The class number one problem for some non-abelian normal CM-fields of degree $24$

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

F. L

EMMERMEYER

S. L

OUBOUTIN

R. O

KAZAKI

The class number one problem for some non-abelian

normal CM-fields of degree 24

Journal de Théorie des Nombres de Bordeaux, tome

11, n

o

_{2 (1999),}

p. 387-406

<http://www.numdam.org/item?id=JTNB_1999__11_2_387_0>

© Université Bordeaux 1, 1999, tous droits réservés.

L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

The class number

one

problem

for

some

non-abelian

normal

CM-fields of

_degree

24

par F.

LEMMERMEYER,

S.

LOUBOUTIN,

et R. OKAZAKI RÉSUMÉ. Nous déterminons tous les corps de nombres de

degré

24,

galoisiens

mais

_{non-abéliens,}

à

_{multiplication complexe}

et tels

que les groupes de Galois de leurs sous-corps totalement réels maximaux soient

isomorphes

à

_A4

_(le

_groupe alterné de

_degré

4 et d’ordre

₁₂₎

_qui

sont de nombres de classes d’idéaux

_égaux

à 1. Nous _prouvons

_(i)

_qu’il

_y a deux tels _corps de nombres

de _groupes de Galois

A4

C2

(voir

Théorème

_{14), (ii)}

_qu’il

_y

a au

_plus

un tel _corps de nombres de _groupe de Galois

_SL2(F3)

(voir

Théorème

_18),

et

_(iii)

_que sous

_{l’hypothèse}

de Riemann

généralisée

ce dernier corps candidat est effectivement de nombre de classes d’idéaux

_égal

à 1.

ABSTRACT. We determine all the non-abelian normal CM-fields

of

_degree

24 with class number _one,

_provided

that the Galois _group of their maximal real subfields is

_isomorphic

to

_A4,

the

alternat-ing

group of

degree

4 and order 12. There are two such fields with Galois _group

_A4

_C2

_(see

Theorem

₁₄₎

and at most one with

Ga-lois _group

_{SL2(F3) (see}

Theorem

_18);

if the Generalized Riemann

Hypothesis

is _true, then this last field has class number 1.

1. INTRODUCTION

Let us first fix the notation for the _groups that will occur:

C,

is the

cyclic

group of order

m, V = Gz

x

_C2

is Klein’s four _group,

_Q8

is the

quaternion

group of order

8, A4

is the

alternating

group of order 12

(the

normal

_subgroup

of index 2 in the

_symmetric

group

S4),

and

SL2(F3)

is

the _group of 2 x 2-matrices with entries in the finite field of 3 elements and

determinant 1.

Let N be a normal CM-field of

degree

2n and Galois _group G =

Gal(N/Q) .

Since the

_complex

_conjugation

J is in the centre

_Z(G)

of

_G,

_then,

its maxi-mal real subfield

N+

is a normal real field of

degree n

and

G+

=

Gal(N+ /Q)

is

_isomorphic

to the

_quotient

group

G / {Id, J}.

As delineated in the

intro-duction in

_[22],

and

_according

to

_[17],

_[18],

_[19]

and

_[23],

the

_completion

of the determination of all the non-abelian normal CM-fields of

_degree

32 with class number one is reduced to the determination of all the

non-abelian,

non-dihedral normal CM-fields of

_degree

24.

_Now,

there are 15

groups G of order

24,

three of them

being

abelian and one of them

having

trivial center

_Z(G)

_(namely,

the

_symmetric

_group

_S4

of

_degree

_4).

_Hence,

there are 11

_possible

Galois _groups for non-abelian normal CM-fields. Let

us also

_point

out that Y. Lefeuvre

_proved

in

_[16]

that there is

_only

one

dihe-dral CM-field of

_degree

24 with relative class number _one, and this dihedral CM-field has class number one. We will focus on two of the

_remaining

ten

non-abelian _groups: those which have the

_alternating

_group

_,A4

as a

quo-tient. There are

only

two _{such groups,}

_namely

_SL2(F3)

and

_A4

x

_Cz

_(see

e.g.

[30]).

Let us also

_point

out that

_.A4

x

_G2,

_SL2(IE3)

and

_S4

are the

only

non-abelian _groups of order 24 whose

_{3-Sylow subgroups}

are not normal.

More-over, if the

3-Sylow subgroup

of the Galois group of a non-abelian normal

CM-field N of

_degree

24 is a normal

subgroup,

then N contains a normal

octic CM-subfield M and the relative class number

_hAt

of M divides the relative class number

_hN

of N

_(see

_[19,

Th.

_5]

or

[14,

Cor.

1]),

which

makes the situation easier. For

_example,

_quaternion

octic CM-fields have

even relative class numbers

([17]),

hence there are no normal CM-fields

with relative class number one with Galois group

_C3

x

_Q8

or

~8.

