Global Units modulo Circular Units : descent without Iwasawa’s Main Conjecture.^{∗}

Jean-Robert Belliard

Université de Franche-Comté, Laboratoire de mathématiques UMR 6623, 16 route de Gray, 25030 Besançon cedex, FRANCE.

                               belliard@math.univ-fcomte.fr

September 2, 2005

Abstract

Iwasawa’s classical asymptotical formula gives 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. It relies on ”good” descent properties satisfied by X_{n}. If F is abelian and F_{∞} is cyclotomic it is known that the p-parts of the order 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 without using Iwasawa’s Main Conjecture.

0 Introduction

Let K be a number field and p an odd prime (p ≠ 2) and let K_{∞}/K be a ℤ_{p}-extension (quite soon K_{∞}/K will be the cyclotomic ℤ_{p}-extension). Recall the usual notations : Γ = Gal(K_{∞}/K) is the Galois group of K_{∞}/K, K_{n} is the n^{th}-layer of K_{∞} (so that [K_{n} : K] = p^{n}), Γ_{n} = Gal(K_{∞}/K_{n}), and G_{n} = Gal(K_{n}/K) ≅ Γ/Γ_{n}. 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_{←} M_{n} 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 M_{n}’s themselves from that structure. For instance, if M_{∞} is Λ-torsion, one can attach two invariants λ and μ to M_{∞}. If we assume further that (i) the Γ_{n} coinvariants (M_{∞})_{Γ_{n}} are finite (ii) the sequence (M_{n})_{n∈ℕ} behaves ”nicely” viz descent one then can prove that the orders of the M_{n} are asymptotically equivalent to p^{λn+μp^{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 tor_{Λ}(𝔛_{∞}) 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_{∞} 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 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 (up to now) that all K_{n} 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 ≠ 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 λ and μ of both \overline{U}_{∞}/\overline{C}_{∞} and of X_{∞} are equal. So that we have easily a somewhat indirect proof of an analogue of Iwasawa theorem for unit class groups. Clearly a direct proof of this fact must exist and one of our present goal is to write it down. As a corollary of Sinnott index formulas we then recover the equality of the λ and μ invariants of the two class groups without using Iwasawa Main Conjecture. 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 his strongest form : that is following the path outlined just above.

∗2000 Mathematics Subject Classification. Primary 11R23.

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 :

U_{F} is the group of units of F.

U′_{F} is the group of S-units of F : that is, elements of F^{×} whose valuations are trivial for all finite places v of F such that v ∉ S.

X_{F} 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 in S.

𝔛_{F} is the Galois group over F of the maximal S-ramified (i.e. unramified outside S) abelian pro-p-extension of F.

N_{F} is the multiplicative group of the semi-local numbers. As a mere ℤ_{p}-module N_{F} = ∏_{v∈S} \overline{F_{v}^{×}} where F_{v} is the completion of F at the place v ∈ S.

U_{F} is the group of semi-local units of F. As a mere ℤ_{p}-module U_{F} = ∏_{v∈S} U_{v}^{1} where U_{v}^{1} is the set of principal units of F_{v}, that is units ≡ 1 modulo the maximal ideal of F_{v}.

For any abelian group A, we’ll denote by \overline{A} the p-completion of A, i.e. the inverse limit \overline{A} = lim A/A^{p^{m}}. If A is finitely generated over ℤ, then Ā ≅ A ⊗ ℤ_{p}.

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

1 Consequences of Leopoldt’s Conjecture

If F 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 place v in S and if we put G_{v} = Gal(F_{v}/ℚ_{p}) then we have (as ℤ_{p}- modules) U_{F} ≅ U_{v}^{1}⊗_{ℤ_{p}[G_{v}]}ℤ_{p}[Gal(F/ℚ)] and N_{F} ≅ \overline{F_{v}^{×}}⊗_{ℤ_{p}[G_{v}]}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 ℚ number field and the n^{th}-finite layers K_{n}/K, i.e. K_{n} is the unique subfield of K_{∞} 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}, U_{n} and so on make sense. For any extension L/F of global fields the formula

N_{L/F} ((x_{w})_{w|p}) = (∏_{w|v} N_{L_{w}/K_{v}}(x_{w}))_{v|p}

defines a Galois equivariant morphism N_{L/F} : N_{L} ⟶ N_{F} which is compatible with the usual norms on global units for instance. As ever \overline{U}_{∞}, \overline{C}_{∞} N_{∞}, U_{∞}, \overline{U}′_{∞} and so on denotes the inverse limit (related to norm maps) of \overline{U}_{n}, \overline{C}_{n}, N_{n}, U_{n}, \overline{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 K_{n} are abelian over ℚ). However a more precise and general result on quotients of semi-local units modulo circular units is 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 𝔹_{∞}/ℚ be the cyclotomic ℤ_{p}-extension of ℚ and G be the Galois group G = Gal(K/ℚ). Recall that Γ = Gal(K_{∞}/K) and that Λ = ℤ_{p}[[Γ]]. For all non-trivial Dirichlet characters ψ of Ĝ, following [T] let us denote by g_{ψ}(T) the Iwasawa power series associated to the Kubota-Leopoldt p-adic L-function L_{p}(ψ, s).

1. Assume K ∩𝔹_{∞} = ℚ; then up to a power of p, the characteristic ideal of the Λ-torsion module U_{∞}/\overline{C}_{∞} is generated by the product ∏_{ψ∈Ĝ,ψ≠1} g_{ψ}(T).

2. For all n ∈ ℕ, the Γ_{n}-coinvariants (U_{∞}/\overline{C}_{∞})_{Γ_{n}} and (\overline{U}_{∞}/\overline{C}_{∞})_{Γ_{n}} are finite.

