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

TL;DR

This paper demonstrates that the unit class groups, formed by global units modulo circular units in abelian cyclotomic extensions, satisfy descent properties similar to ideal class groups without relying on Iwasawa's Main Conjecture.

Contribution

It establishes the descent properties of unit class groups directly, bypassing the need for Iwasawa's Main Conjecture in the analysis.

Findings

Unit class groups satisfy good descent properties.

Asymptotic equivalence of unit class groups and ideal class numbers.

Descent properties proven without Iwasawa's Main Conjecture.

Abstract

Iwasawa's classical asymptotical formula relates the orders of the $p$-parts $X_n$ of the ideal class groups along a $\ZM_p$-extension $F_\infty/F$ of a number field $F$, to Iwasawa structural invariants $\la$ and $\mu$ attached to the inverse limit $X_\infty=\limpro X_n$. It relies on "good" descent properties satisfied by $X_n$. If $F$ is abelian and $F_\infty$ is cyclotomic it is known that the $p$-parts of the orders of the global units modulo circular units $U_n/C_n$ are asymptotically equivalent to the $p$-parts of the ideal class numbers. This suggests that these quotients $U_n/C_n$, so to speak unit class groups, satisfy also good descent properties. We show this directly, i.e. without using Iwasawa's Main Conjecture.

Equations

$${\mathcal{U}}_F\cong U_v^1{\otimes }_{{\mathbb{Z}}_p[G_v]}{\mathbb{Z}}_p[\mathrm{Gal}(F/\mathbb{Q})]\text{and}{\mathcal{N}}_F\cong \overline{F_v^{\times }}{\otimes }_{{\mathbb{Z}}_p[G_v]}{\mathbb{Z}}_p[\mathrm{Gal}(F/\mathbb{Q})].$$

$$N_{L/F}((x_w{)}_{w\mid p})=(\prod _{w\mid v}{N}_{L_w/K_v}(x_w){)}_{v\mid p}$$

$${\mathrm{Char}}_{\Lambda }(({\mathcal{U}}_{\infty }^{\psi }/{\overline{C}}_{\infty }^{\psi }))=(N_{{\mathbb{Q}}_p[\psi ]/{\mathbb{Q}}_p}(g_{\psi }(T)))=\left(\prod _{\sigma \in \mathrm{Gal}({\mathbb{Q}}_p[\psi ]/{\mathbb{Q}}_p)}{g}_{{\psi }^{\sigma }}(T)\right).$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$({\mathcal{N}}_{\infty }{)}_{{\Gamma }_n}⟶{\mathcal{N}}_n$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

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$$\underset{v\in S_d}{⨁}\mathrm{Gal}((K_{d,v}{)}_{\infty }^{\mathrm{cyc}}/K_{d,v})\cong {\mathbb{Z}}_p[S_d].$$

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$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$u\in \bigcap _{m\ge n}{N}_{K_{m,v}/K_{n,v}}(U_{K_{m,v}}^1)⟺N_{K_{n,v}/D_{n,v}}(u)=1$$

$${\tilde{\mathcal{U}}}_n=\mathrm{Ker}(N_{K_n/D_n}:{\mathcal{U}}_n⟶{\mathcal{U}}_{D_n}).$$

$$K_d\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\tilde{\mathcal{U}}}_n=N_{K_d/K_n}({\tilde{\mathcal{U}}}_d)\subset \mathrm{Ker}(N_{K_n/D_n}:{\mathcal{U}}_n⟶{\mathcal{U}}_{D_n}).$$

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$$N_{F/\mathbb{Q}}:{\mathcal{U}}_F⟶{\mathcal{U}}_{\mathbb{Q}}.$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

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$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$({\mathfrak{X}}_{\infty }{)}_{{\Gamma }_n}\cong {\text{Tor}}_{{\mathbb{Z}}_p}{\mathfrak{X}}_n.$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

0 Introduction

Let $K$ be a number field and $p$ an odd prime ($p\neq 2$) and let $K_{\infty}/K$ be a ${\mathbb{Z}}_{p}$-extension (quite soon $K_{\infty}/K$ will be the cyclotomic ${\mathbb{Z}}_{p}$-extension). Recall the usual notations : $\Gamma=\mathop{\mathrm{Gal}}\nolimits(K_{\infty}/K)$ is the Galois group of $K_{\infty}/K$, $K_{n}$ is the $n^{\text{th}}$-layer of $K_{\infty}$ (so that $[K_{n}:K]=p^{n}$), $\Gamma_{n}=\mathop{\mathrm{Gal}}\nolimits(K_{\infty}/K_{n})$, and $G_{n}=\mathop{\mathrm{Gal}}\nolimits(K_{n}/K)\cong\Gamma/\Gamma_{n}$. Let us consider a sequence $(M_{n})_{n\in{\mathbb{N}}}$ of ${\mathbb{Z}}_{p}[G_{n}]$-modules equipped with norm maps $M_{n}\longrightarrow M_{n-1}$ and the inverse limit of this sequence $M_{\infty}=\lim\limits_{\leftarrow}M_{n}$ seen as a $\Lambda={\mathbb{Z}}_{p}[[\Gamma]]$-module. The general philosophy of Iwasawa theory is to study the simpler $\Lambda$-structure of $M_{\infty}$, then to try and recollect information about the $M_{n}$’s themselves from that structure. For instance, if $M_{\infty}$ is $\Lambda$-torsion, one can attach two invariants $\lambda$ and $\mu$ to $M_{\infty}$. If we assume further that

• $(i)$

the $\Gamma_{n}$ coinvariants $(M_{\infty})_{\Gamma_{n}}$ are finite,

• $(ii)$

the sequence $(M_{n})_{n\in{\mathbb{N}}}$ behaves ”nicely” viz descent;

then one can prove that the orders of the $M_{n}$ are asymptotically equivalent to $p^{\lambda n+\mu p^{n}}$. These two assumptions are expected to occur whenever one chooses for $(M_{n})$ significant modules (from the number theoretic point of view). However proving them may require some effort : actually, $(i)$ for the canonical Iwasawa module $\rm{tor}_{\Lambda}({\mathfrak{X}}_{\infty})$ is (one of the many equivalent formulations of) Leopoldt’s conjecture. The historical example occurs when we specialize $M_{n}=X_{n}$, the $p$-part of ideal class group of $K_{n}$. Then the asymptotic formulas are a theorem of Iwasawa. The proof of this theorem uses an auxiliary module $Y_{\infty}$ which is pseudo-isomorphic to $X_{\infty}$ but with better descent properties.

In the present paper we are interested in similar statements for unit class groups, that is for $M_{n}={\overline{U}}_{n}/{\overline{C}}_{n}$, the $p$-part of the quotient of units modulo the subgroup of circular units of $K_{n}$. In order that the latter merely exist we need to assume that all $K_{n}$ are abelian over ${\mathbb{Q}}$ : hence $K$ is abelian and $K_{\infty}/K$ 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 $p\neq 2$. Now by Sinnott index formulas we know that the orders of ${\overline{U}}_{n}/{\overline{C}}_{n}$ are asymptotically equivalent to the orders of $X_{n}$. And as a consequence of Iwasawa’s Main Conjecture, the structural invariants $\lambda$ and $\mu$ of both inverse limits ${\overline{U}}_{\infty}/{\overline{C}}_{\infty}=\lim\limits_{\leftarrow}\overline{U}_{n}/\overline{C}_{n}$ and of $X_{\infty}=\lim\limits_{\leftarrow}X_{n}$ are 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 groups $\overline{U}_{n}/\overline{C}_{n}$ along the finite steps of the ${\mathbb{Z}}_{p}$-extension are those prescribed by the structural Iwasawa invariants of the inverse limit $\overline{U}_{\infty}/\overline{C}_{\infty}$. 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 $\lambda$ and $\mu$ invariants of the two class groups without using Iwasawa Main Conjecture. The equality between these two $\lambda$ 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 $\mu$-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 ${\overline{A}}$ the $p$-completion of $A$, i.e. the inverse limit ${\overline{A}}=\underset{\leftarrow}{\lim}\ A/A^{p^{m}}$. If $A$ is finitely generated over ${\mathbb{Z}}$, then $\bar{A}\cong A\otimes{\mathbb{Z}}_{p}$.
• $U_{F}$ is the group of units of $F$.
• $U^{\prime}_{F}$ is the group of $S$-units of $F$ : that is, elements of $F^{\times}$ whose valuations are trivial for all finite places $v$ of $F$ such that $v\not\in S$.
• $X_{F}$ is the $p$-part of $cl(F)$ which in turn is the ideal class group of $F$.
• $X^{\prime}_{F}$ is the $p$-part of $cl^{\prime}(F)$ which in turn is the quotient of $cl(F)$ modulo the subgroup generated by classes of primes in $S$.
• ${\mathfrak{X}}_{F}$ is the Galois group over $F$ of the maximal $S$-ramified (i.e. unramified outside $S$) abelian pro-$p$-extension of $F$.
• ${\mathcal{N}}_{F}$ is the multiplicative group of the semi-local numbers. As a mere ${\mathbb{Z}}_{p}$-module ${\mathcal{N}}_{F}=\prod_{v\in S}\overline{F_{v}^{\times}}$ where $F_{v}$ is the completion of $F$ at the place $v\in S$.
• ${\mathcal{U}}_{F}$ is the group of semi-local units of $F$. As a mere ${\mathbb{Z}}_{p}$-module ${\mathcal{U}}_{F}=\prod_{v\in S}U^{1}_{v}$ where $U^{1}_{v}$ is the set of principal units of $F_{v}$, that is units $\equiv 1$ modulo the maximal ideal of $F_{v}$.

