Group determinant formulas and class numbers of cyclotomic fields

GROUP DETERMINANT FORMULAS AND CLASS NUMBERS OF CYCLOTOMIC FIELDS

Hwan Yup Jung and Jaehyun Ahn

Reprinted from the

Journal of the Korean Mathematical Society Vol. 44, No. 3, May 2007

c

°2007 The Korean Mathematical Society

GROUP DETERMINANT FORMULAS AND CLASS NUMBERS OF CYCLOTOMIC FIELDS

Hwan Yup Jung and Jaehyun Ahn

Abstract. Let m, n be positive integers or monic polynomials in Fq[T ] with n|m. Let Kmand Km⁺be the m-th cyclotomic field and its maximal real subfield, respectively. In this paper we define two matrices D⁺m,nand D⁻m,nwhose determinants give us the ratios^h(O_h(O^K+m)

K+n) and ^h_h⁻₋^(O_(O^Km)

Kn) with some factors, respectively.

1. Introduction

In [3], Girstmair gave a recursion formula for the relative class numbers in the tower of cyclotomic number fields with prime power conductor. Adopt- ing the idea of Girstmair, Jung and Ahn [4] gave an analogous result in the tower of cyclotomic function fields with prime power conductor. In this paper we generalize the results of Girstmair and Jung-Ahn to the tower of any two cyclotomic fields with arbitrary conductor. We treat both cyclotomic number field and cyclotomic function field cases in a unique way.

The layout of this paper is as follows. In section 2, we obtain a group determinant formula (Theorem 2.4) which is a generalization of [6, Lemma 2], [5, Lemma 2.1] and [1, Prop. 2.1]. Let m, n be positive integers or monic polynomials in Fq[T ] with n|m. In section 3, we define two matrices D⁺_m,nand D⁻_m,n whose determinants give us the ratios ^h(O_h(O^K+m)

K+n) and ^h_h⁻−^(O(O^Km_Kn)⁾ with some factors, respectively. In section 4, we restrict ourselves to the prime power conductor case that m = p^{`</sup>and n = p<sup>`−1}. We show that the recursion formulas of Girstmair ([3, Theorem]) and Jung-Ahn ([4, Thm. 4.1]) are immediately obtained from Theorem 3.3. We also obtain the recursion formula between h(O_K⁺

m) and h(O_K⁺

n), and the one between h(K_m⁺) and h(K_n⁺) in function field case.

Received October 14, 2004.

2000 Mathematics Subject Classification. 11R18, 11R60.

Key words and phrases. cyclotomic unit, cyclotomic function field.

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (R05-2004-1007-0).

c

°2007 The Korean Mathematical Society 499

2. Group determinant formulas

Let G be a finite abelian group of order n. Let L²(G) be the n-dimensional vector space of complex-valued functions on G. Let bG be the character group of G with values in C. Let χ0∈ bG be the trivial character. For any subgroup H of G, we define L²(G)^H = {f ∈ L²(G) : f (στ ) = f (σ) for all σ ∈ G and τ ∈ H}

and bG^H = {χ ∈ bG : χ(σ) = 1 for all σ ∈ H}. For any f ∈ L²(G), we define s^H_f ∈ L²(G)^H by s^H_f(σ) =P

τ ∈Hf (στ ) for σ ∈ G. For any subset M of G, let s(M ) = P

σ∈Mσ and eM = s(M )/|M |. Let eχ = _|G|¹ P

σ∈Gχ(σ)σ⁻¹ ∈ C[G]

be the idempotent element associated to χ ∈ bG.

For a subspace L of C[G] and a subgroup H of G, we define two subspaces L^H= {x ∈ L : eHx = x} and LH= {x ∈ L : eHx = 0}. For any two subgroups H, H⁰ of G, we denote (L^H)_H⁰ by L^H_H0 and (L_H)_H⁰ by L_H,H⁰. It is easy to see that L^H_H0 = L^H_HH0. We are interested in the structures of C[G]^H_H0 and C[G]H,H⁰

as C-vector spaces.

