On the branch order of the ring of integers of an abelian number field

LXII.3 (1992)

On the branch order of the

ring of integers of an abelian number field

by

Kurt Girstmair (Innsbruck)

1. Introduction. Let K be an (absolutely) abelian number field of conductor n, G K its Galois group over Q, and O K the ring of integers of K.

By QG K we denote the rational and by ZG K the integral group ring of G K . The field K is a QG K -module via the usual action of G K on K. Let

R K = {α ∈ QG K ; αO K ⊆ O _K } .

The set R K is a subring of QG ^K that contains ZG ^K . As a ZG ^K -module, O _K is isomorphic to R K . In accordance with Leopoldt [1], we call R K the branch order (“Zweigordnung”) of O K . Let us now describe the order R K , i.e., the structure of O K as a Galois module.

The letter p always means a prime number. We put n ^∗ = {p ; p | n} .

Moreover, if d is a natural number, let

[d] = {q ; q | d, d/q square-free, (q, d/q) = 1} . The set [d] is called the branch class of d, and it is easy to see that

(1) {d ; d | n} =

•

[ {[d] ; n ^∗ | d | n}

(disjoint union). By X K we denote the character group of G K . If χ ∈ X K , f χ means the conductor of χ. Moreover, for α = P{a _σ σ ; σ ∈ G K } ∈ QG K

we put

χ(α) = X

a σ χ(σ) .

For each divisor d of n with n ^∗ | d there is a uniquely determined element ε d,K ∈ QG K with

(2) χ(ε d,K ) =

n 1 if f χ ∈ [d],

0 otherwise.

From (1) and (2) one sees that (ε d,K ; n ^∗ | d | n) is a complete system of orthogonal idempotents of QG K . Hence

QG K = M

{QG K ε d,K ; n ^∗ | d | n} , and it is known that

(3) R K = M

{ZG K ε d,K ; n ^∗ | d | n}

(cf. [1], [2]). For this reason we call ε d,K the branch idempotent of d, n ^∗ | d | n.

The ZG K -modules ZG K ε d,K are indecomposable, and there does not exist a decomposition of R K into indecomposable ZG K -submodules different from (3). Of course, ε d,K can be written as

ε d,K = X

{c _σ σ ; σ ∈ G} ,

with c σ ∈ Q. It seems that an explicit formula for the coefficients c σ has not been given so far. Indeed, in the previous papers [1], [2] the branch idempotent ε d,K only appears in the form

ε d,K = |G K | ⁻¹ X

{χ(σ)σ ; σ ∈ G K , f χ ∈ [d]} ,

which immediately follows from (2). In this note we give an explicit formula for the numbers c σ , σ ∈ G K , in the case K = Q n = Q(e ^2πi/n ). We shall see that this also yields an explicit description of ε d,K in the general case.

2. The main result. The Galois group G n of Q n over Q has the shape G n = {σ k ; 1 ≤ k ≤ n, (k, n) = 1} ,

where σ k is defined by

σ k (e ^2πi/n ) = e ^2πik/n .

It what follows we write ε d instead of ε _d,Q

. Suppose now that K is an arbitrary abelian number field with conductor n.

The restriction map

res : QG n → QG K

is Q-linear and defined by res(σ k ) = σ k | _K . We note the following Proposition. Let n be the conductor of K, and let n ^∗ | d | n. Then

res(ε d ) = ε d,K .

P r o o f. Take a character χ ∈ X K . Then χ = χ ◦ res : QG b ⁿ → C is a character of G n with conductor f χ . Therefore

χ(res(ε d )) = χ(ε b d ) = n 1 if f χ ∈ [d], 0 otherwise.

Thus res(ε d ) satisfies condition (2), which means res(ε d ) = ε d,K .

Due to the Proposition, ε d,K is explicitly known if ε d is. Let us therefore describe ε d . As above, let n ^∗ | d | n and suppose that k ∈ N, 1 ≤ k ≤ n − 1, (k, n) = 1. We define

d k = (d, k − 1) and, provided that n ^∗ | k − 1,

r k = Y

{p ; p | d k /n ^∗ , p - d/d k } . Theorem. In the above situation write

ε d = X

{c k σ k ; 1 ≤ k ≤ n, (k, n) = 1}

with c k ∈ Q for all k. Let ϕ : N → N be the Euler function and µ : N → N the M¨ obius function. Then

c k =







µ(d/d k )d k ϕ(r k ) nr k

if n ^∗ | k − 1,

0 otherwise.

