CYCLOTOMIC UNITS IN Z_{p}-EXTENSIONS

Radan Kučera, Jan Nekovář

1. Introduction

Let p be a prime number, k an abelian field (i.e. a finite abelian extension of Q), and K the cyclotomic Z_{p}-extension of k. For each n ≥ 0, let K_{n} denote the unique extension of k in K of degree p^{n} over k. Let E_{K_{n}} be the group of units of K_{n}, C_{K_{n}} the Sinnott group of circular units of K_{n}, and h_{K_{n}}^{+} the class number of the maximal real subfield of K_{n}. Sinnott (see [S2, p.182]) proved the following

Theorem (Sinnott). There is a rational number c, depending only on K, such that, for all sufficiently large n, we have

[E_{K_{n}} : C_{K_{n}}] = \{ \begin{matrix}c · h_{K_{n}}^{+} & if k is imaginary, \\ c · 2^{[K_{n}:Q]} · h_{K_{n}}^{+} & if k is real. \end{matrix}

Let k be real, and for each n ≥ 0 let C_{1,K_{n}} = {ε ∈ E_{K_{n}}; ε^{2} ∈ C_{K_{n}}}. Then there is a rational number c_{1}, depending only on K, such that

[E_{K_{n}} : C_{1,K_{n}}] = c_{1} · h_{K_{n}}^{+}

for all sufficiently large n.

Let \overline{C}_{K_{n}} ⊇ C_{K_{n}} be the Washington group of cyclotomic units of K_{n} (the definitions of both groups are given below). The motivation of this paper (based on a question of M. Kolster) was to compare projective limits (with respect to norms) C = \underleftarrow{lim}(C_{K_{n}} ⊗ Z_{p}) and \overline{C} = \underleftarrow{lim}(\overline{C}_{K_{n}} ⊗ Z_{p}). Unfortunately, we were not able to decide whether C = \overline{C} in general, but we shall prove the following analogy of Sinnott’s theorem, which implies that C is of finite index in \overline{C}:

Theorem. There are a rational number c′ and an integer d, depending only on K, such that, for all sufficiently large n, we have

[E_{K_{n}} : \overline{C}_{K_{n}}] = c′ · p^{t_{n}} · h_{K_{n}}^{+},

where t_{n} ≥ 0 are integers smaller than d.

It seems to be reasonable to expect that one can always take d = 1 in the previous theorem, but we were unable to prove it.

During the work on this paper, the first named author was supported by a postdoctoral grant from Université Laval in Québec and Centre interuniversitaire en calcul mathématique algébrique; the second named author was supported by a fellowship from the Miller Institute for Basic Research at the University of California at Berkeley. The authors wish to express their gratitude to these institutions for their hospitality.

2. Notations and preliminaries

Let p be a prime number, fixed through the whole paper. Let q = p, if p is odd, and let q = 4 if p = 2. For any positive integer m let ζ_{m} = e^{2πi/m}. For any integer n ≥ 0 we define B_{n} to be the real subfield of Q(ζ_{qp^{n}}) of absolute degree p^{n}. Let B = ⋃_{n=0}^{∞} B_{n}.

We choose an abelian field k, which may be real or imaginary, and for convinience we suppose k to be contained in the complex field C. Let X be the group of characters of Gal(k/Q), considered as a group of Dirichlet characters modulo the conductor of k. For any χ ∈ X there is Iwasawa’s decomposition χ = θψ, where θ is a character of the first kind (i.e. the conductor of θ is not divisible by qp) and ψ is a character of the second kind (i.e. an even character whose conductor and order are powers of p). Let

X′ = {θ; χ = θψ ∈ X, θ of 1^{st} kind, ψ of 2^{nd} kind}

and let k′ be the field corresponding to X′. It is easy to see that kB_{n} = k′B_{n} for all sufficiently large n, so it is enough to prove Theorem for k′. Therefore in the whole paper we shall suppose that the conductor of k is either f or qf, where p |̸ f. For any integer n ≥ 0, let K_{n} = kB_{n} and L_{n} = B_{n}(ζ_{qf}) = Q(ζ_{qp^{n}f}). Similarly, let K = kB and L = B(ζ_{qf}). Let H = Gal(L/K). It is clear that, for any n ≥ 0, the Galois group Gal(L_{n}/K_{n}) is isomorphic to H via restriction. We shall identify these groups.

For any abelian field M, let E_{M} and W_{M} be the group of units of M and the group of roots of unity in M, respectively. The Sinnott group of circular units of M is defined as the intersection C_{M} = D_{M} ∩ E_{M}, where D_{M} is the subgroup of the multiplicative group M^{×} generated by −1 and

{N_{Q(ζ_{m})/M∩Q(ζ_{m})}(1 − ζ_{m}^{a}); a, m ∈ Z, 1 < m |̸ a}

(see [S2, p. 201]). Let \overline{C}_{M} be the Washington group of cyclotomic units of M, namely \overline{C}_{M} = C_{Q(ζ_{m})} ∩ E_{M}, where m is the conductor of M (see [W2, p. 143]).

Let ℓ_{M} : M^{×} → R[Gal(M/Q)] be the logarithmic mapping defined for any α ∈ M^{×} by

ℓ_{M}(α) = −\frac{1}{2} ∑_{σ∈Gal(M/Q)} log |α^{σ}|σ^{−1}.

Let T_{M} = ℓ_{M}(D_{M}) and T_{0,M} = T_{M} ∩ (1 − e_{1,M})T_{M}, where e_{1,M} = \frac{1}{[M:Q]} ∑_{σ∈Gal(M/Q)} σ is the idempotent corresponding to the trivial character on Gal(M/Q).

If M_{1}/M_{2} is an extension of abelian fields, then the restriction map

res_{M_{1}/M_{2}} : R[Gal(M_{1}/Q)] → R[Gal(M_{2}/Q)]

is the ring homomorphism induced by the restriction of automorphisms and the corestriction map

cor_{M_{1}/M_{2}} : R[Gal(M_{2}/Q)] → R[Gal(M_{1}/Q)]

is the homomorphism of additive groups determined by

cor_{M_{1}/M_{2}} σ = ∑_{τ|_{M_{2}} =σ} τ,

for any σ ∈ Gal(M_{2}/Q). The sum is taken over all τ ∈ Gal(M_{1}/Q) whose restriction to M_{2} is σ.

3. The relative index of Washington’s and Sinnott’s units

Our aim is to compute the index [\overline{C}_{K_{n}} : C_{K_{n}}] up to a bounded factor.