For any subgroup H of G, we denote by RG/H any system of representatives of G/H and when RG/H is fixed, we define a function rH : G → G such that rH(σ)H = σH with rH(σ) ∈ RG/H for each σ ∈ G. From [1, Prop. 2.1], we know that

Proposition 2.1. For any two subgroups H, H⁰ of G, we take R_G/H and R_G/HH⁰ with R_G/HH⁰ ⊆ R_G/H. Then we have

(i) dimCC[G]^H_H0 = |G/H| − |G/HH⁰|,

(ii) {eχ : χ ∈ bG^H\ bG^HH0} and {(σ − rHH⁰(σ))s(H) : σ ∈ RG/H\RG/HH⁰} are two C-bases of C[G]^H_H0, and

(iii) for any f ∈ L²(G),

(2.1) Y

χ∈ bG^H\ bG^HH0

X

σ∈G

χ(σ)f (σ) = det¡

εσ,τ: σ, τ ∈ RG/H\RG/HH⁰

¢,

where ε_σ,τ= s^H_f (στ⁻¹) − s^H_f(σr_HH⁰(τ )⁻¹).

Since C[G]H,H⁰ = C[G]H∩ C[G]H⁰ and C[G]H+ C[G]H⁰ = C[G]HH⁰, we have dimCC[G]H,H⁰ = |G| − |G/H| − |G/H⁰| + |G/HH⁰|. In fact, it is easy to see that {e_χ: χ ∈ bG\( bG^H∪ bG^H0)} is a basis for C[G]_H,H⁰. We need another basis of C[G]H,H⁰ to obtain a group determinant formula for^Q_{χ6∈ bG}H∪ bG^H0

P

σ∈Gχ(σ)f (σ).

We need the following lemma. The proof is an elementary one, so we leave it to the readers.

Lemma 2.2. Let H, H⁰ be two subgroups of G. Let R_G/H, R_G/H⁰ and R_G/HH⁰ be systems of representatives of G/H, G/H⁰ and G/HH⁰respectively. Then the following conditions are equivalent.

(i) σ − rH(σ) − rH⁰(σ) + rHH⁰(σ) = 0 for σ ∈ RG/H∪ RG/H⁰.

(ii) rH⁰(σ) = rHH⁰(σ) for σ ∈ RG/H and rH(σ) = rHH⁰(σ) for σ ∈ RG/H⁰. (iii) rH◦ rH⁰ = rH⁰◦ rH= rHH⁰ as functions from G to G.

(iv) For any σ ∈ R_G/H (resp. τ ∈ R_G/H⁰) there exists h⁰ ∈ H⁰ (resp.

h ∈ H) such that σh⁰ ∈ R_G/HH⁰ (resp. τ h ∈ R_G/HH⁰) and R_G/HH⁰ ⊂ R_G/H∩ R_G/H⁰.

Throughout the paper, we say that R_G/H, R_G/H⁰ and R_G/HH⁰ satisfy the condition (∗) if they satisfy any (thus all) conditions of Lemma 2.2.

Proposition 2.3. Let H, H⁰ be two subgroups of G. Suppose that RG/H, RG/H⁰ and RG/HH⁰ satisfy the condition (∗). Then {σ − rH(σ) − rH⁰(σ) + rHH⁰(σ) : σ 6∈ RG/H∪ RG/H⁰} is a C-basis of C[G]H,H⁰.

Proof. Let X = {σ − rH(σ) − rH⁰(σ) + rHH⁰(σ) : σ 6∈ RG/H ∪ RG/H⁰}. By Lemma 2.2 (iii), we have σ −rH(σ) −rH⁰(σ)+rHH⁰(σ) = σ −rH⁰(σ)−(rH(σ)−

rH⁰(rH(σ))) and so eH(σ−rH(σ)−rH⁰(σ)+rHH⁰(σ)) = eH⁰(σ−rH(σ)−rH⁰(σ)+

rHH⁰(σ)) = 0. Thus X ⊂ C[G]H,H⁰. Since dimCC[G]H,H⁰ = |G| − |G/H| −

|G/H⁰| + |G/HH⁰| = |X|, it suffices to show that X generates C[G]H,H⁰. Let x =P

σ∈Gaσσ ∈ C[G]H,H⁰. Since eHx = eH⁰x = 0, we have thatP

τ ∈Haστ = P

τ⁰∈H⁰aστ⁰ = 0 for any σ ∈ G. Let ¯x = P

σ6∈RG/H∪R_G/H0aσ(σ − rH(σ) − rH⁰(σ) + rHH⁰(σ)). By Lemma 2.2 (i), we have

¯

x = X

σ∈G

aσ(σ − rH(σ) − rH⁰(σ) + rHH⁰(σ))

= x − X

σ∈R_G/H

(X

τ ∈H

aστ)σ − X

σ∈R_G/H0

( X

τ⁰∈H⁰

aστ⁰)σ

+ X

σ∈R_G/HH0

( X

µ∈HH⁰

aσµ)σ

= x.