Proof. Let us consider ψ-parts M^{ψ} as defined in [T] for any ℤ_{p}[G]-module M. These ψ parts are naturally ℤ_{p}[ψ]-modules and ℤ_{p}[G]-submodules of M (see also Beliaeva’s thesis [Bt]). From theorem 3.1 of [T] we use that U_{∞}^{ψ}/\overline{C}_{∞}^{ψ} is ℤ_{p}-torsion free and that its characteristic series over Λ[ψ] is g_{ψ}(T). Let σ ∈ Gal(ℚ_{p}[ψ]/ℚ_{p}); then (U_{∞}/\overline{C}_{∞})^{ψ^{σ}} is isomorphic to (U_{∞}/\overline{C}_{∞})^{ψ} and g_{ψ^{σ}}(T) = σ(g_{ψ}(T)) (the last equality can be checked by following Iwasawa’s construction of g_{ψ}, see §7.2 of [W]). For R equal to Λ or to Λ[ψ], let us abbreviate by 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 Char_{Λ[ψ]}(M) then N_{ℚ_{p}[ψ]/ℚ_{p}}(f) generates Char_{Λ}(M). Hence we get

Char_{Λ}((U_{∞}^{ψ}/\overline{C}_{∞})^{ψ}) = (N_{ℚ_{p}[ψ]/ℚ_{p}}(g_{ψ}(T))) = ( ∏_{σ∈Gal(ℚ_{p}[ψ]/ℚ_{p})} g_{ψ^{σ}}(T)) .

Let us fix a set Ψ of representative of Ĝ 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 quotient M/ ⊕_{ψ∈Ψ} M^{ψ} is annihilated by #G. This applies to U_{∞} and \overline{C}_{∞}, so that we may use the inclusions ⨁_{ψ∈Ψ} U_{∞}^{ψ} ⊂ U_{∞} and ⨁_{ψ∈Ψ} \overline{C}_{∞}^{ψ} ⊂ \overline{C}_{∞} to form the snake diagram :

\xymatrix{0 \ar[r] & ⨁_{ψ∈Ψ} \overline{C}_{∞}^{ψ} \ar[r] \ar@{^{(}->}[d] & ⨁_{ψ∈Ψ} U_{∞}^{ψ} \ar[r] \ar@{^{(}->}[d] & ⨁_{ψ∈Ψ} U_{∞}^{ψ}/\overline{C}_{∞}^{ψ} \ar[r] \ar@{-->}[d]^{α} & 0 \\ 0 \ar[r] & _{\overline{C}_{∞}} \ar[r] \ar[d] & U_{∞} \ar[r] \ar[d] & U_{∞}/\overline{C}_{∞} \ar[r] \ar[d] & 0 \\  & ℤ_{p} − torsion \ar[r] & ℤ_{p} − torsion \ar[r] & Coker α \ar[r] & 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}/\overline{C}_{∞}^{ψ} are ℤ_{p}-free. This proves that up to a power of p the characteristic ideal Char_{Λ}(U_{∞}/\overline{C}_{∞}) is equal to Char_{Λ}(⨁_{ψ∈Ψ} U_{∞}/\overline{C}_{∞}), which in turn is what we wanted.

The second claim is equivalent to say that T divides no generator of the two characteristic ideals. From the inclusion \overline{U}_{∞}/\overline{C}_{∞} ⊂ U_{∞}/\overline{C}_{∞} we see that the first characteristic ideal divides the second. Hence 2 will follow from g_{ψ}(0) = L_{p}(ψ, 1) ≠ 0, which is an equivalent formulation of Leopoldt’s conjecture (see [W] corollary 5.30). Therefore in the special case K ∩ 𝔹_{∞} = ℚ the second claim is true. We now want to remove this hypothesis. Let I_{p} ⊂ Gal(K_{∞}/ℚ) be the inertia

subgroup of p, and let \overline{I}_{p} ⊂ I_{p} be its pro-p-part. Then the field L = K_{∞}^{\overline{I}_{p}} is a finite abelian over ℚ number field and p is at most tamely ramified in L. Therefore L ∩ 𝔹_{∞} = ℚ. Since all modules M_{∞} 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. By construction we have L ⊂ K_{∞} and therefore L_{∞} ⊂ K_{∞}. By definition of L, the group Gal(K_{∞}/L_{∞}) is then a subgroup of \overline{I}_{p}, 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_{∞} above p. This shows L_{∞} = K_{∞}.

□ Now, with that lemma we already have proven the finiteness assertion when we take coinvariants

along Γ̃_{n} = Gal(K_{∞}/L_{n}). We conclude the proof of the second part of theorem 1.1 by pointing that even if the first few Γ_{n} may differ from Γ̃_{n}, we get Γ_{n} = Γ̃_{n} as soon as K_{∞}/K_{n} is totally ramified at p. Due to canonical surjections (M_{∞})_{Γ_{n}} ↠ (M_{∞})_{Γ_{n−1}}, this does not change anything to the finiteness of (U_{∞}/\overline{C}_{∞})_{Γ_{n}}.

□

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_{∞}/\overline{C}_{∞})_{Γ_{n}} (resp. (\overline{U}_{∞}/\overline{C}_{∞})_{Γ_{n}}) has been proved, we want to relate these modules with their counterparts U_{n}/\overline{C}_{n} (resp. \overline{U}_{n}/\overline{C}_{n}) at finite level. On the way we will study descent for various other multiplicative Galois modules.

2 Background: easy part of descent

Let M_{∞} = lim_{←}(M_{n}) be a Λ-module. Of course the projections M_{∞} ⟶ M_{n} factor through (M_{∞})_{Γ_{n}} ⟶ M_{n}, but in general these maps have non trivial kernels and cokernels. Let us denote Ker_{n}(M_{∞}) for the kernel of (M_{∞})_{Γ_{n}} ⟶ M_{n}, M̃_{n} ⊂ M_{n} for its image, and Coker_{n}(M_{∞}) = M_{n}/M̃_{n} for its cokernel. Let l 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 E_{n} for the n^{th} finite layers. Let us abbreviate L_{E} = \overline{E^{×}}, consistently L_{n} = \overline{(E_{n})^{×}}, and L_{∞} = lim_{←} L_{n}. For all n ∈ ℕ ∪ ∞, local class field theory identifies L_{n} with Gal(M_{n}/E_{n}), where M_{n} is the maximal abelian p-extension of E_{n}.