For any integral sequences a_{n}, b_{n}, we shall write a_{n} ≈ b_{n} if there exist integers x, y ≥ 1 satisfying a_{n} | xb_{n} and b_{n} | ya_{n} for all n.

Lemma 1.

[\overline{C}_{K_{n}} : C_{K_{n}}] ≈ [T_{0,L_{n}}^{H} : cor_{L_{n}/K_{n}} T_{0,K_{n}}].

Proof. Sinnott proved that

0 ⟶ W_{M} ⟶ C_{M} ⟶^{ℓ_{M}} T_{0,M} ⟶ 0

is an exact sequence for any abelian field M (see Lemma 4.2 of [S2]). Taking H-invariants for M = L_{n} gives the exact sequence

0 ⟶ W_{L_{n}}^{H} ⟶ C_{L_{n}}^{H} ⟶^{ℓ_{Ln}} T_{0,L_{n}}^{H} ⟶ H^{1}(H, W_{L_{n}}).

Of course, H = Gal(L_{n}/K_{n}), so W_{L_{n}}^{H} = W_{K_{n}} and directly from the definition we have C_{L_{n}}^{H} = \overline{C}_{K_{n}}. It is easy to see that ℓ_{L_{n}}(α) = cor_{L_{n}/K_{n}} ℓ_{K_{n}}(α) for any α ∈ K_{n}^{×}, so the following diagram with exact rows commutes

0 ⟶ C_{K_{n}}/W_{K_{n}} ⟶^{ℓ_{Kn}} T_{0,K_{n}} ⟶ 0
⊆↓  ↓^{cor}^{Ln/Kn}
0 ⟶ \overline{C}_{K_{n}}/W_{K_{n}} ⟶^{ℓ_{Ln}} T_{0,L_{n}}^{H} ⟶ H^{1}(H, W_{L_{n}})

and the lemma follows from the following lemma, because W_{K_{n}} ⊆ C_{K_{n}} ⊆ \overline{C}_{K_{n}} ⊆ E_{K_{n}}, so \overline{C}_{K_{n}}/C_{K_{n}} is finite (by [S2]) and isomorphic to (\overline{C}_{K_{n}}/W_{K_{n}})/(C_{K_{n}}/W_{K_{n}}).

Lemma 2. #H^{1}(H, W_{L_{n}}) is bounded by a constant.
Proof. Let μ_{m} = ⟨ζ_{m}⟩ be the group of m-th roots of unity. We have W_{L_{n}} = μ_{f} × μ_{qp^{n}}, so
H^{1}(H, W_{L_{n}}) = H^{1}(H, μ_{f}) × H^{1}(H, μ_{qp^{n}}),

so it is enough to show that H^{1}(H, μ_{qp^{n}}) is bounded. If the action of H on μ_{qp^{n}} is trivial, then

#H^{1}(H, μ_{qp^{n}}) = # Hom(H, μ_{qp^{n}}) ≤ #H_{p^{∞}},

where H_{p^{∞}} is the p-Sylow subgroup of H and we are done. Let us suppose that the action of H on μ_{qp^{n}} is not trivial. Then H can be written as a product of cyclic groups generated, say, by σ_{1}, …, σ_{s}, and σ_{1} has a non-trivial action on μ_{qp^{n}}.

We need to distinguish two cases. At first, let us suppose p ≠ 2. Since H ⊆ Gal(L_{n}/B_{n}), the action of σ_{i}^{p−1} on μ_{p^{1+n}} is trivial for each i ∈ {1, …, s}. For any i ∈ {1, …, s} let us choose c_{i} ∈ Z satisfying ζ_{p^{1+n}}^{σ_{i}} = ζ_{p^{1+n}}^{c_{i}}. We have c_{i}^{p−1} ≡ 1 (mod p^{1+n}) for each i ∈ {1, …, s} and c_{1} ≢ 1 (mod p^{1+n}), therefore c_{1} ≢ 1 (mod p). For any ρ ∈ μ_{p^{1+n}} there is unique ξ ∈ μ_{p^{1+n}} satisfying ξ^{c_{1}−1} = ρ; let us define the coboundary f_{ρ} : H → μ_{p^{1+n}} by f_{ρ}(τ) = ξ^{τ−1} for each τ ∈ H. It is clear that f_{ρ}(σ_{1}) = ρ. Let f : H → μ_{p^{1+n}} be a crossed homomorphism. For any i ∈ {2, …, s} we have

(f(σ_{1}))^{σ_{i}}f(σ_{i}) = f(σ_{1}σ_{i}) = (f(σ_{i}))^{σ_{1}}f(σ_{1}),

so

(f(σ_{i}))^{c_{1}−1} = (f(σ_{1}))^{c_{i}−1}.

Therefore f is uniquely determined by the value ρ = f(σ_{1}), hence f = f_{ρ}. We have proved H^{1}(H, μ_{qp^{n}}) = 0 in this case.

Now, let us suppose p = 2. Since H ⊆ Gal(L_{n}/B_{n}), any σ_{i} acts on μ_{2^{2+n}} like the identity or the complex conjugation. Let c_{i} ∈ {1, −1} satisfy ζ_{2^{2+n}}^{σ_{i}} = ζ_{2^{2+n}}^{c_{i}} for each i ∈ {1, …, s}. Then c_{1} = −1. For any ρ ∈ μ_{2^{2+n}} there is the coboundary f_{ρ} : H → μ_{p^{2+n}} defined by f_{ρ}(τ) = ρ^{τ−1} for each τ ∈ H. Then f_{ρ}(σ_{1}) = ρ^{−2}, so we have 2^{1+n} different coboundaries. Let f : H → μ_{p^{2+n}} be a crossed homomorphism. For any i ∈ {2, …, s} we have

(f(σ_{i}))^{−2} = (f(σ_{1}))^{c_{i}−1},

hence f(σ_{i}) = ±1 if c_{i} = 1 and f(σ_{i}) = ±f(σ_{1}) if c_{i} = −1. Therefore there exist at most 2^{s−1} crossed homomorphisms with the same value f(σ_{1}) and

#H^{1}(H, μ_{qp^{n}}) ≤ \frac{2^{2+n} 2^{s−1}}{2^{1+n}} = 2^{s}.

The lemma is proved.

Remark. In fact, we have reproved a version of Sah’s lemma.

Lemma 3.

[T_{0,L_{n}}^{H} : cor_{L_{n}/K_{n}} T_{0,K_{n}}] ≈ [((1 − e_{1,L_{n}})T_{L_{n}})^{H} : cor_{L_{n}/K_{n}}(1 − e_{1,K_{n}})T_{K_{n}}].

