Fermat quotient of cyclotomic units by

LXXVI.4 (1996)

Fermat quotient of cyclotomic units

by

Tsutomu Shimada (Yokohama)

Introduction. Let p be an odd prime number, K a finite extension of the field of rational numbers Q, O

_K

the ring of integers of K, and E

_K

its group of units. Let N be the set of natural numbers, and Z and Z

p

be the ring of rational integers and the ring of p-adic integers, respectively.

We define

E

_K

(p

ⁿ

) = {u ∈ E

_K

: u ≡ 1 mod p

ⁿ

}, n ∈ N.

For each u ∈ E

K

(p), we call u − 1

p mod p (∈ O

K

/(p))

the Fermat quotient mod p of u. That is, for a unit u = 1 + px

u

, x

u

∈ O

K

, x

_u

mod p is the Fermat quotient mod p of u. From now on, we omit “mod p”

for simplicity.

We define a homomorphism ψ: E

K

(p) → O

K

/(p) by u = 1 + px

u

7→

x

_u

mod p. Let F(K) denote the set of all Fermat quotients of u ∈ E

_K

(p) or the image of ψ. Clearly, F(K) forms a subspace of F

_p

-vector space O

_K

/(p) where F

p

denotes the field with p elements, and kernel of ψ is E

K

(p

²

). So, we have

F(K) ∼ = E

K

(p)/E

_K

(p

²

) as F

_p

-vector spaces.

Now, the following two statements are well known: first, if ψ(u

₁

), . . . , ψ(u

s

) are linearly independent over F

p

then u

1

, . . . , u

s

∈ F

K

(p) are Z

p

- independent, and secondly, the dimension of E

_K

(p

ⁿ

)/E

_K

(p

ⁿ⁺¹

) over F

_p

equals the Z

p

-rank of E

K

(p) for sufficiently large n (see, Levesque [3] and Sands [4]). On the other hand, the Leopoldt conjecture states that Z

_p

-rank of E

_K

(p) equals Z-rank of E

_K

.

The aim of the present article is to study the dimension of F(K) over F

_p

, when K is a cyclotomic field.

[335]

1. Notations and results. Let ζ

n

= exp(2πi/n) for n ∈ N, and let m ∈ N be odd, square free, 3 ≤ m and prime to p. We let K = Q(ζ

_mp

) and let E

_K

(p

ⁿ

), F(K) and ψ be as in the introduction.

Since O

K

= Z[ζ

mp

] = Z[ζ

m

][1 − ζ

p

] and m is square free, the set {ζ

_m^r

(1 − ζ

p

)

^ν

: 1 ≤ r ≤ m, (r, m) = 1, ν = 0, 1, . . . , p − 2}

forms a Z-basis of O

_K

. As is well known,

dim

_Fp

(Z[ζ

_mp

]/(p)) = [Q(ζ

_mp

) : Q] = ϕ(mp), where ϕ denotes the Euler function. Therefore, the set

{ζ

_m^r

(1 − ζ

_p

)

^ν

mod p : 1 ≤ r ≤ m, (r, m) = 1, ν = 0, 1, . . . , p − 2}

forms an F

p

-basis of O

K

/(p).

Representing each x mod p ∈ O

_K

/(p) in this basis, we have x ≡ c

₀

+ c

₁

(1 − ζ

_p

) + c

₂

(1 − ζ

_p

)

²

+ . . . + c

_p−2

(1 − ζ

_p

)

^p−2

mod p with c

i

∈ Z[ζ

m

] (i = 0, 1, . . . , p − 2), determined uniquely modulo p.

Let π denote 1 − ζ

_p

and let

E

_K

(pπ

ⁱ

) = {u ∈ E

_K

: u ≡ 1 mod pπ

ⁱ

} (i = 1, . . . , p − 2).

Since

E

_K

(p) ⊃ E

_K

(pπ) ⊃ . . . ⊃ E

_K

(pπ

^p−2

) ⊃ E

_K

(p

²

), and u

^p

∈ E

_K

(p

²

) for all u ∈ E

_K

(p), we have

F(K) = (E

_K

(p)/E

_K

(pπ)) ⊕ . . . ⊕ (E

_K

(pπ

^k+1

)/E

_K

(pπ

^k+2

)) ⊕ . . .

. . . ⊕ (E

_K

(pπ

^p−2

)/E

_K

(p

²

)).

Thus

dim

_Fp

F(K) = dim

_Fp

(E

_K

(p)/E

_K

(pπ)) + . . .

. . . + dim

_Fp

(E

_K

(pπ

^k+1

)/E

_K

(pπ

^k+2

)) + . . .

. . . + dim

F_p

(E

K

(pπ

^p−2

)/E

K

(p

²

)), −1 ≤ k ≤ p − 3.

We define subsets V

k

(k = −1, 0, 1, . . . , p − 3) of F(K) by V

k

= {x

u

mod p ∈ F(K) : u ∈ E

K

(p),

c

₀

≡ c

₁

≡ . . . ≡ c

_k

≡ 0 mod p, c

_k+1

6≡ 0 mod p}, where u = 1 + px

u

and x

u

≡ c

0

+ c

1

(1 − ζ

p

) + . . . + c

p−2

(1 − ζ

p

)

^p−2

mod p, and e V

k

be the subspace generated by all elements in V

k

over F

p

. Of course,

F(K) = V

₋₁

∪ V

₀

∪ V

₁

∪ . . . ∪ V

_p−3

∪ {0 mod p} (disjoint union).