Lemma 2.1 The natural map (L_{∞})_{Γ_{n}} ⟶ L_{n} fits into an exact sequence

\xymatrix{0 \ar[r] & (L_{∞})_{Γ_{n}} \ar[r] & L_{n} \ar[r]^{Artin} & Gal(E_{∞}/E_{n}) \ar[r] & 0.}

Proof. This is well known to experts. We follow the cohomological approach of [N1]. Consider the inflation-restriction sequence associated to the extension of groups

\xymatrix{1 \ar[r] & H \ar[r] & G_{n} \ar[r] & Γ_{n} \ar[r] & 1,}

where G_{n} is the absolute Galois group of E_{n} :

\xymatrix{0 \ar[r] & H^{1}(Γ_{n}, ℚ_{p}/ℤ_{p}) \ar[r] & H^{1}(G_{n}, ℚ_{p}/ℤ_{p}) \ar[r] & H^{1}(H, ℚ_{p}/ℤ_{p})^{Γ_{n}} \ar[r] & H^{2}(Γ_{n}, ℚ_{p}/ℤ_{p}) = 0.}

The last equality occurs because Γ_{n} ≃ ℤ_{p} is pro-p-free. Applying Pontryagin duality and class field theory we get

\xymatrix{0 \ar[r] & (L_{∞})_{Γ_{n}} \ar[r] & L_{n} \ar[r]^{Artin} & Gal(E_{∞}/E_{n}) \ar[r] & 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 and v ∈ S_{n} put K_{n,v} for the completion at v of K_{n} and (K_{n,v})_{∞}^{cyc}/K_{n,v} for its cyclotomic ℤ_{p}-extension. For all n the natural map

(N_{∞})_{Γ_{n}} ⟶ N_{n}

fits into an exact sequence

\xymatrix{0 \ar[r] & (N_{∞})_{Γ_{n}} \ar[r] & N_{n} \ar[r]^{⊕_{v∈Sn} Artin at v} & ⨁_{v∈S_{n}} Gal((K_{n,v})_{∞}^{cyc}/K_{n,v}) \ar[r] & 0.}

Proof. Let us examine the places above p along K_{∞}/K. These primes have non trivial conjugated (hence equal) decomposition subgroups. There exists an n_{0} such that (i) no primes above p splits anymore in K_{∞}/K_{n_{0}} and (ii) all primes above p are totally split in K_{n_{0}}/K. For all n ≥ n_{0} we then have S_{n} ≅ S_{n_{0}}, N_{n} = ⨁_{v∈S_{n_{0}}} L_{K_{n,v}} and Gal(K_{∞}/K_{n}) identifies with the local Galois groups

Gal((K_{n,v})_{∞}^{cyc}/K_{n,v}) (at every v ∈ S_{n}). So in the case n ≥ n_{0}, the proposition follows from lemma 2.1. Next suppose that 0 ≤ n < n_{0}, and let G_{n}^{m} be the Galois group Gal(K_{m}/K_{n}). Of course we have (N_{∞})_{Γ_{n}} ≅ ((N_{∞})_{Γ_{n_{0}}} )_{G_{n}^{n_{0}}} . Using the previous case we have an exact sequence

(†) \xymatrix{0 \ar[r] & (N_{∞})_{Γ_{n_{0}}} \ar[r] & N_{n_{0}} \ar[r] & ⨁_{v∈S_{n_{0}}} Gal((K_{n_{0},v})_{∞}^{cyc}/K_{n_{0},v}) \ar[r] & 0}

As (K_{n_{0},v})_{∞}^{cyc}/K_{n_{0},v} is cyclotomic, the action of the local Galois group Gal(K_{n_{0},v}/ℚ_{p}) on the group Gal((K_{n_{0},v})_{∞}^{cyc}/K_{n_{0},v}) is trivial. Hence we have an isomorphism of Galois modules

⨁_{v∈S_{n_{0}}} Gal((K_{n_{0},v})_{∞}^{cyc}/K_{n_{0},v}) ≅ ℤ_{p}[S_{n_{0}}].

Since the places above p split totally in K_{n_{0}}/K, these two modules are G_{0}^{n_{0}}-cohomologically trivial, and the same is true for N_{n_{0}} ≅ N_{0} ⊗_{ℤ_{p}} ℤ_{p}[G_{0}^{n_{0}}] (as G_{0}^{n_{0}}-modules). Therefore in the sequence (†) two (hence all) terms are G_{0}^{n_{0}}-cohomologically trivial. The triviality of Ĥ^{1}(G_{n}^{n_{0}}, (N_{∞})_{Γ_{n_{0}}} ) proves the injectivity of (N_{∞})_{Γ_{n}} ⟶ N_{n}. It then suffices to apply N_{K_{n_{0}} /K_{n}} to the sequence (†) and use the triviality of the three Ĥ^{0}(G_{n}^{n_{0}}, −−) to get the full sequence

\xymatrix{0 \ar[r] & (N_{∞})_{Γ_{n}} \ar[r] & N_{n} \ar[r]^{⊕_{v∈Sn} Artin at v} & ⨁_{v∈S_{n}} Gal((K_{n,v})_{∞}^{cyc}/K_{n,v}) \ar[r] & 0}.

□ Let n_{0} 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 ≥ n_{0} we have D_{n} = D_{n_{0}}. Let us put D = D_{n_{0}} for the decomposition subfield for the place p in K_{∞}.

Proposition 2.3 Recall that for a given Λ-module M_{∞} = lim_{←}(M_{n}) we denote Ker_{n}(M_{∞}) for the

kernel, M̃_{n} for the image and Coker_{n}(M_{∞}) for the cokernel of the natural map (M_{∞})_{Γ_{n}} ⟶ M_{n}.