Proof. We have the following commutative diagram with exact rows:

0 ⟶ T_{0,K_{n}} ⟶^{⊆} (1 − e_{1,K_{n}})T_{K_{n}} ⟶ ((1 − e_{1,K_{n}})T_{K_{n}})/T_{0,K_{n}} ⟶ 0
cor^{Ln/Kn} ↓  ↓^{cor}^{Ln/Kn}  ↓^{f_{n}}
0 ⟶ T_{0,L_{n}}^{H} ⟶^{⊆} ((1 − e_{1,L_{n}})T_{L_{n}})^{H} ⟶  X_{n}  ⟶ 0

where

X_{n} = (((1 − e_{1,L_{n}})T_{L_{n}})^{H} + T_{0,L_{n}})/T_{0,L_{n}} ⊆ (((1 − e_{1,L_{n}})T_{L_{n}})/T_{0,L_{n}})^{H}

and f_{n} is determined by the corestriction. Since cor_{L_{n}/K_{n}} is injective, the Snake lemma gives the following exact sequence:

0 ⟶ Ker f_{n} ⟶ T_{0,L_{n}}^{H}/ cor_{L_{n}/K_{n}} T_{0,K_{n}} ⟶
⟶ ((1 − e_{1,L_{n}})T_{L_{n}})^{H}/ cor_{L_{n}/K_{n}}(1 − e_{1,K_{n}})T_{K_{n}} ⟶ Coker f_{n} ⟶ 0.

Proposition 4.1 of [S2] gives that

#((1 − e_{1,K_{n}})T_{K_{n}})/T_{0,K_{n}} ≈ p^{n},
#((1 − e_{1,L_{n}})T_{L_{n}})/T_{0,L_{n}} ≈ p^{n}.

Because T_{0,L_{n}}^{H}/ cor_{L_{n}/K_{n}} T_{0,K_{n}} is finite by Lemma 1, all groups in the previous exact sequence are finite and the lemma will be proved if we show

# Ker f_{n} ≈ # Coker f_{n}.

Of course,

\frac{# Ker f_{n}}{# Coker f_{n}} = \frac{#((1 − e_{1,K_{n}})T_{K_{n}})/T_{0,K_{n}}}{#X_{n}} .

Since X_{n} ⊆ ((1 − e_{1,L_{n}})T_{L_{n}})/T_{0,L_{n}}, it is enough to show that there is a positive integer c satisfying p^{n} | c(#X_{n}). Let

α_{n} = N_{Q(ζ_{qp^{n}})/B_{n}}(1 − ζ_{qp^{n}}).

It is easy to see that α_{n} ∈ D_{L_{n}}^{H}, so ℓ_{L_{n}}(α_{n}) ∈ T_{L_{n}}^{H} and (1 − e_{1,L_{n}})ℓ_{L_{n}}(α_{n}) + T_{0,L_{n}} ∈ X_{n}. We shall use the isomorphism

((1 − e_{1,L_{n}})T_{L_{n}})/T_{0,L_{n}} ≅ (e_{1,L_{n}}T_{L_{n}})/T_{L_{n}}^{Gal(L_{n}/Q)}

(see the proof of Proposition 4.1 of [S2]). This isomorphism sends β_{n} = (1 − e_{1,L_{n}})ℓ_{L_{n}}(α_{n}) + T_{0,L_{n}} to γ_{n} = e_{1,L_{n}}ℓ_{L_{n}}(α_{n}) + T_{L_{n}}^{Gal(L_{n}/Q)}. We have

e_{1,L_{n}}ℓ_{L_{n}}(α_{n}) = −\frac{1}{2}e_{1,L_{n}} ∑_{σ∈Gal(L_{n}/Q)} log |α_{n}^{σ}| = −\frac{1}{2}e_{1,L_{n}} log | N_{L_{n}/Q}(α_{n})|

and

N_{L_{n}/Q}(α_{n}) = N_{B_{n}/Q}(N_{L_{n}/B_{n}}(α_{n})) = N_{Q(ζ_{qp^{n}})/Q}(1 − ζ_{qp^{n}})^{[L_{n}:B_{n}]} = p^{φ(qf)},

so γ_{n} = −\frac{φ(qf)}{2}e_{1,L_{n}} log p + T_{L_{n}}^{Gal(L_{n}/Q)}. Sinnott proved (see [S2, the last paragraph of p. 204]) that

2T_{L_{n}}^{Gal(L_{n}/Q)} ⊆ ℓ_{L_{n}}(Q^{×}) = −\frac{[L_{n}:Q]}{2}e_{1,L_{n}} log |Q^{×}|.

Let b be the order of β_{n} in X_{n}. Then b is also the order of γ_{n}, so \frac{1}{2}φ(qf)b is a multiple of \frac{1}{4}[L_{n} : Q] = \frac{1}{4}φ(qp^{n}f). We have proved that p^{n} divides 2b, so p^{n} | 2(#X_{n}). The lemma is proved.

4. Sinnott’s modules

For the following lemma we shall need Sinnott’s module U. Let m be the product of all primes dividing f. Let M be either K_{n} or L_{n}, n ≥ 0, and G = Gal(M/Q). We put s(X) = ∑_{τ∈X} τ ∈ Z[G] for any subgroup X of G and e_{M}^{+} = \frac{1+j}{2} , where j is the restriction of the complex conjugation in G. For any prime l | pm let I_{l,M} ⊆ G be the inertia group of l and λ_{l} be a Frobenius automorphism for l in G; λ_{l} is well-defined modulo I_{l,M}. We define

e_{l,M} = \frac{1}{#I_{l,M}} s(I_{l,M})  and  (l, M)^{∗} = λ_{l}^{−1}e_{l,M}.

For each positive divisor r | pm let I_{r,M} be the subgroup of G generated by all I_{l,M} with l | r. Then U_{M} can be defined as the Z[G]-module generated in Q[G] by the elements

s(I_{r,M}) ∏_{l| \frac{pm}{r}}(1 − (l, M)^{∗}),

where r varies over the positive divisors of pm (see Proposition 2.3 of [S2]). Similarly, we define V_{M} as the Z[G]-module generated in Q[G] by the elements

s(I_{r,M}) ∏_{l| \frac{m}{r}}(1 − (l, M)^{∗}),

where r runs through the set of the positive divisors of m (the module V_{M} is denoted by U_{m} in [S2]).

