Cyclotomic units, Gauss sums, and Iwasawa invariants of certain real abelian fields

Hiroki Sumida^{∗}

Abstract Let p be an odd prime number and k a finite extension of Q. It is conjectured that Iwasawa invariants λ_{p}(k) and μ_{p}(k) vanish for all p and totally real fields k. Using cyclotomic units and Gauss sums, we compute Iwasawa invariants, especially ν_{p}(k) of certain real abelian fields.

Mathematics Subject Classification (2000): 11R23

1 Introduction

Let p be a prime number and k a finite extension of the rational number field Q. Denote by k_{∞} the cyclotomic Z_{p}-extension of k and by k_{n} the subfield of k_{∞} with [k_{n} : k] = p^{n}. Let A_{n}(k) be the p-Sylow subgroup of the ideal class group of k_{n} and p^{e_{n}} the order of A_{n}(k). Iwasawa proved that there are three invariants λ_{p}(k), μ_{p}(k) and ν_{p}(k) such that e_{n} = λ_{p}(k)n + μ_{p}(k)p^{n} + ν_{p}(k) for all sufficiently large n (cf. [12]).

It is conjectured that μ-invariant vanishes for all p and k. In fact, Ferrero and Washington proved that μ_{p}(k) = 0 for all p and abelian number fields k in [2]. On the other hand, there are many number fields k with λ_{p}(k) > 0 (cf. [6, p. 266]). However we cannot find any totally real number field with λ_{p}(k) > 0 at present (cf. [3, 4, 8, 13, 15]), and it is conjectured that λ_{p}(k) = 0 for all p and totally real number fields k. This is a version of generalized Greenberg’s conjecture on ideal class groups of multiple Z_{p}^{d}-extensions for total real number fields (cf. [17]). Particularly Vandiver’s conjecture claims that λ_{p}(k) = μ_{p}(k) = ν_{p}(k) = 0 for any maximal real subfields k of pcyclotomic fields. This conjecture is verified for p < 4000000 in [1].

In [9], having made use of cyclotomic units, we gave a criterion for verification of λ_{p}(k) = μ_{p}(k) = 0. We applied it to p = 3, 5, 7 and some quadratic fields k, and verified λ_{p}(k) = μ_{p}(k) = 0 for them. Moreover we gave an upper bound for ν_{p}(k). In this paper, by making use of both cyclotomic units and Gauss sums, we give a criterion for the determination of the exact value of ν_{p}(k) under some conditions. These conditions make the situation simple, but the importance of the problem seems to remain. For a generalization of our criterion, we need to investigate the structure of A_{n}(k) in general case. It will be possible but much more complicated to give such an explicit criterion or an algorithm.

∗Partially supported by the Grants-in-Aid for Encouragement of Young Scientists (No.11740020), The Ministry of Education, Science, Sports and Culture of Japan.

We hope that our explicit criterion will give some concrete information on the ideal class group of real abelian fields, especially the maximal subfields of p-cyclotomic fields.

2 Theorems

Let p be an odd prime number, χ a nontrivial even primitive Dirichlet character and k the abelian extension of Q associated to ⟨χ⟩. We fix a n-th root of unity ζ_{n} so that ζ_{nm}^{m} = ζ_{n} for all n, m ≥ 1. Put K = k(ζ_{p}), Δ = Gal(K/Q), Γ = Gal(K_{∞}/K) and G_{∞} = Gal(K_{∞}/Q). Let f_{n} be the conductor of K_{n} and f′ = f_{0}/p. For simplicity, we assume the following condition:

(C1) χ^{p−1} is the trivial character.

We fix an inclusion map \overline{Q} ↪ \overline{Q}_{p}, where \overline{F} is the algebraic closure of a field F. By (C1), the values of χ are contained in Z_{p}. Put χ^{∗} = χ^{−1}ω, where ω is the Teichmüller character Z/pZ → Z_{p}. Following [9] (cf. [8]), we assume:

(C2) χ(p) ≠ 1 and χ^{∗}(p) ≠ 1.