Here is the

_plan

of our

_{investigation: first,}

we will characterize the

nor-mal CM-fields with Galois _groups

_SL2(F3)

or

_.~44

x

_C2

and odd relative class number

_(see

Theorems 11 and

_10).

Then we will reduce the determination

of all the normal CM-fields of Galois group

A4

x

_CZ

to the determination of all the

_imaginary

sextic

_cyclic

fields of relative class number

_4,

and

us-ing

results from

_[26]

we will

_complete

the determination of all the normal

CM-fields with Galois group

.A4

x

_C2

with relative and absolute classs

num-bers

_equal

to one

(see

Theorem

14).

Next we will

_give

lower bounds on

relative class numbers of normal CM-fields of

_degree

24 with Galois group

SL2(F3)

and odd relative class number

_(see

Theorem

_16).

Since the

_explicit

construction of normal CM-fields of

_degree

24 with Galois _group

_SL2(F3)

and odd relative class number is far from

_being

easy, we will focus on the

class number one

problem

for these fields and

_quote

a result from

[23]

(see

Proposition

17)

which will

_provide

us with a list of 23

_cyclic

cubic fields

such that _any normal CM-field of

_degree

24 with Galois group

SL2(F3)

and

class number one is the Hilbert 2-class field of a field in that list.

_Finally,

we will

complete

the determination of all the normal CM-fields of

_degree

24 with Galois _group

_SL2(F3)

and class number one modulo GRH

(see

Theorem

_18).

1.1.

_Diagrams

of subfields. Let

N+

be a normal field of

_degree

12 with Galois _group

_,.44.

This _{group contains} four

_conjugacy

classes:

_Cll

=

{Id},

DIAGRAM 1

C12

=

{(123), (142),

(134),

(243)},

Cl3

=

{(132),

(124), (143),

(234)},

and

_Cl4

=

{(12)(34), (13)(24), (14)(23)}.

Set V =

Cil

U

C14.

Then V is _a

normal

_subgroup

of

_A4,

the

_quotient

_group

_A41V

is

_cyclic

of order 3 and V is

_isomorphic

to Klein’s four _group. Note that V is a normal

2-Sylow

subgroup

of A4.

Let F be the

_cyclic

cubic subfield

of N+

fixed

_by

V. Then

N+/F

is

_{biquadratic bicyclic}

and F is a

_cyclic

cubic field whose conductor we will denote

_{by f .}

Let N be a normal CM-field of

degree

24 with Galois _group

SL2(F3).

There are

_eight

elements of order 3 in

SL2(F3),

hence four

_subgroups

of

order 3 in

_SL2(F3).

These four _groups are

_pairwise

_conjugate.

Let K

be the fixed field of any of these four

subgroups.

Then K is a non-normal

octic CM-subfield of N whose normal closure is

_N,

and

K+

is a non-normal

totally

real

_{primitive quartic}

field whose normal closure is

N+.

The same

holds for _any of the four fields

_conjugate

to K over

_Q.

The

_(incomplete)

lattice of subfields is

_given

in

_Diagram

1.

Let N be a normal CM-field of

_degree

24 with Galois _group G

= A4

x

C2.

Note that

_{Z(G) =}

_{{(I, 1),}

_(1,

_-1)1

so that the

complex conjugation

J of

G = _is

_equal

_to

(1, -1).

Let

Tl =

(12)(34),

_T2 =

(13)(24)

and T3 =

(14)(23)

denote the three elements of order 2 of

.A.4.

Set

Gi

=

{(l,1),

(Ti,1),

(Tj7-1),

(Tk, -1)~

with 1 i 3 and =

{1,2,3}.

Then

each

_Gz

is a

subgroup

of G

= A4

x

_CZ

and is

_isomorphic

to

_(7G/27G)2

and

=

Cl4

being

a

conjugacy

class in

A4

the three _groups

Gi,

1

i 3 are

_conjugate

in G.

DIAGRAM 2

let

_Ki

be the fixed field of

Gi,

and F be the fixed field of V x

_{CZ .}

The

(incomplete)

lattice of subfields is

_given

in

_Diagram

2.

1.2. Factorizations of Dedekind zeta functions.

For a CM-field N we let

_~N

denote its Dedekind zeta

_function,

_hN

its relative class

_{number, C~N}

E

_{1~2}

its Hasse unit

_index,

_WN

_{its group} of

roots of

_unity

and zvN the order of

WN.

Note that

WN = WA

where A

denotes the maximal abelian subfield of N. We have

Let us make some more

general

remarks.

Using

[7]

one can

_easily

_prove

that we have the

following

factorizations of Dedekind zeta functions into

products

of Artin’s L-functions :

where X

ranges over all the irreducible characters of G =

Gal(N//)

which

satisfy

X(Id).

In the same _way, for _any subfield F of

N+,

we have

where X

ranges over all the irreducible characters of G =

Gal(N/F)

are

_meromorphic

and that Artin’s L-functions associated with characters

induced

_by

linear characters of

_subgroups

are entire.

