Applying Pontryagin duality and class field theory we get 0 ^//(L∞)Γn

//Ln Artin//Gal(E∞/En) ^//0.

Let us come back to our global field K and its cyclotomic Z_p-extension.

Proposition 2.2 Let S_n be the set of places of K_n dividing p. For all n andv ∈S_n put Kn,v for the completion at v of Kn and (Kn,v)^cyc_∞ /Kn,v for its cyclotomic Z_p- extension. For all n the natural map

(N_∞)_Γn −→ N_n fits into an exact sequence

0 ^//(N∞)Γn

//N_n ^{⊕v∈SnArtin atv//}L

v∈SnGal((Kn,v)^cyc_∞/Kn,v) ^//0.

Proof. Let us examine the places above p along K∞/K. These primes have non trivial conjugated (hence equal) decomposition subgroups. There exists a d ∈ N such that (i) no primes abovep splits anymore in K_∞/K_d and (ii) all primes above pare totally split inKd/K. For alln ≥dwe then haveSn ∼=Sd,Nn= M

v∈Sd

LKn,v and Gal(K∞/Kn) identifies with the local Galois groups Gal((Kn,v)^cyc_∞/Kn,v) (at every v ∈Sn). So in the casen≥d, the proposition follows from lemma 2.1. Next suppose that 0 ≤ n < d, and let G^m_n be the Galois group Gal(K_m/K_n). Of course we have (N∞)Γn ∼= ((N∞)Γ_d)_G^d_n. Using the previous case we have an exact sequence

(†) 0 ^//(N∞)Γ_d //N_d ^//L

v∈S_dGal((Kd,v)^cyc_∞ /Kd,v) ^//0

As (Kd,v)^cyc_∞/Kd,v is cyclotomic, the action of the local Galois group Gal(Kd,v/Qp) on the group Gal((Kd,v)^cyc_∞ /Kd,v) is trivial. Hence we have an isomorphism of Galois

modules M

v∈Sd

Gal((Kd,v)^cyc_∞ /Kd,v)∼=Z_p[Sd].

Since the places above p split totally in Kd/K, over G^d₀ these two modules are cohomologically trivial, and the same is true forN_d∼=N₀⊗Z_pZ_p[G^d₀] (asG^d₀-modules).

Therefore in the sequence (†) two (hence all) terms are G^d₀-cohomologically trivial.

The triviality of Hb¹(G^d_n,(N_∞)Γd) proves the injectivity of (N_∞)Γn −→ N_n. It then suffices to apply N_Kd/Kn to the sequence (†) and use the triviality of the three Hb⁰(G^d_n,−−) to get the full sequence

0 ^//(N_∞)Γn

//N_n ^{⊕v∈SnArtin atv//}L

v∈SnGal((K_n,v)^cyc_∞ /K_n,v) ^//0.

Let d be as before. Let Dn be the decomposition subfield for the place p in Kn

(i.e p is totally split in Dn and no place of Dn above p splits anymore in Kn/K).

Then for all n ≥ d we have Dn = Dd. Let us put D = Dd for the decomposition subfield for the place p inK∞.

Proposition 2.3 Recall that for a Λ-module M∞= lim

←(Mn) we denote Kern(M∞) for the kernel, Mf_n for the image and Coker_n(M_∞) for the cokernel of the natural map (M∞)Γn −→Mn.

1. For all n≥0, Kern(U∞) is isomorphic to Z_p[Sn], hence its Z_p-rank is #Sn. 2. For alln ≥0, Coker_n(U_∞)∼=N_Kn/Dn(U_n), hence its Z_p-rank is #S_n. In other

words we have an exact sequence

0 ^//Ue_n ^//Un N_Kn/Dn

//U_Dn.

Proof. For any local fieldEthe (normalized) valuationvofEgives an exact sequence 0→ U_E¹ → E^× →^v Z_p →0, where U_E¹ is the set of principal units of E. Taking the semi-local version of this sequence and projective limits we obtain

0 ^//U_∞ ^//N_∞ ^//Z_p[Sd] ^//0.