So for instance the ${\overline{U}}_{n}$’s above could have been understood as $U_{K_{n}}\otimes{\mathbb{Z}}_{p}$ and $X_{F}=\overline{cl(F)}$.

1 Consequences of Leopoldt’s Conjecture

If $F$ is Galois over ${\mathbb{Q}}$, the groups ${\mathcal{U}}_{F}$ and ${\mathcal{N}}_{F}$ come equipped with the induced action of $\mathop{\mathrm{Gal}}\nolimits(F/{\mathbb{Q}})$. In other words if we fix one place $v$ in $S$ and if we put $G_{v}=\mathop{\mathrm{Gal}}\nolimits(F_{v}/{\mathbb{Q}}_{p})$ then we have (as ${\mathbb{Z}}_{p}$-modules) :

[TABLE]

These isomorphisms define the action of $\mathop{\mathrm{Gal}}\nolimits(F/{\mathbb{Q}})$ on ${\mathcal{U}}_{F}$ and on ${\mathcal{N}}_{F}$. Let us now consider the cyclotomic ${\mathbb{Z}}_{p}$-extension $K_{\infty}/K$ of our totally real abelian over ${\mathbb{Q}}$ number field $K$ and the $n^{\text{th}}$-finite layers $K_{n}/K$, i.e. $K_{n}$ is the unique subfield of $K_{\infty}$ with $[K_{n}:K]=p^{n}$. We will denote by $C_{n}$ (and of course we’ll be more interested in ${\overline{C}}_{n}$) the group of circular units of $K_{n}$ as defined by Sinnott ([Si]). We will indicate consistently by the subscript n arithmeticall object related to $K_{n}$. So ${\overline{U}}_{n}$, ${\mathcal{U}}_{n}$ and so on make sense. For any extension $L/F$ of global fields the formula

[TABLE]

defines a Galois equivariant morphism $N_{L/F}\colon{\mathcal{N}}_{L}\longrightarrow{\mathcal{N}}_{F}$ which is compatible with the usual norms on global units for instance. As ever ${\overline{U}}_{\infty}$, ${\overline{C}}_{\infty}$ ${\mathcal{N}}_{\infty}$, ${\mathcal{U}}_{\infty}$, ${\overline{U}}^{\prime}_{\infty}$ and so on denotes the inverse limit (related to norm maps) of ${\overline{U}}_{n}$, ${\overline{C}}_{n}$, ${\mathcal{N}}_{n}$, ${\mathcal{U}}_{n}$, ${\overline{U}}^{\prime}_{n}$ 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 $K_{n}$ are abelian over ${\mathbb{Q}}$). 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 $G=\mathop{\mathrm{Gal}}\nolimits(K/{\mathbb{Q}})$. Recall that $\Gamma=\mathop{\mathrm{Gal}}\nolimits(K_{\infty}/K)$ and that $\Lambda={\mathbb{Z}}_{p}[[\Gamma]]$. Fix a generator $\gamma$ of $\Gamma$. For all non-trivial Dirichlet characters of the first kind $\psi$ of $\widehat{G}$, following [T] let us denote by $g_{\psi}(T)$ the Iwasawa power series associated to the Kubota-Leopoldt $p$-adic $L$-function $L_{p}(\psi,s)$ (with our fixed choice of $\gamma$).

Assume $p$ is tamely ramified in $K$ so that every characters of $G$ are of the first kind. Then up to a power of $p$, the characteristic ideal of the $\Lambda$-torsion module ${\mathcal{U}}_{\infty}/{\overline{C}}_{\infty}$ is generated by the product $\displaystyle\prod_{\psi\in\widehat{G},\psi\neq 1}g_{\psi}(T)$. 2. 2.

For all $n\in{\mathbb{N}}$, the $\Gamma_{n}$-coinvariants $({\mathcal{U}}_{\infty}/{\overline{C}}_{\infty})_{\Gamma_{n}}$ and $({\overline{U}}_{\infty}/{\overline{C}}_{\infty})_{\Gamma_{n}}$ are finite.

Proof. Let us consider $\psi$-parts $M^{\psi}$ as defined in [T] for any ${\mathbb{Z}}_{p}[G]$-module $M$. These $\psi$ parts are naturally ${\mathbb{Z}}_{p}[\psi]$-modules and ${\mathbb{Z}}_{p}[G]$-submodules of $M$. For technicalities about $\psi$-components which are often left to readers see Beliaeva’s thesis [Bt]. From theorem 3.1 of [T] we use that ${\mathcal{U}}_{\infty}^{\psi}/{\overline{C}}_{\infty}^{\psi}$ is ${\mathbb{Z}}_{p}$-torsion free and that its characteristic series over $\Lambda[\psi]$ is $g_{\psi}(T)$. Let $\sigma\in\mathop{\mathrm{Gal}}\nolimits({\mathbb{Q}}_{p}[\psi]/{\mathbb{Q}}_{p})$; then $({\mathcal{U}}_{\infty}/{\overline{C}}_{\infty})^{\psi^{\sigma}}$ is isomorphic to $({\mathcal{U}}_{\infty}/{\overline{C}}_{\infty})^{\psi}$ and $g_{\psi^{\sigma}}(T)=\sigma(g_{\psi}(T))$. For $R$ equal to $\Lambda$ or to $\Lambda[\psi]$, let us abbreviate by ${\rm Char}_{R}(M)$ the characteristic ideal of the $R$-module $M$. It is an easy linear algebra exercise to check that if a power series $f$ generates ${\rm Char}_{\Lambda[\psi]}(M)$ then $N_{{\mathbb{Q}}_{p}[\psi]/{\mathbb{Q}}_{p}}(f)$ generates ${\rm Char}_{\Lambda}(M)$. Hence we get

[TABLE]

Let us fix a set $\Psi$ of representatives of $\widehat{G}$ up to ${\mathbb{Q}}_{p}$-conjugation classes. If a ${\mathbb{Z}}_{p}[G]$-module $M$ is ${\mathbb{Z}}_{p}$-torsion free then the submodules of $M$, $(M^{\psi})_{\psi\in\Psi}$, are mutually direct summands and the quotient $M/\oplus_{\psi\in\Psi}M^{\psi}$ is annihilated by $\#G$. This applies to ${\mathcal{U}}_{\infty}$ and ${\overline{C}}_{\infty}$, so that we may use the inclusions $\bigoplus_{\psi\in\Psi}{\mathcal{U}}_{\infty}^{\psi}\subset{\mathcal{U}}_{\infty}$ and $\bigoplus_{\psi\in\Psi}{\overline{C}}^{\psi}_{\infty}\subset{\overline{C}}_{\infty}$ to form the snake diagram :

[TABLE]

Now the kernel of $\alpha$ must be ${\mathbb{Z}}_{p}$-torsion, as a submodule of the cokernel of the first inclusion, hence is trivial because all ${\mathcal{U}}_{\infty}^{\psi}/{\overline{C}}_{\infty}^{\psi}$ are ${\mathbb{Z}}_{p}$-free. This proves that up to a power of $p$ the characteristic ideal ${\rm Char}_{\Lambda}({\mathcal{U}}_{\infty}/{\overline{C}}_{\infty})$ is equal to ${\rm Char}_{\Lambda}(\bigoplus_{\psi\in\Psi}{\mathcal{U}}_{\infty}/{\overline{C}}_{\infty})$, which in turn is what we wanted.