1. For all n ≥ 0, Ker_{n}(U_{∞}) is isomorphic to ℤ_{p}[S_{n}], hence its ℤ_{p}-rank is #S_{n}.

2. For all n ≥ 0, Coker_{n}(U_{∞}) ≅ N_{K_{n}/D_{n}}(U_{n}), hence its ℤ_{p}-rank is #S_{n}. In other words we have an exact sequence

\xymatrix{0 \ar[r] & Ũ_{n} \ar[r] & U_{n} \ar[r]^{N_{Kn/Dn}} & U_{D_{n}}} .

Proof. For any local field E the (normalized) valuation v of E gives an exact sequence 0 → U_{E}^{1} → \overline{E^{×}} →^{v} ℤ_{p} → 0, where U_{E}^{1} is the set of principal units of E. Taking the semi-local counterpart of this sequence and projective limits we obtain

\xymatrix{0 \ar[r] & U_{∞} \ar[r] & N_{∞} \ar[r] & ℤ_{p}[S_{n_{0}}] \ar[r] & 0}.

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

\xymatrix{0 \ar[r] & ℤ_{p}[S_{n_{0}}]^{Γ_{n}} \ar[r] & (U_{∞})_{Γ_{n}} \ar[r] & (N_{∞})_{Γ_{n}} \ar[r] & (ℤ_{p}[S_{n_{0}}])_{Γ_{n}} \ar[r] & 0}

Now since (N_{∞})_{Γ_{n}} ⟶ N_{n} is a monomorphism, Ker_{n}(U_{∞}) identifies with ℤ_{p}[S_{n_{0}}]^{Γ_{n}} ≅ ℤ_{p}[S_{n}]. This proves 1.

For 2. we use the notations n_{0}, D_{n} and D of the proof of proposition 2.2. We first prove the case n ≥ n_{0}. By compactness we have Ũ_{n} = ⋂_{m≥n} N_{K_{m}/K_{n}}(U_{m}). Because n ≥ n_{0} the global norm N_{K_{m}/K_{n}}: U_{m} ⟶ 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 ∈ S_{m} above a fixed v ∈ S_{n}). Fix a place v ∈ S_{n} and for all m ≥ 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} = ℚ_{p}. Let u ∈ U_{K_{n,v}}^{1}, then using local class field theory we see that u ∈ ⋂_{m≥n} N_{K_{m,v}/K_{n,v}}(U_{K_{m,v}}^{1}) if and only if N_{K_{n,v}/ℚ_{p}}(u) ∈ ⋂_{n∈ℕ} N_{(ℚ_{p})_{n}^{cyc}/ℚ_{p}}(U_{(ℚ_{p})_{n}^{cyc}}^{1}), where (ℚ_{p})_{n}^{cyc} denotes the n^{th}-step of the cyclotomic (hence totally ramified) ℤ_{p}-extension of ℚ_{p}. But (., (ℚ_{p})_{∞}^{cyc}/ℚ_{p}) gives an isomorphism between U_{ℚ_{p}}^{1} and Gal((ℚ_{p})_{∞}^{cyc}/ℚ_{p}). We have proven the equivalence u ∈ ⋂_{m≥n} N_{K_{m,v}/K_{n,v}}(U_{K_{m,v}}^{1}) ⟺ N_{K_{n,v}/D_{n,v}}(u) = 1. As no place above p splits in K_{n}/D_{n} this gives

Ũ_{n} = Ker(N_{K_{n}/D_{n}}: U_{n} ⟶ U_{D_{n}}).

Next pick an n such that 0 ≤ n < n_{0}. Consider the diagram of fields

[figure: K_{n_{0}}; K_{n}; D_{n_{0}}; D_{n}]

There every prime above p is totally split in extensions n_{0}/n and no prime above p splits at all in extensions K/D. It follows that N_{K_{n_{0}} /K_{n}} is surjective onto U_{n} and that

Ũ_{n} = N_{K_{n_{0}} /K_{n}}(Ũ_{n_{0}}) ⊂ Ker(N_{K_{n}/D_{n}}: U_{n} ⟶ U_{D_{n}}).

Conversely pick u = (u_{v})_{v∈S_{n}} ∈ Ker(N_{K_{n}/D_{n}}: U_{n} ⟶ U_{D_{n}}). At each v ∈ S_{n} choose a single w(v) ∈ S_{n_{0}} above v and define t = (t_{w})_{w∈S_{n_{0}}} ∈ U_{n_{0}} with for all v t_{w}(v) = u_{v} and t_{w} = 1 if there does not exist v such that w = w(v). Then we have N_{K_{n_{0}} /K_{n}}(t) = u and t ∈ Ũ_{n_{0}} which shows that u ∈ Ũ_{n}.

□ 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}′_{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 : ℚ], r_{2} = 0, and all \overline{U}′_{n} are ℤ_{p}-torsion free. To avoid ugly notations ˜ \overline{M} we will denote Ũ′_{n} for the image of \overline{U}′_{∞} in \overline{U}′_{n}.

Proposition 2.4

1. The Λ-module \overline{U}′_{∞} is free of rank [K : ℚ].

2. For all n the natural map (\overline{U}′_{∞})_{Γ_{n}} ⟶ \overline{U}′_{n} is injective.

3. Ũ′_{n} ≅ (\overline{U}′_{∞})_{Γ_{n}} is a free ℤ_{p}[G_{n}]-module of rank [K : ℚ] (hence is ℤ_{p} free of rank p^{n}[K : ℚ] = [K_{n} : 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 to use 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 𝔛_{∞} be the standard Iwasawa module 𝔛_{∞} = lim_{←}(𝔛_{n}), where 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 = ∞). Recall from the end of the introduction the notation X′_{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

\xymatrix{0 \ar[r] & \overline{U}′_{∞} \ar[r] & N_{∞} \ar[r] & 𝔛_{∞} \ar[r] & X′_{∞} \ar[r] & 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 (\overline{U}′_{∞})_{Γ} ⟶ (N_{∞})_{Γ} is a monomorphism. Now 2. follows from proposition 2.2.