But for all n, N_n^Γ =N0 contains no infinitely p-divisible element. Therefore N_∞^Γ = {0}and taking Γn-cohomology on this sequence we get :

0 ^//Z_p[Sd]^{Γn //}(U_∞)Γn

//(N_∞)Γn

//(Z_p[Sd])Γn

//0

Now since (N∞)Γn −→ Nnis a monomorphism, Kern(U∞) identifies withZ_p[Sd]^Γn ∼= Z_p[Sn]. This proves 1.

For 2. we use the notations d, D_n and D of the proof of proposition 2.2. We first prove the case n ≥ d. By compactness we have Uen = T

m≥nNKm/Kn(Um).

Because n ≥ d the global norm NKm/Kn: Um −→ Un is nothing else but the direct sum place by place of the local norms of the extensions Km,w/Kn,v (at each unique w ∈ Sm above a fixed v ∈ Sn). Fix a place v ∈ Sn and for all m ≥ n still call v the unique place of Sm above v and also v the unique place of Dn under v. Note that Dn,v = Q_p. Let u ∈ U_K¹_n,v, then using local class field theory we see that u ∈ T

m≥nN_Km,v/Kn,v(U_K¹_m,v) if and only if N_Kn,v/Qp(u) ∈ T

n∈NN_(Qp)^cyc_{n /Qp}(U_(Q¹

p)^cycn ), where (Qp)^cyc_n denotes the n^th-step of the cyclotomic (hence totally ramified) Z_p- extension of Q_p. By local class field theory (.,(Q_p)^cyc_∞ /Q_p) is an isomorphism from U_Q¹_p to Gal((Q_p)^cyc_∞/Q_p). The equivalence

u∈ \

m≥n

NKm,v/Kn,v(U_K¹_m,v) ⇐⇒ NKn,v/Dn,v(u) = 1 follows. As no place above p splits inKn/Dn this gives

Uen= Ker(NKn/Dn:Un−→ UDn).

Next pick ann such that 0≤n < d. Consider the diagram of fields Kd

CC CC CC CC

Kn

CC CC CC CC

Dd

Dn

There every prime abovepis totally split in the extensions Kd/Kn, andDd/Dnand no prime above p splits at all in Kn/Dn nor in Kd/Dd. It follows that NKd/Kn is surjective onto U_n and that

Uen =NK_d/Kn(Ued)⊂Ker(NKn/Dn: Un−→ UDn).

Conversely pick u = (uv)v∈Sn ∈ Ker(NKn/Dn: Un −→ UDn). At each v ∈ Sn choose a single w(v)∈ Sd above v and define t = (tw)w∈Sd ∈ U_d by putting tw = 1 if there does not exist v such that w = w(v) and t_w(v) = u_v for all v in S_n. Then we have NK_d/Kn(t) =u and t ∈Ued which shows that u∈Uen.

Similar but not so precise statements about the sequence of global units could be deduced from the proposition 2.3 and from the following proposition 2.4. As they are not needed we don’t state them. To end this section we recall the analogous proposition for the sequence U^′_n of (p)-units, which is a result of Kuz^′min ([K], theorem 7.2 and theorem 7.3). Recall that K is totally real, so that r1 = [K : Q], r2 = 0, and allU^′_nare Z_p-torsion free. To avoid ugly notationsMfwe will denoteUe_n^′ for the image of U^′_∞ inU^′_n.

Proposition 2.4

1. The Λ-module U^′_∞ is free of rank [K :Q].

2. For all n the natural map (U^′_∞)Γn −→U^′_n is injective.

3. Ue_n^′ ∼= (U^′_∞)Γn is a free Z_p[Gn]-module of rank [K :Q] (hence isZ_p free of rank pⁿ[K :Q] = [Kn :Q]).

Proof. 1 is theorem 7.2 and 2 is theorem 7.3 of [K]. There the number field K is arbitrary and a considerable amount of effort is made to avoid using Leopoldt’s conjecture. Another proof (also without using Leopoldt’s conjecture) is in [KNF].

