Computation of Iwasawa ν-invariants of certain real abelian fields

Hiroki Sumida^{∗}

Abstract

Let p be a 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 number fields k. Using cyclotomic units and Gauss sums, we give an effective method for computing the other Iwasawa invariants ν_{p}(k) of certain real abelian fields. As numerical examples, we compute Iwasawa invariants associated to k = Q(\sqrt{f}, ζ_{p} + ζ_{p}^{−1}) in the range 1 < f < 200, 5 ≤ p < 10000.

1 Introduction

Let p be a prime number and k a finite extension of Q. Denote by k_{∞} the cyclotomic Z_{p}-extension of k and by k_{n} the subfield of k_{∞} such that [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}(k)} 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}(k) = λ_{p}(k)n + μ_{p}(k)p^{n} + ν_{p}(k) for all sufficiently large n (cf. [15]).

It is conjectured that μ-invariants vanish 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 [4]. For λ-invariants, there are many (non-totally real) number fields k with λ_{p}(k) > 0 (cf. [7, p. 266]). On the other hand, it is conjectured that λ_{p}(k) = μ_{p}(k) = 0 for all p and totally real number fields k, which is called as Greenberg’s conjecture. At present, we have some numerical examples supporting this conjecture for abelian fields (e.g. [10, 12, 17, 20]). In their methods, cyclotomic units and prime numbers are used.

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

Mathematics Subject Classification (2000): 11R23

Key words and phrases: Iwasawa invariant, ideal class group, cyclotomic unit, Gauss sum, Vandiver’s conjecture, Greenberg’s conjecture

In [11], by a similar method, we verify λ_{p}(k) = μ_{p}(k) = 0 for p = 3, 5, 7 and quadratic fields k with small discriminants, and give an upper bound for ν_{p}(k). However it is difficult to determine the exact value by their methods, because we need to find a p-th root of a cyclotomic unit (cf. [24, pp.420–423]). In this paper, by using cyclotomic units, Gauss sums and prime numbers, we give an effective method for computing the exact values of ν_{p}(k) under some technical conditions.

Using our method, we can settle the values of tables in [11] by computer calculation. Moreover we compute Iwasawa invariants associated to k = Q(\sqrt{f}, ζ_{p} + ζ_{p}^{−1}) in the range 1 < f < 200 and 5 ≤ p < 10000, where f is the discriminant of a real quadratic field. We can obtain some examples such that λ_{p}(k) = μ_{p}(k) = 0 and ν_{p}(k) > 0 for large primes p. The number of such cases seems to follow the deduction in [24, pp.158–159].

2 Result

2.1 General case

We fix an inclusion map \overline{Q} ↪ \overline{Q}_{p}, where \overline{F} is the algebraic closure of a field F. Further we fix a primitive n-th root of unity ζ_{n} so that ζ_{nm}^{m} = ζ_{n} for all m, n ≥ 1. Let p be an odd prime number and f_{0} an integer such that p|f_{0} and p^{2} ∤ f_{0}. Let K be a subfield of Q(ζ_{f_{0}}) containing ζ_{p} whose conductor is f_{0}. For simplicity, we assume the following condition:

(C1) p does not divide |Gal(K/Q)|.

Put f_{n} = f_{0}p^{n}. Then K_{n} = KQ_{n} ⊆ Q(ζ_{f_{n}}). Further put Δ = Gal(K_{∞}/Q_{∞}), Γ = Gal(K_{∞}/K) and G_{∞} = Gal(K_{∞}/Q) = Δ × Γ.

Let Φ be a Q_{p}-valued character of Δ which is irreducible over Q_{p}. Put

e_{Φ} = \frac{1}{|Δ|} ∑_{δ∈Δ} Φ(δ)δ^{−1}

the idempotent of the group ring Q_{p}[Δ]. This is an element of Z_{p}[Δ] by (C1). Let ϕ be an irreducible component of Φ over \overline{Q}_{p}. We say Φ is even (resp. odd) when ϕ(ρ) = 1 (resp. ϕ(ρ) = −1), where ρ is the complex conjugate in Δ. Let O_{ϕ} be the subring of \overline{Q}_{p} generated by all ϕ(δ) (δ ∈ Δ) over Z_{p}. We identify e_{Φ}Z_{p}[Δ] with O_{ϕ} by δ ↦ ϕ(δ).