Let L_{p}(s, χ) be the p-adic L-function associated to χ which is constructed in [14]. By [11, §6], there uniquely exists G_{χ}(T) ∈ Z_{p}[[T]] satisfying G_{χ}((1 + f_{0})^{1−s} − 1) = L_{p}(s, χ) for all s ∈ Z_{p}. In [2], it is shown that p does not divide G_{χ}(T). Therefore, by the p-adic Weierstrass preparation theorem, we can uniquely write G_{χ}(T) = g_{χ}(T)u_{χ}(T), where g_{χ}(T) is a distinguished polynomial of Z_{p}[T] and u_{χ}(T) is an invertible element of Z_{p}[[T]]. If g_{χ}(T) = 1, then the χ-part of A_{n}(K) vanishes for all n ≥ 0 (cf. [9, pp. 213-214]). As a next step, we consider the following case:

(C3) g_{χ}(T) = T − α for some α ∈ pZ_{p}.

Put α^{∗} = \frac{f_{0} − α}{1 + α} ∈ pZ_{p}, e = v_{p}(α) and e^{∗} = v_{p}(α^{∗}), where v_{p} is the p-adic

valuation with v_{p}(p) = 1.

In [9], we define the following polynomials X_{n}(T) ∈ Z_{p}[T] and Y_{n}(T) ∈ Z[T]:

X_{n}(T) = \frac{ω_{n}(T) − ω_{n}(α)}{T − α} and Y_{n}(T) ≡ X_{n}(T)mod p^{n+e},

where ω_{n}(T) = (1 + T)^{p^{n}} − 1. Similarly we define X_{n}^{∗}(T) ∈ Z_{p}[T] and Y_{n}^{∗}(T) ∈ Z[T]:

X_{n}^{∗}(T) = \frac{ω_{n}(T) − ω_{n}(α^{∗})}{T − α^{∗}} and Y_{n}^{∗}(T) ≡ X_{n}^{∗}(T)mod p^{n+e^{∗}}.

For a character ψ of Δ, put e_{ψ} = \frac{1}{♯Δ} ∑_{δ∈Δ} ψ(δ)δ^{−1} ∈ Z_{p}[Δ] the idempotent

of the group ring Q_{p}[Δ]. We can regard χ as an even character of Δ. Let e_{χ,n} (resp. e_{χ^{∗},n}) be an element of Z[Δ] such that e_{χ,n} ≡ e_{χ}mod p^{n+e} (resp. e_{χ^{∗},n} ≡ e_{χ^{∗}}mod p^{n+e^{∗}}) and the sum of all the coefficients is zero.

We fix the topological generator γ_{0} of Γ such that ζ_{f_{n}}^{γ_{0}} = ζ_{f_{n}}^{1+f_{0}} for all n ≥ 0. We identify, as usual, the complete group ring Z_{p}[[Γ]] with the power series ring Λ = Z_{p}[[T]] by γ_{0} = 1 + T. Thus, for a Z_{p}[[G_{∞}]]-module M, M(ψ) = e_{ψ}M is regarded as a Λ-module, where we identify Δ with Gal(K_{∞}/Q_{∞}). Let X_{∞} = X_{∞}(K) = lim_{←} A_{n}(K), where the inverse limit is taken with respect to relative norm maps. Then X_{∞}(ψ) is regarded as a Λmodule in a natural way. We can define Iwasawa invariants λ_{p}(ψ), μ_{p}(ψ) and ν_{p}(ψ) for X_{∞}(ψ). Since [K : k] divides p−1, N_{K_{n}/k_{n}} : A_{n}(K)(χ) → A_{n}(k)(χ) is an isomorphism. In this setting, since μ_{p}(K) = 0, Greenberg’s conjecture claims that λ_{p}(χ) = 0 for all even characters χ. We put

♯A_{n}(K)(χ) = p^{e_{n}(χ)}.