Lemma 4. For any n ≥ 0

[((1 − e_{1,L_{n}})T_{L_{n}})^{H} : cor_{L_{n}/K_{n}}(1 − e_{1,K_{n}})T_{K_{n}}] = [(e_{L_{n}}^{+}U_{L_{n}})^{H} : cor_{L_{n}/K_{n}} e_{K_{n}}^{+}U_{K_{n}}].

Proof. Corollary on page 207 of [S2] and the last but three line of the same page give

(1 − e_{1,M})T_{M} = ω′_{M}U_{M} = ω′_{M}e_{M}^{+}U_{0,M}

for any abelian field M. Here

U_{0,M} = U_{M} ∩ (1 − e_{1,M})U_{M}

and

ω′_{M} = ∑_{χ} L′(0, \overline{χ})e_{χ,M},

where the sum is taken over all even non-trivial characters χ of G = Gal(M/Q), e_{χ,M} = \frac{1}{[M:Q]} ∑_{σ∈G} χ(σ)σ^{−1} is the idempotent and L(s, χ) the L-series corresponding to χ. Since L′(0, \overline{χ}) ≠ 0 for all such characters, multiplying by ω′_{M} is a bijection on

(e_{M}^{+}U_{0,M}) ⊗ R = e_{M}^{+}(1 − e_{1,M})R[G].

Moreover

res_{L_{n}/K_{n}} ω′_{L_{n}} = ∑_{χ} L′(0, \overline{χ}) res_{L_{n}/K_{n}} e_{χ,L_{n}} = ω′_{K_{n}},

because res_{L_{n}/K_{n}} e_{χ,L_{n}} is e_{χ,K_{n}} if χ is also a character of Gal(K_{n}/Q) and 0 otherwise. Hence

ω′_{L_{n}} cor_{L_{n}/K_{n}} α = cor_{L_{n}/K_{n}}(α res_{L_{n}/K_{n}} ω′_{L_{n}}) = cor_{L_{n}/K_{n}}(ω′_{K_{n}}α)

for any α ∈ Q[Gal(K_{n}/Q)] and the following diagram commutes

e_{K_{n}}^{+}U_{0,K_{n}} ⟶^{ω′_{Kn}} (1 − e_{1,K_{n}})T_{K_{n}}
cor^{Ln/Kn} ↓  ↓^{cor}^{Ln/Kn}
e_{L_{n}}^{+}U_{0,L_{n}} ⟶^{ω′_{Ln}} (1 − e_{1,L_{n}})T_{L_{n}}

The rows of this diagram are isomorphisms, so we have obtained that

[((1 − e_{1,L_{n}})T_{L_{n}})^{H} : cor_{L_{n}/K_{n}}(1 − e_{1,K_{n}})T_{K_{n}}] = [(e_{L_{n}}^{+}U_{0,L_{n}})^{H} : cor_{L_{n}/K_{n}} e_{K_{n}}^{+}U_{0,K_{n}}].

We have a direct sum U_{M} = U_{0,M} ⊕ s(Gal(M/Q))Z (see [S2, p. 193]) for any abelian field M, so

cor_{L_{n}/K_{n}} e_{K_{n}}^{+}U_{K_{n}} = (cor_{L_{n}/K_{n}} e_{K_{n}}^{+}U_{0,K_{n}}) ⊕ s(Gal(L_{n}/Q))Z,
(e_{L_{n}}^{+}U_{L_{n}})^{H} = (e_{L_{n}}^{+}U_{0,L_{n}})^{H} ⊕ s(Gal(L_{n}/Q))Z.

Hence

[(e_{L_{n}}^{+}U_{0,L_{n}})^{H} : cor_{L_{n}/K_{n}} e_{K_{n}}^{+}U_{0,K_{n}}] = [(e_{L_{n}}^{+}U_{L_{n}})^{H} : cor_{L_{n}/K_{n}} e_{K_{n}}^{+}U_{K_{n}}]

and the lemma follows.

Lemma 5.

[(e_{L_{n}}^{+}U_{L_{n}})^{H} : cor_{L_{n}/K_{n}} e_{K_{n}}^{+}U_{K_{n}}] ≈ [(e_{L_{n}}^{+}V_{L_{n}})^{H} : cor_{L_{n}/K_{n}} e_{K_{n}}^{+}V_{K_{n}}].

Proof. By Lemma 1.1 of [S2] we have

[(e_{L_{n}}^{+}U_{L_{n}})^{H} : cor_{L_{n}/K_{n}} e_{K_{n}}^{+}U_{K_{n}}] = ((e_{L_{n}}^{+}U_{L_{n}})^{H} : (e_{L_{n}}^{+}V_{L_{n}})^{H})[(e_{L_{n}}^{+}V_{L_{n}})^{H} : cor_{L_{n}/K_{n}} e_{K_{n}}^{+}V_{K_{n}}]
· (cor_{L_{n}/K_{n}} e_{K_{n}}^{+}V_{K_{n}} : cor_{L_{n}/K_{n}} e_{K_{n}}^{+}U_{K_{n}}),

where ( : ) means the generalized index (see [S2], p. 187). The lemma will be proved if we show that

((e_{L_{n}}^{+}U_{L_{n}})^{H} : (e_{L_{n}}^{+}V_{L_{n}})^{H}) ≈ 1  and  (cor_{L_{n}/K_{n}} e_{K_{n}}^{+}V_{K_{n}} : cor_{L_{n}/K_{n}} e_{K_{n}}^{+}U_{K_{n}}) ≈ 1.

Since cor_{L_{n}/K_{n}} is an injective homomorphism,

(cor_{L_{n}/K_{n}} e_{K_{n}}^{+}V_{K_{n}} : cor_{L_{n}/K_{n}} e_{K_{n}}^{+}U_{K_{n}}) = (e_{K_{n}}^{+}V_{K_{n}} : e_{K_{n}}^{+}U_{K_{n}}),

which is independent of n for n sufficiently large (the p -part of the index is described in Proposition 6.2 of [S2], while for the non-p -part we can use the “p ≠ l” case of Sinnott’s proof of Theorem 6.1 of [S2]). Lemma 1.2(a) of [S2] gives

((e_{L_{n}}^{+}U_{L_{n}})^{H} : (e_{L_{n}}^{+}V_{L_{n}})^{H}) = ((1 − e_{p,L_{n}})(e_{L_{n}}^{+}U_{L_{n}})^{H} : (1 − e_{p,L_{n}})(e_{L_{n}}^{+}V_{L_{n}})^{H})
· ((e_{L_{n}}^{+}U_{L_{n}})^{HI_{p,Ln}} : (e_{L_{n}}^{+}V_{L_{n}})^{HI_{p,Ln}}).