For each k (−1 ≤ k ≤ p − 3), we define a mapping π

k

: e V

k

→ Z[ζ

m

]/(p)

by x

_u

mod p 7→ c

_k+1

mod p and let V

_k

= π

_k

( e V

_k

). Then, since V

_k

∼ =

E

K

(pπ

^k+1

)/E

K

(pπ

^k+2

) for −1 ≤ k ≤ p − 3, we have dim

_Fp

F(K) =

p−3

X

k=−1

dim

_Fp

V

_k

.

We now take polynomials S

t

(X) ∈ Q[X], Z 3 t ≥ 0, such that S

t

(n) = 1

^t

+ 2

^t

+ . . . + n

^t

for all n ∈ N. For example,

S

0

(X) = X, S

1

(X) =

¹₂

X(X + 1), S

2

(X) =

¹₆

X(X + 1)(2X + 1), etc.

As is well known, (k + 1)!S

_k

(X) ∈ Z[X], deg S

_k

(X) = k + 1, and S

_k

(−1) = 0 for k ≥ 1.

We define I

_k

(n) =

X

n 0 l=1

S

_k



− l n



ζ

_n^l

, 0 ≤ k ≤ p − 2, n ∈ N, where P

_n0

l=1

denotes the sum taken over all l = 1, . . . , n that are prime to n.

Let G be the Galois group of Q(ζ

m

) over Q and b G its character group. As is well known, G is isomorphic to (Z/mZ)

^×

(the multiplicative group of all residue classes prime to m) by assigning σ

_µ

: ζ

_m

7→ ζ

_m^µ

to µ mod m.

Note that ψ is a Gal(K/Q)-homomorphism. We now state our results, Theorems 1–4.

Theorem 1. (1) There exist units α

k

∈ E

K

(p), 1 ≤ k ≤ p − 3, such that π

_k

(ψ(α

_k

)) = I

_k

(m) mod p.

(2) For k = 0, there exists α

0

∈ E

K

(p) such that π

₀

(ψ(α

₀

)) = (1 − σ

_p

)I

₀

(m) mod p.

(3) For k = −1, there exist β

_ν

∈ E

_K

(p), 2 ≤ ν ≤ m/2 and (ν, m) = 1, such that

π

₋₁

(ψ(β

_ν

)) = (1 − σ

_ν

)I

_p−2

(m) mod p.

(4) If B

_k+1

6≡ 0 mod p for some k = 1, 3, . . . , p − 4, then there exists u

k

∈ E

K

(p) such that

π

k

(ψ(u

k

)) = 1 mod p,

where B

n

denote the Bernoulli numbers (the definition will be given in Section 4).

For any a ∈ Z, let M (a) ∈ Z denote the non-negative minimal residue

of a mod m, that is, a ≡ M (a) mod m and 0 ≤ M (a) ≤ m − 1. When

b ∈ Z is prime to m, we let M (1/b) ∈ Z be the integer such that b ×

M (1/b) ≡ 1 mod m and 0 ≤ M (1/b) ≤ m − 1. Also, we take M (a/b) for

M (M (a) × M (1/b)).

For any σ

r

∈ G and k (0 ≤ k ≤ p − 2), I

_k

(m)

^σr

=

X

m ₀ l=1

S

_k



− l m

 ζ

_m^lr

=

X

m ₀ l=1

S

_k



− M (l/r) m

 ζ

_m^l

.

We then define for each k, 0 ≤ k ≤ p − 2, the matrix A

_k

(m) =

 S

_k



− M (l/r) m



1≤r,l≤m (r,m)=(l,m)=1

,

where we index the rows by r, and the columns by l. Since I

_k

(m)

^σ−1

= (−1)

^k+1

I

k

(m) for 1 ≤ k ≤ p − 2 (see Lemma 3.2), we have rank A

k

(m) ≤

1

2

ϕ(m) for such k.

In addition, we define B

_k

(m) =

 S

_k



− M (l/r) m



1≤r,l≤m/2 (r,m)=(l,m)=1

, 0 ≤ k ≤ p − 2.

Theorem 2. For each k (1 ≤ k ≤ p − 3), we have det B

_k

(m) =

 −1 2m

^k



_ϕ(m)/2

ζ

_Q(ζm)+

(−k) Y

χ∈G

b

id.6=χ:even

Y

q|m

(1 − χ

₁

(q)q

^k

)

× Y

q|m

(1 − q

^k

) − m

^k

ϕ(m)

 ,

if k is odd, and det B

_k

(m) =

 1 2m

^k



_ϕ(m)/2

ζ

_Q(ζm)

ζ

_Q(ζm)+

(−k) Y b

G3χ:odd

Y

q|m

(1 − χ

₁

(q)q

^k

),

if k is even. Moreover , if det B

_k

(m) 6≡ 0 mod p, then dim

_Fp

V

_k

≥

¹₂

ϕ(m).

Here ζ

_Q(ζm)

and ζ

_Q(ζm)+

denote the Dedekind zeta functions of Q(ζ

_m

) and Q(ζ

m

)

⁺

(the maximal real subfield of Q(ζ

m

)), respectively, and χ

1

the primitive Dirichlet character associated with χ. Also, Q

q|m

denotes the product taken over all distinct primes q which divide m.

B

_k

(m) is, in some sense, a generalization of matrices defined by Carlitz [1] and Tateyama [5].

Let f be the order of the element p mod m in (Z/mZ)

^×

, and g = ϕ(m)/f .

