Appendix : about the Λ-freeness of \overline{C}_{∞}.

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

Let F be an abelian totally real number field (without any further restriction). In the paper [Bel98] we gave a sufficient condition to Λ-freeness of the Λ-module \overline{C}_{∞} associated to the cyclotomic ℤ_{p}-extension F_{∞}/F. There amongst other hypotheses we assumed that p was unramified in the base field denoted K there and F here. Now due to the applications given in the main part of this paper, it seems worthwhile to write down precisely the relevant condition without assuming anything about ramification at p.

Lemma 0.1 Let L ⊂ F_{∞} be the maximal subfield of F_{∞} such that p is (at most) tamely ramified in L (under hypothesis (R), L is the field F^{tam} of ??). Put Λ_{L} := ℤ_{p}[[Gal(L_{∞}/L)]], where L_{∞}/L is the cyclotomic ℤ_{p}-extension of L. Then we have :

1. As a group, \overline{C}_{∞} only depends on F_{∞} not on F.

2. L_{∞} = F_{∞}.

3. \overline{C}_{∞} is Λ-free if and only if \overline{C}_{∞} is Λ_{L}-free.

Proof. 1 is clear. 2 is lemma 1.2 of [Bel05]. By theorem 2.2 of [Bel02], the Λ-freeness is equivalent to an asymptotic condition (namely the ”asymptotic Galois Descent” property for the \overline{C}_{n}’s). But there exists some N ∈ ℕ and some b ∈ ℤ such that for all n ≥ N we have F_{n} = L_{n+b} : the equivalence in 3 follows.

□ In the light of the lemma 0.1, as far as the Λ-freeness of \overline{C}_{∞} is concerned, we can (but shall

not) in the sequel assume without loss of generality that p is (at most) tamely ramified in F. We could then just refer to [Bel98] for all proofs. But we thought it would be better to give them here including most of the details, taking this opportunity to simplify the arguments of [Bel98] by translating them at infinite level.

We need a few more notations. They are close to those used in [Bel98] but formally not exactly same. Let P be the set of rational primes l ≠ p which are ramified in F/ℚ. The case P = ∅ is obvious but allowed. For all supernatural numbers t we put ℚ(t) = ℚ(ζ_{t}). For all J ⊂ P and all n ∈ ℕ ∪ {∞}, we put F_{n}(J) := ℚ(∏_{l∈J} l^{∞}p^{∞}) ∩ F_{n}. We shall abbreviate by F (J) := F_{0}(J), G(J) = Gal(F_{∞}(J)/𝔹_{∞}), where 𝔹_{∞} is the cyclotomic ℤ_{p}-extension of ℚ, and G = G(P). For any abelian number field K we write cond(K) for its conductor, that is the minimal n ∈ ℕ such that K ⊂ ℚ(n). For all n ∈ ℕ and all J ⊂ P we will denote by ε_{n}(J) the following cyclotomic number (unit if J ≠ ∅) :

ε_{n}(J) := N_{ℚ(cond(F_{n}(J)))/F_{n}(J)}(1 − ζ_{cond(F_{n}(J))}).

It follows from the classical distribution relations that for n ≥ 1 the ε_{n}(J)’s form a norm coherent sequence. Therefore, fixing a γ generating Gal(𝔹_{∞}/ℚ) we may define ε_{∞}(J) ∈ \overline{C}_{∞} by the following formula :

ε_{∞}(J) := \{ \begin{matrix}(ε_{n}(J))_{n≥1} if J ≠ ∅ \\ (ε_{n}(∅)^{(γ−1)})_{n∈ℕ} else. \end{matrix}

We shall examine the Λ_{L}[G]-module structure of \overline{C}_{∞} (recall that G = G(P) = Gal(F_{∞}/𝔹_{∞})), and for that it is more convenient to write additively the multiplication in \overline{C}_{∞} and to keep multiplication for the action of Λ[G]. Consistently for 𝔹_{∞} ⊂ N ⊂ K ⊂ F_{∞} we will denote by T_{K/N} the

trace (actually the algebraic norm map) acting on elements x ∈ \overline{C}_{∞}^{Gal(F_{∞}/K)}. The elements ε_{∞}(J) are subject to the following distribution relations :