The second claim is equivalent to the fact that both characteristic ideals are prime to all $(T+1)^{p^{n}}-1$ for all $n\in{\mathbb{N}}$. From the inclusion ${\overline{U}}_{\infty}/{\overline{C}}_{\infty}\subset{\mathcal{U}}_{\infty}/{\overline{C}}_{\infty}$ we see that the first characteristic ideal divides the second. Let $\zeta$ be a $p$-power order root of unity and let $\rho$ be the unique character of the second kind such that $\rho(\gamma)=\zeta^{-1}$. Then we have $g_{\psi}(\zeta-1)=L_{p}(1,\psi\rho)$. By Leopoldt’s conjecture (see [W] corollary 5.30) $L_{p}(1,\psi\rho)\neq 0$. Hence in the special case where $p$ is tame in $K$, 2 follows from 1. We now want to remove this hypothesis. Let $I_{p}\subset\mathop{\mathrm{Gal}}\nolimits(K_{\infty}/{\mathbb{Q}})$ be the inertia subgroup of $p$, and let ${\overline{I}}_{p}\subset I_{p}$ be its pro-$p$-part. Then the field $L=K_{\infty}^{{\overline{I}}_{p}}$ is a finite abelian over ${\mathbb{Q}}$ number field and $p$ is at most tamely ramified in $L$. Since all modules $M_{\infty}$ depend on $K_{\infty}$ 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

$L_{\infty}=K_{\infty}$.

Proof. Let ${\mathbb{B}}_{\infty}/{\mathbb{Q}}$ be the ${\mathbb{Z}}_{p}$-extension of ${\mathbb{Q}}$. By construction we have $L\subset K_{\infty}$ and therefore $L_{\infty}\subset K_{\infty}$. By definition of $L$, the group $\mathop{\mathrm{Gal}}\nolimits(K_{\infty}/L_{\infty})$ is then a subgroup of ${\overline{I}}_{p}$, hence $K_{\infty}/L_{\infty}$ is a $p$-extension totally ramified at $p$. But on the other hand ${\mathbb{B}}_{\infty}\subset L_{\infty}$ and $K_{\infty}$ is abelian over ${\mathbb{Q}}$. Let ${\mathbb{Q}}^{\rm{ab}}$ be the maximal abelian extension of ${\mathbb{Q}}$. Then ${\mathbb{Q}}^{\rm{ab}}/{\mathbb{B}}_{\infty}$ is tamely ramified at (the only) prime of ${\mathbb{B}}_{\infty}$ above $p$. This shows $L_{\infty}=K_{\infty}$.

$\quad\square$

Now, with that lemma we already have proven the finiteness assertion when we take coinvariants along $\widetilde{\Gamma}_{n}=\mathop{\mathrm{Gal}}\nolimits(K_{\infty}/L_{n})$. We conclude the proof of the second part of theorem 1.1 by pointing that even if the first few $\Gamma_{n}$ may differ from $\tilde{\Gamma}_{n}$, we get $\Gamma_{n}=\tilde{\Gamma}_{n}$ as soon as $K_{\infty}/K_{n}$ is totally ramified at $p$. Due to canonical surjections $(M_{\infty})_{\Gamma_{n}}\twoheadrightarrow(M_{\infty})_{\Gamma_{n-1}}$, this does not change anything to the finiteness of $({\mathcal{U}}_{\infty}/{\overline{C}}_{\infty})_{\Gamma_{n}}$.

$\quad\square$

• 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 $({\mathcal{U}}_{\infty}/{\overline{C}}_{\infty})_{\Gamma_{n}}$ (resp. $({\overline{U}}_{\infty}/{\overline{C}}_{\infty})_{\Gamma_{n}}$) has been proved, we want to relate these modules with their counterparts ${\mathcal{U}}_{n}/{\overline{C}}_{n}$ (resp. ${\overline{U}}_{n}/{\overline{C}}_{n}$) at finite levels. On the way we will study descent for various other multiplicative Galois modules.

2 Background: easy part of descent

Let $M_{\infty}=\lim\limits_{\leftarrow}(M_{n})$ be a $\Lambda$-module. Of course the projections $M_{\infty}\longrightarrow M_{n}$ factor through $(M_{\infty})_{\Gamma_{n}}\longrightarrow M_{n}$, but in general these maps have non trivial kernels and cokernels. Let us denote $\mathop{\mathrm{Ker}}\nolimits_{n}(M_{\infty})$ for the kernel of $(M_{\infty})_{\Gamma_{n}}\longrightarrow M_{n}$, ${\widetilde{M}}_{n}\subset M_{n}$ for its image, and $\mathop{\mathrm{Coker}}\nolimits_{n}(M_{\infty})=M_{n}/{\widetilde{M}}_{n}$ for its cokernel. Let $l$ be a prime number ($l=p$ is allowed) and $E$ be a finite extension of ${\mathbb{Q}}_{l}$. Put $E_{\infty}/E$ for a ${\mathbb{Z}}_{p}$-extension of $E$, and $E_{n}$ for the $n^{\rm th}$ finite layers. Let us abbreviate $L_{E}=\overline{E^{\times}}$, consistently $L_{n}=\overline{(E_{n})^{\times}}$, and $L_{\infty}=\lim\limits_{\leftarrow}L_{n}$. For all $n\in{\mathbb{N}}\cup{\infty}$, local class field theory identifies $L_{n}$ with $\mathop{\mathrm{Gal}}\nolimits(M_{n}/E_{n})$, where $M_{n}$ is the maximal abelian $p$-extension of $E_{n}$.

Lemma 2.1

The natural map $(L_{\infty})_{\Gamma_{n}}\longrightarrow L_{n}$ fits into an exact sequence

[TABLE]

Proof. This is well known to experts. We follow the cohomological short cut of [N1]. The group $\Gamma_{n}\simeq{\mathbb{Z}}_{p}$ is pro-$p$-free hence we have $H^{2}(\Gamma_{n},{\mathbb{Q}}_{p}/{\mathbb{Z}}_{p})=0$. Consider the inflation-restriction sequence associated to the extension of groups

[TABLE]

where ${\mathcal{G}}_{n}$ is the absolute Galois group of $E_{n}$ :

[TABLE]

Applying Pontryagin duality and class field theory we get

[TABLE]

$\quad\square$

Let us come back to our global field $K$ and its cyclotomic ${\mathbb{Z}}_{p}$-extension.