Then, by (C3) (cf. [9, p. 213]), we have

0 ≤ e_{n}(χ) ≤ n + e.

We define a cyclotomic unit c_{n} of K_{n} by

c_{n} = (N_{F_{n}/K_{n}} (1 − ζ_{f_{n}}))^{(r−1)e_{χ,n}} ,

where F_{n} = Q(ζ_{f_{n}}) and r = N𝔭 the cardinality of the residue class field of a prime ideal 𝔭 of K over p.

We can directly obtain the following theorem by the argument of [9, pp. 211-213].

Theorem 1. Assume (C1)-(C3). We have λ_{p}(χ) = 0 if and only if c_{n}^{Y_{n}(T)} ∉ (K_{n}^{×})^{p^{n+e}} holds for some n ≥ 0. Further, for n ≥ 0, e_{n}(χ) satisfies

c_{n}^{Y_{n}(T)} ∈ (K_{n}^{×})^{p^{en(χ)}} and c_{n}^{Y_{n}(T)} ∉ (K_{n}^{×})^{p^{en(χ)+1}}.

Proof. Let K_{n,𝔭_{n}} be the completion of K_{n} at 𝔭_{n}|p and U_{𝔭_{n}} the group of principal units of K_{n,𝔭_{n}}. Put U_{n} = ∏_{𝔭_{n}|p}U_{𝔭_{n}}. Let d_{n} be the diagonal map:

d_{n} : K_{n}^{×} → ∏_{𝔭_{n}|p}K_{n,𝔭_{n}}^{×} (x ↦ (x, x, ..., x)).

We denote by E_{n} the group of units of K_{n} and by C_{n} the group of cyclotomic units of K_{n} in the sense of [5]. Put

E_{n} = \overline{U_{n} ∩ d_{n}(E_{n})} and C_{n} = \overline{U_{n} ∩ d_{n}(C_{n})},

where \overline{A} is the topological closure in U_{n}. By (C2) and [5, Theorem 2], we have the following Λ-isomorphisms:

U_{n}(χ) ≃  Λ/(ω_{n}(T))
⋃ |  ⋃ |
E_{n}(χ) ≃ (T − α, p^{n+e−x})/(ω_{n}(T))
⋃ |  ⋃ |
C_{n}(χ) ≃ (T − α, p^{n+e})/(ω_{n}(T))

when c_{n}^{Y_{n}(T)} ∈ (K_{n}^{×})^{p^{x}} and c_{n}^{Y_{n}(T)} ∉ (K_{n}^{×})^{p^{x+1}}. By the Iwasawa main conjecture proved in [16], we have

p^{e_{n}(χ)} = ♯A_{n}(K)(χ) = ♯(E_{n}(χ)/C_{n}(χ)) = p^{x}.

In particular, λ_{p}(χ) > 0 if and only if e_{n}(χ) = n + e for all n ≥ 0 (cf. [9, pp. 213-214]). Hence we obtain the theorem.

Put N = n + e − 1. Let Õ_{N} be the ring of integers of F_{N}, 𝔏̃ a prime ideal of F_{N} with 𝔏̃ ∤ f_{N} and l = 𝔏̃ ∩ Q the prime number contained in 𝔏̃. Denote by χ𝔏̃ the character of (Õ_{N}/𝔏̃)^{×} with values in μ_{f_{N}} ⊂ F_{N}, defined by the following congruence:

χ𝔏̃(y) = χ𝔏̃(ymod 𝔏̃) ≡ y^{(N𝔏̃−1)/f_{N}} mod 𝔏̃, y ∈ Õ_{N} with 𝔏̃ ∤ y.