Thus X generates C[G]H,H⁰. ¤

Theorem 2.4. For any two subgroups H, H⁰ of G, suppose that R_G/H, R_G/H⁰ and R_G/HH⁰ satisfy the condition (∗). For any f ∈ L²(G), we have

(2.2) Y

χ6∈ bG^H∪ bG^H0

X

σ∈G

χ(σ)f (σ) = det¡

ησ,τ : σ, τ /∈ R_G/H∪ R_G/H⁰¢ ,

where ησ,τ = f (στ⁻¹) − f (σrH(τ )⁻¹) − f (σrH⁰(τ )⁻¹) + f (σrHH⁰(τ )⁻¹).

Proof. Let Ψ be a linear transformation on C[G]H,H⁰ defined as multiplication by θ =P

σ∈Gf (σ)σ. Clearly the matrix of Ψ with respect to the basis X1= {eχ : χ 6∈ bG^H∪ bG^H0} is a diagonal matrix diag¡ P

σ∈Gχ(σ)f (σ) : χ 6∈ bG^H∪ Gb^H0¢

. To compute the matrix of Ψ with respect to the basis X2= {σ −rH(σ)−

rH⁰(σ) + rHH⁰(σ) : σ 6∈ RG/H∪ RG/H⁰}, we consider θ(σ − rH(σ) − rH⁰(σ) +

rHH⁰(σ)) with σ 6∈ R_G/H∪ R_G/H⁰. Then we have θ(σ − rH(σ) − rH⁰(σ) + rHH⁰(σ))

= X

g∈G

f (g)(gσ − grH(σ) − grH⁰(σ) + grHH⁰(σ))

= X

τ ∈G

(f (σ⁻¹τ ) − f (rH(σ)⁻¹τ ) − f (rH⁰(σ)⁻¹τ ) + f (rHH⁰(σ)⁻¹τ ))τ

= X

τ 6∈R_G/H∪R_G/H0

(f (σ⁻¹τ ) − f (rH(σ)⁻¹τ ) − f (rH⁰(σ)⁻¹τ )

+f (rHH⁰(σ)⁻¹τ )) × (τ − rH(τ ) − rH⁰(τ ) + rHH⁰(τ )).

The last equality follows from the same argument in proof of Proposition 2.3.

By comparing determinants of matrices of Ψ with respect to X₁ and X₂, we

prove the theorem. ¤

Remark 2.5. For a subset M of G, we denote M⁻¹ = {σ⁻¹ : σ ∈ M }. For any RG/H, RG/H⁰ and RG/HH⁰ satisfying the condition (∗), R⁻¹_G/H, R⁻¹_G/H0 and R⁻¹_G/HH0 also satisfy the condition (∗). As in the proof of [5, Lemma 2.1], for any f ∈ L²(G), we have

Y

χ6∈ bG^H∪ bG^H0

X

σ∈G

χ(σ)f (σ) = ± det¡

˜

η_σ,τ : σ, τ ∈ G\(R_G/H∪ R_G/H0)¢ ,

where ˜ησ,τ = f (στ ) − f (σrH(τ )) − f (σrH⁰(τ )) + f (σrHH⁰(τ )).

3. Ratio of class numbers of cyclotomic fields

In this section we give determinant formulas for the ratio of class numbers of cyclotomic fields in both number field and function field cases. We handle both number field and function field cases in a unique way.

First we denote k = Q, A = Z or k = Fq(T ), A = Fq[T ], where Fq is a finite field with q elements. Let A⁺= {a ∈ A : a is positive if A = Z or a is monic if A = Fq[T ]}. For any m ∈ A⁺, we denote by Km the m-th cyclotomic field in both number field and function field cases and by K_m⁺ its maximal real subfield. Let Gm = Gal(Km/k), G⁺_m = Gal(K_m⁺/k) and J = Gal(Km/K_m⁺).