On the other hand our abelian number fieldK does satisfy Leopoldt’s conjecture so we may give, for the convenience of the reader, the following shorter proof. We may suppose n = 0 (else replace K by Kn). Let X_∞ be the standard Iwasawa module X_∞ = lim

← (X_n), where X_n is the Galois group over Kn of its maximal abelian S- ramified p-extension (Nota : of course this definition works also forn =∞). Recall from the end of the introduction the notationX_n^′ for thep-part of the (p)-class group of Kn. From class field theory we have the (some time called decomposition) exact sequence

0 ^//U^′_∞ ^//N_∞ ^//X_∞ ^//X_∞^{′ //}0.

Put D∞ = Im(N∞ −→ X_∞). By Leopoldt’s conjecture for K we have X^Γ

∞ = 0 therefore D^Γ_∞ = 0. This implies that the induced map (U^′_∞)Γ −→ (N∞)Γ is a monomorphism. Now 2. follows from proposition 2.2.

K is totally real and p 6= 2, so (U^′_∞)Γ is Z_p-free as a submodule of U^′₀. As for N_∞, we have (U^′_∞)^Γ = 0 (same argument applies). These two facts suffice to show the Λ-freeness of U^′_∞. To compute the rank we consider the sequence

0 ^//(U^′_∞)_Γ ^//(N∞)Γ //(D∞)Γ //0.

As D∞ is a torsion-Λ-module (as a submodule of X_∞) with trivial Γ-invariants, its Γ-coinvariants are finite. Hence (U^′_∞)Γ has the same Z_p-rank as (N_∞)Γ. By proposition 2.2 this rank is rankZp(N₀)−#S0 = [K :Q]. By Nakayama’s lemma the Λ-rank ofU^′_∞ is also [K :Q]. This concludes the proof of 1.

3 is an immediate corollary of 1 and 2.

3 From semi-local to global and vice-versa

We now state and proceed to prove our main result in this paper. We want to show that descent works asymptotically well forM∞ =U∞/C∞ or (which will be proven to be equivalent) forM∞ =U∞⁽⁰⁾/C∞ (see explanations and notations below for the symbol ⁽⁰⁾). This means that in both cases above, Kern(M∞) and Cokern(M∞) are finite of bounded orders. Our strategy of proof is the following. First we will show that bounding the kernels and cokernels associated to both modules is equivalent.

Then we will use the injectivity of descent on U^′_∞ (proposition 2.4) to bound the kernels of descent forU_∞/C_∞. Then we use local class field theory (proposition 2.3) to bound the cokernels of descent for U∞⁽⁰⁾/C∞.

But before this we have to slightly change the sequence Un/Cn. Indeed, for all n, the module U_n/Cn is (by Leopoldt’s conjecture) of Z_p-rank 1 while (U_∞/C∞)Γn

is torsion. This rank 1 comes fromNKn/Q(Un) = {0}, or if we adopt the class field theory point of view, it represents the rank of the Z_p-extension K∞/K. Let us put the

Notation: LetF be a number field. In the sequel U_F⁽⁰⁾ will denote the kernel of NF/Q: U_F −→ U_Q.

Consistently Un⁽⁰⁾ will denote the kernel of NKn/Q and U∞⁽⁰⁾ = lim

← Un⁽⁰⁾. By proposition 2.3 we have Uen ⊂ Un⁽⁰⁾ and therefore U∞ = lim

← (Un⁽⁰⁾). More- over Un⁽⁰⁾/Cn is torsion and since NKn/Q(Un) is Z_p-torsion free we have Un⁽⁰⁾/Cn = TorZ_p(U_n/Cn), and accordingly Un⁽⁰⁾/Un = TorZ_p(U_n/Un). For all these reasons, it is clearly more convenient to (and we will from now) use the sequence (Un⁽⁰⁾)n∈N

instead of (U_n)_n∈N. This convention gives sense to the notations Uen⁽⁰⁾, Ker_n(U∞⁽⁰⁾), Coker_n(U∞⁽⁰⁾) and consistently to the same notations associated to sequencesUn⁽⁰⁾/C_n, Un⁽⁰⁾/Un and so on. Of course, due to proposition 2.3, only the various Cokern will actually change when we replace U_n by Un⁽⁰⁾.