Let g(m) and g

⁺

(m) denote the number of distinct prime ideals which divide

m in Q(ζ

m

) and Q(ζ

m

)

⁺

, respectively. We write h

⁻

(Q(ζ

m

)) for the relative

class number of Q(ζ

_m

). By the theorem of Tateyama [5], we can prove

Theorem 3. Assume g(m) = g

⁺

(m) and p does not divide h

⁻

(Q(ζ

m

)).

Then

dim

_Fp

V

₀

≥ (

1

2

ϕ(m) if p does not decompose in Q(ζ

_m

)/Q(ζ

_m

)

⁺

,

1

2

(ϕ(m) − g) if p decomposes in Q(ζ

_m

)/Q(ζ

_m

)

⁺

.

Note that when the right-hand side in the inequality above is not positive, or p ≡ 1 mod m, our theorem says nothing. Finally, by the same calculation as in the proof of Theorem 2 we obtain:

Theorem 4. We have det B

_p−2

(m) =

 −1 2m

^p−2



_ϕ(m)/2

ζ

_Q(ζm)+

(2 − p) Y

χ∈G

b

id.6=χ:even

Y

q|m

(1 − χ

₁

(q)q

^p−2

)

×  Y

q|m

(1 − q

^p−2

) − m

^p−2

ϕ(m)



and if det B

_p−2

(m) 6≡ 0 mod p, then

dim

_Fp

V

₋₁

≥

¹₂

ϕ(m) − 1.

The rest of the article will be devoted to the proofs of the theorems stated above. In Section 2, we discuss some elementary properties of the Fermat quotient of cyclotomic units. The main result here is Lemma 2.5. In Section 3, introducing I

_k

(m), we prove Theorem 1(1). In Section 4, we prove Theorem 2 and the first part of Theorem 4. In Section 5, discussing I

₀

(m) and the rank of A

0

(m), we prove Theorem 1(2) and Theorem 3. In Section 6, by the argument in Q(ζ

_m

), we prove Theorem 1(3) and the second part of Theorem 4. Finally, the proof of Theorem 1(4), which is obtained essentially in Washington [6], will be given in Section 7.

Before concluding this section, we classify typical generators of cyclo- tomic units (in the sense of Sinnot) into three types:

I. 1 − ζ

_d

ζ

_p

(1 6= d | m),

II. 1 − ζ

_d

(1 6= d | m, d is composite), 1 − ζ

_q^ν

1 − ζ

_q

(q | m, q is a prime, 2 ≤ ν ≤ q − 1), III. 1 − ζ

_p^ν

1 − ζ

_p

(2 ≤ ν ≤ p − 1).

We shall use cyclotomic units of type I to prove Theorem 1(1), (2), type II (units in Q(ζ

_m

)) to prove Theorem 1(3), and type III (units in Q(ζ

_p

)) to prove Theorem 1(4).

Also, see Leopoldt [2] for units of type II.

2. Fermat quotient of units of type I. We first prove a preliminary lemma:

Lemma 2.1. For each ν ∈ N, we have (1 − X)

^pν

≡ 1 − X

^pν

− p



X

^pν−1

+ X

^2pν−1

2 + . . . + X

^{(p−1)pν−1}

p − 1



mod p

²

. P r o o f. We have

(1 − X)

^pν

= 1 − X

^pν

+

p

X

^ν−1 i=1

 p

^ν

i



(−X)

ⁱ

. Since

(p

^ν

− 1) . . . (p

^ν

− i + 1)

(i − 1)! ≡ (−1)

ⁱ⁻¹

mod p

^ν

, we have

 p

^ν

i



≡ (−1)

ⁱ⁻¹

i p

^ν

mod p

^2ν−ordpi

for all i = 1, . . . , p

^ν

− 1,

where ord

p

i denotes the exact exponent of the power of p dividing i. Now, we have 2ν − ord

_p

i ≥ 2ν − (ν − 1) ≥ 2 and ord

_p

(p

^ν

/i) ≥ ν − (ν − 1) = 1.

So,

^p_i^ν

6≡ 0 mod p

²

if and only if ord

_p

i = ν − 1, i.e. i = jp

^ν−1

for some j (1 ≤ j ≤ p − 1). If i = jp

^ν−1

(1 ≤ j ≤ p − 1), then

 p

^ν

i



(−X)

ⁱ

≡ (−1)

ⁱ⁻¹

j p(−X)

ⁱ

= − p

j X

^jpν−1

mod p

²

. This completes the proof.

We define

f (X) = 1 1 − X

^p



X + X

²

2 + . . . + X

^p−1

p − 1



∈ Q(X).

Using Lemma 2.1, we have

(1 − ζ

_m

ζ

_p

)

^{pf +1}

≡ 1 − ζ

_m^{pf +1}

− p



ζ

_m^pf

+ ζ

_m^2pf

2 + . . . + ζ

m^(p−1)pf

p − 1



= 1 − ζ

_m^p

− p



ζ

_m

+ ζ

_m²

2 + . . . + ζ

_m^p−1

p − 1



= (1 − ζ

_m^p

)(1 − pf (ζ

m

)) mod p

²

, (1 − ζ

_m

ζ

_p

)

^p

≡ 1 − ζ

_m^p

− p



ζ

_m

ζ

_p

+ ζ

_m²

ζ

_p²

2 + . . . + ζ

_m^p−1

ζ

_p^p−1

p − 1



= (1 − ζ

_m^p

)(1 − pf (ζ

_m

ζ

_p

)) mod p

²

,

and so

(1 − ζ

m

ζ

p

)

^{pf +1−p}

≡ 1 − p(f (ζ

m

) − f (ζ

m

ζ

p

)) mod p

²

.