Proposition 2.2

Let $S_{n}$ be the set of places of $K_{n}$ dividing $p$. For all $n$ and $v\in S_{n}$ put $K_{n,v}$ for the completion at $v$ of $K_{n}$ and $(K_{n,v})^{\rm cyc}_{\infty}/K_{n,v}$ for its cyclotomic ${\mathbb{Z}}_{p}$-extension. For all $n$ the natural map

[TABLE]

fits into an exact sequence

[TABLE]

Proof. Let us examine the places above $p$ along $K_{\infty}/K$. These primes have non trivial conjugated (hence equal) decomposition subgroups. There exists a $d\in{\mathbb{N}}$ such that $(i)$ no primes above $p$ splits anymore in $K_{\infty}/K_{d}$ and $(ii)$ all primes above $p$ are totally split in $K_{d}/K$. For all $n\geq d$ we then have $S_{n}\cong S_{d}$, $\displaystyle{\mathcal{N}}_{n}=\bigoplus_{v\in S_{d}}L_{K_{n,v}}$ and $\mathop{\mathrm{Gal}}\nolimits(K_{\infty}/K_{n})$ identifies with the local Galois groups $\mathop{\mathrm{Gal}}\nolimits((K_{n,v})^{\rm cyc}_{\infty}/K_{n,v})$ (at every $v\in S_{n}$). So in the case $n\geq d$, the proposition follows from lemma 2.1. Next suppose that $0\leq n<d$, and let $G^{m}_{n}$ be the Galois group $\mathop{\mathrm{Gal}}\nolimits(K_{m}/K_{n})$. Of course we have $({\mathcal{N}}_{\infty})_{\Gamma_{n}}\cong(({\mathcal{N}}_{\infty})_{\Gamma_{d}})_{G^{d}_{n}}$. Using the previous case we have an exact sequence

[TABLE]

As $(K_{d,v})^{\rm cyc}_{\infty}/K_{d,v}$ is cyclotomic, the action of the local Galois group $\mathop{\mathrm{Gal}}\nolimits(K_{d,v}/{\mathbb{Q}}_{p})$ on the group $\mathop{\mathrm{Gal}}\nolimits((K_{d,v})^{\rm cyc}_{\infty}/K_{d,v})$ is trivial. Hence we have an isomorphism of Galois modules

[TABLE]

Since the places above $p$ split totally in $K_{d}/K$, over $G^{d}_{0}$ these two modules are cohomologically trivial, and the same is true for ${\mathcal{N}}_{d}\cong{\mathcal{N}}_{0}\otimes_{{\mathbb{Z}}_{p}}{\mathbb{Z}}_{p}[G^{d}_{0}]$ (as $G^{d}_{0}$-modules). Therefore in the sequence $({\dagger})$ two (hence all) terms are $G^{d}_{0}$-cohomologically trivial. The triviality of $\widehat{H}^{1}(G^{d}_{n},({\mathcal{N}}_{\infty})_{\Gamma_{d}})$ proves the injectivity of $({\mathcal{N}}_{\infty})_{\Gamma_{n}}\longrightarrow{\mathcal{N}}_{n}$. It then suffices to apply $N_{K_{d}/K_{n}}$ to the sequence $({\dagger})$ and use the triviality of the three $\widehat{H}^{0}(G^{d}_{n},--)$ to get the full sequence

[TABLE]

$\quad\square$

Let $d$ be as before. Let $D_{n}$ be the decomposition subfield for the place $p$ in $K_{n}$ (i.e $p$ is totally split in $D_{n}$ and no place of $D_{n}$ above $p$ splits anymore in $K_{n}/K$). Then for all $n\geq d$ we have $D_{n}=D_{d}$. Let us put $D=D_{d}$ for the decomposition subfield for the place $p$ in $K_{\infty}$.

Proposition 2.3

Recall that for a $\Lambda$-module $M_{\infty}=\lim\limits_{\leftarrow}(M_{n})$ we denote $\mathop{\mathrm{Ker}}\nolimits_{n}(M_{\infty})$ for the kernel, ${\widetilde{M}}_{n}$ for the image and $\mathop{\mathrm{Coker}}\nolimits_{n}(M_{\infty})$ for the cokernel of the natural map $(M_{\infty})_{\Gamma_{n}}\longrightarrow M_{n}$.

For all $n\geq 0$, $\mathop{\mathrm{Ker}}\nolimits_{n}({\mathcal{U}}_{\infty})$ is isomorphic to ${\mathbb{Z}}_{p}[S_{n}]$, hence its ${\mathbb{Z}}_{p}$-rank is $\#S_{n}$. 2. 2.

For all $n\geq 0$, $\mathop{\mathrm{Coker}}\nolimits_{n}({\mathcal{U}}_{\infty})\cong N_{K_{n}/D_{n}}({\mathcal{U}}_{n})$, hence its ${\mathbb{Z}}_{p}$-rank is $\#S_{n}$. In other words we have an exact sequence

[TABLE]

Proof. For any local field $E$ the (normalized) valuation $v$ of $E$ gives an exact sequence $0\rightarrow U^{1}_{E}\rightarrow\overline{E^{\times}}\overset{v}{\rightarrow}{\mathbb{Z}}_{p}\rightarrow 0$, where $U^{1}_{E}$ is the set of principal units of $E$. Taking the semi-local version of this sequence and projective limits we obtain

[TABLE]

But for all $n$, ${\mathcal{N}}_{n}^{\Gamma}={\mathcal{N}}_{0}$ contains no infinitely $p$-divisible element. Therefore ${\mathcal{N}}_{\infty}^{\Gamma}=\{0\}$ and taking $\Gamma_{n}$-cohomology on this sequence we get :

[TABLE]

Now since $({\mathcal{N}}_{\infty})_{\Gamma_{n}}\longrightarrow{\mathcal{N}}_{n}$ is a monomorphism, $\mathop{\mathrm{Ker}}\nolimits_{n}({\mathcal{U}}_{\infty})$ identifies with ${\mathbb{Z}}_{p}[S_{d}]^{\Gamma_{n}}\cong{\mathbb{Z}}_{p}[S_{n}]$. 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 $n\geq d$. By compactness we have ${\widetilde{\mathcal{U}}}_{n}=\bigcap_{m\geq n}N_{K_{m}/K_{n}}({\mathcal{U}}_{m})$. Because $n\geq d$ the global norm $N_{K_{m}/K_{n}}\colon{\mathcal{U}}_{m}\longrightarrow{\mathcal{U}}_{n}$ is nothing else but the direct sum place by place of the local norms of the extensions $K_{m,w}/K_{n,v}$ (at each unique $w\in S_{m}$ above a fixed $v\in S_{n}$). Fix a place $v\in S_{n}$ and for all $m\geq n$ still call $v$ the unique place of $S_{m}$ above $v$ and also $v$ the unique place of $D_{n}$ under $v$. Note that $D_{n,v}={\mathbb{Q}}_{p}$. Let $u\in U^{1}_{K_{n,v}}$, then using local class field theory we see that $u\in\bigcap_{m\geq n}N_{K_{m,v}/K_{n,v}}(U^{1}_{K_{m,v}})$ if and only if $N_{K_{n,v}/{\mathbb{Q}}_{p}}(u)\in\bigcap_{n\in{\mathbb{N}}}N_{({\mathbb{Q}}_{p})^{\rm cyc}_{n}/{\mathbb{Q}}_{p}}(U^{1}_{({\mathbb{Q}}_{p})^{\rm cyc}_{n}})$, where $({\mathbb{Q}}_{p})^{\rm cyc}_{n}$ denotes the $n^{\text{th}}$-step of the cyclotomic (hence totally ramified) ${\mathbb{Z}}_{p}$-extension of ${\mathbb{Q}}_{p}$. By local class field theory $(.,({\mathbb{Q}}_{p})^{\rm cyc}_{\infty}/{\mathbb{Q}}_{p})$ is an isomorphism from $U^{1}_{{\mathbb{Q}}_{p}}$ to $\mathop{\mathrm{Gal}}\nolimits(({\mathbb{Q}}_{p})^{\rm cyc}_{\infty}/{\mathbb{Q}}_{p})$. The equivalence

[TABLE]

follows. As no place above $p$ splits in $K_{n}/D_{n}$ this gives

[TABLE]

Next pick an $n$ such that $0\leq n<d$. Consider the diagram of fields

[TABLE]

There every prime above $p$ is totally split in the extensions $K_{d}/K_{n}$, and $D_{d}/D_{n}$ and no prime above $p$ splits at all in $K_{n}/D_{n}$ nor in $K_{d}/D_{d}$. It follows that $N_{K_{d}/K_{n}}$ is surjective onto ${\mathcal{U}}_{n}$ and that

[TABLE]

Conversely pick $u=(u_{v})_{v\in S_{n}}\in\mathop{\mathrm{Ker}}\nolimits(N_{K_{n}/D_{n}}\colon{\mathcal{U}}_{n}\longrightarrow{\mathcal{U}}_{D_{n}})$. At each $v\in S_{n}$ choose a single $w(v)\in S_{d}$ above $v$ and define $t=(t_{w})_{w\in S_{d}}\in{\mathcal{U}}_{d}$ by putting $t_{w}=1$ 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 $N_{K_{d}/K_{n}}(t)=u$ and $t\in{\widetilde{\mathcal{U}}}_{d}$ which shows that $u\in{\widetilde{\mathcal{U}}}_{n}$.