We have

((1 − e_{p,L_{n}})e_{L_{n}}^{+}U_{L_{n}})^{H} = ((1 − e_{p,L_{n}})e_{L_{n}}^{+}V_{L_{n}})^{H}

(see [S2, last line of p. 210]), so

((1 − e_{p,L_{n}})(e_{L_{n}}^{+}U_{L_{n}})^{H} : (1 − e_{p,L_{n}})(e_{L_{n}}^{+}V_{L_{n}})^{H}) = \frac{[((1 − e_{p,L_{n}})e_{L_{n}}^{+}V_{L_{n}})^{H} : (1 − e_{p,L_{n}})(e_{L_{n}}^{+}V_{L_{n}})^{H}]}{[((1 − e_{p,L_{n}})e_{L_{n}}^{+}U_{L_{n}})^{H} : (1 − e_{p,L_{n}})(e_{L_{n}}^{+}U_{L_{n}})^{H}]}.

To prove that the numerator and the denominator of the last expression are bounded, we define mappings

F_{1} : ((1 − e_{p,L_{n}})e_{L_{n}}^{+}V_{L_{n}})^{H}/(1 − e_{p,L_{n}})(e_{L_{n}}^{+}V_{L_{n}})^{H} → H^{1}(H, (e_{L_{n}}^{+}V_{L_{n}})^{I_{p,Ln}})
F_{2} : ((1 − e_{p,L_{n}})e_{L_{n}}^{+}U_{L_{n}})^{H}/(1 − e_{p,L_{n}})(e_{L_{n}}^{+}U_{L_{n}})^{H} → H^{1}(H, (e_{L_{n}}^{+}U_{L_{n}})^{I_{p,Ln}})

as follows. For any u ∈ V_{L_{n}} (resp. u ∈ U_{L_{n}}) let F_{1} (resp. F_{2}) maps the class containing (1 − e_{p,L_{n}})e_{L_{n}}^{+}u to the cohomology class of the crossed homomorphism sending h ∈ H to (h − 1)e_{L_{n}}^{+}u. It is easy to check that F_{1} and F_{2} are well-defined injective homomorphisms. Proposition 6.2 (and Lemma 5.4 for the non-p-part) of [S2] gives that there is n_{0} such that for all n ≥ n_{0} we have

cor_{L_{n}/L_{n_{0}}} (e_{L_{n_{0}}}^{+} V_{L_{n_{0}}} )^{I_{p,Ln_{0}}} = (e_{L_{n}}^{+}V_{L_{n}})^{I_{p,Ln}} .

Directly from the definition we have res_{L_{n}/L_{n_{0}}} (e_{L_{n}}^{+}V_{L_{n}}) = e_{L_{n_{0}}}^{+} V_{L_{n_{0}}} , so cor_{L_{n}/L_{n_{0}}} s(I_{p,L_{n_{0}}} ) = s(I_{p,L_{n}}) implies

cor_{L_{n}/L_{n_{0}}} (s(I_{p,L_{n_{0}}} )e_{L_{n_{0}}}^{+} V_{L_{n_{0}}} ) = s(I_{p,L_{n}})e_{L_{n}}^{+}V_{L_{n}}.

Since L_{n} is a cyclotomic field, we have by Proposition 5.1 of [S1] that U_{L_{n}} = V_{L_{n}} · U_{p,L_{n}}, where U_{p,L_{n}} is the Z[Gal(L_{n}/Q)]-module generated in Q[Gal(L_{n}/Q)] by s(I_{p,L_{n}}) and 1 − (p, L_{n})^{∗}. Lemma 5.2 of [S1] gives

(e_{L_{n}}^{+}U_{L_{n}})^{I_{p,Ln}} = ((e_{L_{n}}^{+}V_{L_{n}})U_{p,L_{n}})^{I_{p,Ln}} = s(I_{p,L_{n}})e_{L_{n}}^{+}V_{L_{n}} + (1 − λ_{p,L_{n}}^{−1})(e_{L_{n}}^{+}V_{L_{n}})^{I_{p,Ln}}

and for any n ≥ n_{0} we have

(e_{L_{n}}^{+}U_{L_{n}})^{I_{p,Ln}} = cor_{L_{n}/L_{n_{0}}} (s(I_{p,L_{n_{0}}} )e_{L_{n_{0}}}^{+} V_{L_{n_{0}}} ) + (1 − λ_{p,L_{n}}^{−1}) cor_{L_{n}/L_{n_{0}}} (e_{L_{n_{0}}}^{+} V_{L_{n_{0}}} )^{I_{p,Ln_{0}}}
= cor_{L_{n}/L_{n_{0}}} (e_{L_{n_{0}}}^{+} U_{L_{n_{0}}} )^{I_{p,Ln_{0}}} .

Since cor_{L_{n}/L_{n_{0}}} is an injective homomorphism, we have obtained for each n ≥ n_{0}

H^{1}(H, (e_{L_{n}}^{+}V_{L_{n}})^{I_{p,Ln}}) ≅ H^{1}(H, (e_{L_{n_{0}}}^{+} V_{L_{n_{0}}} )^{I_{p,Ln_{0}}} )
H^{1}(H, (e_{L_{n}}^{+}U_{L_{n}})^{I_{p,Ln}}) ≅ H^{1}(H, (e_{L_{n_{0}}}^{+} U_{L_{n_{0}}} )^{I_{p,Ln_{0}}} )

and

((1 − e_{p,L_{n}})(e_{L_{n}}^{+}U_{L_{n}})^{H} : (1 − e_{p,L_{n}})(e_{L_{n}}^{+}V_{L_{n}})^{H}) ≈ 1.

The injectivity of cor_{L_{n}/L_{n_{0}}} also implies

(e_{L_{n}}^{+}U_{L_{n}})^{I_{p,Ln} H} = (cor_{L_{n}/L_{n_{0}}} (e_{L_{n_{0}}}^{+} U_{L_{n_{0}}} )^{I_{p,Ln_{0}}} )^{H} = cor_{L_{n}/L_{n_{0}}} (e_{L_{n_{0}}}^{+} U_{L_{n_{0}}} )^{I_{p,Ln_{0}} H}
(e_{L_{n}}^{+}V_{L_{n}})^{I_{p,Ln} H} = (cor_{L_{n}/L_{n_{0}}} (e_{L_{n_{0}}}^{+} V_{L_{n_{0}}} )^{I_{p,Ln_{0}}} )^{H} = cor_{L_{n}/L_{n_{0}}} (e_{L_{n_{0}}}^{+} V_{L_{n_{0}}} )^{I_{p,Ln_{0}} H}.