We fix a 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 O_{ϕ}[[Γ]] with the power series ring Λ = O_{ϕ}[[T]] by γ_{0} = 1 + T. Thus, for a Z_{p}[[G_{∞}]]-module M, we regard M(Φ) = e_{Φ}M as a Λ-module. Let X_{∞} = X_{∞}(K) = lim_{←} A_{n}(K), where the inverse limit is taken with respect to relative norm maps. Then X_{∞}(Φ) is regarded as a Λ-module in the above way. We can define e_{n}(Φ), λ_{p}(Φ), μ_{p}(Φ) and ν_{p}(Φ) for A_{n}(Φ) and X_{∞}(Φ). Let k be a subfield of K and ϕ(δ) = 1 for any δ ∈ Gal(K/k). Then, since p does not divide [K : k], N_{K_{n}/k_{n}} : A_{n}(K)(Φ) → A_{n}(k)(Φ) is an isomorphism.

From now on, let Ψ be a non-trivial even Q_{p}-valued character of Δ which is irreducible over Q_{p} and ψ its fixed irreducible component of Ψ over \overline{Q}_{p}. Put Ψ^{∗} = Ψ^{−1}ω and ψ^{∗} = ψ^{−1}ω, where ω is the Teichmüller character Z/pZ → Z_{p} (ω(a) ≡ a mod p). We denote by O both O_{ψ} and O_{ψ^{∗}}. We fix a prime element π of O.

Let L_{p}(s, ψ) be the p-adic L-function associated to ψ which is constructed in [18]. By [14, §6], there uniquely exists G_{Ψ}(T) ∈ O[[T]] satisfying G_{Ψ}((1 + f_{0})^{1−s} − 1) = L_{p}(s, ψ) for all s ∈ Z_{p}. In [4], it is shown that π does not divide G_{Ψ}(T). Therefore, by the p-adic Weierstrass preparation theorem, we can uniquely write G_{Ψ}(T) = g_{Ψ}(T)u_{Ψ}(T), where g_{Ψ}(T) is a distinguished polynomial of O[T] and u_{Ψ}(T) is an invertible element of O[[T]]. In a similar way, we can define g_{Ψ^{∗}}(T) ∈ O[T] from the power series G_{Ψ^{∗}}(T) ∈ O[[T]] satisfying G_{Ψ^{∗}}((1+f_{0})^{s}−1) = L_{p}(s, ψ). Put λ̃(Ψ) = deg g_{Ψ}(T) = deg g_{Ψ^{∗}}(T).

Further, for simplicity, we assume

(C2) ψ(p) ≠ 1 and ψ^{∗}(p) ≠ 1.

For a non-negative integer n, we denote by p^{a_{n}} (resp. p^{a_{n}^{∗}}) the exponent of Λ/(ω_{n}, g_{Ψ}(T)) (resp. Λ/(ω_{n}, g_{Ψ^{∗}}(T))), where ω_{n} = (1 + T)^{p^{n}} − 1. Since Leopoldt’s conjecture holds for K_{n} and p ([1, 2]), we have a_{n} < ∞. Further we have a_{n}^{∗} < ∞ by (C2). Let e_{Ψ,n} (resp. e_{Ψ^{∗},n}) be an element of Z[Δ] such that e_{Ψ,n} ≡ e_{Ψ} mod p^{a_{n}} (resp. e_{Ψ^{∗},n} ≡ e_{Ψ^{∗}} mod p^{a_{n}^{∗}}) and the sum of all the coefficients is zero. Put N = N(n) = max{n, a_{n} − 1} and N^{∗} = N^{∗}(N) = max{N, a_{N}^{∗} − 1}.

In [10], we define the following polynomials X_{n}(T) ∈ O[T] and Y_{n}(T) ∈ O[T]:

X_{n}(T)g_{Ψ}(T) = p^{a_{n}} and Y_{n}(T) ≡ X_{n}(T) mod p^{a_{n}}.

In a similar way, we define X_{N}^{∗}(T) ∈ O[T] and Y_{N}^{∗}(T) ∈ O[T]:

X_{N}^{∗}(T)g_{Ψ^{∗}}(T) = p^{a_{N}^{∗}} and Y_{N}^{∗}(T) ≡ X_{N}^{∗}(T) mod p^{a_{N}^{∗}} .

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

c_{n} = (N_{Q(ζ_{fn})/K_{n}} (1 − ζ_{f_{n}}))^{e_{Ψ,n}} .

Let Õ_{N} be the ring of integers of Q(ζ_{f_{N}}) and 𝔏̃ a prime ideal of Q(ζ_{f_{N}}) with 𝔏̃ ∤ f_{N} which is lying above a prime number l. Denote by χ𝔏̃ the character of (Õ_{N}/𝔏̃)^{×} with values in ⟨ζ_{f_{N}} ⟩, defined by the following congruence:

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

We define a Gauss sum g′_{N}(𝔏̃) ∈ Q(ζ_{f_{N} 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. Let 𝔄 be an ideal of K_{N} with 𝔄Õ_{N} = ∏_{i=1}^{r} 𝔏̃_{i}^{e_{i}}, where 𝔏̃_{i} is a prime ideal of Q(ζ_{f_{N}}) lying above l_{i}. We define a Gauss sum

g_{N}(𝔄) = (∏_{i=1}^{r} g′_{N}(𝔏̃_{i})^{e_{i}})^{f′e_{Ψ^{∗},N}} ∈ K_{N}^{×},

where f′ = f_{0}/p (cf. [22, Proposition 3.1] and [9, pp.44-45]).