Consequently, f (ζ

_m

)−f (ζ

_m

ζ

_p

) mod p belongs to F(K). Note that (O

_K

/(p))

^×

(the multiplicative group of O

_K

/(p)) has the exponent p

^{f +1}

− p. We first prove the existence of a “canonical” element in F(K), a linear combination over F

_p

of conjugates of f (ζ

_m

) − f (ζ

_m

ζ

_p

) mod p (see Lemma 2.3), and next determine the coefficients modp of its image by π

k

(see Lemma 2.5). From ζ

_p^j

= ((ζ

_p

− 1) + 1)

^j

, it can be easily seen that

1 − ζ

_p^j

= X

j i=1

 j i



(−1)

ⁱ⁺¹

(1 − ζ

_p

)

ⁱ

. Hence

f (ζ

_m

) − f (ζ

_m

ζ

_p

) = 1 1 − ζ

m^p

p−1

X

j=1

ζ

_m^j

j (1 − ζ

_p^j

) (1)

= 1

1 − ζ

m^p p−1

X

j=1

X

j i=1

ζ

_m^j

j

 j i



(−1)

ⁱ⁺¹

(1 − ζ

p

)

ⁱ

= 1

1 − ζ

m^p p−1

X

i=1 p−1

X

j=i

ζ

_m^j

j

 j i



(−1)

ⁱ⁺¹

(1 − ζ

p

)

ⁱ

, that is,

f (ζ

_m

) − f (ζ

_m

ζ

_p

) = 1 1 − ζ

m^p

p−1

X

i=1



p−1

X

j=1 j i

j ζ

_m^j



(−1)

ⁱ⁺¹

(1 − ζ

_p

)

ⁱ

,

where

^j_i

= 0 if j < i.

Lemma 2.2. For each ν = 1, . . . , p − 1 we have f (ζ

m

) − f (ζ

m

ζ

_p^ν

) ≡ 1

1 − ζ

m^p p−2

X

i=1



p−1

X

j=1 νj

i

j ζ

_m^j



(−1)

ⁱ⁺¹

(1 − ζ

p

)

ⁱ

mod p.

P r o o f. Taking ζ

_p^ν

for ζ

_p

in both sides of (1), we get

f (ζ

m

) − f (ζ

m

ζ

_p^ν

) = 1 1 − ζ

m^p

p−1

X

j=1

ζ

_m^j

j (1 − ζ

_p^νj

).

We denote by P (a) (a ∈ Z) the non-negative minimal residue of a mod p

and use it similarly to M (a) introduced in the previous section. Then

p−1

X

j=1

ζ

_m^j

j (1 − ζ

_p^νj

)

≡

p−1

X

j=1

ζ

m^{P (j/ν)}

j/ν (1 − ζ

_p^j

)

= ν

p−1

X

j=1

ζ

m^{P (j/ν)}

j (1 − ζ

_p^j

) = ν

p−1

X

j=1

ζ

m^{P (j/ν)}

j

 X

j i=1

 j i



(−1)

ⁱ⁺¹

(1 − ζ

p

)

ⁱ



= ν

p−1

X

i=1



p−1

X

j=1 j i

j ζ

_m^{P (j/ν)}



(−1)

ⁱ⁺¹

(1 − ζ

_p

)

ⁱ

mod p.

Now

^j_i

≡

^{νP (j/ν)}_i

mod p, as νP (j/ν) ≡ j mod p. It follows that

p−1

X

j=1 j i

j ζ

_m^{P (j/ν)}

≡

p−1

X

j=1 1

ν νP (j/ν) i

j/ν ζ

_m^{P (j/ν)}

≡

p−1

X

j=1 1

ν νP (j/ν) i

P (j/ν) ζ

_m^{P (j/ν)}

=

p−1

X

j=1 1 ν νj

i

j ζ

_m^j

mod p.

We then have

p−1

X

j=1

ζ

_m^j

j (1 − ζ

_p^νj

) ≡

p−1

X

i=1



p−1

X

j=1 νj

i

j ζ

_m^j



(−1)

ⁱ⁺¹

(1 − ζ

p

)

ⁱ

mod p.

This completes the proof.

Let g

i

(X) (1 ≤ i ≤ p − 1) be a polynomial with coefficients in F

p

such that g

i

(0) = 0 and of degree ≤ i. We define a map e τ

ν

(1 ≤ ν ≤ p − 1) from F

_p

[X] into itself by e τ

_ν

(g(X)) = g(νX) for all g(X) ∈ F

_p

[X]. Let N

_i

(1 ≤ i ≤ p − 1) be the i × i matrix

N

_i

=



 



1 1 . . . 1 2

ⁱ

2

ⁱ⁻¹

. . . 2 .. . .. . . .. ...

i

ⁱ

i

ⁱ⁻¹

. . . i



 

 ,

where each component represents a corresponding residue class mod p. Then it is clear that

N

_i

=



 



1 0

2 . ..