The coefficient c k of the branch idempotent can also be described in a somewhat different way. For m ∈ N ∪ {0} put

v p (m) =

 max{j ; p ^j | m} if m 6= 0,

∞ otherwise

(so v p (m) is the p-exponent of m).

Corollary. In the context of the Theorem, c k = 0 if n ^∗ - k − 1 or if there is a p with v p (d) ≥ v p (k − 1) + 2. Otherwise

c k = d n

Y  1 − 1

p ; 2 ≤ v p (d) ≤ v p (k − 1)



× Y



− 1

p ; v p (d) > v p (k − 1)

 . P r o o f o f t h e T h e o r e m. If n ^∗ | d | n put

(4) γ d = X

{ε _q ; n ^∗ | q | d} . For a character χ ∈ X n = X _Q

,

(5) χ(γ d ) = n 1 if f χ | d,

0 otherwise.

This follows from (1) and (2). The condition (5) determines γ d uniquely.

Put

(6) e γ d = ϕ(d)

ϕ(n)

X {σ _k ; k ≡ 1 mod d} .

Let χ ∈ X n be such that f χ - d. Then there exists a number j, 1 ≤ j ≤ n, j ≡ 1 mod d, with χ(σ j ) 6= 1. But e γ d = σ j e γ d , hence

χ( e γ d ) = χ(σ j )χ( γ e d ) ,

which implies χ( e γ d ) = 0. On the other hand, let f χ divide d. Then χ(σ k ) = 1 for all k ≡ 1 mod d and

χ( γ e d ) = (ϕ(d)/ϕ(n))|{k ; 1 ≤ k ≤ n, k ≡ 1 mod d}| = 1 .

Since γ d is determined by (5), we have shown γ d = γ e d , i.e., (6) is the explicit form of γ d .

By means of the M¨ obius inversion formula we obtain from (4)

(7) ε d = X

{µ(d/q)γ q ; n ^∗ | q | d} . On inserting (6) into (7) we get

(8) ε d = X

(k,n)=1

X {µ(d/q)ϕ(q)/ϕ(n) ; n ^∗ | q | d, q | k − 1}  σ k .

If n ^∗ | q, ϕ(q)/ϕ(n) equals q/n. Hence (8) yields

(9) c k =  P{µ(d/q)q/n ; n ^∗ | q | d _k } if n ^∗ | k − 1,

0 otherwise.

For the remainder of the proof assume that n ^∗ | k − 1. Then c k = µ(d/d k ) X

{µ(d _k /q)q/n ; n ^∗ | q | d _k , (d/d k , d k /q) = 1} . The substitution l = d k /q yields

c k = µ(d/d k )d k n ⁻¹ X

{µ(l)/l ; l | d k /n ^∗ , (d/d k , l) = 1} .

But µ(l) is different from 0 if and only if l is square-free. For a number l of this kind the assertions

l | d k /n ^∗ , (d/d k , l) = 1 and l | r k

are equivalent. Therefore we get

c k = µ(d/d k )d k n ⁻¹ r ⁻¹ _k X

{µ(l)r _k /l ; l | r k }

= µ(d/d k )d k n ⁻¹ r ⁻¹ _k ϕ(r k ) , which we had to show.

Example. Let n = p ^m and d = p ^q , 2 ≤ q ≤ m. Then the Corollary yields

ε d = p ^q−m X

{(−1/p)σ _1+dj/p ; 1 ≤ j < p ^m−q+1 , p - j}

+ X

{(1 − 1/p)σ _1+dj/p ; 0 ≤ j < p ^m−q+1 , p | j}



.

References

[1] H. W. L e o p o l d t, ¨ Uber die Hauptordnung der ganzen Elemente eines abelschen Zahl- k¨ orpers, J. Reine Angew. Math. 201 (1959), 119–149.

[2] G. L e t t l, The ring of integers of an abelian number field , ibid. 404 (1990), 162–170.

INSTITUT F ¨ UR MATHEMATIK UNIVERSIT ¨ AT INNSBRUCK TECHNIKERSTR. 25/7 A-6020 INNSBRUCK OSTERREICH ¨

Received on 23.12.1991 (2209)