Lemma 0.2 Let (\frac{𝔄}{N/K} ) be the (global) Artin symbol for any abelian extension N/K and (frac-

tional) ideal 𝔄 of K. For all J ⊂ P and all I ⊂ J, put P_{I,J} for the set of rational primes l dividing cond(F_{∞}(J)) but not dividing cond(F_{∞}(I)). We have :

T_{F_{∞}(J)/F_{∞}(I)}(ε_{∞}(J)) = ( ∏_{l∈P_{I,J}} (1 − (\frac{(l)}{F_{∞}(I)/ℚ})^{−1})) ε_{∞}(I).

Proof. The analogous relations at finite levels are well known. We then just take inverse limits. □

Lemma 0.3 The system \{ε_{∞}(J), J ⊂ P\} generates \overline{C}_{∞} over Λ_{L}[Gal(F_{∞}/𝔹_{∞})].

Proof. This follows from distribution relations (e.g. lemma 0.2) and lemma 2.3 of [Gre92].

□ We now state the ad hoc hypothesis that will ensure the freeness of \overline{C}_{∞}. This ”generalized

hypothesis (B)” is an immediate generalization of (HB) in [Bel98].

Definition 0.4 For all J ⊂ P we call norm ideal of J and denote by N(J) the ideal of ℤ_{p}[G(J)] generated by traces

N(J) := ⟨T_{F_{∞}(J)/F_{∞}(J−{l})}; l ∈ J⟩ .

We say that the pair (F, p) satisfies ”generalized Hypothesis B” ((gHB) for short) if and only if for all J ⊂ P the quotient ℤ_{p}[G(J)]/N(J) is torsion free.

This hypothesis is quite technical, but very natural regarding the proof of theorem 0.6. The following proposition shows that (gHB) holds true in many usual cases. Indeed all known cases of freedom of \overline{C}_{∞} are consequence of (gHB) via theorem 0.6.

Proposition 0.5 For all l ∈ P let us denote by I_{l} ⊂ G the inertia subgroup for l in F_{∞}/𝔹_{∞}. Recall that for any finite abelian group H, we write \overline{H} for its p-Sylow subgroup (considered also as the maximal p-quotient of H). The pair (F, p) satisfies (gHB) as soon as one of the following properties holds :

1. The \overline{I}_{l}, l ∈ P are mutually direct summands. In other words the natural map ⨁_{l∈P} \overline{I}_{l} ⟶ \overline{G} is injective.

2. dim_{𝔽_{p}}(G/pG) ≤ 1, or equivalently \overline{G} is cyclic.

3. #P ≤ 2

Proof. These sufficient conditions are the exact analogues of those in section IV.1 of [Bel98]. Note that they concern only the maximal p-quotient of the Galois groups involved. Since the (unique) prime above p of 𝔹_{∞} is tamely ramified in F_{∞}/𝔹_{∞}, everything goes as if p were unramified in the base field F. Taking this into account, we can repeat \underline{verbatim} the proofs in [Bel98]. □

Theorem 0.6 Let us denote by r(I) the ℤ_{p}-rank of ℤ_{p}[G(I)]/N(I). Assume that (F, p) satisfies (gHB). Fix any total order ≤ on the set of all subsets of P extending the inclusion. Then :

1. ∀J ⊂ P we have isomorphisms of Λ_{L}[G]-modules :

\frac{ℤ_{p}[G(J)]}{N(J)} ⊗_{ℤ_{p}} Λ_{L} ≅ \frac{⟨ε_{∞}(J)⟩}{⟨ε_{∞}(I); I ⊂ J, I ≠ J⟩ ⋂⟨ε_{∞}(J)⟩} ≅ \frac{⟨ε_{∞}(J)⟩}{⟨ε_{∞}(I); I ≤ J, I ≠ J⟩ ⋂⟨ε_{∞}(J)⟩};

hence all three Λ_{L}[G]-modules are free over Λ_{L} (of ranks r(J)).

2. \overline{C}_{∞} ≅ ⨁_{I⊂P} Λ_{L}^{r(I)} as Λ_{L}-modules; in particular \overline{C}_{∞} is Λ_{L}-free of rank [L : ℚ] and Λ-free of rank [F : ℚ].