Let 𝔏 (resp. 𝔏^{∗}) be a prime ideal of K_{N} (resp. K_{N^{∗}}) of degree 1 lying above a prime number l (resp. l^{∗}). Put F_{𝔏} = (O_{N}/𝔏)^{×}/((O_{N}/𝔏)^{×})^{p^{an}}, F_{𝔏^{∗}} = (O_{N^{∗}}/𝔏^{∗})^{×}/((O_{N^{∗}}/𝔏^{∗})^{×})^{p^{a^{∗}_{N}}}, F_{l} = ∏_{𝔏|l} F_{𝔏} and F_{l^{∗}} = ∏_{𝔏^{∗}|l^{∗}} F_{𝔏^{∗}}, where O_{N} is the ring of integers of K_{N}. For L = {l_{i}|1 ≤ i ≤ s_{0}} and L^{∗} = {l_{j}^{∗}|1 ≤ j ≤ s^{∗}}, let K_{N}(L) (resp. K_{N^{∗}}(L^{∗})) be the group of all elements of K_{N}^{×} (resp. K_{N^{∗}}^{×}) which are prime to all l_{i} ∈ L (resp. l_{j}^{∗} ∈ L^{∗}). We define the following maps:

D_{N,L} : K_{N}(L) → ∏_{i=1}^{s_{0}} F_{l_{i}} (x ↦ ((x mod 𝔏) mod ((O_{N}/𝔏)^{×})^{p^{an}})_{𝔏,i})

and

D_{N^{∗},L^{∗}}^{∗} : K_{N^{∗}}(L^{∗}) → ∏_{j=1}^{s^{∗}} F_{l_{j}^{∗}}, (x ↦ ((x mod 𝔏^{∗}) mod ((O_{N^{∗}}/𝔏^{∗})^{×})^{p^{a^{∗}_{N}}} )_{𝔏^{∗},j}) .

We fix a prime ideal 𝔏_{i}|l_{i} for each i and put L′ = {𝔏_{i}|1 ≤ i ≤ s_{0}}. We define

C′_{n} = ⟨c_{n}^{T^{m}Y_{n}(T)}|0 ≤ m ≤ s′_{n}⟩_{Z[Δ]}

and

G′_{N,L′} = ⟨g_{N}(L_{i})^{T^{m}Y_{N}^{∗} (T)}|1 ≤ m ≤ s′_{N}, 1 ≤ i ≤ s_{0}⟩_{Z[Δ]},

where s′_{n} = min{λ̃(Ψ), p^{n}} − 1 and s_{0} = dim_{O/π} A_{0}(Ψ^{∗})/πA_{0}(Ψ^{∗}). Put L′_{0} = {𝔏_{i,0}|𝔏_{i,0} = N_{K_{N} /K_{0}}𝔏_{i}, 𝔏_{i} ∈ L′}.

Theorem 1. Assume (C1) and (C2). Then

|A_{n}(Ψ)| ≤ \frac{|Λ/(g_{Ψ}(T), ω_{n})|}{|D_{N,L}(C′_{n})|}.

If |D_{N^{∗}(0),L_{N^{∗}(0)}^{∗}}^{∗}(G′_{0,L′_{0}})| = |Λ/(g_{Ψ^{∗}}(T), ω_{0})| for some L′_{0} and L_{N^{∗}(0)}^{∗}, then the equality holds. If |A_{m}(Ψ)| = |A_{m+1}(Ψ)| for some m, then λ_{p}(Ψ) = μ_{p}(Ψ) = 0 and ν_{p}(Ψ) = e_{m}(Ψ).

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 [6]. 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 [6, Theorem 2], we have the following Λ-isomorphisms:

U_{n}(Ψ) ≃  Λ/(ω_{n})
⋃ |  ⋃ |
E_{n}(Ψ) ≃  V_{n}
⋃ |  ⋃ |
C_{n}(Ψ) ≃ (g_{Ψ}(T), ω_{n})/(ω_{n}).