Therefore

((e_{L_{n}}^{+}U_{L_{n}})^{HI_{p,Ln}} : (e_{L_{n}}^{+}V_{L_{n}})^{HI_{p,Ln}}) = (cor_{L_{n}/L_{n_{0}}} (e_{L_{n_{0}}}^{+} U_{L_{n_{0}}} )^{I_{p,Ln_{0}} H} : cor_{L_{n}/L_{n_{0}}} (e_{L_{n_{0}}}^{+} V_{L_{n_{0}}} )^{I_{p,Ln_{0}} H})
= ((e_{L_{n_{0}}}^{+} U_{L_{n_{0}}} )^{I_{p,Ln_{0}} H} : (e_{L_{n_{0}}}^{+} V_{L_{n_{0}}} )^{I_{p,Ln_{0}} H})

and the lemma is proved.

5. Λ-modules

Lemma 6. There are μ, ν_{1}, ν_{2} ∈ Z such that μ ≥ 0 and
p^{μp^{n}+ν_{1}} | [(e_{L_{n}}^{+}V_{L_{n}})^{H} ⊗ Z_{p} : (cor_{L_{n}/K_{n}} e_{K_{n}}^{+}V_{K_{n}}) ⊗ Z_{p}] | p^{μp^{n}+ν_{2}}

for each n ≥ 0.

Proof. Up to now we have worked in the Galois group rings of L_{n} and K_{n}, but for the computation of the p-part of [(e_{L_{n}}^{+}V_{L_{n}})^{H} : cor_{L_{n}/K_{n}} e_{K_{n}}^{+}V_{K_{n}}] (which is precisely [(e_{L_{n}}^{+}V_{L_{n}})^{H} ⊗Z_{p} : (cor_{L_{n}/K_{n}} e_{K_{n}}^{+}V_{K_{n}})⊗Z_{p}]) we have to work in the inverse limits of these group rings (with respect to the restrictions). Let

R_{L} = \underleftarrow{lim} Z_{p}[Gal(L_{n}/Q)],  R_{K} = \underleftarrow{lim} Z_{p}[Gal(K_{n}/Q)],  Λ = \underleftarrow{lim} Z_{p}[Gal(L_{n}/Q(ζ_{qf})] ≅ Z_{p}[[T]].

The previous isomorphism can be described as follows: 1 + T is the image of γ = (…, γ_{n}, …), where γ_{n} ∈ Gal(L_{n}/Q(ζ_{qf})) is determined by γ_{n}(ζ_{qp^{n}}) = ζ_{qp^{n}}^{1+q}. We have

res_{L_{n}/L_{n−1}} ∘ cor_{L_{n}/K_{n}} = cor_{L_{n−1}/K_{n−1}} ∘ res_{K_{n}/K_{n−1}}

for each n ≥ 1, so there is a well-defined injective homomorphism cor_{L/K} : R_{K} → R_{L}. Similarly, we have a homomorphism res_{L/K} : R_{L} → R_{K}. It is easy to see that R_{L} and R_{K} are finitely generated Λ-modules without Λ-torsion (Λ acts on R_{K} via res_{L/K}). Let ω_{n} = γ^{p^{n}} − 1.

For each integer n ≥ 0 we have natural projections

π_{L}^{(n)} : R_{L} ⊗_{Z_{p}} Q_{p} → Q_{p}[Gal(L_{n}/Q)]  and  π_{K}^{(n)} : R_{K} ⊗_{Z_{p}} Q_{p} → Q_{p}[Gal(K_{n}/Q)]

with kernels ω_{n}(R_{L} ⊗_{Z_{p}} Q_{p}) and ω_{n}(R_{K} ⊗_{Z_{p}} Q_{p}), respectively.

Let A be a finitely generated Λ-module in R_{L} ⊗_{Z_{p}} Q_{p} (resp. R_{K} ⊗_{Z_{p}} Q_{p}). We say, after Sinnott, that A satisfies (∗) if, for any integer n ≥ 0, Λ-module ω_{n}A has finite index in A ∩ Ker π_{L}^{(n)} (resp. A ∩ Ker π_{K}^{(n)}).

Let A, B be finitely generated Λ-modules in R_{L} ⊗_{Z_{p}} Q_{p}. We say that A and B are comparable if there are positive integers a, b such that p^{a}A ⊆ B and p^{b}B ⊆ A.

For any positive divisor r of m we can consider

α_{r} = (…, s(I_{r,L_{n}}) ∏_{l| \frac{m}{r}}(1 − (l, L_{n})^{∗}), …) ∈ R_{L} ⊗_{Z_{p}} Q_{p}.

We define V_{L} to be the R_{L}-module generated in R_{L} ⊗_{Z_{p}} Q_{p} by the elements α_{r}, where r runs through the set of the positive divisors of m. Similarly we define R_{K}-module V_{K} ⊆ R_{K} ⊗_{Z_{p}} Q_{p}. Let e_{L}^{+} = (…, e_{L_{n}}^{+}, …) and e_{K}^{+} = (…, e_{K_{n}}^{+}, …). It is easy to see that

(e_{L_{n}}^{+}V_{L_{n}}) ⊗ Z_{p} = π_{L}^{(n)}(e_{L}^{+}V_{L})  and  (cor_{L_{n}/K_{n}} e_{K_{n}}^{+}V_{K_{n}}) ⊗ Z_{p} = π_{L}^{(n)}(cor_{L/K} e_{K}^{+}V_{K}).

Proposition 6.1 of [S2] gives that e_{L}^{+}V_{L} and e_{K}^{+}V_{K} satisfy (∗), hence cor_{L/K} e_{K}^{+}V_{K} also satisfies (∗). Lemma 5.4 of [S2] implies that

(#H)(e_{L}^{+}V_{L})^{H} ⊆ e_{L}^{+}s(H)V_{L} ⊆ cor_{L/K} e_{K}^{+}V_{K} ⊆ (e_{L}^{+}V_{L})^{H},

so (e_{L}^{+}V_{L})^{H} and cor_{L/K} e_{K}^{+}V_{K} are comparable. By Lemma 6.2 of [S2], (e_{L}^{+}V_{L})^{H} satisfies (∗) and there are μ, ν ∈ Z such that μ ≥ 0 and, for all sufficiently large n,

(π_{L}^{(n)}(e_{L}^{+}V_{L})^{H} : (cor_{L_{n}/K_{n}} e_{K_{n}}^{+}V_{K_{n}}) ⊗ Z_{p}) = p^{μp^{n}+ν}.