We define a Gauss sum g′_{N}(𝔏̃) ∈ F_{N}(ζ_{l}) by

g′_{N}(𝔏̃) = − ∑_{y∈(Õ_{N} /𝔏̃)^{×}} χ𝔏̃(y)ζ_{l}^{T r(y)},

where T r denotes the trace map Õ_{N}/𝔏̃ → Z/lZ (cf. [7, 18]). Let 𝔄 be an ideal of K_{N} with 𝔄Õ_{N} = ∏_{i=1}^{r} 𝔏̃_{i}^{e_{i}}, where 𝔏̃_{i} is an prime ideal of F_{N} lying above l_{i}. We define g_{N}(𝔄) ∈ F_{N}(ζ_{l_{1}l_{2}..l_{r}}) by

g_{N}(𝔄) = (∏_{i=1}^{r} g′_{N}(𝔏̃_{i})^{e_{i}})^{e_{χ^{∗},N} f′} .

By [18, Proposition 3.1], we have g_{N}(𝔄) ∈ K_{N}.

We investigate on the cyclotomic units of Theorem 1 by using Gauss sums in the following way.

Theorem 2. Assume (C1)-(C3). Let 𝔏 be a prime ideal of K_{N} with 𝔏 ∤ f_{0} satisfying

g_{N}(𝔏)^{Y_{N}^{∗} (T)} ∉ (K_{N}^{×})^{p}.

Then, for n ≥ 0 and 0 ≤ x ≤ n + e,

c_{n}^{Y_{n}(T)} ∈ (K_{n}^{×})^{p^{x}}

if and only if

c_{n}^{Y_{n}(T)}mod 𝔏 ∈ ((O_{N}/𝔏)^{×})^{p^{x}}.

Proof. Put

θ_{N} = ∑_{1≤m<f_{N} , (m,f_{N})=1} (1 − \frac{m}{f_{N}} ) σ_{m} ∈ Q[Gal(F_{N}/Q)].

By [18, Proposition 3.1], since θ_{N}e_{χ^{∗},N}f′ ∈ Z[Gal(F_{N}/Q)], we have

(g_{N}(𝔏)) = 𝔏^{f′e_{χ^{∗},N} θ_{N}}

and

e_{χ^{∗},N}θ_{N}Y_{N}^{∗}(T)

= (e_{χ^{∗}} + p^{N+e^{∗}}Z)θ_{N}Y_{N}^{∗}(T)
≡ (T − α^{∗})u(T)(X_{N}^{∗}(T) + p^{N+e^{∗}}W (T))e_{χ^{∗}} + p^{N+e^{∗}}Zθ_{N}Y_{N}^{∗}(T)mod (ω_{N}(T))
≡ p^{N+e^{∗}}(u′ + W (T)(T − α^{∗}))u(T)e_{χ^{∗}} + p^{N+e^{∗}}Zθ_{N}Y_{N}^{∗}(T)mod (ω_{N}(T))

for some Z ∈ Z_{p}[Δ], W (T) ∈ Z_{p}[T], u(T) ∈ Z_{p}[[T]]^{×} and u′ ∈ Z_{p}^{×}. Let L_{N} be

the maximal unramified abelian p-extension of K_{N}. We denote by (\frac{L_{N} /K_{N}}{𝔏}) the Artin symbol. Put

σ_{𝔏} = (\frac{L_{N}/K_{N}}{𝔏} )^{e_{χ^{∗}}} .

Then we have σ_{𝔏}^{T} = σ_{𝔏}^{α^{∗}} by the main conjecture and σ_{𝔏}^{δ} = σ_{𝔏}^{χ^{∗}(δ)} for δ ∈ Δ. Therefore, by the class field theory, g_{N}(𝔏)^{Y_{N}^{∗} (T)} ∉ (K_{N}^{×})^{p} if and only if σ_{𝔏} generates Gal(L_{N}/K_{N})(χ^{∗}) ≃ Λ/(T − α^{∗}, ω_{N}(T)) ≃ Z/p^{N+e^{∗}}Z. Let L_{N}(χ^{∗}) be the subfield of L_{N} corresponding to ∏_{ψ≠χ^{∗}} Gal(L_{N}/K_{N})(ψ). Further since

c_{n}^{Y_{n}(T)} ∈ U_{n}(χ)^{p^{N+1}},

K_{N} ( \sqrt[p^{N+1}]{c_{n}^{Y_{n}(T)}} ) ⊆ L_{N}(χ^{∗}).