Proposition 3.1 Recall that Xn stands for the p-part of the class group of Kn, and that Kern(X∞) is the kernel of the natural map (X∞)Γn −→Xn.

1. The map (U∞⁽⁰⁾/U∞)Γn −→ Un⁽⁰⁾/Un fits into an exact sequence

0 ^//(X∞)^{Γn //}(U∞⁽⁰⁾/U_∞)_Γn ^//Un⁽⁰⁾/U_n ^//Kern(X∞) ^//0 2. Kern(U∞⁽⁰⁾/U∞) and Cokern(U∞⁽⁰⁾/U∞) are finite and of bounded orders.

Proof. By global class field theory we have an (some time called inertia) exact sequence :

(R) 0 ^//U_n/Un //X_n ^//Xn //0.

Taking the Z_p-torsion counterpart of the sequence (R) we have : 0 ^//Un⁽⁰⁾/Un //TorZp(X_n) ^//Xn.

On the other hand if we take limits up to K∞ on (R), then apply Γn-cohomology, we obtain (by Leopoldt,X^Γ_∞ⁿ = 0 and therefore X_∞^Γn is finite):

0 ^//X_∞^{Γn //}(U∞⁽⁰⁾/U∞)Γn

//(X_∞)Γn

//(X∞)Γn

//0.

Lemma 2.1 has a global analogue which is the following :

Lemma 3.2 The natural map (X_∞)Γn −→X_n fits into the exact sequence 0 ^//(X_∞)Γn

//X_n ^{res //}Gal(K∞/Kn) ^//0.

In other words, descent provides an isomorphism (X_∞)Γn ∼= TorZpX_n.

Proof. The proof of (X_∞)Γn ֒→ X_n is exactly the same as for 2.1 : we only need to replace the G_n there by the group GS(Kn) which is the Galois group of the maximal S-ramified extension of Kn. The remaining part of the exact sequence comes from maximality properties defining (X_∞)Γn andX_n. Let Mnbe the maximal abelian S ramified p-extension of Kn. Then, by Leopoldt’s conjecture, one has TorZp(X_n) = Gal(M_n/K_∞), which gives the isomorphism.

Let us resume the proof of 1 of 3.1. Putting together the three last sequences we obtain

0

(X∞)^{Γn  //}(U∞⁽⁰⁾/U∞)Γn

//(X_∞)_Γn ^//

(X_∞)_Γn ^//

0

0 ^//Un⁽⁰⁾/Un //TorZp(X_n) ^//

Xn

0 1 of 3.1 follows then from the snake lemma.

By Leopoldt’s conjecture the (X∞)^Γn’s are finite. As X∞ is a noetherian Λ- module the ascending union S

n∈N(X∞)^Γn stabilizes. This shows that the orders of Ker_n(U∞⁽⁰⁾/U_∞) stabilize. As for Coker_n(U∞⁽⁰⁾/U_∞)∼= Kern(X_∞), the mapsX_n+1 −→

Xn (and consequently X∞ −→Xn) are surjective as soon as Kn+1/Kn does ramify.

By the classical Iwasawa theorems (see §13 of [W]), the orders of (X_n) is asymp- totically equivalent to p^λXn+µXpn, where λ_X and µ_X are the structural invariants of X∞. Since the order of (X∞)^Γ is finite the orders of (X∞)Γn are also asymptotically equivalent to p^λXn+µXpn. This proves that the orders of Kern(X∞), hence those of Cokern(U∞⁽⁰⁾/U∞), are bounded and concludes the proof of 2.

By Leopoldt’s Conjecture we have for all n : (U∞⁽⁰⁾/U∞)^Γn ⊂ X^Γn

∞ = 0. And therefore an exact sequence

(U∞/C∞)Γn ֒→(U_∞⁽⁰⁾/C∞)Γn ։(U_∞⁽⁰⁾/U∞)Γn.