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

\xymatrix{0 \ar[r] & (\overline{U}′_{∞})_{Γ} \ar[r] & (N_{∞})_{Γ} \ar[r] & (D_{∞})_{Γ} \ar[r] & 0}.

As D_{∞} is a torsion-Λ-module (as a submodule of 𝔛_{∞}) with trivial Γ-invariants, its Γ-coinvariants are finite. Hence (\overline{U}′_{∞})_{Γ} has the same ℤ_{p}-rank as (N_{∞})_{Γ}. By proposition 2.2 this rank is rank_{ℤ_{p}}(N_{0})− #S_{0} = [K : ℚ]. By Nakayama’s lemma the Λ-rank of \overline{U}′_{∞} 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 goal in this paper. We want to show that descent works asymptotically well for M_{∞} = \overline{U}_{∞}/\overline{C}_{∞} or (which will be proven to be equivalent) for M_{∞} = U_{∞}^{(0)}/\overline{C}_{∞} (see explanations and notations below for the symbol ^{(0)}). This means that in both cases above, Ker_{n}(M_{∞}) and Coker_{n}(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 \overline{U}′_{∞} (proposition 2.4) to bound the kernels of descent for \overline{U}_{∞}/\overline{C}_{∞}. Then we use local class field theory (proposition 2.3) to bound the cokernels of descent for U_{∞}^{(0)}/\overline{C}_{∞}.

But before this we have to slightly change the sequence U_{n}/\overline{C}_{n}. Indeed, for all n, the module U_{n}/\overline{C}_{n} is (by Leopoldt’s conjecture) of ℤ_{p}-rank 1 while (U_{∞}/\overline{C}_{∞})_{Γ_{n}} is torsion. This rank 1 comes from N_{K_{n}/ℚ}(\overline{U}_{n}) = \{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: Let F be a number field. In the sequel U_{F}^{(0)} will denote the kernel of

N_{F/ℚ}: U_{F} ⟶ U_{ℚ}.

Consistently U_{n}^{(0)} will denote the kernel of N_{K_{n}/ℚ} and U_{∞}^{(0)} = lim_{←} U_{n}^{(0)}.

By proposition 2.3 we have Ũ_{n} ⊂ U_{n}^{(0)} and therefore U_{∞} = lim_{←}(U_{n}^{(0)}). Moreover U_{n}^{(0)}/\overline{C}_{n} is

torsion and since N_{K_{n}/ℚ}(U_{n}) is ℤ_{p}-torsion free we have U_{n}^{(0)}/\overline{C}_{n} = Tor_{ℤ_{p}}(U_{n}/\overline{C}_{n}), and accordingly U_{n}^{(0)}/\overline{U}_{n} = Tor_{ℤ_{p}}(U_{n}/\overline{U}_{n}). For all these reasons, it is clearly more convenient to (and we will from now) use the sequence (U_{n}^{(0)})_{n∈ℕ} instead of (U_{n})_{n∈ℕ}. This sequence gives sense to the notations Ũ_{n}^{(0)}, Ker_{n}(U_{∞}^{(0)}), Coker_{n}(U_{∞}^{(0)}) and consistently to the same notations associated to sequences U_{n}^{(0)}/\overline{C}_{n}, U_{n}^{(0)}/\overline{U}_{n} and so on. Of course, due to proposition 2.3, only the various Coker_{n} will actually change when we replace U_{n} by U_{n}^{(0)}.

Proposition 3.1 Recall that X_{n} stands for the p-part of the class group of K_{n}, and that Ker_{n}(X_{∞}) is the kernel of the natural map (X_{∞})_{Γ_{n}} ⟶ X_{n}.

1. The map (U_{∞}^{(0)}/\overline{U}_{∞})_{Γ_{n}} ⟶ U_{n}^{(0)}/\overline{U}_{n} fits into an exact sequence

\xymatrix{0 \ar[r] & (X_{∞})^{Γ_{n}} \ar[r] & (U_{∞}^{(0)}/\overline{U}_{∞})_{Γ_{n}} \ar[r] & U_{n}^{(0)}/\overline{U}_{n} \ar[r] & Ker_{n}(X_{∞}) \ar[r] & 0}

2. Ker_{n}(U_{∞}^{(0)}/\overline{U}_{∞}) and Coker_{n}(U_{∞}^{(0)}/\overline{U}_{∞}) are finite and of bounded orders.

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

(R)  \xymatrix{0 \ar[r] & U_{n}/\overline{U}_{n} \ar[r] & 𝔛_{n} \ar[r] & X_{n} \ar[r] & 0.}

Taking the ℤ_{p}-torsion counterpart of the sequence (R) we have :

\xymatrix{0 \ar[r] & U_{n}^{(0)}/\overline{U}_{n} \ar[r] & Tor_{ℤ_{p}}(𝔛_{n}) \ar[r] & X_{n}.}

On the other hand if we take limits up to K_{∞} on (R), then apply Γ_{n}-cohomology, we obtain (by Leopoldt, 𝔛_{∞}^{Γ_{n}} = 0):

\xymatrix{0 \ar[r] & X_{∞}^{Γ_{n}} \ar[r] & (U_{∞}^{(0)}/\overline{U}_{∞})_{Γ_{n}} \ar[r] & (𝔛_{∞})_{Γ_{n}} \ar[r] & (X_{∞})_{Γ_{n}} \ar[r] & 0.}

Lemma 2.1 has a global analogue which is the following :

Lemma 3.2 The natural map (𝔛_{∞})_{Γ_{n}} ⟶ 𝔛_{n} fits into the exact sequence

\xymatrix{0 \ar[r] & (𝔛_{∞})_{Γ_{n}} \ar[r] & 𝔛_{n} \ar[r]^{res} & Gal(K_{∞}/K_{n}) \ar[r] & 0.}

In other words, descent provides an isomorphism

(𝔛_{∞})_{Γ_{n}} ≅ Tor_{ℤ_{p}}𝔛_{n}.