Hence we have

[K_{N} ( \sqrt[p^{N+1}]{c_{n}^{Y_{n}(T)}} ) : K_{N}] = [K_{N} ( \sqrt[p^{N+1}]{c_{n}^{Y_{n}(T)}} )_{𝔏′′} : K_{N,𝔏}] ,

where 𝔏′′ is the prime ideal lying above 𝔏. Since c_{n}^{Y_{n}(T)} ∈ (K_{N}^{×})^{p^{x}} if and only if c_{n}^{Y_{n}(T)} ∈ (K_{n}^{×})^{p^{x}}, this implies the theorem.

We easily obtain the following corollary from Theorem 2. Corollary 1. Assume (C1)-(C3). Let 𝔏 be an ideal of K_{N}, l = 𝔏 ∩ Q and 𝔏′ an ideals of F_{N}(ζ_{l}) with 𝔏 ∤ f_{0} and 𝔏′ ∤ f_{0}l satisfying

g_{N}(𝔏)^{Y_{N}^{∗} (T)}mod 𝔏′ ∉ ((O′_{N}/𝔏′)^{×})^{p},

where O′_{N} is the ring of integers of F_{N}(ζ_{l}). Then, for n ≥ 0 and 0 ≤ x ≤ n+e,

c_{n}^{Y_{n}(T)} ∈ (K_{n}^{×})^{p^{x}}

if and only if

c_{n}^{Y_{n}(T)}mod 𝔏 ∈ ((O_{N}/𝔏)^{×})^{p^{x}}.

Whether g_{N}(𝔏)^{Y_{N}^{∗} (T)}mod 𝔏′ ∉ ((O′_{N}/𝔏′)^{×})^{p} holds or not is depends only on l and l′ = 𝔏′ ∩ Q but not on the choice of prime ideals 𝔏, 𝔏′. This fact holds because K_{N} (\sqrt[p]{g_{N}(𝔏)^{Y_{N}^{∗} (T)}}) is a Galois extension of Q. (Consider

(g_{N}(𝔏)^{Y_{N}^{∗} (T)})^{τ} for τ ∈ Gal(K_{N}/Q) (cf. the proof of Theorem 2)). Hence we can denote the above condition by (G_{n,l,l′}).

Combining Theorem 1 and Corollary 1, we can compute Iwasawa invariants (if Greenberg’s conjecture is valid for p and k). The computation includes only integer calculations. For another computations, see [3] and [13].

3 Examples

Let the situation be the same as in the previous section. We assume that (C1)-(C3) are satisfied for p and χ. In this section, we explain how to check the condition (G_{n,l,l′}) on Gauss sums using computer calculation. Concerning computation of α, α^{∗} and c_{n}^{Y_{n}(T)}, see [9, pp. 206-208].

Let l (resp. l′) be a prime number with l ≡ 1mod f_{N} (resp. l′ ≡ 1mod f_{N}l) and g (resp. g′) a primitive root in Z/lZ (resp. Z/l′Z). Let s (resp. t) be an integer satisfying s ≡ g′^{(l′−1)/f_{N}} mod l′ (resp. t ≡ g′^{(l′−1)/l}mod l′). Then for some ideal 𝔏̃|l of F_{N} and 𝔏′|l′ of F_{N}(ζ_{l}), s ≡ χ𝔏̃(g)mod 𝔏′ and t ≡ ζ_{l}mod 𝔏′.