$\quad\square$

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 ${\overline{U}}^{\prime}_{n}$ 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 $r_{1}=[K:{\mathbb{Q}}]$, $r_{2}=0$, and all $\overline{U}^{\prime}_{n}$ are ${\mathbb{Z}}_{p}$-torsion free. To avoid ugly notations $\widetilde{{\overline{M}}}$ we will denote ${\widetilde{U}}^{\prime}_{n}$ for the image of $\overline{U}^{\prime}_{\infty}$ in $\overline{U}^{\prime}_{n}$.

Proposition 2.4

The $\Lambda$-module ${\overline{U}}_{\infty}^{\prime}$ is free of rank $[K:{\mathbb{Q}}]$. 2. 2.

For all $n$ the natural map $(\overline{U}^{\prime}_{\infty})_{\Gamma_{n}}\longrightarrow\overline{U}^{\prime}_{n}$ is injective. 3. 3.

${\widetilde{U}}^{\prime}_{n}\cong(\overline{U}^{\prime}_{\infty})_{\Gamma_{n}}$* is a free ${\mathbb{Z}}_{p}[G_{n}]$-module of rank $[K:{\mathbb{Q}}]$ (hence is ${\mathbb{Z}}_{p}$ free of rank $p^{n}[K:{\mathbb{Q}}]=[K_{n}:{\mathbb{Q}}]$).*

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 field $K$ 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 $K_{n}$). Let ${\mathfrak{X}}_{\infty}$ be the standard Iwasawa module ${\mathfrak{X}}_{\infty}=\lim\limits_{\leftarrow}({\mathfrak{X}}_{n})$, where ${\mathfrak{X}}_{n}$ is the Galois group over $K_{n}$ of its maximal abelian $S$-ramified $p$-extension (Nota : of course this definition works also for $n=\infty$). Recall from the end of the introduction the notation $X^{\prime}_{n}$ for the $p$-part of the $(p)$-class group of $K_{n}$. From class field theory we have the (some time called decomposition) exact sequence

[TABLE]

Put ${\mathcal{D}}_{\infty}=\mathop{\mathrm{Im}}\nolimits({\mathcal{N}}_{\infty}\longrightarrow{\mathfrak{X}}_{\infty})$. By Leopoldt’s conjecture for $K$ we have ${\mathfrak{X}}_{\infty}^{\Gamma}=0$ therefore ${\mathcal{D}}_{\infty}^{\Gamma}=0$. This implies that the induced map $(\overline{U}^{\prime}_{\infty})_{\Gamma}\longrightarrow({\mathcal{N}}_{\infty})_{\Gamma}$ is a monomorphism. Now 2. follows from proposition 2.2.

$K$ is totally real and $p\neq 2$, so $(\overline{U}^{\prime}_{\infty})_{\Gamma}$ is ${\mathbb{Z}}_{p}$-free as a submodule of $\overline{U}^{\prime}_{0}$. As for ${\mathcal{N}}_{\infty}$, we have $(\overline{U}^{\prime}_{\infty})^{\Gamma}=0$ (same argument applies). These two facts suffice to show the $\Lambda$-freeness of $\overline{U}^{\prime}_{\infty}$. To compute the rank we consider the sequence

[TABLE]

As ${\mathcal{D}}_{\infty}$ is a torsion-$\Lambda$-module (as a submodule of ${\mathfrak{X}}_{\infty}$) with trivial $\Gamma$-invariants, its $\Gamma$-coinvariants are finite. Hence $(\overline{U}^{\prime}_{\infty})_{\Gamma}$ has the same ${\mathbb{Z}}_{p}$-rank as $({\mathcal{N}}_{\infty})_{\Gamma}$. By proposition 2.2 this rank is $\rm{rank}_{{\mathbb{Z}}_{p}}({\mathcal{N}}_{0})-\#S_{0}=[K:{\mathbb{Q}}]$. By Nakayama’s lemma the $\Lambda$-rank of $\overline{U}^{\prime}_{\infty}$ is also $[K:{\mathbb{Q}}]$. This concludes the proof of 1.

3 is an immediate corollary of 1 and 2.

$\quad\square$

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 for $M_{\infty}={\overline{U}}_{\infty}/\overline{C}_{\infty}$ or (which will be proven to be equivalent) for $M_{\infty}={\mathcal{U}}^{(0)}_{\infty}/\overline{C}_{\infty}$ (see explanations and notations below for the symbol (0)). This means that in both cases above, $\mathop{\mathrm{Ker}}\nolimits_{n}(M_{\infty})$ and $\mathop{\mathrm{Coker}}\nolimits_{n}(M_{\infty})$ 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 ${\overline{U}}^{\prime}_{\infty}$ (proposition 2.4) to bound the kernels of descent for $\overline{U}_{\infty}/\overline{C}_{\infty}$. Then we use local class field theory (proposition 2.3) to bound the cokernels of descent for ${\mathcal{U}}^{(0)}_{\infty}/\overline{C}_{\infty}$.

But before this we have to slightly change the sequence ${\mathcal{U}}_{n}/\overline{C}_{n}$. Indeed, for all $n$, the module ${\mathcal{U}}_{n}/\overline{C}_{n}$ is (by Leopoldt’s conjecture) of ${\mathbb{Z}}_{p}$-rank $1$ while $({\mathcal{U}}_{\infty}/\overline{C}_{\infty})_{\Gamma_{n}}$ is torsion. This rank $1$ comes from $N_{K_{n}/{\mathbb{Q}}}(\overline{U}_{n})=\{0\}$, or if we adopt the class field theory point of view, it represents the rank of the ${\mathbb{Z}}_{p}$-extension $K_{\infty}/K$. Let us put the

Notation: Let $F$ be a number field. In the sequel ${\mathcal{U}}^{(0)}_{F}$ will denote the kernel of

[TABLE]

Consistently ${\mathcal{U}}^{(0)}_{n}$ will denote the kernel of $N_{K_{n}/{\mathbb{Q}}}$ and ${\mathcal{U}}_{\infty}^{(0)}=\lim\limits_{\leftarrow}{\mathcal{U}}_{n}^{(0)}$.

By proposition 2.3 we have ${\widetilde{\mathcal{U}}}_{n}\subset{\mathcal{U}}^{(0)}_{n}$ and therefore ${\mathcal{U}}_{\infty}=\lim\limits_{\leftarrow}({\mathcal{U}}^{(0)}_{n})$. Moreover ${\mathcal{U}}^{(0)}_{n}/\overline{C}_{n}$ is torsion and since $N_{K_{n}/{\mathbb{Q}}}({\mathcal{U}}_{n})$ is ${\mathbb{Z}}_{p}$-torsion free we have ${\mathcal{U}}^{(0)}_{n}/\overline{C}_{n}=\text{\rm Tor}_{{\mathbb{Z}}_{p}}({\mathcal{U}}_{n}/\overline{C}_{n})$, and accordingly ${\mathcal{U}}^{(0)}_{n}/\overline{U}_{n}=\text{\rm Tor}_{{\mathbb{Z}}_{p}}({\mathcal{U}}_{n}/\overline{U}_{n})$. For all these reasons, it is clearly more convenient to (and we will from now) use the sequence $({\mathcal{U}}_{n}^{(0)})_{n\in{\mathbb{N}}}$ instead of $({\mathcal{U}}_{n})_{n\in{\mathbb{N}}}$. This convention gives sense to the notations ${\widetilde{\mathcal{U}}}^{(0)}_{n}$, $\mathop{\mathrm{Ker}}\nolimits_{n}({\mathcal{U}}_{\infty}^{(0)})$, $\mathop{\mathrm{Coker}}\nolimits_{n}({\mathcal{U}}_{\infty}^{(0)})$ and consistently to the same notations associated to sequences ${\mathcal{U}}^{(0)}_{n}/\overline{C}_{n}$, ${\mathcal{U}}^{(0)}_{n}/\overline{U}_{n}$ and so on. Of course, due to proposition 2.3, only the various $\mathop{\mathrm{Coker}}\nolimits_{n}$ will actually change when we replace ${\mathcal{U}}_{n}$ by ${\mathcal{U}}_{n}^{(0)}$.