By the snake lemma with the analogue exact sequence at finite level we obtain the sequence

0 ^//Kern(U∞/C∞) ^//Kern(U∞⁽⁰⁾/C∞) ^//Kern(U∞⁽⁰⁾/U∞)

ED BC GF

@A

++

WW WW WW W

Cokern(U∞/C∞) ^//Cokern(U∞⁽⁰⁾/C∞) ^//Cokern(U∞⁽⁰⁾/U∞) ^//0.

Now, using this sequence and the proposition 3.1 we prove our first key lemma :

Lemma 3.3

1. Kern(U∞/C∞)andKern(U∞⁽⁰⁾/C∞) are finite. Their orders are simultaneously bounded or not.

2. Cokern(U∞/C∞) and Cokern(U∞⁽⁰⁾/C∞) are finite. Their orders are simulta- neously bounded or not.

Remark : We have proven that, even if not bounded, the sequences of orders would have been asymptotically equivalent. We will not use this, because we will now proceed in bounding these orders !

4 Descent kernels

The second key lemma is Lemma 4.1

1. The orders of the kernels of the natural maps (U∞/C∞)Γn −→ Un/Cn are bounded.

2. The orders of the kernels of the natural maps (U∞⁽⁰⁾/C_∞)_Γn −→ Un⁽⁰⁾/C_n are bounded.

Proof. From the commutative diagram : (C∞)Γn

//

(U∞)Γn

//

(U∞/C∞)Γn

//

0

0 ^//C_n ^//U_n ^//U_n/C_n ^//0 we deduce the exact sequence :

0 ^// _Im(Ker^Kern(U_n(C^∞)

∞))

//Kern(U∞/C∞) ^//Cn/eCn

By the part 2 of theorem 1.1 Kern(U∞/C∞) is finite. To control Cokern(C∞) we use the lemma

Lemma 4.2 There exists an N such that for all n ≥N we have Cn/Cen∼=CN/CeN

Proof. LetI be the inertia subfield ofpforK_∞/Q. By lemma 2.5 of [Bjr], fornlarge enough (n such that I ⊂ Kn is large enough), we have Cn = CenCI. It follows that Coker_n(C_∞)∼=C_I/(Ce_nT

C_I). Now the increasing sequence Ce_nT

C_I has to stabilize because CI is of finiteZ_p-rank. This shows the lemma 4.2

To prove the lemma 4.1 it then suffices to bound the orders of Kern(U∞)/Im(Kern(C∞)).

For that, we prove that this sequence of quotients stabilizes :

Proposition 4.3 For any noetherian Λ-module M∞ let Minv(M∞)denote the sub- module of M∞ defined as follows :

Minv(M∞) = [

n∈N

(M∞)^Γn. Exists N such that for all n≥N we have

Kern(U∞)/Im(Kern(C∞))∼= Minv(U^′_∞/U∞) Im(Minv(U^′_∞/C∞)) Proof. Starting from the sequence :

0 ^//C∞ //U^′_∞ ^//U^′_∞/C∞ //0 one gets the diagram

0 ^//(U^′_∞/C∞)^{Γn //}

(C∞)Γn

//

(U^′_∞)Γn

0 ^//Cn //U^′_n

By the proposition 2.4 the kernels Kern(U^′_∞) are trivial. Hence we have an isomorphism Kern(C∞) ≃ (U^′_∞/C∞)^Γn. Since U^′_∞ is a noetherian module, so is U^′_∞/C_∞and therefore the increasing sequence (U^′_∞/C_∞)^Γn stabilizes and fornlarge we have Kern(C∞)≃Minv(U^′_∞/C∞). The same arguments withU∞ instead ofC∞

proves that (providedngreater than someN) we have Kern(U∞)∼= Minv(U^′_∞/U∞).

This shows the proposition and also (putting everything together) the first part of lemma 4.1. The second part of 4.1 follows then from lemma 3.3.