By the Iwasawa main conjecture proved in [8, 21, 25], we have

p^{e_{n}(Ψ)} = |A_{n}(Ψ)| = |E_{n}(Ψ)/C_{n}(Ψ)| = |U_{n}(Ψ)/C_{n}(Ψ)|/|U_{n}(Ψ)/E_{n}(Ψ)|
= |Λ/(g_{Ψ}(T), ω_{n})|/|U_{n}(Ψ)^{p^{an}}/E_{n}(Ψ)^{p^{an}}|
= |Λ/(g_{Ψ}(T), ω_{n})|/|C′_{n}(Ψ)/E_{n}(Ψ)^{p^{an}}|,

where C′_{n} = \overline{U_{n} ∩ (d_{n}(C′_{n})C_{n}^{p^{an}})}. We will show that the surjective map:

C′_{n}(Ψ)/E_{n}(Ψ)^{p^{an}} → D_{N,L}(C′_{n})

is an isomorphism if |D_{N^{∗},L^{∗}}^{∗}(G′_{N,L′})| = |Λ/(g_{Ψ^{∗}}(T), ω_{N})|. Denote by τ the canonical isomorphism:

τ : (Z/f_{N}lZ)^{×} → Gal(Q(ζ_{f_{N} l})/Q) (m mod f_{N}l ↦ τ_{m} (ζ_{f_{N} l}^{τ_{m}} = ζ_{f_{N} l}^{m})) .

Put

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

By Stickelberger’s theorem ([22, Proposition 3.1]), for any sufficiently large m, there exists g′ ∈ K_{N}^{×} such that

(g_{N}(𝔏_{i})) = 𝔏_{i}^{f′e_{Ψ^{∗},m}θ_{N}} (g′^{p^{a^{∗}_{N}}}).

We here have

e_{Ψ^{∗},m}θ_{N}Y_{N}^{∗}(T)
= (e_{Ψ^{∗}} + p^{m}Z)θ_{N}Y_{N}^{∗}(T)
≡ g_{Ψ^{∗}}(T)u(T)(X_{N}^{∗}(T) + p^{a_{N}^{∗}} W (T))e_{Ψ^{∗}} + p^{m}Zθ_{N}Y_{N}^{∗}(T) mod ω_{N}
≡ p^{a_{N}^{∗}} (1 + W (T)g_{Ψ^{∗}}(T))u(T)e_{Ψ^{∗}} + p^{m}Zθ_{N}Y_{N}^{∗}(T) mod ω_{N}

for some Z ∈ Z_{p}[Δ], W (T) ∈ O[T] and u(T) ∈ O[[T]]^{×}. Let L_{N} be the maximal unramified abelian p-extension of K_{N}. Let L_{N}(Ψ^{∗}) be the subfield of L_{N} corresponding to ∏_{Φ≠Ψ^{∗}} Gal(L_{N}/K_{N})(Φ), where Φ runs over every Q_{p}-

valued character of Δ which is irreducible over Q_{p}. We denote by (\frac{L_{N} /K_{N}}{𝔏_{i}}) the Artin symbol. Put

σ_{𝔏_{i}} = (\frac{L_{N}(Ψ^{∗})/K_{N}}{𝔏_{i}} ) = (\frac{L_{N}(Ψ^{∗})/K_{N}}{𝔏_{i}} )^{e_{Ψ^{∗}}} = (\frac{L_{N}/K_{N}}{𝔏_{i}} )_{|L_{N} (Ψ^{∗})}^{e_{Ψ^{∗}}} .

Then, by Stickelberger’s theorem, we have σ_{𝔏_{i}}^{g_{Ψ^{∗}}(T)} = 1. Let us consider the following map:

A_{N}(Ψ^{∗}) → K_{N}^{×}/(K_{N}^{×})^{p^{a^{∗}_{N}}} (Cl(𝔄) ↦ α^{e_{Ψ^{∗},N}} mod (K_{N}^{×})^{p^{a^{∗}_{N}}}),

where α ∈ K_{N}^{×} such that 𝔄^{p^{a^{∗}_{N}}} = (α). By the proposition below,

|D_{N^{∗}(0),L_{N^{∗}(0)}^{∗}}^{∗}(G′_{0,L′_{0}})| = |Λ/(g_{Ψ^{∗}}(T), ω_{0})|

implies that

⟨Cl(𝔏_{i})^{qe_{Ψ^{∗}}T^{m}}|1 ≤ m ≤ s′_{N}, 1 ≤ i ≤ s_{0}⟩_{Z[Δ]} = A_{N}(Ψ^{∗})

for some integer q such that p ∤ q. By the class field theory, we have

⟨σ_{𝔏_{i}}^{T^{m}}|1 ≤ m ≤ s′_{N}, 1 ≤ i ≤ s_{0}⟩_{Z[Δ]} = Gal(L_{N}(Ψ^{∗})/K_{N}).