There is a Λ-module A′ ⊆ R_{L} ⊗_{Z_{p}} Q_{p} such that e_{L}^{+}V_{L} ⊆ A′ and, for all sufficiently large n, the kernel of π_{L}^{(n)} in e_{L}^{+}V_{L} is equal to ω_{n}A′ (see the proof of Lemma 6.2 of [S2] for A = e_{L}^{+}V_{L}). Therefore we have an exact sequence

0 ⟶ ω_{n}A′ ⟶^{⊆} e_{L}^{+}V_{L} ⟶^{π_{L}^{(n)}} (e_{L_{n}}^{+}V_{L_{n}}) ⊗ Z_{p} ⟶ 0.

Taking H-invariants gives the exact sequence

0 ⟶ (ω_{n}A′)^{H} ⟶^{⊆} (e_{L}^{+}V_{L})^{H} ⟶^{π_{L}^{(n)}} (e_{L_{n}}^{+}V_{L_{n}})^{H} ⊗ Z_{p} ⟶ Y_{n} ⟶ 0,

where Y_{n} is the kernel of the homomorphism H^{1}(H, ω_{n}A′) → H^{1}(H, e_{L}^{+}V_{L}) induced by the inclusion ω_{n}A′ ⊆ e_{L}^{+}V_{L}. Since A′ ⊆ R_{L} ⊗_{Z_{p}} Q_{p}, the action of ω_{n} is an injective mapping on A′, so H^{1}(H, A′) ≅ H^{1}(H, ω_{n}A′). For any Λ-module M, let M_{ω_{n}} be the submodule of M which is killed by the action of ω_{n}. As e_{L}^{+}V_{L} ⊆ A′, we have Y_{n} ⊆ Ker(H^{1}(H, ω_{n}A′) → H^{1}(H, A′)) and the latter group is isomorphic to H^{1}(H, A′)_{ω_{n}} (here we have used the fact that A′_{ω_{n}} = 0, as A′ ⊆ R_{L} ⊗_{Z_{p}} Q_{p}). Because A′ is a finitely generated Λ-module and H is finite, H^{1}(H, A′) is also a finitely generated Λ-module. Moreover, H^{1}(H, A′) is (#H)-torsional, so, due to the well-known theorem on the structure of finitely generated Λ-modules, there are finite Λ-modules A, B and Λ-module M ≅ ⨁_{i=1}^{s}(Λ/p^{t_{i}}Λ), where s, t_{1}, …, t_{s} are positive integers, such that

0 ⟶ A ⟶ H^{1}(H, A′) ⟶ M ⟶ B ⟶ 0

is an exact sequence of Λ-modules. Taking the kernels of ω_{n}-action gives the following exact sequence

0 ⟶ A_{ω_{n}} ⟶ H^{1}(H, A′)_{ω_{n}} ⟶ M_{ω_{n}}

It follows easily from the structure of M that M_{ω_{n}} = 0, so

H^{1}(H, A′)_{ω_{n}} ≅ A_{ω_{n}}.

Hence

[(e_{L_{n}}^{+}V_{L_{n}})^{H} ⊗ Z_{p} : π_{L}^{(n)}((e_{L}^{+}V_{L})^{H})] = #Y_{n} | #(H^{1}(H, A′)_{ω_{n}}) | #A

is bounded. The lemma is proved.

6. Back to units

Proposition. There are μ, ν_{1}, ν_{2} ∈ Z such that μ ≥ 0 and
p^{μp^{n}+ν_{1}} | [\overline{C}_{K_{n}} : C_{K_{n}}]_{p} | p^{μp^{n}+ν_{2}}

for each n ≥ 0, where [\overline{C}_{K_{n}} : C_{K_{n}}]_{p} means the p-part of the index [\overline{C}_{K_{n}} : C_{K_{n}}].

Proof. This follows from Lemmas 1, 3, 4, 5 and 6.

Lemma 7. Let M be a real abelian field and C_{1,M} = \{ε ∈ E_{M}; ε^{2} ∈ C_{M}\}. Then [C_{1,M} : \overline{C}_{M} ∩ C_{1,M}] is 1 or 2 and [C_{1,M} : C_{M}] divides 2[\overline{C}_{M} : C_{M}].

Proof. Let m be the conductor of M. Since M is real, we have M ⊆ Q(ζ_{m})^{+} and C_{M} ⊆ C_{Q(ζ_{m})^{+}}. Any ε ∈ C_{Q(ζ_{m})^{+}} is (up to a sign) a combination of elements of the form N(1 − ζ_{n}^{a}), such that 1 < n | m and (a, n) = 1, where N means the norm from Q(ζ_{n}) to Q(ζ_{m})^{+} ∩ Q(ζ_{n}) = Q(ζ_{n})^{+} (see [L]). But

N(1 − ζ_{n}^{a}) = (1 − ζ_{n}^{a})(1 − ζ_{n}^{−a}) = −ζ_{n}^{−a}(1 − ζ_{n}^{a})^{2}.

Therefore C_{M} ⊆ W_{Q(ζ_{m})}C_{Q(ζ_{m})}^{2}.

Let f : C_{1,M} → W_{Q(ζ_{m})}/W_{Q(ζ_{m})}^{2} be defined as follows. For any ε ∈ C_{1,M} there are ρ ∈ W_{Q(ζ_{m})} and η ∈ C_{Q(ζ_{m})} such that ε^{2} = ρη^{2}; let f(ε) = ρW_{Q(ζ_{m})}^{2}. It is easy to see that f is a well-defined homomorphism and that [C_{1,M} : Ker f] is 1 or 2. If ε ∈ Ker f then ε^{2} = ρ^{2}η^{2} for some ρ ∈ W_{Q(ζ_{m})} and η ∈ C_{Q(ζ_{m})}, so ε = ±ρη ∈ C_{Q(ζ_{m})} ∩ E_{M} = \overline{C}_{M}. Therefore [C_{1,M} : \overline{C}_{M} ∩ C_{1,M}] is 1 or 2. It is easy to see that

[\overline{C}_{M} : C_{M}] = \frac{[\overline{C}_{M} : \overline{C}_{M} ∩ C_{1,M}][C_{1,M} : C_{M}]}{[C_{1,M} : \overline{C}_{M} ∩ C_{1,M}]}

and the lemma follows.