For the cyclotomic theory of function field, we refer the reader Rosen’s book ([8, Chap. 12]). It is well known that Gm ∼= (A/mA)^∗ in both number field and function field cases. For any a ∈ A with (a, m) = 1, we denote by σa the element of Gm which corresponds to a mod m under the above isomorphism.

For any subset S of A, we define σ(S) = {σa ∈ Gm: a ∈ S with (a, m) = 1}.

Let bGmbe the group of characters of Gmwith values in C. A character χ ∈ bGm

is called real if χ(J) = {1} and non-real otherwise. Thus bG^J_mis the subgroup of all real characters of Gm and bGm\ bG^J_m is the set of all non-real characters of Gm. For χ ∈ bGm and a ∈ A, we define χ(a) as follows. Let fχ ∈ A⁺

be the conductor of χ. If (a, fχ) = 1, we define χ(a) = χ((a, Kfχ/k)), where (a, Kfχ/k) is the Artin symbol. If (a, fχ) 6= 1, we put χ(a) = 0.

For a, b ∈ Fq[T ], a < b means that deg b > deg a if a is nonzero or that b is nonzero if a = 0. For any m ∈ A⁺, we define Mm = {a ∈ A : 0 <

a < m with (a, m) = 1} and M⁺_m = {a ∈ Mm : a < m/2} if k = Q and M⁺_m= Mm∩ A⁺ if k = Fq(T ). We also define M⁻_m= Mm\M⁺_m. Since Mmis a set of representatives of (A/mA)^∗, we have σ(Mm) = Gm. It is easy to see that σ(M⁺_m) is a set of representatives of Gm/J. For any a ∈ A with (a, m) = 1, let (a)mbe the unique element of Mm such that a ≡ (a)m mod m. We define sgn_m(a) for (a, m) = 1 as follows. If k = Q, define sgn_m(a) = 1 if (a)m∈ M⁺_m and −1 otherwise. If k = F_q(T ), define sgn_m(a) as the leading coefficient of (a)m. We also define deg_m(a) = deg((a)m).

For any nonzero a, f ∈ A with (a, f ) = 1, we define g_f⁺(a) =

(

log |1 − ζ_f^a| if k = Q, ϕ(a/f )/(q − 1) if k = Fq(T ), and

g_f⁻(a) = (1

2− h^a_fi if k = Q, Zf(0, a) if k = Fq(T ),

where ζ_f = exp(2πi/f ), ha/f i is the fractional part of a/f if k = Q, and ϕ(a/f ) = (q − 1)(deg f − 1 − deg_f(a)) − 1, Zf(s, a) is the partial zeta func- tion associated to the class of a in Cl(A) if k = Fq(T ). The notation “f |n”

means that f divides n with f, n ∈ A⁺. Assume that we are given {af,σ ∈ C : f |m with f 6= 1 and σ ∈ Gm}. For any nonzero a ∈ A, let Φ(a) be the order of (A/aA)^∗. For χ ∈ bG_m and 1 6= f ∈ A⁺ with f |m, we define bf,χ = ^Φ(m)_{Φ(f )} P

σ∈Gmaf,σχ(σ) and cχ = P

16=f |m fχ|f bf,χ

Q

p|f(1 − χ(p)) where p ranges over all prime numbers (or all monic irreducible polynomials) dividing f . For any σ ∈ Gm, we define

(3.1) f_m^±(σ) = X

16=f |m

X

a∈Mm

a_f,σaσ⁻¹g_f^±(a).

Lemma 3.1. For χ ∈ bGm, we have X

σ∈Gm

χ(σ)f_m^±(σ) = X

a∈M_fχ

χ(a)g^±_fχ(a)cχ.

Proof. If k = Fq(T ), the proof follows from [5, Thms. 3.1, 3.6]. If k = Q, the proof for f_m⁻ and f_m⁺ follows from [6, Thm. 1] and [9, Lemma 8.4-8.7],

respectively. ¤

We note thatP

a∈M_fχχ(a)g⁻_f

χ(a) = −B1,χif k = Q andP

a∈M_fχχ(a)g⁻_f

χ(a) = L(0, χ) if k = Fq(T ), where B1,χ is the first generalized Bernoulli number and L(s, χ) is the Artin L-function associated to character χ.