Proof. \overline{C}_{∞} has the same Λ-rank (resp. Λ_{L}-rank) as \overline{U}_{∞}, namely [F : ℚ] (resp. [L : ℚ] = [F_{∞} : 𝔹_{∞}]) : see e.g. [Kuz72], [Bel02] ... By (gHB) the modules \frac{ℤ_{p}[G(J)]}{N(J)} ⊗_{ℤ_{p}} Λ_{L} are Λ_{L}-free. We first prove that 1 implies 2. From the tautological (split over Λ_{L} under (gHB) by 1) exact sequences :

ES(J) \xymatrix{⟨ε_{∞}(I); I ≤ J, I ≠ J⟩ \ar@{^{(}->}[r] & ⟨ε_{∞}(I); I ≤ J⟩ \ar@{->>}[r] & \frac{⟨ε_{∞}(J)⟩}{⟨ε_{∞}(I); I ≤ J, I ≠ J⟩ ⋂⟨ε_{∞}(J)⟩}}

we see that 1 implies 2 by induction. Let us prove 1.

By lemma 0.2 and by the inclusion ⟨ε_{∞}(I); I ⊂ J, I ≠ J⟩ ⊂ ⟨ε_{∞}(I); I ≤ J, I ≠ J⟩ we have surjective morphisms :

(Sur) \frac{ℤ_{p}[G(J)]}{N(J)} ⊗_{ℤ_{p}} Λ_{L} ↠ \frac{⟨ε_{∞}(J)⟩}{⟨ε_{∞}(I); I ⊂ J, I ≠ J⟩ ⋂⟨ε_{∞}(J)⟩} ↠ \frac{⟨ε_{∞}(J)⟩}{⟨ε_{∞}(I); I ≤ J, I ≠ J⟩ ⋂⟨ε_{∞}(J)⟩}.

Therefore the required isomorphisms will follow from equalities of Λ_{L}-ranks. Now :

Lemma 0.7 Recall that for all I, r(I) is the ℤ_{p}-rank of ℤ_{p}[G(I)]/N(I). Put d(I) = [F_{∞}(I) : 𝔹_{∞}].

1. For all J ⊂ P we have

r(J) = ∑_{I⊂J}(−1)^{#J−#I}d(I)

2.

[L : ℚ] = [F_{∞} : 𝔹_{∞}] = d(P) = ∑_{J⊂P} r(J)

Proof. 1 is proven using character theory and following the same way as proposition 2.11 of [Bel98] : just replace the function f_{n}(I) there by d(I) here. 2 follows from 1 and the combinatorial lemma 2.14 of [Bel98]. Details are left to the reader.

□ We resume the proof of theorem 0.6. For any Λ_{L}-module M let us denote its Λ_{L}-rank by

rank_{Λ_{L}}(M). Put

t(J) := rank_{Λ_{L}} (\frac{⟨ε_{∞}(J)⟩}{⟨ε_{∞}(I); I ≤ J, I ≠ J⟩ ⋂⟨ε_{∞}(J)⟩}) .

From the surjections (Sur) we see that r(J) ≥ t(J). On the other hand, summing ranks in all the sequences (ES(J)), we recover that ∑_{J} t(J) = rank_{Λ_{L}}(\overline{C}_{∞}) = d(P) = ∑_{J} r(J) by 2 of lemma 0.7. Therefore all inequalities r(J) ≥ t(J) are actually equalities. This proves theorem 0.6.

□

References

[Bel98] J.-R. Belliard, Sur la structure galoisienne des unités circulaires dans les ℤ_{p}-extensions, J. Number Theory 69 (1998), no. 1, 16–49.

[Bel02] , Sous-modules d’unités en théorie d’Iwasawa, Théorie des nombres, Années 1998/2001, Publ. Math. UFR Sci. Tech. Besançon, 2002, p. 12.

[Bel05] , Global units modulo circular units : descent without Iwasawa’s Main Conjecture, preprint (2005).

[Gre92] C. Greither, Class groups of abelian fields, and the main conjecture, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 449–499.

[Kuz72] L. V. Kuz′min, The Tate module of algebraic number fields, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 267–327.