Proof. The proof of 𝔛_{∞} ↪ 𝔛_{n} is exactly the same as for 2.1 : we only need to replace the 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 (𝔛_{∞})_{Γ_{n}} and 𝔛_{n}. Let M_{n} be the maximal abelian S ramified p-extension of K_{n}. Then, by Leopoldt’s conjecture, one has Tor_{ℤ_{p}}(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

\xymatrix{ &  & 0 \ar[d] \\ (X_{∞})^{Γ_{n}} \ar@{^{(}->}[r] & (U_{∞}^{(0)}/\overline{U}_{∞})_{Γ_{n}} \ar[d] \ar[r] & (𝔛_{∞})_{Γ_{n}} \ar[r] \ar[d] & (X_{∞})_{Γ_{n}} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & U_{n}^{(0)}/\overline{U}_{n} \ar[r] & Tor_{ℤ_{p}}(𝔛_{n}) \ar[r] \ar[d] & X_{n} \\  &  & 0}

1 of 3.1 follows then from the snake lemma.

By Leopoldt conjecture the (X_{∞})^{Γ_{n}}’s are finite. As X_{∞} is a noetherian Λ-module the ascending union ⋃_{n∈ℕ}(X_{∞})^{Γ_{n}} stabilizes. This shows that the orders of Ker_{n}(U_{∞}^{(0)}/\overline{U}_{∞}) stabilize. As for Coker_{n}(U_{∞}^{(0)}/\overline{U}_{∞}) ≅ Ker_{n}(X_{∞}), the maps X_{n+1} ⟶ X_{n} (and consequently X_{∞} ⟶ 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^{λ_{X}n+μ_{X}p^{n}}, where λ_{X} and μ_{X} are the structural invariants of X_{∞}. Since the order of (X_{∞})^{Γ} is finite the orders of (X_{∞})_{Γ_{n}} are also asymptotically equivalent to p^{λ_{X}n+μ_{X}p^{n}}. This proves that the orders of Ker_{n}(X_{∞}), hence those of Coker_{n}(U_{∞}^{(0)}/\overline{U}_{∞}), are bounded and concludes the proof of 2.

□ By Leopoldt’s Conjecture we have for all n : (U_{∞}^{(0)}/\overline{U}_{∞})^{Γ_{n}} ⊂ 𝔛_{∞}^{Γ_{n}} = 0. And therefore an exact

sequence

(\overline{U}_{∞}/\overline{C}_{∞})_{Γ_{n}} ↪ (U_{∞}^{(0)}/\overline{C}_{∞})_{Γ_{n}} ↠ (U_{∞}^{(0)}/\overline{U}_{∞})_{Γ_{n}}.

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

[figure: 0; Ker_{n}(\overline{U}_{∞}/\overline{C}_{∞}); Ker_{n}(U_{∞}^{(0)}/\overline{C}_{∞}); Ker_{n}(U_{∞}^{(0)}/\overline{U}_{∞}); Coker_{n}(\overline{U}_{∞}/\overline{C}_{∞}); Coker_{n}(U_{∞}^{(0)}/\overline{C}_{∞}); Coker_{n}(U_{∞}^{(0)}/\overline{U}_{∞}); 0.]

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

Lemma 3.3

1. Ker_{n}(\overline{U}_{∞}/\overline{C}_{∞}) and Ker_{n}(U_{∞}^{(0)}/\overline{C}_{∞}) are finite. Their orders are simultaneously bounded or not.

2. Coker_{n}(\overline{U}_{∞}/\overline{C}_{∞}) and Coker_{n}(U_{∞}^{(0)}/\overline{C}_{∞}) are finite. Their orders are simultaneously 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 (\overline{U}_{∞}/\overline{C}_{∞})_{Γ_{n}} ⟶ \overline{U}_{n}/\overline{C}_{n} are bounded.

2. The orders of the kernels of the natural maps (U_{∞}^{(0)}/\overline{C}_{∞})_{Γ_{n}} ⟶ U_{n}^{(0)}/\overline{C}_{n} are bounded.

Proof. From the commutative diagram :

\xymatrix{ & (\overline{C}_{∞})_{Γ_{n}} \ar[r] \ar[d] & (\overline{U}_{∞})_{Γ_{n}} \ar[r] \ar[d] & (\overline{U}_{∞}/\overline{C}_{∞})_{Γ_{n}} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & _{\overline{C}_{n}} \ar[r] & _{\overline{U}_{n}} \ar[r] & \overline{U}_{n}/\overline{C}_{n} \ar[r] & 0}

we deduce the exact sequence :

[figure: 0; \frac{Ker_{n}(\overline{U}_{∞})}{Im(Ker_{n}(\overline{C}_{∞}))}; Ker_{n}(\overline{U}_{∞}/\overline{C}_{∞}); \overline{C}_{n}/C̃_{n}]

By the part 2 of theorem 1.1 Ker_{n}(\overline{U}_{∞}/\overline{C}_{∞}) is finite. To control Coker_{n}(\overline{C}_{∞}) we use the lemma

Lemma 4.2 There exists an N such that for all n ≥ N we have \overline{C}_{n}/C̃_{n} ≅ \overline{C}_{N}/C̃_{N}

Proof. Let I be the inertia subfield of p for K_{∞}/ℚ. By lemma 2.5 of [Bjr], for n large enough (n such that I ⊂ K_{n} is large enough), we have \overline{C}_{n} = C̃_{n}C_{I}. It follows that Coker_{n}(\overline{C}_{∞}) ≅ C_{I}/(C̃_{n} ⋂ C_{I}). Now the increasing sequence C̃_{n} ⋂ \overline{C}_{I} has to stabilize because \overline{C}_{I} is of finite ℤ_{p}-rank. This shows the lemma 4.2

□ To prove the lemma 4.1 it then suffices to bound the orders of Ker_{n}(\overline{U}_{∞})/ Im(Ker_{n}(\overline{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 submodule of M_{∞} defined as follows :

Minv(M_{∞}) = ⋃_{n∈ℕ}(M_{∞})^{Γ_{n}}.