Let F be a finite abelian extension of k. We let OF be the integral closure of A in F with its unit group O_F^∗. Let h(OF) be the ideal class number of OF and h⁻(OF) = h(OF)/h(O_F⁺), the relative ideal class number of OF. Let R(F ) be the regulator of F (for k = Fq(T ), we follow the definition in [10, Sect. 1]). Let WF be the group of all roots of unity in F and wF = |WF| be its order. Let QF = [O_F^∗ : WFO^∗_F+]. Note that QKm = 1 if m is a prime power and QKm = |J| if m is not a prime power (see [9, Coro. 4.13], [2, Prop. 1.14]).

We denote by χ0 the trivial character of Gm. Proposition 3.2. For any m ∈ A⁺, we have

(3.2) Y

χ06=χ∈ bG^J_m

X

a∈M_fχ

χ(a)g⁺_fχ(a) = ±|J|^[Km+:k]−1h(O_K⁺_m)R(K_m⁺)

and

(3.3) Y

χ∈ bGm\ bG^J_m

X

a∈M_fχ

χ(a)g_f⁻_χ(a) = ±|J|^[Km+:k]h⁻(OKm) QKmwKm

.

Proof. In case k = Q, it is known ([9, page 42], [9, Thm. 4.9]) that 2^[Km+:Q]−1h(O_K⁺

m)R(K_m⁺) q

|d(Km⁺)|

= Y

χ06=χ∈ bG^J_m

L(1, χ), Y

χ06=χ∈ bG^J_m

L(1, χ) = ± Y

χ06=χ∈ bG^J_m

(τ (χ)/fχ) Y

χ06=χ∈ bG^J_m

X

a∈M_fχ

χ(a)g⁺_fχ(a),

where d(K_m⁺) be the discriminant of K_m⁺ and τ (χ) =P_fχ

a=1χ(a)ζ_f^a_χ is a Gauss sum. Then (3.2) follows from the fact that Q

χτ (χ)/fχ = 1/

q

|d(Km⁺)| ([9, Thm. 3.11, Coro. 4.6]). Since P

a∈M_fχχ(a)g⁻_fχ(a) = −B1,χ, (3.3) immediately follows from [9, Thm. 4.17].

Suppose that k = Fq(T ). Let h(Km) and h(K_m⁺) be the divisor class numbers of Km and K_m⁺, respectively. Let h⁻(Km) = h(Km)/h(K_m⁺) be the relative divisor class number of Km. It is known [7, Corollary after Lemma 4.1] that h(K_m⁺) = R(K_m⁺)h(O_K⁺

m). Then (3.2) immediately follows from the fact that ([5, (3.4)])

h(K_m⁺) = (q − 1)^−[Km+:k]+1 Y

χ06=χ∈ bG^J_m

X

a∈M_fχ

χ(a)g_f⁺

χ(a).

From [12, Lemma 3, Prop. 4], we have Y

χ∈ bGm\ bG^J_m

X

a∈M_fχ

χ(a)g_f⁻_χ(a) = h⁻(Km) = (q − 1)^emh⁻(OKm),

where em is [K_m⁺ : k] − 1 if m is a prime power and [K_m⁺ : k] − 2 otherwise.

Since wKm = q − 1 = |J|, we get (3.3). ¤

We now fix n, m ∈ A⁺ with n|m. Let H = Gal(Km/Kn). We identify bG^H_m with bGn. Then bG^JH_m is the group of all real characters of Gn. Let M^∗_m,n = {a ∈ Mm: (a)n∈ M⁺_n}. For a ∈ A with (a, m) = 1, we define f_m^±(a) = f_m^±(σa).