We write

Y_{N}^{∗}(T) = ∑_{j=0}^{p^{N} −1} a_{j}(1 + T)^{j} = ∑_{j=0}^{p^{N} −1} a_{j}γ_{0}^{j}, a_{j} ∈ Z.

Then, from the definition of Y_{N}^{∗}(T), we have a_{j} ≡ (1 + α^{∗})^{p^{N} −1−j}mod p^{N+e^{∗}}. Denote by σ the canonical isomorphism

σ : (Z/f_{N}lZ)^{×} → Gal(Q(F_{N}(ζ_{l})/Q), m ↦ σ_{m}.

From now we identify Gal(K_{N}/K_{0}) (resp. Gal(F_{N}/Q_{N})) with Gal(F_{N}(ζ_{l})/F_{0}(ζ_{l})) (resp. Gal(F_{N}(ζ_{l})/Q_{N}(ζ_{l}))). Let 𝔄_{N} be the subgroup of (Z/f_{N}lZ)^{×} which is corresponding to Gal(F_{N}(ζ_{l})/Q_{N}(ζ_{l})) by the isomorphism σ. We write

( ∑_{τ∈Gal(F_{N} /K_{N})} τ) e_{χ^{∗},N} = ∑_{m∈𝔄_{N}} b_{m}σ_{m}, b_{m} ∈ Z.

Then we have

g_{N}(𝔏)^{Y_{N}^{∗} (T)} = ( ∏_{j=1}^{p^{N} −1} ( ∏_{m∈𝔄_{N}} (− ∑_{y∈(Õ_{N} /𝔏̃)^{×}} χ𝔏̃(y)ζ_{l}^{y})^{b_{m}σ_{m}} )^{a_{j}γ_{0}^{j}} )^{f′}

≡ (  ∏_{1≤j<p^{N} , m∈𝔄_{N}} (− ∑_{i=1}^{l−1} s^{m(1+f_{0})^{j}}t^{g^{i}})^{b_{m}a_{j}} )^{f′} mod 𝔏′.

By this formula, we can check the condition (G_{n,l,l′}) by computer calculation.

We explain how to use the criterion in Corollary 1. Let p = 3 and χ an even primitive quadratic Dirichlet character whose conductor is f. Then k = Q(\sqrt{f}) is the real quadratic field associated to ⟨χ⟩. By [10], we have A_{n}(k) = A_{n}(k)(χ) ≅ A_{n}(K)(χ). By the argument in [9, p. 208], we can show whether c_{n}^{Y_{n}(T)}mod 𝔏 ∈ ((O_{n}/𝔏)^{×})^{p^{x}} holds or not depends only on l = L ∩ Q but not on the choice of L. So we can denote the condition:

c_{n}^{Y_{n}(T)}mod 𝔏 ∈ ((O_{n}/𝔏)^{×})^{p^{x}} and c_{n}^{Y_{n}(T)}mod 𝔏 ∉ ((O_{n}/𝔏)^{×})^{p^{x+1}}

by (H_{n,l,x}).

f = 29. By computation we have

\{ \begin{matrix}α ≡ 18mod p^{3}, & e = 2 \\ α^{∗} ≡ 15mod p^{3}, & e^{∗} = 1. \end{matrix}

Let

n = 0, l = 523, g = 2, l′ = 1911043 and g′ = 3.

Then

g_{1}(𝔏)^{Y_{1}^{∗}(T)}mod 𝔏′ ∉ ((O′_{N}/𝔏′)^{×})^{p},

i.e. (G_{0,l,l′}) holds. Further since (H_{0,l,0}) holds, we obtain

e_{0} = e_{1} = e_{2} = ... = 0 and λ_{p}(k) = μ_{p}(k) = ν_{p}(k) = 0.

f = 761. By computation we have

\{ \begin{matrix}α ≡ 3mod p^{2}, & e = 1 \\ α^{∗} ≡ 3mod p^{2}, & e^{∗} = 1. \end{matrix}

Let

n = 0, l = 4567, g = 3, l′ = 62558767 and g′ = 3.