For c ∈ C′_{n}, since d_{n}(c) ∈ U_{n}^{p^{an}}, K_{N} ( \sqrt[p^{an}]{c} ) ⊆ L_{N}(Ψ^{∗}). Hence K_{N} ( \sqrt[p^{an}]{c} ) = K_{N} (i.e. c ∈ (K_{N}^{×})^{p_{n}^{a}}) if and only if c mod 𝔏 ∈ ((O_{N}/𝔏)^{×})^{p^{an}} for all 𝔏|l_{i} and l_{i} ∈ L. This implies that D_{N,L}(C′_{n}) ≃ C′_{n}(Ψ)/E_{n}(Ψ)^{p^{an}}. Therefore the above assertion follows.

By (C2), we have A_{n}(Ψ) ≃ X_{∞}(Ψ)/ω_{n}X_{∞}(Ψ) (cf.[10, p. 731]). If |A_{m}(Ψ)| = |A_{m+1}(Ψ)|, then ω_{m}X_{∞}(Ψ) = ω_{m+1}X_{∞}(Ψ) ⊆ (p, T)ω_{m}X_{∞}(Ψ). By Nakayama’s lemma, we have ω_{m}X_{∞}(Ψ) = 0. Therefore A_{n}(Ψ) ≃ X_{∞}(Ψ) for n ≥ m and the last assertion follows.

Proposition 1. In the above setting, |D_{N^{∗},L^{∗}}^{∗}(G′_{N,L′})| = |Λ/(g_{Ψ^{∗}}(T), ω_{N})| holds for some L^{∗} if and only if |D_{N^{∗}(0),L_{N^{∗}(0)}^{∗}}^{∗}(G′_{0,L′_{0}})| = |Λ/(g_{Ψ^{∗}}(T), ω_{0})| holds for some L_{N^{∗}(0)}^{∗}.

Proof. We here have

σ_{𝔏_{i}}|_{L_{0}(Ψ^{∗})} = (\frac{L_{0}(Ψ^{∗})/K_{0}}{N_{K_{N} /K_{0}}𝔏_{i}} ) = (\frac{L_{0}(Ψ^{∗})/K_{0}}{𝔏_{i,0}} ) = σ_{𝔏_{i,0}}.

By Nakayama’s lemma,

⟨σ_{𝔏_{i}}^{T^{m}}|1 ≤ m ≤ s′_{N}, 1 ≤ i ≤ s_{0}⟩_{Z[Δ]} = Gal(L_{N}(Ψ^{∗})/K_{N})

if and only if

⟨σ_{𝔏_{i,0}}|1 ≤ i ≤ s_{0}⟩_{Z[Δ]} = Gal(L_{0}(Ψ^{∗})/K_{0}).

By the Chebotarev density theorem, the assertion follows.

2.2 Special case

We explain how to use the criterion in practice. Following [11], we consider the following case.

(D1) The exponent of Gal(K/Q) divides p − 1.

(D2) ψ(p) ≠ 1 and ψ^{∗}(p) ≠ 1.

(D3) g_{ψ}(T) = T − α for some α ∈ pZ_{p}.

By (D1), the values of ψ are contained in Z_{p} and Ψ = ψ. 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

such that v_{p}(p) = 1. Then we have a_{n} = n + e, a_{n}^{∗} = n + e^{∗}, N = N(n) = n + e − 1, N^{∗} = N^{∗}(N) = N + e^{∗} − 1 = n + e + e^{∗} − 2 and 0 ≤ e_{n}(ψ) ≤ n + e. Further

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

and

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

where u = \frac{p^{n+e}}{1 − (1 + α)^{p^{n}}} and u^{∗} = \frac{p^{N+e^{∗}}}{1 − (1 + α^{∗})^{p^{N}}}.

Let 𝔏 (resp. 𝔏^{∗}) be a prime ideal of K_{N} (resp. K_{N^{∗}}) of degree 1 lying above a prime number l (resp. l^{∗}). Put

D_{N,𝔏} : K_{N}(𝔏) → F_{𝔏} and D_{N^{∗},𝔏^{∗}}^{∗} : K_{N^{∗}}(𝔏^{∗}) → F_{𝔏^{∗}}.

By (D3), we have σ_{𝔏}^{γ_{0}} = σ_{𝔏}^{1+α^{∗}}. Further we have σ_{𝔏}^{δ} = σ_{𝔏}^{ψ^{∗}(δ)} for δ ∈ Δ. Hence the orders of D_{N,𝔏}(C′_{n}) and D_{N^{∗},𝔏^{∗}}^{∗}(G′_{N,𝔏}) depend only on l and l^{∗}. So we can write

|D_{N,l}(C′_{n})| = |D_{N,𝔏}(C′_{n})| and |D_{N^{∗},l^{∗}}^{∗}(G′_{N,l})| = |D_{N^{∗},𝔏^{∗}}^{∗}(G′_{N,𝔏})|.

In this case, we can write Theorem 1 as follows.

Theorem 2. Assume (D1)-(D3). For any prime number l which splits completely in K_{N}/Q,

|A_{n}(ψ)| ≤ \frac{p^{n+e}}{|D_{N,l}(C′_{n})|}.