7. The proof of the Theorem

At first, let us show that [\overline{C}_{K_{n}} : C_{K_{n}}] can be divisible only by finitely many primes, namely by primes dividing #H. Indeed,

N_{L_{n}/K_{n}}(C_{L_{n}}) ⊆ C_{K_{n}} ⊆ C_{L_{n}}^{H} = \overline{C}_{K_{n}}

and

C_{L_{n}}^{H}/ N_{L_{n}/K_{n}}(C_{L_{n}}) = Ĥ^{0}(H, C_{L_{n}})

is finite and (#H)-torsional. Hence, due to Sinnott’s theorem (see Introduction), it is enough to prove the Theorem locally. We shall do it by the following two lemmas.

For any prime l, the symbol [ : ]_{l} means the l-part of the index [ : ].

Lemma 8. There are integers c_{1} and c_{2} such that

p^{c_{1}}(h_{K_{n}}^{+})_{p} | [E_{K_{n}} : \overline{C}_{K_{n}}]_{p} | p^{c_{2}}(h_{K_{n}}^{+})_{p},

where (h_{K_{n}}^{+})_{p} means the p-part of h_{K_{n}}^{+}.

Proof. The theorems of Iwasawa and of Ferrero–Washington (e.g., see Theorem 13.13 and Theorem 7.15 of [W2]) give that there exist integers λ ≥ 0 and ν such that

(h_{K_{n}}^{+})_{p} = p^{λn+ν}

for all sufficiently large n. We need to distinguish two cases.

At first, let us suppose that p ≠ 2 or k is imaginary. Sinnott’s theorem gives that there is an integer ν′ such that

[E_{K_{n}} : C_{K_{n}}]_{p} = p^{λn+ν′}

for all sufficiently large n. Then the Proposition implies μp^{n} + ν_{1} ≤ λn + ν′ for all sufficiently large n, hence μ = 0. The lemma (in this case) follows from Sinnott’s theorem and the Proposition.

Now, let us suppose that p = 2 and k is real. By Sinnott’s theorem there are integers ν′ and ν′′ such that

[E_{K_{n}} : C_{K_{n}}]_{2} = 2^{[K_{n}:Q]+λn+ν′}  and  [E_{K_{n}} : C_{1,K_{n}}]_{2} = 2^{λn+ν′′}

for all sufficiently large n. Comparing with the Proposition gives

μ2^{n} + ν_{1} ≤ [k : Q] · 2^{n} + λn + ν′

for all sufficiently large n, so μ ≤ [k : Q]. On the other hand, Lemma 7 and the Proposition imply

[k : Q] · 2^{n} + ν′ − ν′′ ≤ 1 + μ2^{n} + ν_{2},

for all sufficiently large n, hence [k : Q] ≤ μ. Therefore μ = [k : Q] and Sinnott’s theorem and the Proposition give the lemma.

Lemma 9. Let l be a prime different from p. Then [E_{K_{n}} : \overline{C}_{K_{n}}]_{l} is independent of n, for n sufficiently large.

Proof. By Washington’s theorem, the l-part (h_{K_{n}}^{+})_{l} of the class number h_{K_{n}}^{+} is independent of n, for n sufficiently large (see [W1]).

At first, let us suppose that p ≠ 2 or k is imaginary. Since C_{K_{n}} ⊆ C_{K_{n+1}} and \overline{C}_{K_{n}} ⊆ \overline{C}_{K_{n+1}}, we can consider the homomorphism of finite groups \overline{C}_{K_{n}}/C_{K_{n}} → \overline{C}_{K_{n+1}}/C_{K_{n+1}} induced by the inclusion. The restriction of this homomorphism to the l-Sylow subgroups is injective, since ε ∈ \overline{C}_{K_{n}} ∩ C_{K_{n+1}} implies ε^{p} = N_{K_{n+1}/K_{n}}(ε) ∈ C_{K_{n}}. Therefore

[\overline{C}_{K_{n}} : C_{K_{n}}]_{l} | [\overline{C}_{K_{n+1}} : C_{K_{n+1}}]_{l}.

Of course, [\overline{C}_{K_{n}} : C_{K_{n}}]_{l} divides [E_{K_{n}} : C_{K_{n}}]_{l}, which is bounded by a constant by Sinnott’s and Washington’s theorems. Hence [\overline{C}_{K_{n}} : C_{K_{n}}]_{l} is independent of n, for n sufficiently large, and the lemma follows in this case.

Now, let us suppose that p = 2 and k is real. Similarly as in the previous part we can prove that

[C_{1,K_{n}} : \overline{C}_{K_{n}} ∩ C_{1,K_{n}}]_{2} | [C_{1,K_{n+1}} : \overline{C}_{K_{n+1}} ∩ C_{K_{n+1}}]_{2}
[\overline{C}_{K_{n}} : \overline{C}_{K_{n}} ∩ C_{1,K_{n}}]_{2} | [\overline{C}_{K_{n+1}} : \overline{C}_{K_{n+1}} ∩ C_{K_{n+1}}]_{2}.

Since [C_{1,K_{n}} : \overline{C}_{K_{n}} ∩ C_{1,K_{n}}]_{2} is 1 or 2 by Lemma 7, [C_{1,K_{n}} : \overline{C}_{K_{n}} ∩ C_{1,K_{n}}]_{2} is independent of n, for n sufficiently large. Hence, by Sinnott’s theorem, we have the same for [E_{K_{n}} : \overline{C}_{K_{n}} ∩ C_{1,K_{n}}]_{2}. Therefore [\overline{C}_{K_{n}} : \overline{C}_{K_{n}} ∩ C_{K_{n}}]_{2} is independent of n, for n sufficiently large. The lemma follows.

References

[L] G. Lettl, A note on Thaine’s circular units, J. Number Theory 35 (1990), 224–226.

[S1] W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field, Ann. of Math. 108 (1978), 107–134.

[S2] W. Sinnott, On the Stickelberger ideal and the circular units of an abelian field, Inv. Math. 62 (1980), 181–234.

[W1] L. C. Washington, The non-p-part of the class number in a cyclotomic Z_{p}-extension, Inv. Math. 49 (1979), 87–97.

[W2] L. C. Washington, “Introduction to cyclotomic fields”, Springer–Verlag, New York, 1982.

Radan Kučera  Jan Nekovář^{1}
Department of Mathematics  Department of Mathematics
Faculty of Science  University of California
Masaryk University  Berkeley
Janáčkovo nám. 2a  CA 94720
662 95 Brno  U.S.A.

Czech Republic

1 Address after August 1, 1993: Mathematical Institute, Charles University, Sokolovská 83, 186 00 Praha 8, Czech Republic