0 i



 





 



1 1 . . . 1 2

ⁱ⁻¹

2

ⁱ⁻²

. . . 1 .. . .. . . .. ...

i

ⁱ⁻¹

i

ⁱ⁻²

. . . 1



 

 ,

therefore det N

i

6= 0, and

N

_i



 



a

_i

X

ⁱ

a

_i−1

X

ⁱ⁻¹

.. . a

1

X



 

 =



 

 e

τ

₁

(g

_i

(X)) e

τ

₂

(g

_i

(X)) .. . e

τ

_i

(g

_i

(X))



 

 ,

where g

i

(X) = a

i

X

ⁱ

+ a

i−1

X

ⁱ⁻¹

+ . . . + a

1

X.

Let (c

i,1

c

i,2

. . . c

i,i

) be the first row of N

⁻¹_i

. Then

a

i

X

ⁱ

= c

i,1

τ e

1

(g

i

(X)) + c

i,2

τ e

2

(g

i

(X)) + . . . + c

i,i

τ e

i

(g

i

(X)).

Letting e C

_i

= c

_i,1

τ e

₁

+ . . . + c

_i,i

e τ

_i

, we write a

_i

X

ⁱ

= e C

_i

(g

_i

(X)), that is C e

i

(g

i

(X)) =

 0 if deg g

_i

(X) < i, a

i

X

ⁱ

if deg g

i

(X) = i.

Now we take

^j_i

, 1 ≤ i ≤ p − 2, for a polynomial in an indeterminate j with coefficients in F

_p

.

Since

^j_i

= (1/i!)j

ⁱ

+ (polynomial with degree ≤ i − 1), for k (0 ≤ k ≤ p − 3) we have

C e

_k+1

 j i



=

 



 

0 if i < k + 1, 1

(k + 1)! j

^k+1

if i = k + 1,

∗ if i > k + 1,

where ∗ means some polynomial in F

p

[j], irrelevant for our purpose.

Let τ

_ν

(1 ≤ ν ≤ p − 1) be an automorphism of Q(ζ

_mp

) over Q such that τ

ν

: ζ

p

7→ ζ

_p^ν

and τ

ν

: ζ

m

7→ ζ

m

, and C

i

(1 ≤ i ≤ p − 1) be the element c

_i,1

τ

₁

+ c

_i,2

τ

₂

+ . . . + c

_i,i

τ

_i

of the group ring of Gal(Q(ζ

_mp

)/Q) over F

_p

(we write its action on F(K) additively). Then we obtain the following:

Lemma 2.3. For each k, 0 ≤ k ≤ p − 3, we have C

_k+1

(f (ζ

_m

) − f (ζ

_m

ζ

_p

))

≡ 1

(k + 1)!

 1

1 − ζ

m^p p−1

X

j=1

j

^k

ζ

_m^j



(−1)

^k

(1 − ζ

_p

)

^k+1

+ 1

1 − ζ

m^p p−2

X

i=k+2



p−1

X

j=1

C e

k+1 j i

j ζ

_m^j



(−1)

ⁱ⁺¹

(1 − ζ

p

)

ⁱ

mod p.

P r o o f. By Lemma 2.2,

τ

ν

(f (ζ

m

) − f (ζ

m

ζ

p

)) = f (ζ

m

) − f (ζ

m

ζ

_p^ν

)

≡ 1

1 − ζ

m^p p−2

X

i=1



p−1

X

j=1

e τ

ν j

i

j ζ

_m^j



(−1)

ⁱ⁺¹

(1 − ζ

p

)

ⁱ

mod p.

Consequently,

C

k+1

(f (ζ

m

) − f (ζ

m

ζ

p

))

≡ 1

1 − ζ

m^p p−2

X

i=1



p−1

X

j=1

C e

_k+1 ^j_i

j ζ

_m^j



(−1)

ⁱ⁺¹

(1 − ζ

_p

)

ⁱ

≡ 1

1 − ζ

m^p



p−1

X

j=1

C e

k+1 j k+1

j ζ

_m^j



(−1)

^k

(1 − ζ

_p

)

^k+1

+ 1

1 − ζ

m^p p−2

X

i=k+2



p−1

X

j=1

C e

k+1 j i

j ζ

_m^j



(−1)

ⁱ⁺¹

(1 − ζ

p

)

ⁱ

≡ 1

1 − ζ

m^p



p−1

X

j=1

1

(k + 1)! j

^k

ζ

_m^j



(−1)

^k

(1 − ζ

p

)

^k+1

+ 1

1 − ζ

m^p p−2

X

i=k+2



p−1

X

j=1

C e

_k+1 ^j_i

j ζ

_m^j



(−1)

ⁱ⁺¹

(1 − ζ

p

)

ⁱ

mod p, as desired.

We define, for k = 0, 1, . . . , p − 2 and n (∈ N) prime to p, J

k

(n) = 1

1 − ζ

n^p p−1

X

j=1

j

^k

ζ

_n^j

when n ≥ 2 and J

k

(1) = 0.

We can deduce that

J

₀

(n) = 1 1 − ζ

n

− 1

1 − ζ

n^p

= (1 − σ

_p

) 1 1 − ζ

n

and

J

_k

(n)

^σ−1

≡ (−1)

^k+1

J

_k

(n) mod p

for n ≥ 2 by easy calculation, where σ

₋₁

denotes the automorphism which sends ζ

n

to ζ

_n⁻¹

.

Lemma 2.4. For each k (1 ≤ k ≤ p − 2) we have J

_k

(m) ≡

m−1

X

l=1



^p−1

X

j=1, M (l/p)≥M (j/p)6=0

j

^k



ζ

_m^l

mod p.