Theorem 3.3. Let m, n ∈ A⁺ with n|m. Assume that we are given {af,σ ∈ C : f |m, f 6= 1, σ ∈ G}, and let f_m⁺, f_m⁻ be defined as in (3.1). We define two matrices

D_m,n⁺ =¡

f_m⁺(ab) − f_m⁺(a (b)n) : a, b ∈ M^∗_m,n\M⁺_n¢ , D⁻_m,n=¡

α_c,d: c, d ∈ M_m\(M^∗_m,n∪ M_n)¢ ,

where α_c,d= f_m⁻(cd)−f_m⁻(cd/sgn_n(d))−f_m⁻(c (d)_n)+f_m⁻(c (d)_n/ sgn_n(d)). Then we have

det(D_m,n⁺ ) = ±h(O_Km⁺)R(K_m⁺) h(O_K⁺

n)R(Kn⁺) × C_m,n⁺ and

det(D⁻_m,n) = ±|J|^{[Km+:k]−[Kn+:k]}h⁻(OKm)QKnwKn

h⁻(O_Kn)Q_Kmw_Km × C_m,n⁻ , where C_m,n⁺ =Q

χ∈ bG^J\ bG^JHcχ and C_m,n⁻ =Q

χ6∈ bG^J∪ bG^Hcχ. Proof. By Lemma 3.1 and Proposition 3.2, we have

Y

χ∈ bG^J\ bG^JH

X

σ∈G

χ(σ)f_m⁺(σ) = Y

χ∈ bG^J\ bG^JH

X

a∈M_fχ

χ(a)g_f⁺_χ(a) × C_m,n⁺

= Q

χ06=χ∈ bG^J

P

a∈M_fχχ(a)g⁺_f

χ(a) Q

χ06=χ∈ bG^JH

P

a∈M_fχχ(a)g_f⁺

χ(a)× C_m,n⁺ (3.4)

= ±|J|^{[Km+:k]−[Kn+:k]}h(O_K+

m)R(K_m⁺)

h(O_Kn⁺)R(Kn⁺)× C_m,n⁺ . Similarly, we have

(3.5) Y

χ6∈ bG^J∪ bG^H

X

σ∈G

χ(σ)f_m⁻(σ) = ±|J|^{[Km+:k]−[Kn+:k]}h⁻(OKm)QKnwKn

h⁻(OKn)QKmwKm

× C_m,n⁻ .

We note that σ(M^∗_m,n), σ(Mn) and σ(M⁺_n) are systems of representatives of G/J, G/H and G/JH, respectively and they satisfy conditions for Proposition 2.1 and Theorem 2.4. It is easy to see that rJ(σa) = σ_{a/ sgnn(a)} and rH(σa) = σ(a)n for any a ∈ Mm. Since f_m⁺ ∈ L²(G)^J, we have s^J_f+

m = |J|f_m⁺. Thus by combining (2.1) and (2.2) with (3.4) and (3.5) respectively, we get the

results. ¤

Remark 3.4. (i) Let M^∗∗_m,n= Mm\M^∗_m,n= {a ∈ Mm: (a)n∈ M⁻_n}. When k = Q, we note that σ(M^∗∗_m,n), σ(Mn) and σ(M⁻_n) are systems of representatives of G/J, G/H and G/JH, respectively and they satisfy conditions for Proposition 2.1 and Theorem 2.4. Thus Theorem 3.3 still holds when we replace M^∗_m,n\M⁺_n

(resp. Mm\(M^∗_m,n∪ Mn)) by M^∗∗_m,n\M⁻_n (resp. Mm\(M^∗∗_m,n∪ Mn)) in the definition of D_m,n⁺ (resp. D⁻_m,n).

(ii) Assume that k = Fq(T ). It is known [5, (3.2)] that

|J|^[K+m:k]h⁻(OKm) QKmwKm

= h⁻(Km).

Thus by Theorem 3.3, we have

det(D_m,n⁻ ) = ±h⁻(K_m)

h⁻(Kn) × C_m,n⁻ .

We note that C_m,n^± is dependent on the choice of {af,σ : 1 6= f |m and σ ∈ G}. For some special choices of them, we compute C_m,n^± .

Example 3.5. For a prime (or monic irreducible) p in A, let ep, fpand gp(resp.