Proposition 3.1

Recall that $X_{n}$ stands for the $p$-part of the class group of $K_{n}$, and that $\mathop{\mathrm{Ker}}\nolimits_{n}(X_{\infty})$ is the kernel of the natural map $(X_{\infty})_{\Gamma_{n}}\longrightarrow X_{n}$.

The map $({\mathcal{U}}_{\infty}^{(0)}/\overline{U}_{\infty})_{\Gamma_{n}}\longrightarrow{\mathcal{U}}^{(0)}_{n}/\overline{U}_{n}$ fits into an exact sequence

[TABLE] 2. 2.

$\mathop{\mathrm{Ker}}\nolimits_{n}({\mathcal{U}}_{\infty}^{(0)}/\overline{U}_{\infty})$* and $\mathop{\mathrm{Coker}}\nolimits_{n}({\mathcal{U}}_{\infty}^{(0)}/\overline{U}_{\infty})$ are finite and of bounded orders.*

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

[TABLE]

Taking the ${\mathbb{Z}}_{p}$-torsion counterpart of the sequence (R) we have :

[TABLE]

On the other hand if we take limits up to $K_{\infty}$ on (R), then apply $\Gamma_{n}$-cohomology, we obtain (by Leopoldt, ${\mathfrak{X}}_{\infty}^{\Gamma_{n}}=0$ and therefore $X_{\infty}^{\Gamma_{n}}$ is finite):

[TABLE]

Lemma 2.1 has a global analogue which is the following :

Lemma 3.2

The natural map $({\mathfrak{X}}_{\infty})_{\Gamma_{n}}\longrightarrow{\mathfrak{X}}_{n}$ fits into the exact sequence

[TABLE]

In other words, descent provides an isomorphism

[TABLE]

Proof. The proof of $({\mathfrak{X}}_{\infty})_{\Gamma_{n}}\hookrightarrow{\mathfrak{X}}_{n}$ is exactly the same as for 2.1 : we only need to replace the ${\mathcal{G}}_{n}$ there by the group $G_{S}(K_{n})$ which is the Galois group of the maximal $S$-ramified extension of $K_{n}$. The remaining part of the exact sequence comes from maximality properties defining $({\mathfrak{X}}_{\infty})_{\Gamma_{n}}$ and ${\mathfrak{X}}_{n}$. Let $M_{n}$ be the maximal abelian $S$ ramified $p$-extension of $K_{n}$. Then, by Leopoldt’s conjecture, one has $\text{\rm Tor}_{{\mathbb{Z}}_{p}}({\mathfrak{X}}_{n})=\mathop{\mathrm{Gal}}\nolimits(M_{n}/K_{\infty})$, which gives the isomorphism. $\quad\square$

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

[TABLE]

1 of 3.1 follows then from the snake lemma.

By Leopoldt’s conjecture the $(X_{\infty})^{\Gamma_{n}}$’s are finite. As $X_{\infty}$ is a noetherian $\Lambda$-module the ascending union $\bigcup_{n\in{\mathbb{N}}}(X_{\infty})^{\Gamma_{n}}$ stabilizes. This shows that the orders of $\mathop{\mathrm{Ker}}\nolimits_{n}({\mathcal{U}}_{\infty}^{(0)}/\overline{U}_{\infty})$ stabilize. As for $\mathop{\mathrm{Coker}}\nolimits_{n}({\mathcal{U}}_{\infty}^{(0)}/\overline{U}_{\infty})\cong\mathop{\mathrm{Ker}}\nolimits_{n}(X_{\infty})$, the maps $X_{n+1}\longrightarrow X_{n}$ (and consequently $X_{\infty}\longrightarrow X_{n}$) are surjective as soon as $K_{n+1}/K_{n}$ does ramify. By the classical Iwasawa theorems (see §13 of [W]), the orders of $(X_{n})$ is asymptotically equivalent to $p^{\lambda_{X}n+\mu_{X}p^{n}}$, where $\lambda_{X}$ and $\mu_{X}$ are the structural invariants of $X_{\infty}$. Since the order of $(X_{\infty})^{\Gamma}$ is finite the orders of $(X_{\infty})_{\Gamma_{n}}$ are also asymptotically equivalent to $p^{\lambda_{X}n+\mu_{X}p^{n}}$. This proves that the orders of $\mathop{\mathrm{Ker}}\nolimits_{n}(X_{\infty})$, hence those of $\mathop{\mathrm{Coker}}\nolimits_{n}({\mathcal{U}}_{\infty}^{(0)}/\overline{U}_{\infty})$, are bounded and concludes the proof of 2.

$\quad\square$

By Leopoldt’s Conjecture we have for all $n$ : $({\mathcal{U}}^{(0)}_{\infty}/\overline{U}_{\infty})^{\Gamma_{n}}\subset{\mathfrak{X}}_{\infty}^{\Gamma_{n}}=0$. And therefore an exact sequence

[TABLE]

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

[TABLE]

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

Lemma 3.3

$\mathop{\mathrm{Ker}}\nolimits_{n}(\overline{U}_{\infty}/\overline{C}_{\infty})$* and $\mathop{\mathrm{Ker}}\nolimits_{n}({\mathcal{U}}^{(0)}_{\infty}/\overline{C}_{\infty})$ are finite. Their orders are simultaneously bounded or not.* 2. 2.

$\mathop{\mathrm{Coker}}\nolimits_{n}(\overline{U}_{\infty}/\overline{C}_{\infty})$* and $\mathop{\mathrm{Coker}}\nolimits_{n}({\mathcal{U}}^{(0)}_{\infty}/\overline{C}_{\infty})$ are finite. Their orders are simultaneously bounded or not.*

$\quad\square$

• 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

The orders of the kernels of the natural maps $({\overline{U}}_{\infty}/{\overline{C}}_{\infty})_{\Gamma_{n}}\longrightarrow{\overline{U}}_{n}/{\overline{C}}_{n}$ are bounded. 2. 2.

The orders of the kernels of the natural maps $({\mathcal{U}}^{(0)}_{\infty}/{\overline{C}}_{\infty})_{\Gamma_{n}}\longrightarrow{\mathcal{U}}^{(0)}_{n}/{\overline{C}}_{n}$ are bounded.

Proof. From the commutative diagram :

[TABLE]

we deduce the exact sequence :

[TABLE]

By the part 2 of theorem 1.1 $\mathop{\mathrm{Ker}}\nolimits_{n}({\overline{U}}_{\infty}/{\overline{C}}_{\infty})$ is finite. To control $\mathop{\mathrm{Coker}}\nolimits_{n}(\overline{C}_{\infty})$ we use the lemma

Lemma 4.2

There exists an $N$ such that for all $n\geq N$ we have ${\overline{C}}_{n}/{\widetilde{C}}_{n}\cong{\overline{C}}_{N}/{\widetilde{C}}_{N}$

Proof. Let $I$ be the inertia subfield of $p$ for $K_{\infty}/{\mathbb{Q}}$. By lemma 2.5 of [Bjr], for $n$ large enough ($n$ such that $I\subset K_{n}$ is large enough), we have $\overline{C}_{n}={\widetilde{C}}_{n}C_{I}$. It follows that $\mathop{\mathrm{Coker}}\nolimits_{n}(\overline{C}_{\infty})\cong C_{I}/({\widetilde{C}}_{n}\bigcap C_{I})$. Now the increasing sequence ${\widetilde{C}}_{n}\bigcap\overline{C}_{I}$ has to stabilize because $\overline{C}_{I}$ is of finite ${\mathbb{Z}}_{p}$-rank. This shows the lemma 4.2

$\quad\square$

To prove the lemma 4.1 it then suffices to bound the orders of

[TABLE]