P r o o f. From the equality −m/(1 − ζ

_m

) = P

_m−1

l=1

lζ

_m^l

, we obtain

−m 1 − ζ

m^p

=

m−1

X

l=1

lζ

_m^lp

and

−mζ

_m^j

1 − ζ

_m^p

=

m−1

X

l=1

lζ

_m^lp+j

=

m−1

X

l=1

lζ

_m^{M (lp+j)}

, 1 ≤ j ≤ p − 1,

where M (∗) is the same as in Section 1. Now, since M ((M (lp + j) − j)/p)

= l and

{M (lp + j) : l = 1, . . . , m − 1} = {0, 1, . . . , m − 1}\{M (j)}, we obtain

−mζ

_m^j

1 − ζ

m^p

=

m−1

X

l=1

M

 M (lp + j) − j p



ζ

_m^{M (lp+j)}

=

m−1

X

l=0

M

 l − j p

 ζ

_m^l

=

m−1

X

l=1

M

 l − j p



ζ

_m^l

+ M

 −j p



(−ζ

_m

− ζ

_m²

− . . . − ζ

_m^m−1

)

=

m−1

X

l=1

 M

 l − j p



− M

 −j p



ζ

_m^l

.

Therefore

J

_k

(m) = − 1 m

p−1

X

j=1

−mζ

_m^j

1 − ζ

m^p

j

^k

= − 1 m

p−1

X

j=1

j

^k



_m−1

X

l=1

 M

 l − j p



− M

 −j p



ζ

_m^l

 ,

that is,

(2) J

k

(m) = − 1 m

m−1

X

l=1



p−1

X

j=1

j

^k

 M

 l − j p



− M

 −j p



ζ

_m^l

.

It is clear that M

 l − j p



=

 M (l/p) − M (j/p) if M (l/p) ≥ M (j/p), M (l/p) − M (j/p) + m if M (l/p) < M (j/p), and

M

 l − j p



−M

 −j p



=

 



M (l/p) if M (j) = 0,

M ((l − j)/p) − {m − M (j/p)}

= M ((l − j)/p) + M (j/p) − m if M (j) 6= 0.

For these reasons, M

 l − j p



−M

 −j p



=

 M (l/p) − m if M (l/p) ≥ M (j/p) 6= 0,

M (l/p) if M (l/p) < M (j/p) or M (j) = 0.

Substituting this in (2), we have J

_k

(m) = − 1

m

m−1

X

l=1

n

^p−1

X

j=1, M (l/p)≥M (j/p)6=0

j

^k

(M (l/p) − m)

+

p−1

X

j=1, M (l/p)<M (j/p) or M (j)=0

j

^k

M (l/p) o

ζ

_m^l

= − 1 m

m−1

X

l=1

n

M (l/p)

p−1

X

j=1

j

^k

− m

p−1

X

j=1, M (l/p)≥M (j/p)6=0

j

^k

o

ζ

_m^l

. The result follows from a well known fact that P

_p−1

j=1

j

^k

≡ 0 mod p.

Lemma 2.5. We have J

k

(m) ≡ (−m)

^k

m−1

X

l=1

S

k



− l m



ζ

_m^l

mod p, 1 ≤ k ≤ p − 2.

P r o o f. It is sufficient to prove (3)

p−1

X

j=1, M (l/p)≥M (j/p)6=0

j

^k

≡ (−m)

^k

S

_k



− l m



mod p.

We deal first with the case where m < p. Let x

_i

= P (−i/m) and y

_i

= (i + x

i

m)/p ∈ Z for each i (1 ≤ i ≤ p − 1), where P (∗) is the same as in the proof of Lemma 2.2. Then we have 1 ≤ y

_i

≤ m.

Suppose that x

_i

< x

_j

for i 6= j, 1 ≤ i, j ≤ p − 1. Then, as j − i = (y

_j

− y

_i

)p − (x

_j

− x

_i

)m and −(p − 2) ≤ j − i, we have

y

_j

− y

_i

≥ −(p − 2) + m

p = −1 + m + 2 p > −1.

Therefore, y

_j

≥ y

_i

if x

_j

> x

_i

. For each l (1 ≤ l ≤ m − 1), let

A

_l

= {j : 1 ≤ j ≤ p − 1, M (l/p) ≥ M (j/p) 6= 0}.

Since y

_l

6= m, or 1 ≤ y

_l

≤ m − 1, and M (j/p) = M (y

_j

), we have A

_l

= {j : 1 ≤ j ≤ p − 1, M (y

_l

) ≥ M (y

_j

) 6= 0}

= {j : 1 ≤ j ≤ p − 1, M (y

_l

) ≥ M (y

_j

), y

_j

6= m}

= {j : 1 ≤ j ≤ p − 1, y

_l

≥ y

_j

}.

If m < i ≤ p − 1, we have i − m = y

i

p − (x

i

+ 1)m; therefore x

i−m

= x

i

+ 1

and y

_i−m

= y

_i

. For that reason, any index i, with common value y

_i

, can be

represented in the form i = j + mν (1 ≤ j ≤ m, Z 3 ν ≥ 0) and, for such

i’s, the maximal value of x

_i

equals x

_j

.

Consequently,

A

_l

= {j : 1 ≤ j ≤ p − 1, x

_j

≤ x

_l

} and

X

p−1

j=1, M (l/p)≥M (j/p)6=0

j

^k

= (−m)

^k

X

j∈A_l



− j m



_k

≡ (−m)

^k

X

j∈A_l

x

^k_j

= (−m)

^k

(1

^k

+ 2

^k

+ . . . + x

^k_l

) = (−m)

^k

S

_k

(x

_l

)

≡ (−m)

^k

S

k



− l m



mod p, as desired.