˜

ep, ˜fp and ˜gp) be the ramification index of p, the residue class degree of p and the number of primes lying above p in K_m(resp. in K_n), respectively. We also define e⁺_p, f_p⁺ and g_p⁺ (resp. ˜e⁺_p, ˜f_p⁺ and ˜g_p⁺) the corresponding ones for K_m⁺ (resp. for K_n⁺). Put af,σ= 1 if f = m, σ = 1 and 0 otherwise. Then we have f_m^± = g^±_mand cχ=Q

p|m(1 − ¯χ(p)). Let C_m,n⁺ (z) =Q

χ∈ bG^J\ bG^JH

Q

p|m(z − ¯χ(p)) and C_m,n⁻ (z) = Q

χ6∈ bG^J∪ bG^H

Q

p|m(z − ¯χ(p)) as in [5, Lemma 3.2, 3.7]. By [9, Thm. 3.7], we have

C_m,n⁺ (z) =Y

p|m

(z^fp+− 1)^g+p (z^f˜p+− 1)^˜g+p, C_m,n⁻ (z) =Y

p|m

(z^fp− 1)^gp (z^fp+− 1)^g+p

³ (z^f˜p− 1)^g˜p (z^f˜p+− 1)^˜g+p

´₋₁ .

By considering the number of the factor (z − 1) in numerator and denominator of C_m,n^± (z), we have

C_m,n⁺ = (Q

p|m(f_p⁺/ ˜f_p⁺)^gp+ if g_p⁺= ˜g⁺_p for all p|m,

0 otherwise

and

C_m,n⁻ =



 Q

p|m

³ fp^gp

f_p^+g+p

´ /³ _˜

fp^˜gp

f˜_p^+˜g+p

´

if g_p− g⁺_p = ˜g_p− ˜g⁺_p for all p|m,

0 otherwise.

Example 3.6. Since C_m,n^± in Example 3.5 are zero for some cases, we make a choice of {af,σ∈ C : 1 6= f |m, σ ∈ G} which gives nonzero C_m,n^± as [6, Example 4]. Let m =Q_s

i=1p^r_iⁱ and S = {1, . . . , s}. For I ⊆ S, define mI =Q

i∈Ip^r_iⁱ. For 1 6= f |m and σ ∈ G, we define

af,σ= (Q

i6∈I(−Φ(p^r_iⁱ)⁻¹) if f = mI for some I ⊆ S and σ = 1,

0 otherwise.

Then, as Kuˇcera [6], we have C_m,n⁺ = ±1 and C_m,n⁻ = ±1.

4. Cyclotomic fields of prime power conductor cases

For any prime (or monic irreducible) p in A and ` ≥ 2 (` ≥ 3 if p = 2), put m = p^{`</sup> and n = p<sup>`−1}. In this section we show that recursion formulas of Girstmair [3, Theorem] and Jung-Ahn [4, Thm. 4.1] are immediately obtained from Theorem 3.3. We also obtain a similar recursion formula for the real class numbers h(OKm) and h(OKn).

Assume that k = Q. For j ∈ Z with (p, j) = 1, let j^∗ ∈ {0, . . . , p − 1} such that (j − (j)n)/n ≡ j^∗ mod p and hjiG = j^∗/p if (j)n< n/2 and 0 otherwise.

Proposition 4.1 ([3], Theorem). For prime p ≥ 3 with ` ≥ 2 (or p = 2 with

` ≥ 3), let m = p<sup>`</sup> and n = p^`−1. Define the matrix C =¡

hcdiG− h−cdiG− hc (d)niG+ h−c (d)niG: c, d ∈ Mm\(M^∗∗_m,n∪ Mn)¢ . Then we have

h⁻(OKm) = p | det C| h⁻(OKn).

Proof. We note that QKm = QKn = 1, wKm = 2m and wKn = 2n (resp.

wKm = m and wKn= n) for odd p (resp. p = 2). Put af,σ= 1 if f = m, σ = 1 and 0 otherwise. Then we have C_m,n⁻ = 1 and f_m⁻(a) = g⁻_m(a) = ¹₂− h_m^ai for a ∈ A with (a, m) = 1. Thus by Theorem 3.3 and Remark 3.4 (i), we have (4.1) det¡

εc,d: c, d ∈ Mm\(M^∗∗_m,n∪ Mn)¢

= ±2^{[Km+:k]−[Kn+:k]} h⁻(O_Km) p h⁻(OKn), where εc,d= f_m⁻(cd) − f_m⁻(cd/ sgn_n(d)) − f_m⁻(c (d)n) + f_m⁻(c (d)n/ sgn_n(d)). We claim that