If |D_{e^{∗}−1,l_{e^{∗}−1}^{∗}}^{∗}(G′_{0,l})| = p^{e^{∗}} for some l_{e^{∗}−1}^{∗} which splits completely in K_{e^{∗}−1}/Q, then the equality holds. If |A_{m}(ψ)| = p^{m+e−1} for some m, then λ_{p}(ψ) = μ_{p}(ψ) = 0 and ν_{p}(ψ) = m + e − 1.

3 Computation of Gauss sums modulo prime ideals

Let l (resp. l^{∗}) be a prime number such that l ≡ 1 mod f_{N} (resp. l^{∗} ≡ 1 mod 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 𝔏|l of K_{N} and 𝔏^{∗}|l^{∗} of K_{N}(ζ_{l}), s ≡ χ_{𝔏}(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, by the definition of Y_{N}^{∗}(T), we have a_{j} ≡ u^{∗}(1+α^{∗})^{p^{N} −1−j} mod p^{N+e^{∗}}.

We here identify Gal(K_{N}/K_{0}) (resp. Gal(K_{N}/Q_{N})) with Gal(K_{N}(ζ_{l})/K_{0}(ζ_{l})) (resp. Gal(K_{N}(ζ_{l})/Q_{N}(ζ_{l}))). Let 𝔄_{N} be the subgroup of (Z/f_{N}lZ)^{×} which is corresponding to Gal(Q(ζ_{f_{N} l})/Q_{N}(ζ_{l})) by the isomorphism τ. Taking an extension to Gal(Q(ζ_{f_{N} l})/Q_{N}(ζ_{l})), we write

( ∑_{τ∈Gal(Q(ζ_{f_{N}})/K_{N})} τ) e_{Ψ^{∗},N} = ∑_{m∈𝔄_{N}} b_{m}τ_{m}, b_{m} ∈ Z.

Then we have

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

≡ (  ∏_{0≤j<p^{N} , m∈𝔄_{N}} (− ∑_{i=0}^{l−2} s^{m(1+f_{0})^{j}i}t^{g^{i}})^{b_{m}a_{j}} )^{f′} mod 𝔏^{∗}.

In order to compute the above sums, it suffices to compute the following discrete Fourier transforms:

F (w) = ∑_{0≤v<z} ζ_{z}^{wv}f(v) = ζ_{2z}^{w^{2}} ∑_{0≤v<z} ζ_{2z}^{−(v−w)^{2}}(ζ_{2z}^{v^{2}}f(v))

for 0 ≤ w < z, where z = f_{N} and f(v) = ∑_{0≤i<l−1, i≡v mod f_{N}} t^{g^{i}}. For l^{∗} ≡ 1 mod 2f_{N}l, by the famous algorithm of cyclic convolution, we can effectively compute Gauss sums modulo prime ideals (cf. [16, 6.4]).

There are another computations to find a p-th root of a cyclotomic unit in [5] and [17]. These computations become much more difficult as the extension degree of the field containing the cyclotomic unit is higher. It is necessary to do the computations with extremely accurate approximations of cyclotomic units.

4 Numerical examples

4.1 Q(\sqrt{f})

Let p be an odd prime number and k the real quadratic field whose discriminant is f. Let φ_{f} be the non-trivial Dirichlet character associated to k = Q(\sqrt{f}). Put K = Q(\sqrt{f}, ζ_{p}). Then we have A_{n}(k) = A_{n}(k)(φ_{f}) ≃ A_{n}(K)(φ_{f}) by [13]. The following examples satisfy (D1)-(D3) for p = 3.

f = 29. By computation, we have

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

Since |D_{1,l}(C′_{0})| = p^{2} for l = 523, we have |A_{0}(φ_{f})| ≤ p^{2−2} = 1 by Theorem 2. Therefore we obtain

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

In this case, we don’t need to calculate a Gauss sum.

f = 761. By computation, we have

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

Though |D_{0,l}(C′_{0})| = 1 for some primes l, |D_{1,l}(C′_{1})| = p for l = 237397. Hence we have |A_{n}(φ_{f})| ≤ p^{2−1} = p for n ≥ 1 by Theorem 2.

Since |D_{0,l_{0}^{∗}}^{∗}(G_{0,l})| = p and |D_{0,l}(C′_{0})| = 1 for l = 4567 and l_{0}^{∗} = 62558767, we have |A_{0}(φ_{f})| = p by Theorem 2. 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 ≡ 30 mod p^{5}, & e = 1, \\ α^{∗} ≡ 84 mod p^{5}, & e^{∗} = 1. \end{matrix}

Though |D_{n,l}(C′_{n})| = 1 for n ≤ 4 and some primes l, |D_{5,l}(C′_{5})| = p for l = 2068903. Hence we have |A_{n}(φ_{f})| ≤ p^{6−1} = p^{5} for n ≥ 5 by Theorem 2.