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

Proposition 4.3

For any noetherian $\Lambda$-module $M_{\infty}$ let ${\rm Minv}(M_{\infty})$ denote the submodule of $M_{\infty}$ defined as follows :

[TABLE]

Exists $N$ such that for all $n\geq N$ we have

[TABLE]

Proof. Starting from the sequence :

[TABLE]

one gets the diagram

[TABLE]

By the proposition 2.4 the kernels $\mathop{\mathrm{Ker}}\nolimits_{n}({\overline{U}}^{\prime}_{\infty})$ are trivial. Hence we have an isomorphism $\mathop{\mathrm{Ker}}\nolimits_{n}({\overline{C}}_{\infty})\simeq({\overline{U}}^{\prime}_{\infty}/{\overline{C}}_{\infty})^{\Gamma_{n}}$. Since $\overline{U}^{\prime}_{\infty}$ is a noetherian module, so is ${\overline{U}}^{\prime}_{\infty}/{\overline{C}}_{\infty}$ and therefore the increasing sequence $({\overline{U}}^{\prime}_{\infty}/{\overline{C}}_{\infty})^{\Gamma_{n}}$ stabilizes and for $n$ large we have $\mathop{\mathrm{Ker}}\nolimits_{n}({\overline{C}}_{\infty})\simeq{\rm Minv}(\overline{U}^{\prime}_{\infty}/\overline{C}_{\infty})$. The same arguments with ${\overline{U}}_{\infty}$ instead of ${\overline{C}}_{\infty}$ proves that (provided $n$ greater than some $N$) we have $\mathop{\mathrm{Ker}}\nolimits_{n}({\overline{U}}_{\infty})\cong{\rm Minv}(\overline{U}^{\prime}_{\infty}/\overline{U}_{\infty})$. 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.

$\quad\square$

5 Descent cokernels

The third and final key lemma is

Lemma 5.1

The orders of the cokernels of the natural maps $({\overline{U}}_{\infty}/{\overline{C}}_{\infty})_{\Gamma_{n}}\longrightarrow{\overline{U}}_{n}/{\overline{C}}_{n}$ are bounded. 2. 2.

The orders of the cokernels of the natural maps $({\mathcal{U}}^{(0)}_{\infty}/{\overline{C}}_{\infty})_{\Gamma_{n}}\longrightarrow{\mathcal{U}}^{(0)}_{n}/{\overline{C}}_{n}$ are bounded.

Proof. Starting with the snake diagram

[TABLE]

one gets the sequence

[TABLE]

Recall that $D$ is the maximal subfield of $K_{\infty}$ such that $p$ is totally split in $D$. We may assume without loss of generality that $D\subset K_{n}$ (else enlarge $n$). By lemma 3.3, $\mathop{\mathrm{Ker}}\nolimits_{n}({\mathcal{U}}^{(0)}_{\infty}/{\overline{C}}_{\infty})$ is bounded. By proposition 2.3 we have $\widetilde{\mathcal{U}}_{n}=\mathop{\mathrm{Ker}}\nolimits(N_{K_{n}/D}\colon{\mathcal{U}}_{n}^{(0)}\longrightarrow{\mathcal{U}}_{D})$. One then gets an isomorphism ${\mathcal{U}}_{n}^{(0)}/\widetilde{{\mathcal{U}}}_{n}\cong N_{K_{n}/D}({\mathcal{U}}_{n}^{(0)})$ and using this isomorphism the preceding sequence reads

[TABLE]

Now, 2 in lemma 5.1 follows from the :

Lemma 5.2

Let $c_{n}$ denotes $[K_{n}:D]$ (asymptotically $c_{n}$ is equivalent to $p^{n}$).

$({\overline{C}}_{D})^{c_{n}}$* is a submodule of bounded finite index in $N_{K_{n}/D}({\overline{C}}_{n})$* 2. 2.

$({\mathcal{U}}^{(0)}_{D})^{c_{n}}$* is a submodule of bounded finite index in $N_{K_{n}/D}({\mathcal{U}}_{n}^{(0)})$* 3. 3.

$({\mathcal{U}}^{(0)}_{D})^{c_{n}}/{\overline{C}}_{D}^{c_{n}}$* is asymptotically equivalent to ${\mathcal{U}}^{(0)}_{D}/{\overline{C}}_{D}$.*

Proof. Assertion 3 is immediate. The finite constant group ${\mathcal{U}}^{(0)}_{D}/{\overline{C}}_{D}$ maps onto $({\mathcal{U}}^{(0)}_{D})^{c_{n}}/({\overline{C}}_{D})^{c_{n}}$. Since the norm $N_{K_{n}/D}$ acts as $c_{n}$ on ${\overline{C}}_{D}$ and ${\mathcal{U}}^{(0)}_{D}$ 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 ${\overline{C}}_{n}=\widetilde{C}_{n}{\overline{C}}_{I}$. Moreover, as ${\widetilde{C}}_{n}\subset{\widetilde{\mathcal{U}}}_{n}^{(0)}$ we have $N_{K_{n}/D}(\widetilde{C}_{n})=0$ by proposition 2.3 (Without using semi-local units, $N_{K_{n}/D}(\widetilde{C}_{n})=0$ can be checked directly using distribution relations on a generating system of ${\widetilde{C}}_{n}$). Hence we get $N_{K_{n}/D}({\overline{C}}_{n})=N_{K_{n}/D}(\widetilde{C}_{n}{\overline{C}}_{I})=N_{K_{n}/D}({\overline{C}}_{I})=N_{I/D}({\overline{C}}_{I})^{[K_{n}:I]}$. This gives 1 because, as $c_{n}$ itself, $[K_{n}:I]$ is asymptotically equivalent to $p^{n}$ and $N_{I/D}({\overline{C}}_{I})$ is of (constant) finite index in ${\overline{C}}_{D}$.

Assertion 2 is an easy exercise using local class field theory. Indeed, recall that ${\mathcal{U}}_{n}=\oplus_{v\mid p}{\overline{U}}^{1}_{K_{n,v}}$. Then the global norm $N_{K_{n}/D}$ acts on each summand as the local norm $N_{K_{n,v}/{\mathbb{Q}}_{p}}$. By local class field theory, the quotient ${\overline{U}}^{1}_{{\mathbb{Q}}_{p}}/N_{K_{n,v}/{\mathbb{Q}}_{p}}({\overline{U}}^{1}_{K_{n,v}})$ is isomorphic to the $p$-part of the ramification subgroup of $\mathop{\mathrm{Gal}}\nolimits(K_{n,v}/{\mathbb{Q}}_{p})$. These wild ramification subgroups are cyclic with orders asymptotically equivalent to $p^{n}$. Summing up, it follows that $N_{K_{n}/D}({\mathcal{U}}_{n})$ contains ${\mathcal{U}}_{D}^{c_{n}}$ with bounded finite index. A fortiori $N_{K_{n}/D}({\mathcal{U}}_{n}^{(0)})$ contains $({\mathcal{U}}^{(0)}_{D})^{c_{n}}$ 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.

$\quad\square$

With lemmas 4.1 and 5.1 we have fullfilled our goal. We have directly proved that the natural descent homomorphisms $(\overline{U}_{\infty}/\overline{C}_{\infty})_{\Gamma_{n}}\longrightarrow\overline{U}_{n}/\overline{C}_{n}$ have bounded kernels and cokernels. As a consequence, we get without using Iwasawa’s Main Conjecture nor Sinnott’s Index Formula an analogue for unit classes of Iwasawa’s theorem. Recall that any torsion $\Lambda$-module $M_{\infty}$ has an invariant $\lambda$ which is the Weierstraß degree of (any) generators of its characteristic ideal and an invariant $\mu$ which is the maximal power of $p$ dividing (any) generators of its characteristic ideal. By purely abstract algebra it is classical and easy to prove that if they are finite the orders of $(M_{\infty})_{\Gamma_{n}}$ are asymptotically equivalent to $p^{\lambda n+\mu p^{n}}$.