Then (G_{0,l,l′}) holds. Further since (H_{0,l,1}) holds, we have e_{0} = 1. Moreover since (H_{1,27397,1}) holds, e_{1} ≤ 1. Therefore we obtain

e_{0} = e_{1} = e_{2} = ... = 1, λ_{p}(k) = μ_{p}(k) = 0 and ν_{p}(k) = 1.

f = 473. By computation we have

\{ \begin{matrix}α ≡ 30mod p^{5}, & e = 1 \\ α^{∗} ≡ 84mod p^{5}, & e^{∗} = 1. \end{matrix}

Let

n = 4, l = 1609147, g = 3, l′ = 369907494067 and g′ = 19.

Then (G_{4,l,l′}) holds. Further since (H_{4,l,5}) holds, we have e_{4} = 5. Moreover since (H_{5,2068903,5}) holds, e_{5} ≤ 5. Therefore we obtain

e_{n} = n+1 for n ≤ 4, e_{n} = 5 for n ≥ 5, λ_{p}(k) = μ_{p}(k) = 0 and ν_{p}(k) = 5.

References

[1] J. Buhler, R. Crandall, R. Ernvall, and Metsänkylä, Irregular primes and cyclotomic invariants up to four million, Math. Comp. 61 (1993), 151–153.

[2] B. Ferrero and L. Washington, The Iwasawa invariant μ_{p} vanishes for abelian number fields, Ann. of Math. 109 (1979), 377–395.

[3] T. Fukuda and K. Komatsu, A capitulation problem and Greenberg’s conjecture on real quadratic fields, Math. Comp. 65 (1996), 313–318.

[4] T. Fukuda and H. Taya, The Iwasawa λ-invariants of Z_{p}-extensions of real quadratic fields, Acta Arith. 69 (1995), 277–292.

[5] R. Gillard, Remarques sur les unités cyclotomiques et les unités elliptiques, J. Number Theory 11 (1979), 21–48.

[6] R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), 263–284.

[7] H. Ichimura, Local units modulo Gauss sums, J. Number Theory 68 (1998), 36–56.

[8] H. Ichimura and H. Sumida, On the Iwasawa invariants of certain real abelian fields II, Internat. J. Math. 7 (1996), 721–744.

[9] , On the Iwasawa invariants of certain real abelian fields, Tôhoku Math. J. 49 (1997), 203–215.

[10] K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257–258.

[11] , Lectures on p-adic L-functions, Ann. of Math. Stud., vol. 74, Princeton Univ. Press: Princeton, N.J., 1972.

[12] , On Z_{l}-extensions of algebraic number fields, Ann. of Math. 98 (1973), 246–326.

[13] J. S. Kraft and R. Schoof, Computing Iwasawa modules of real quadratic number fields, Compositio Math. 97 (1995), 135–155.

[14] T. Kubota and H.W. Leopoldt, Eine p-adische Theorie der Zetawerte, I. Einführung der p-adischen Dirichetschen l-Funktionen, J. reine angew. Math. 214/215 (1964), 328–339.

[15] M. Kurihara, The Iwasawa λ invariants of real abelian fields and the cyclotomic elements, Tokyo J. Math. 22 (1999), 259–277.

[16] B. Mazur and A. Wiles, Class fields of abelian extensions of Q, Invent. Math. 76 (1984), 179–330.

[17] J. Minardi, Iwasawa modules for Z_{p}^{d}-extensions of algebraic number fields, Thesis, University of Washington (1986).

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

Hiroki SUMIDA

Faculty of Integrated Arts and Sciences

Hiroshima University

Kagamiyama

Higashi-Hiroshima 739-8521, Japan sumida@mis.hiroshima-u.ac.jp