Since |D_{0,l_{0}^{∗}}^{∗}(G_{0,l})| = p and |D_{4,l}(C′_{4})| = 1 for l = 1609147 and l_{0}^{∗} = 22833795931, we have |A_{4}(φ_{f})| = p^{5} by Theorem 2. 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.

By the above procedure, we can settle the value in [11].

4.2 Q(\sqrt{f}, ζ_{p} + ζ_{p}^{−1})

Vandiver’s conjecture claims that λ_{p}(k) = μ_{p}(k) = ν_{p}(k) = 0 for all k = Q(ζ_{p} + ζ_{p}^{−1}). By computer calculation, this conjecture is verified for p < 12000000 ([3]). We here replace Q with Q(\sqrt{f}) and consider Iwasawa invariants associated to k = Q(\sqrt{f}, ζ_{p} + ζ_{p}^{−1}) and K = Q(\sqrt{f}, ζ_{p}), where f is the discriminant of a real quadratic field.

In the range 1 < f < 200, 5 ≤ p < 10000 and 2 ≤ 2i ≤ p − 3, there are 171981262 pairs of (φ_{f}ω^{2i}, p) which satisfy (D1) and (D2). Among them, there are 37140 pairs with λ̃_{p}(φ_{f}ω^{2i}) = 1, 46 pairs with λ̃_{p}(φ_{f}ω^{2i}) = 2, and two pairs with λ̃_{p}(φ_{f}ω^{2i}) = 3. We verified Greenberg’s conjecture λ_{p}(φ_{f}ω^{2i}) = μ_{p}(φ_{f}ω^{2i}) = 0 for each of them by the method in [10]. Except for (φ_{177}ω^{6}, 17), we verified it by using cyclotomic units of k_{0} or k_{1}. For (φ_{177}ω^{6}, 17), we used cyclotomic units of k_{2} to verify it. In the range above, there are 38 pairs which don’t satisfy (D2). We also verified the conjecture for each of them.

Proposition 2. λ_{p}(φ_{f}ω^{2i}) = μ_{p}(φ_{f}ω^{2i}) = 0 for all 1 < f < 200, 5 ≤ p < 10000 and 2 ≤ 2i ≤ p − 3.

In the following tables, we give lists of (f, p, 2i) such that λ̃_{p}(φ_{f}ω^{2i}) = 1 and ν_{p}(φ_{f}ω^{2i}) > 0 (v_{p}(α) = e > 1 or v_{p}(α^{∗}) = e^{∗} > 1) or λ̃_{p}(φ_{f}ω^{2i}) = 2, 3. For p = 7, k = Q(\sqrt{m}, ζ_{p} + ζ_{p}^{−1}) and 1 < m < 1000, a list of non-trivial cases (ν_{p} > 0) was given in [20]. Our extended computation coincides with it.

\underline{λ̃_{p}(φ_{f}ω^{2i}) = 1}

\underline{ν_{p}(φ_{f}ω^{2i}) = 1}

\begin{matrix}f & p & 2i & f & p & 2i & f & p & 2i \\ 12 & 701 & 542 & 21 & 199 & 150 & 33 & 53 & 30 \\ 37 & 43 & 32 & 53 & 1033 & 564 & 69 & 19 & 14 \\ 85 & 3697 & 3086 & 88 & 71 & 26 & 101 & 5333 & 2770 \\ 113 & 43 & 32 & 113 & 3373 & 1602 & 124 & 197 & 126 \\ 124 & 239 & 48 & 129 & 67 & 28 & 140 & 4751 & 120 \\ 141 & 5431 & 4826 & 149 & 43 & 32 & 149 & 71 & 16 \\ 149 & 229 & 182 & 157 & 401 & 56 & 161 & 101 & 22 \\ 168 & 37 & 22 & 172 & 73 & 10 & 173 & 43 & 32 \\ 173 & 101 & 42 & 181 & 71 & 52 & 181 & 6991 & 1628 \\ 185 & 827 & 354 & 188 & 1621 & 168 & 197 & 521 & 372 \end{matrix}

\underline{ν_{p}(φ_{f}ω^{2i}) = 2}

\begin{matrix}f & p & 2i \\ 177 & 17 & 6 \end{matrix}

\underline{v_{p}(α) = e > 1}

f  p  2i  f  p  2i  f  p  2i
8 59 36  17 61 32  21 149 128
28 977 828  33 59 42  37 1091 812
41 7  4  41 283 102  44 787 148
53 7  2  53 1879 1158  57 2161 758
61 17  4  61 1747 1270  76 191 84
89 41 10  92 181 124  97 17  4
105 769 524  105 1453 162  120 2749 2196
124 41 30  140 107 74  149 797 140
149 2767 2178  152 17 12  168 43 10
173 13  4  177 31 24  184 373 72
193 7873 1886