Next we consider the case where p < m. Let again x

_i

= P (−i/m) and y

_i

= (i + x

_i

m)/p ∈ Z for i (0 ≤ i ≤ m − 1). Since 0 ≤ y

_i

p = i + x

_i

m ≤ m − 1 + (p − 1)m = pm − 1, we have 0 ≤ y

i

≤ m − 1/p, that is, 0 ≤ y

_i

≤ m − 1. It is clear that {x

_i

: 0 ≤ i ≤ p − 1} = {0, 1, . . . , p − 1}. Since i − j = (y

_i

− y

_j

)p − (x

_i

− x

_j

)m for 0 ≤ i, j ≤ m − 1, we see that x

_i

= x

_j

if and only if i ≡ j mod p, and that y

i

< y

j

if x

i

< x

j

. There exist i

0

and ν

_i

(0 ≤ i

₀

≤ p − 1, 1 ≤ ν

_i

) such that i = i

₀

+ ν

_i

p for each i (p ≤ i ≤ m − 1).

Since i = y

_i0

p − x

_i0

m + ν

_i

p = (y

_i0

+ ν

_i

)p − x

_i0

m, we have y

_i

= y

_i0

+ ν

_i

and x

i

= x

i₀

.

Consequently, for l (1 ≤ l ≤ m − 1), we have

A

_l

= {j : 1 ≤ j ≤ p − 1, M (l/p) ≥ M (j/p)} = {j : 1 ≤ j ≤ p − 1, y

_l

≥ y

_j

}

= {j : 1 ≤ j ≤ p − 1, y

l₀

≥ y

j

} = {j : 1 ≤ j ≤ p − 1, x

l₀

≥ x

j

}.

Now, clearly, −l/m ≡ −l

₀

/m ≡ x

_l0

mod p. Hence X

j∈A_l

j

^k

=

p−1

X

j=1,x_j≤x_l0

j

^k

= (−m)

^k

p−1

X

j=1,x_j≤x_l0



− j m



_k

≡ (−m)

^k

p−1

X

j=1,x_j≤x_l0

x

^k_j

= (−m)

^k

(1

^k

+ 2

^k

+ . . . + x

^k_l0

) = (−m)

^k

S

_k

(x

_l0

)

≡ (−m)

^k

S

_k



− l m



mod p.

We have just completed the proof of Lemma 2.5.

3. I

_k

(m) as a partial sum of J

_k

(m). For any divisor d of m we define M

_d

=



l : l = m

d ν, 1 ≤ ν ≤ d, (ν, d) = 1



.

It is clear that ϕ(d) = ]M

d

(the cardinality of M

d

), M

1

= {m}, {1, 2, . . . , m}

= S

d|m

M

_d

(disjoint union), and M

_d

= {l : 1 ≤ l ≤ m, (l, m) = m/d}. Now we have, from Lemma 2.5, for k = 1, . . . , p − 2,

J

_k

(m) ≡ (−m)

^k

X

m

l=1

S

_k



− l m



ζ

_m^l

= (−m)

^k

X

d|m

X

d 0 ν=1

S

_k



−

m d

ν m

 ζ

_m^(m/d)ν

= (−m)

^k

X

d|m

X

d 0 ν=1

S

_k



− ν d



ζ

_d^ν

mod p,

where P

_d0

ν=1

is the same as in Section 1. So, we have (−m)

^−k

J

_k

(m) ≡ X

d|m

I

_k

(d) mod p.

By the M¨obius inversion formula, it follows that I

_k

(m) ≡ X

d|m

{µ(m/d)(−d)

^−k

J

_k

(d)} mod p

for k (1 ≤ k ≤ p − 2), where µ is the M¨obius function. Each J

_k

(d) mod p (d | m) belongs to V

_k

by the definition, so is I

_k

(m) mod p.

We have thus proved Theorem 1(1).

Lemma 3.1. For all k ∈ N, we have

S

_k

(−X) = (−1)

^k+1

S

_k

(X − 1).

P r o o f. From the definition of S

_k

(X), we have S

_k

(n + 1) = (n + 1)

^k

+ S

_k

(n) for all n ∈ N. Hence

S

_k

(X + 1) = (X + 1)

^k

+ S

_k

(X).

Using this formula, we see that

S

k

(−n) = −(−n + 1)

^k

+ S

k

(−n + 1)

= −(−n + 1)

^k

− (−n + 2)

^k

+ S

_k

(−n + 2)

= . . . = −(−n + 1)

^k

− (−n + 2)

^k

− . . . − (−1)

^k

+ S

_k

(−1)

= (−1)

^k+1

{1

^k

+ 2

^k

+ . . . + (n − 1)

^k

} = (−1)

^k+1

S

_k

(n − 1), for all n ∈ N. This proves our lemma.

The next two lemmas, on properties of I

k

(m), will be used in the follow- ing section.

Lemma 3.2. For all k (1 ≤ k ≤ p − 2), we have

I

_k

(m)

^σ−1

= (−1)

^k+1

I

_k

(m).

P r o o f.

I

k

(m)

^σ−1

= X

m ₀ l=1

S

k



− l m



ζ

_m^m−l

= X

m₀ l=1

S

k



− m − l m

 ζ

_m^l

= X

m ₀ l=1

S

_k

 l m − 1



ζ

_m^l

= (−1)

^k+1

X

m ₀ l=1

S

_k



− l m

 ζ

_m^l

= (−1)

^k+1

I

k

(m).