5 Descent cokernels

The third and final key lemma is Lemma 5.1

1. The orders of the cokernels of the natural maps (U∞/C∞)Γn −→ Un/Cn are bounded.

2. The orders of the cokernels of the natural maps (U∞⁽⁰⁾/C∞)Γn −→ Un⁽⁰⁾/Cn are bounded.

Proof. Starting with the snake diagram (C∞)Γn

//

(U_∞)_Γn ^//

(U_∞/C∞)Γn

//

0

0 ^//Cn //

Un⁽⁰⁾

//U⁽⁰⁾n /Cn //0

one gets the sequence

Kern(U∞⁽⁰⁾/C∞) ^//C_n/Ce_n ^//Un⁽⁰⁾/Ue_n ^//Cokern(U∞⁽⁰⁾/C∞) ^//0.

Recall that D is the maximal subfield of K∞ such that p is totally split in D. We may assume without loss of generality thatD⊂Kn(else enlarge n). By lemma 3.3, Ker_n(U∞⁽⁰⁾/C_∞) is bounded. By proposition 2.3 we have Ue_n = Ker(N_Kn/D: Un⁽⁰⁾ −→

U_D). One then gets an isomorphism Un⁽⁰⁾/Ue_n ∼=NKn/D(Un⁽⁰⁾) and using this isomor- phism the preceding sequence reads

0 ^//N_Kn/D(C_n) ^//N_Kn/D(Un⁽⁰⁾) ^//Cokern(U∞⁽⁰⁾/C∞) ^//0.

Now, 2 in lemma 5.1 follows from the :

Lemma 5.2 Let cn denotes [Kn :D] (asymptotically cn is equivalent to pⁿ).

1. (CD)^cn is a submodule of bounded finite index in NKn/D(Cn) 2. (U_D⁽⁰⁾)^cn is a submodule of bounded finite index in NKn/D(Un⁽⁰⁾) 3. (U_D⁽⁰⁾)^cn/C^c_Dⁿ is asymptotically equivalent to U_D⁽⁰⁾/C_D.

Proof. Assertion 3 is immediate. The finite constant group U_D⁽⁰⁾/CD maps onto (U_D⁽⁰⁾)^cn/(C_D)^cn. Since the norm N_Kn/D acts as c_n onC_D and U_D⁽⁰⁾ themselves, the inclusions and finiteness of indices in 1 and 2 are clear. We have to show that these finite indices are bounded.

For assertion 1 we use again lemma 2.5 of [Bjr], that is C_n = Ce_nC_I. Moreover, asCe_n ⊂Uen⁽⁰⁾ we have N_Kn/D(Ce_n) = 0 by proposition 2.3 (Without using semi-local units, N_Kn/D(Cen) = 0 can be checked directly using distribution relations on a gen- erating system of Cen). Hence we get NKn/D(Cn) = NKn/D(CenCI) = NKn/D(CI) = NI/D(CI)^[Kn:I]. This gives 1 because, as cn itself, [Kn :I] is asymptotically equiva- lent to pⁿ and N_I/D(CI) is of (constant) finite index in CD.

Assertion 2 is an easy exercise using local class field theory. Indeed, recall that U_n = ⊕_v|pU¹_Kn,v. Then the global norm NKn/D acts on each summand as the lo- cal norm NKn,v/Qp. By local class field theory, the quotient U¹_Qp/NKn,v/Qp(U¹_Kn,v) is isomorphic to the p-part of the ramification subgroup of Gal(Kn,v/Q_p). These wild ramification subgroups are cyclic with orders asymptotically equivalent to pⁿ. Summing up, it follows that NKn/D(U_n) contains U_D^cn with bounded finite index. A fortioriNKn/D(Un⁽⁰⁾) contains (U_D⁽⁰⁾)^cn with bounded finite index. This concludes the proof of lemma 5.2 and therefore of the second claim in lemma 5.1. The first claim in 5.1 then follows from lemma 3.3.

With lemmas 4.1 and 5.1 we have fullfilled our goal. We have directly proved that the natural descent homomorphisms (U∞/C∞)Γn −→ Un/Cn have bounded kernels and cokernels. As a consequence, we get without using Iwasawa’s Main