Theorem 5.3

Let $\lambda_{1}$ and $\mu_{1}$ denote the structural invariants of the Iwasawa module ${\overline{U}}_{\infty}/{\overline{C}}_{\infty}$. Then the orders of ${\overline{U}}_{n}/{\overline{C}}_{n}$ are asymptotically equivalent to $p^{\lambda_{1}n+\mu_{1}p^{n}}$. 2. 2.

Let $\lambda_{2}$ and $\mu_{2}$ denotes the structural invariants of the Iwasawa module ${\mathcal{U}}_{\infty}/{\overline{C}}_{\infty}$. Then the orders of ${\mathcal{U}}^{(0)}_{n}/{\overline{C}}_{n}$ are asymptotically equivalent to $p^{\lambda_{2}n+\mu_{2}p^{n}}$.

Proof. By propositions 4.1 and 5.1 the orders of ${\overline{U}}_{n}/{\overline{C}}_{n}$ are asymptotically equivalent to the orders of $({\overline{U}}_{\infty}/{\overline{C}}_{\infty})_{\Gamma_{n}}$. As they are finite the last orders are equivalent to what we need. This shows 1. Same argument proves 2 as well.

$\quad\square$

6 Two applications of Iwasawa’s theorem for unit classes

Our first application is a structural link between unit and ideal classes at infinity.

Theorem 6.1

The $\Lambda$-modules $\overline{U}_{\infty}/{\overline{C}}_{\infty}$ and $X_{\infty}$ share the same structural invariants $\lambda_{1}=\lambda_{X}$ and $\mu_{1}=\mu_{X}$. 2. 2.

The $\Lambda$-modules ${\mathcal{U}}_{\infty}^{(0)}/{\overline{C}}_{\infty}$ and ${\mathfrak{X}}_{\infty}$ share the same structural invariants $\lambda_{2}=\lambda_{\mathfrak{X}}$ and $\mu_{2}=\mu_{\mathfrak{X}}$.

Proof. Consider the exact sequence of $\Lambda$-torsion modules

[TABLE]

The invariants $\lambda$ and $\mu$ are additive in exact sequences. Therefore, going through the above (DNA) sequence, we see that 1 is equivalent to 2. Now, by Sinnott’s index formula, the orders of $X_{n}$ are asymptotically equivalent to the orders of $\overline{U}_{n}/\overline{C}_{n}$. Using theorem 5.3 and Iwasawa’s theorem we get that the orders of $(X_{\infty})_{\Gamma_{n}}$ and $(\overline{U}_{\infty}/\overline{C}_{\infty})_{\Gamma_{n}}$ are finite and asymptotically equivalent. Therefore the sequence $\lambda_{1}n+\mu_{1}p^{n}$ is equivalent to the sequence $\lambda_{X}n+\mu_{X}p^{n}$ : assertion 1 follows.

$\quad\square$

As explained in the introduction, theorem 6.1 is an immediate consequence of the Iwasawa Main Conjecture. However the point here is that we achieved a direct proof by only making use of Iwasawa classical theorem, Sinnott’s index formula, Coleman’s morphism, and Leopoldt’s conjecture (the last two via theorem 1.1).

Conversely, theorem 6.1 could be used to simplify the (now classical) proof of the Main Conjecture via Euler systems and the ”class number trick”. Let us recall the main lines, skipping technical details on characters. The $p$-adic $L$-functions are related via Coleman’s theory to the characteristic series of ${\mathcal{U}}^{(0)}_{\infty}/\overline{C}_{\infty}$, and one version of the Main Conjecture asserts that ${\mathcal{U}}^{(0)}_{\infty}/\overline{C}_{\infty}$ and ${\mathfrak{X}}_{\infty}$ have the same characteristic series (up to power of $p$, and $\theta$-componentwise for all Dirichlet characters $\theta$ of the first kind). This is done in two steps :

Use the Euler System of cicular units to ”bound class groups” and to show that the characteristic series of $X_{\infty}$ divide that of $\overline{U}_{\infty}/\overline{C}_{\infty}$. Hence, following the sequence $(DNA)$, to show that the characteristic series of ${\mathfrak{X}}_{\infty}$ divides that of ${\mathcal{U}}^{(0)}_{\infty}/\overline{C}_{\infty}$. For full details, see [G].

• -

To show the converse property, it suffices to prove the equality of the relevant $\lambda$-invariants. For this, one uses the ”class number trick”. Suppose that $K$ is the maximal real subfield of $M:=K(\zeta_{p})$, which is not a loss of generality. By Kummer duality the Iwasawa invariants of ${\mathfrak{X}}_{\infty}(K)$ and of $X_{\infty}^{-}(M)$ are equals. Then Iwasawa’s asymptotical formula and the (minus part) of the class number formula show what we want.

It is this ”class number trick” which could be advantageously replaced by theorem 6.1.

Up to now we did not use Kummer duality nor any knowledge about the minus part of class groups, nor Ferrero-Washington’s Theorem. We use them now to show the vanishing of the $\mu$-invariant and thus remove the assertions ”up to power of $p$” in all the discussions just above.

Theorem 6.2

The structural invariant $\mu$ of the module ${\overline{U}}_{\infty}/{\overline{C}}_{\infty}$ is trivial. 2. 2.

The structural invariant $\mu$ of the module ${\mathcal{U}}_{\infty}^{(0)}/{\overline{C}}_{\infty}$ is trivial.

Proof. We claim that all four modules in the above (DNA) sequence have trivial $\mu$-invariant and we only need to prove it for three out of them (actually two well chosen would be enough). By 1 of theorem 6.1 $\mu_{1}=\mu_{X}$ and by theorem 7.15 of [W] $\mu_{X}=0$. Let us draw the main lines of the proof written in [W] of the triviality of $\mu_{X}$.

• step 1

The main ingredient is Ferrero-Washington’s theorem [FW] which claims that the power series $g_{\psi}(T)$ of our first section is prime to $p$.

• step 2

Over $K(\zeta_{p})$, using step 1 and the analytic class number formula for the minus part, one deduces that the sequence of orders of $X_{n}^{-}$ is equivalent to $p^{\lambda n}$, where $\lambda$ is the Weierstrass degree of the product of relevant $g_{\psi}(T)$’s. By Iwasawa’s theorem, this implies the triviality of the structural $\mu$-invariant of $X_{\infty}^{-}$.

• step 3

Using the classical mirror inequality $\mu^{+}\leq\mu^{-}$ derived from Kummer duality, one recovers the triviality of the $\mu$-invariants of the plus part $X^{+}_{\infty}$ over $K(\zeta_{p})$, which in turn implies the triviality of the $\mu$-invariant of $X_{\infty}$ for our base field $K$.

For the remaining module ${\mathfrak{X}}_{\infty}$, a possible proof follows the above first two steps by just replacing the analytic class number formula for the minus part by Leopoldt’s formula for the order of the even part of the ${\mathfrak{X}}_{n}$ in terms of products of values at $1$ of $p$-adic $L$-functions. Alternatively let us just examine more carefully the third step. Actually Kummer duality (we are only making use here of corollary 11.4.4 of [NSW] but full original Kummer duality is in [I2]) gives that $\mu({\mathfrak{X}}_{\infty}^{+})=\mu(X_{\infty}^{-})$ over $K(\zeta_{p})$ and the inequality in the third step follows from ${\mathfrak{X}}_{\infty}\twoheadrightarrow X_{\infty}$. Hence step 2 gives directly $\mu({\mathfrak{X}}_{\infty}^{+})=0$ over $K(\zeta_{p})$, which is all we need. The triviality of $\mu_{2}$ follows and this concludes the proof of 2 of theorem 6.2.

$\quad\square$

• Remark : Assertion 1 of theorem 6.2 is also proven by Greither in the appendix of [FG]. Greither’s proof is slightly different because it makes no use of Leopoldt’s conjecture, and for that reason needs to work with maybe infinite $|({\overline{U}}_{\infty}/{\overline{C}}_{\infty})_{\Gamma_{n}}|$. To deal with that difficulty, Greither introduces the notion of ”tame” sequences of modules (roughly speaking these are sequences of modules whose inverse limits are without $\mu$).

Acknowledgement : I thank professor Th´ông Nguyên Quang ¯Dô for many instructive conversations and helpful comments on this paper.