Lemma 3.3. For k = 0, we have

I

₀

(m)

^σ−1

+ I

₀

(m) = −µ(m).

P r o o f.

I

₀

(m)

^σ−1

= X

m 0 l=1



− l m

 ζ

_m^−l

=

X

m0 l=1



− m − l m

 ζ

_m^l

= − X

m 0

l=1

ζ

_m^l

− X

m 0 l=1



− l m



ζ

_m^l

= −µ(m) − I

₀

(m).

This completes the proof of the lemma.

4. Determinant of B

k

(m). Let A

k

(m) and B

k

(m) (0 ≤ k ≤ p − 2) be as in Section 1. Recall that

G = Gal(Q(ζ

_m

)/Q) = (Z/mZ)

^×

= {σ

_r

: ζ

_m

7→ ζ

_m^r

: 1 ≤ r < m, (r, m) = 1};

we shall identify G with {l : 1 ≤ l < m, (l, m) = 1}.

Let H = Gal(Q(ζ

m

)

⁺

/Q) = (Z/mZ)

^×

/{±1}; we shall identify H with {l : 1 ≤ l < m/2, (l, m) = 1}. By Lemma 3.1,

(4) S

k



− m − l m



= (−1)

^k+1

S

k



− l m



, k ∈ N.

Now, let us prove Theorem 2 and the first part of Theorem 4. First we assume k (1 ≤ k ≤ p − 2) is odd. Then, by means of (4), we can take S

_k

(−l/m) for a function on H. Then it is easily verified, by a result on the group determinant, that

det B

_k

(m) = Y

χ∈H

b X

l∈H

S

_k



− l m

 χ(l).

Moreover, by (4), X

l∈H

S

k



− l m



χ(l) = 1 2

X

l∈G

S

k



− l m



χ(l)

for all χ ∈ b H (the set of all even characters in b G). Therefore, (5) det B

k

(m) =

 1 2



_ϕ(m)/2

Y b

G3χ:even

X

l∈G

S

k



− l m

 χ(l).

Next we assume k is even. We fix an arbitrary odd character ξ in b G.

Then, from (4), we have S

k



− m − l m



= −S

k



− l m

 ,

so that S

_k

(−l/m)ξ(l) can be regarded as a function on H. Hence, similarly to the case of k odd, we have

det

 S

k



− M (l/r) m

 ξ

 M

 l r



l,r∈H

= Y

χ∈H

b X

l∈H

S

_k



− l m



ξ(l)χ(l)

= Y

χ∈H

b 1 2

X

l∈G

S

k



− l m



ξ(l)χ(l)

=

 1 2



_ϕ(m)/2

Y b

G3χ:even

X

l∈G

S

k



− l m



ξ(l)χ(l)

=

 1 2



_ϕ(m)/2

Y b

G3χ:odd

X

l∈G

S

k



− l m

 χ(l).

On the other hand, as ξ(M (l/r)) = ξ(l)ξ(r

⁻¹

),



 

 



.. . . . . S

k



− M (l/r) m

 ξ

 M

 l r



. . . .. .



 

 



r,l∈H

=



 



. .. 0

ξ(r

⁻¹

)

0 . ..



 



r∈H



 

 



.. . . . . S

_k



− M (l/r) m

 . . . .. .



 

 



r,l∈H



 



. .. 0 ξ(l) 0 . ..



 



l∈H

.

Consequently,

(6) det B

_k

(m) =

 1 2



_ϕ(m)/2

Y b

G3χ:odd

X

l∈G

S

_k



− l m

 χ(l).

We recall the Bernoulli numbers and polynomials defined by t

e

^t

− 1 = X

∞ n=0

B

_n

t

ⁿ

/n! and B

_n

(X) = X

n i=0

 n i



B

_i

X

ⁿ⁻ⁱ

,

respectively. It is well known that S

k

(X − 1) = 1

k + 1 (B

k+1

(X) − B

k+1

) for any k ∈ N.

Since S

_k



− l m



= (−1)

^k+1

S

_k

 l m − 1



= (−1)

^k+1

k + 1

 B

_k+1

 l m



− B

_k+1

 , we have

X

l∈G

S

k



− l m

 χ(l)

= (−1)

^k+1

k + 1

X

m l=1

B

_k+1

 l m



χ(l) − (−1)

^k+1

k + 1 B

_k+1

X

m l=1

χ(l), for any χ ∈ b G, where χ is understood to be a Dirichlet character defined modulo m. Clearly, B

_k+1

P

_m

l=1

χ(l) 6= 0 if and only if k is odd and χ is the principal character 1

m

∈ b G, where 1

m

(l) = 1 if (l, m) = 1, and 1

m

(l) = 0 if (l, m) > 1. Moreover, in that case, B

_k+1

P

_m

l=1

χ(l) = B

_k+1

ϕ(m).

We also recall the generalized Bernoulli numbers defined by B

n,χ

= m

ⁿ⁻¹

X

m a=1

χ(a)B

n

 a m



, n ∈ N, χ ∈ b G,

satisfying

L(1 − n, χ) = − B

_n,χ

n ,

where L(s, χ) denotes the L-function attached to χ. By elementary results on L-functions, we have, for k (1 ≤ k ≤ p − 2),

L(−k, χ) = L(−k, χ

₁

) Y

q|m

(1 − χ

₁

(q)q

^k

), where Q

q|m

and χ

₁

are as in Section 1. It follows that