\underline{v_{p}(α^{∗}) = e^{∗} > 1}

f  p  2i  f  p  2i  f  p  2i
8 2221 1600  13 109 6  17 1319 88
28 223 126  33 31 24  33 1777 1184
41 19 12  41 421 126  60 19 14
61 7481 3516  73 11  2  73 1487 808
76 1451 418  76 4283 3484  97 367 26
109 41 32  133 1061 446  136 449 284
152 41  2  152 4027 3108  156 4637 2280
157 8221 582  165 29 26  165 89 66
165 1229 48  172 11  4  172 1487 900
177 337 74  184 1171 464  185 167 68
188 89 76

\underline{λ̃_{p}(φ_{f}ω^{2i}) = 2}
f  p  2i  f  p  2i  f  p  2i
8 1151 842  21 11  4  24 29  4
24 181 84  29 569 64  37 5  2
37 89 66  37 3251 1094  40 257 232
44 653 448  53 193 14  56 1663 616
60 1277 582  60 1481 986  92 5  2
97 271 94  104 19 14  104 7919 4386
105 373 340  109 131 100  109 293 132
109 373 128  124 733 58  124 2111 1480
129 23 4  133 911 196  136 71 20
137 17 8  140 23 10  140 367 292
141 113 108  141 5939 2938  145 43 28
145 61 58  145 167 128  145 4157 3528
149 5  2  149 509 426  161 2389 646
165 11 2  172 13 10  172 47 38
173 7  4  177 157 48  181 223 26
185 17 6

\underline{λ̃_{p}(φ_{f}ω^{2i}) = 3}

\begin{matrix}f & p & 2i & f & p & 2i \\ 165 & 23 & 6 & 185 & 17 & 10 \end{matrix}

As in the argument of [24, pp.158–159], we can deduce that the number of exceptions to Vandiver’s conjecture for x_{0} ≤ p ≤ x_{1} should be approximately \frac{1}{2}(log log x_{1} − log log x_{0}).

We here use a similar naive approach. For G_{φ_{f} ω^{2i}}(T) = ∑_{j=0}^{∞} a_{j}T^{j} and an integer c, we assume that the probability of a_{j} ≡ c mod p^{k} is 1/p^{k}. Further, for a cyclotomic unit c_{0} with d_{0}(c_{0}) ∈ U_{0}^{p^{e}}, we assume that the probability of c_{0} ∈ E_{0}^{p^{k}} is 1/p^{k} if k ≤ e. Then, for a fixed conductor f and x_{0} ≤ p ≤ x_{1}, we can deduce that the number of pairs (φ_{f}ω^{2i}, p) such that λ̃_{p} = 1 and ν_{p} > 0 (e > 1 or e^{∗} > 1) should be

∑_{x_{0}≤p≤x_{1}} \frac{p − 3}{2} (\frac{1}{p^{2}} − \frac{1}{p^{3}} ) .

Let us apply this to our examples with x_{0} = 37 and x_{1} = 10000. Since the number of f’s is 60, there should be about 60 × 0.44645 · ·· ‘exceptions’. Our lists seem to support the deduction.

For f = 1 (i.e. ψ = ψ^{0}), the numerators of Bernoulli numbers B_{2i} are relatively small. We had better take this fact into consideration. By calculating their numerators, we easily see that there is no example such that A_{0}(ω^{2i}) ≠ {0} for i = 1, 2, 3, 4, 5 and 7. Further there is no example such that A_{0}(ω^{2i}) ≠ \{0\} for i = \frac{p−3}{2} (see [19, 23]). Therefore, excluding these i’s, we can deduce that ‘the number of exceptions’ to Vandiver’s conjecture for 37 ≤ p ≤ q (q′) may be smaller than E in the following table.

E  q  q′
0.1  113  149
0.2  419  450
0.3  1667  1737
0.4  8963  9058
0.5 67807  68096
0.6 799417  800172
0.7 16224227 16224227
0.8 640612703 640492743
0.9  10ˆ 10.75
1.0  10ˆ 13.13
1.5  10ˆ 35.71
2.0  10ˆ 97.07

Put E_{1}(x) = ∑_{37≤p≤x} \frac{p − 17}{2} \frac{1}{p^{2}} and E_{2}(x) = \frac{1}{2}log log x−0.70476207···. Then q

(resp. q′) is the first prime (resp. integer) with E_{1}(p) > E (resp. E_{2}(n) > E).

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Hiroki SUMIDA

Faculty of Integrated Arts and Sciences

Hiroshima University

Kagamiyama

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