Exists N such that for all n ≥ N we have

Ker_{n}(\overline{U}_{∞})/ Im(Ker_{n}(\overline{C}_{∞})) ≅ \frac{Minv(\overline{U}′_{∞}/\overline{U}_{∞})}{Im(Minv(\overline{U}′_{∞}/\overline{C}_{∞}))}

Proof. Starting from the sequence :

\xymatrix{0 \ar[r] & \overline{C}_{∞} \ar[r] & \overline{U}′_{∞} \ar[r] & \overline{U}′_{∞}/\overline{C}_{∞} \ar[r] & 0}

one gets the diagram

\xymatrix{0 \ar[r] & (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{n}} \ar[r] \ar[d] & (\overline{C}_{∞})_{Γ_{n}} \ar[r] \ar[d] & (\overline{U}′_{∞})_{Γ_{n}} \ar[d] \\  & 0 \ar[r] & _{\overline{C}_{n}} \ar[r] & _{\overline{U}{′ \atop n}}}

By the proposition 2.4 the kernels Ker_{n}(\overline{U}′_{∞}) are trivial. Hence we have an isomorphism Ker_{n}(\overline{C}_{∞}) ≃ (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{n}}. Since \overline{U}′_{∞} is a noetherian module, so is \overline{U}′_{∞}/\overline{C}_{∞} and therefore the increasing sequence (\overline{U}′_{∞}/\overline{C}_{∞})^{Γ_{n}} stabilizes and for n large we have Ker_{n}(\overline{C}_{∞}) ≃ Minv(\overline{U}′_{∞}/\overline{C}_{∞}). The same arguments with \overline{U}_{∞} instead of \overline{C}_{∞} proves that (provided n greater than some N) we have Ker_{n}(\overline{U}_{∞}) ≅ Minv(\overline{U}′_{∞}/\overline{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.

□ Rephrasing 1 of lemma 4.1 one has :

Corollary 4.4 ∃c, N ∈ ℕ such that for all n ≥ N the inequalities |(\overline{U}_{∞}/\overline{C}_{∞})_{Γ_{n}}| ≤ c|\overline{U}_{n}/\overline{C}_{n}| hold.

□

Theorem 4.5 The structural invariant μ of the module \overline{U}_{∞}/\overline{C}_{∞} is trivial.

Proof. Let λ and μ denote the structural invariants of some Λ-module M_{∞}. As soon as they are finite, the sequence of orders |(M_{∞})_{Γ_{n}}| is equivalent to p^{μp^{n}+λn}. For the module \overline{U}_{∞}/\overline{C}_{∞}, theorem 1.1 gives the required finiteness, hence we have |(\overline{U}_{∞}/\overline{C}_{∞})_{Γ_{n}}| ∼ p^{μp^{n}+λn}, where the symbol ∼ means that quotient of one sequence by the other is bounded (above and below) by non zero positive constants independent of n. On the other hand the orders of \overline{U}_{n}/\overline{C}_{n} are equivalent to the orders of X_{n} by Sinnott’s index formula ([Si]). By Iwasawa theorem, the orders of X_{n} are asymptotically equivalent to p^{λ_{X}n+μ_{X}p^{n}} where λ_{X} and μ_{X} are the structural invariants attached to X_{∞}. By the now well known theorem 7.15 of [W] μ_{X} = 0. Using corollary 4.4 one then get the asymptotical inequality : p^{λ_{X}n} ≥ p^{λn+μp^{n}}. This forces μ to be zero and λ_{X} ≥ λ.

□

Remark : Theorem 4.5 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}_{∞}/\overline{C}_{∞})_{Γ_{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 μ).

5 Descent cokernels

The third and final key lemma is

Lemma 5.1

1. The orders of the cokernels of the natural maps (\overline{U}_{∞}/\overline{C}_{∞})_{Γ_{n}} ⟶ \overline{U}_{n}/\overline{C}_{n} are bounded.

2. The orders of the cokernels of the natural maps (U_{∞}^{(0)}/\overline{C}_{∞})_{Γ_{n}} ⟶ U_{n}^{(0)}/\overline{C}_{n} are bounded.

Proof. Starting with the snake diagram

\xymatrix{ & (\overline{C}_{∞})_{Γ_{n}} \ar[r] \ar[d] & (U_{∞})_{Γ_{n}} \ar[r] \ar[d] & (U_{∞}/\overline{C}_{∞})_{Γ_{n}} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & _{\overline{C}_{n}} \ar[r] & U_{n}^{(0)} \ar[r] & U_{n}^{(0)}/\overline{C}_{n} \ar[r] & 0}

one gets the sequence

\xymatrix{Ker_{n}(U_{∞}^{(0)}/\overline{C}_{∞}) \ar[r] & \overline{C}_{n}/C̃_{n} \ar[r] & U_{n}^{(0)}/Ũ_{n} \ar[r] & Coker_{n}(U_{∞}^{(0)}/\overline{C}_{∞}) \ar[r] & 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 that D ⊂ K_{n} (else enlarge n). By lemma 3.3, Ker_{n}(U_{∞}^{(0)}/\overline{C}_{∞}) is bounded. By proposition 2.3 we have Ũ_{n} = Ker(N_{K_{n}/D}: U_{n}^{(0)} ⟶ U_{D}). One then gets an isomorphism U_{n}^{(0)}/Ũ_{n} ≅ N_{K_{n}/D}(U_{n}^{(0)}) and using this isomorphism the preceding sequence reads

\xymatrix{0 \ar[r] & N_{K_{n}/D}(\overline{C}_{n}) \ar[r] & N_{K_{n}/D}(U_{n}^{(0)}) \ar[r] & Coker_{n}(U_{∞}^{(0)}/\overline{C}_{∞}) \ar[r] & 0.}

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}).