1 2

¡hcd/mi − h−cd/mi − hc (d)n/mi + h−c (d)n/mi¢

= hcdiG− h−cdiG− hc (d)niG+ h−c (d)niG

for any c, d ∈ Mm\(M^∗∗_m,n∪ Mn). Since h−j/mi = 1 − hj/mi if hj/mi 6= 0, we have

1 2

¡hcd/mi − h−cd/mi − hc (d)n/mi + h−c (d)n/mi¢

= hcd/mi − hc (d)n/mi.

We note that j^∗/p = (j − (j)n)/m if j ∈ Mm. Suppose that (cd)n< n/2. Since (−cd)n> n/2 and (c (d)n)n= (cd)n< n/2, we have

hcdiG− h−cdiG− hc (d)niG+ h−c (d)niG

= (cd)m− ((cd)m)n

m −(c (d)n)m− ((c (d)n)m)n

m

= (cd)m

m −(c (d)n)m

m = hcd/mi − hc (d)n/mi

because ((cd)m)n= ((c (d)n)m)n = (cd)n. Thus the claim holds when (cd)n<

n/2. Similarly, we show that the claim holds when (cd)n > n/2. Thus we get

the result from (4.1). ¤

Assume that k = Fq(T ). For a ∈ A with (a, m) = 1, we define haiJA= 1 if sgn_m(a) = 1 and 0 otherwise.

Proposition 4.2 ([4], Theorem 4.1). For a monic irreducible p and ` ≥ 2, let m = p<sup>`</sup> and n = p^`−1. Define the matrix

D = ¡

hcdiJA− hcd/ sgn_n(d)iJA− hc (d)niJA+ hc (d)n/ sgn_n(d)iJA: c, d ∈ Mm\(M^∗_m,n∪ Mn)¢

. Then we have

h⁻(Km) = | det D| h⁻(Kn).

Proof. In this case, we have that QKm = QKn = 1 and wKm = wKn = q − 1.

Put af,σ = 1 if f = m, σ = 1 and 0 otherwise. Then C_m,n⁻ = 1 and f_m⁻(a) = Zm(0, a) = haiJA−_q−1¹ for a ∈ A with (a, m) = 1 ([4, Lemma 3.1]). Thus the result immediately follows from Theorem 3.3 and Remark 3.4 (ii). ¤ Finally we obtain a recursion formula for the real class numbers h(O_K⁺_m) and h(O⁺_Kn) from Theorem 3.3.

Proposition 4.3. Let p ∈ A be a prime (or monic irreducible) and ` ≥ 2 integer (if p = 2, we assume ` ≥ 3).

(i) When k = Q, det¡

log | sin(πab/m)/ sin(πa (b)n/m)| : a, b ∈ M^∗_m,n\M⁺_n¢

= ±h(O_K⁺

m)R(K_m⁺) h(O_Kn⁺)R(Kn⁺). (ii) When k = Fq(T ),

det¡

deg_m(ab) − deg_m(a (b)n) : a, b ∈ M^∗_m,n\M⁺_n¢

= ±h(O_K⁺

m)R(K_m⁺)

h(O_Kn⁺)R(Kn⁺) = ±h(K_m⁺) h(Kn⁺).

Proof. Put af,σ= 1 if f = m, σ = 1 and 0 otherwise. Then we have C_m,n⁺ = 1.

When k = Q, we have f_m⁺(a) = g_m⁺(a) = log |1 − ζ_m^a| = log | sin(πa/m)|.

When k = Fq(T ), we have f_m⁺(a) = g⁺_m(a) = deg(m) − 1 − deg_m(a) − _q−1¹ and so f_m⁺(ab) − f_m⁺(a (b)n) = −(deg_m(ab) − deg_m(a (b)n)). Thus the result immediately follows from Theorem 3.3 and Remark 3.4. ¤

References

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Hwan Yup Jung

Department of Mathematics Education Chungbuk National University

Cheongju 361-763, Korea

E-mail address: [email protected] Jaehyun Ahn

Department of Mathematics Chungnam National University Daejon 305-764, Korea

E-mail address: [email protected]