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

2. (U_{D}^{(0)})^{c_{n}} is a submodule of bounded finite index in N_{K_{n}/D}(U_{n}^{(0)})

3. (U_{D}^{(0)})^{c_{n}}/\overline{C}_{D}^{c_{n}} is asymptotically equivalent to U_{D}^{(0)}/\overline{C}_{D}.

Proof. 3 is immediate : the finite constant group U_{D}^{(0)}/\overline{C}_{D} maps onto (U_{D}^{(0)})^{c_{n}}/(\overline{C}_{D})^{c_{n}}. Since the norm N_{K_{n}/D} acts as c_{n} on \overline{C}_{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 1 we use again lemma 2.5 of [Bjr] that is \overline{C}_{n} = C̃_{n}\overline{C}_{I}. Moreover, as C̃_{n} ⊂ Ũ_{n}^{(0)} we have N_{K_{n}/D}(C̃_{n}) = 0 by proposition 2.3 (Without using semi-local units, N_{K_{n}/D}(C̃_{n}) = 0 can be checked directly using distribution relations on a generating system of C̃_{n}). Hence we get N_{K_{n}/D}(\overline{C}_{n}) = N_{K_{n}/D}(C̃_{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}.

2 is an easy exercise using local class field theory. Indeed, recall that U_{n} = ⊕_{v|p}\overline{U}_{K_{n,v}}^{1}. Then the global norm N_{K_{n}/D} acts on each summand as the local norm N_{K_{n,v}/ℚ_{p}}. By local class field theory,

the quotient \overline{U}_{ℚ_{p}}^{1}/N_{K_{n,v}/ℚ_{p}}(\overline{U}_{K_{n,v}}^{1}) is isomorphic to the p-part of the ramification subgroup of Gal(K_{n,v}/ℚ_{p}). These wild ramification subgroups are cyclic with orders asymptotically equivalent to p^{n}. Summing up, it follows that N_{K_{n}/D}(U_{n}) contains U_{D}^{c_{n}} with bounded finite index. A fortiori N_{K_{n}/D}(U_{n}^{(0)}) contains (U_{D}^{(0)})^{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.

□

6 Iwasawa theorem for unit classes

Theorem 6.1

1. Let λ_{1} and μ_{1} denote the structural invariants of the Iwasawa module \overline{U}_{∞}/\overline{C}_{∞}. Then μ_{1} = 0 and the orders of \overline{U}_{n}/\overline{C}_{n} are asymptotically equivalent to p^{λ_{1}n}.

2. Let λ_{2} and μ_{2} denotes the structural invariants of the Iwasawa module U_{∞}/\overline{C}_{∞}. Then μ_{2} = 0 and the orders of U_{n}^{(0)}/\overline{C}_{n} are asymptotically equivalent to p^{λ_{2}n}.

Proof. By theorem 4.5 we have μ_{1} = 0. 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}_{∞}/\overline{C}_{∞})_{Γ_{n}}. As they are finite the last orders are equivalent to p^{λ_{1}n}. This shows 1.

For 2 the same argument as for 1 will give |U_{n}^{(0)}/\overline{C}_{n}| ∼ p^{λ_{2}n+μ_{2}p^{n}}. It only remains to show that μ_{2} = 0. Consider the exact sequence of Λ-torsion modules

(DNA) \xymatrix{0 \ar[r] & \overline{U}_{∞}/\overline{C}_{∞} \ar[r] & U_{∞}^{(0)}/\overline{C}_{∞} \ar[r] & 𝔛_{∞} \ar[r] & X_{∞} \ar[r] & 0.}

We claim that all four above modules have trivial μ-invariant and we only need to prove it for three out of them (actually two well chosen would be enough). For \overline{U}_{∞}/\overline{C}_{∞} this is part 1. For X_{∞} this is theorem 7.15 of [W]. Let us draw the main lines of the proof written in [W] of this theorem :

step 1 The main ingredient is Ferrero-Washington theorem [FW] which claims that the power series g_{ψ}(T) of our first section is prime to p.

step 2 Over K(ζ_{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^{λn}, where λ is the Weierstrass degree of the product of relevant g_{ψ}(T)’s. By Iwasawa theorem, this implies the triviality of the structural μ-invariant of X_{∞}^{−}.

step 3 Using the classical mirror inequality μ^{+} ≤ μ^{−} derived from Kummer duality, one recovers the triviality of the μ-invariants of the plus part X_{∞}^{+} over K(ζ_{p}), which in turn implies the triviality of the μ-invariant of X_{∞} for our base field K.

For the remaining module 𝔛_{∞}, 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 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 μ(𝔛_{∞}^{+}) = μ(X_{∞}^{−}) over K(ζ_{p}) and the inequality in the third step follows from 𝔛_{∞} ↠ X_{∞}. Hence step 2 gives directly μ(𝔛_{∞}^{+}) = 0 over K(ζ_{p}), which is all we need.

The triviality of μ_{2} follows and this conclude the proof of 2 of theorem 6.1.

□

Corollary 6.2

1. The Λ-modules \overline{U}_{∞}/\overline{C}_{∞} and X_{∞} share the same structural invariant λ.

2. The Λ-modules U_{∞}^{(0)}/\overline{C}_{∞} and 𝔛_{∞} share the same structural invariant λ.

Proof. The invariants λ and μ 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 as used in the proof of 4.5, the orders of X_{n} are asymptotically equivalent to the orders of \overline{U}_{n}/\overline{C}_{n}. These orders are related to the corresponding λ invariant by Iwasawa’s theorem for X_{n} and theorem 6.1 for \overline{U}_{n}/\overline{C}_{n}. This proves 1.

□

Remark : As explained in the introduction the corollary 6.2 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’s classical theorems, Sinnott’s index formula, Coleman’s morphisms, and Leopoldt’s conjecture (last two via theorem 1.1).

Acknowledgement : I thank professor Th´ông Nguy˜ên-Quang-D̄˜ô for many instructive conversations about and helpful comments on this paper.

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