1.3.

_{Prerequisites.}

Here we record some results that will be used later.

Proposition

1.

1. Let N be a

CM-field

with maximal real

_subfield

N+

and let t denote the number

_of

_prime

ideals

_of

N+

which

_ramify

in the

_quadratic

extension

N/N+.

Then

2t-l

divides

_hN.

In

_{particular, if}

_hN

is odd then at most

one

_prime

ideal

_of

N is

_rarrzified

irc the

_quadratic

extensiorc

_N/N+.

Moreover, if

the narrow class number

hN+

of

N+

is odd then the

2-rank

_{r2 (N)}

_of

the ideal class _group

_of

N is

_equal

to t - 1.

2. Let

_K/L

be a

finite

extension

_of

number

fields. If

at least one

_prime

ideal

_of

L is

_{totally ramified}

in the

_K/L

then the norrrt _map

froyn

the narrow ideal class _group

_of

K to that

_of

L is

_surjective.

_If

_K/L

is

quadratic,

L is

_totally

real and K is

_{totally imaginary}

then the norm

map

from

the wide ideal class group

of

K to that

of

L is

surjective.

3. Let N be a

CM-field

with maximal real

subfield

N+

and assume

_hN

is

odd. Let

_H2(N),

_H2(N+)

and

_{Hz (N+)}

denote the 2-class

_{field of}

_N,

the 2-class

_{field of}

N+

and the narrow 2-class

field of

N+,

_{respectively.}

(a)

The 2-class _group

_of

N+

is

_cyclic.

(b)

H2(N)

=

NH2(N+)

and

H2(N)

is _a

CM-field

with maximal real

subfield

H2(N+).

Hence,

the 2-class _group

_of

N is

_{cyclic, for}

it is

_isomorphic

to the 2-class _group

_of

N+.

(c)

If

N/N+

is

_unramified

at all the

_{finite places}

then

_{H2 (N)}

H2 (N+)

and

_{hN+ =}

_2hN+.

(d)

If N/N+

is

_ramified

at some

finite place,

then

Hi(N+)

=

H2(N+)

hN+

=

hN+

and the narrow 2-class _group

of

N+

is

cyclic.

4. Let G denote the wide or the narrow ideal class _group

of

_any

cyclic

cubic

_field.

_Then,

_for

_any n &#x3E; 0 the 2-ranks

_of

the _groups

G 2n

{g2n ;

g E

_G~

are even.

Moreover,

the narrow class number

_of

a

cyclic

cubic

_field

is either

_equal

to its wide class number or

equal

to

four

times its wide class number.

Proof.

1. follows from the

_ambiguous

class number formula

_([13,

Ch.

_13,

Lemma

_4.1]:

2t-l(VN+ : U2N+) hN+

Here

_UN+

denotes the unit _group of the

_ring

of

_integers

of

_N+,

and

_VN+

its

_{subgroup consisting}

of all units that are norms from N.

_Now,

assume

h9+

is odd. Since

_{h9+ ~ (UN+ : UN+)hN}

and

since this index is a _power

of

_2,

we

_get

_hN+

=

h9+, UN+

=

Ur,+

and

_VN+

=

UN+,

which

_yields

C1N

are the same when

hN+

is

odd,

and we find

2t-1 =

#ANN+ = 2’’2r’

which is what we wanted to _prove.

NIN+

2. follows from the

_proof

of Theorem 10.8 in

_[31].

3.(a):

The norm _map N from the ideal class group of N to that of

N+

is

onto

_{(point 2),}

hence its kernel has order

_hN.

The kernel of the canonical

map j

from the ideal class group of

N+

to that of N has order 2

([31,

Th.

_10.3~)

and

_{No j}

sends an ideal class to its _square.

_Therefore,

2’’2~~’+~-1

divides

_hN,

where

_r2(N+)

_{denotes the 2-rank of the ideal class group of}

N+.

_Therefore,

if

_hN

is odd then the 2-class _group of

N+

is

_cyclic.

(b), (c)

and

_(d)

follow from the very definitions of these class

fields,

from

the inclusions N C

_NH2(N+)

C

_{NH2 (N+)}

_g

_H2(N)

and from the fact that

_hN

odd

_implies

_NH2(N+)

=

H2(N).

4.

_Adapt

the statement and

_proof

of

_[31,

Th.

_10.1].

D

Proposition

2. Let be an

..4¢-extension,

and assume that

N/N+

and

_M/N+

are

quadratic

extensions such that

Gal(N/Q) rri Gal(M/Q) ri

SL2(F3).

Then the

_quadratic

subextension

_{M’ / N+}

contained in

_MN/N+

and

_{different from}

M and N is formal

_{over Q}

with

_,,44

x

_G2.

Proof.

Write M = and N =

N+( f )

for some _1£, v E

N+.

Since

M/Q

and are

_normal,

we know that =

a 2

and

v 1

3a2

are _squares in

N+

for _every (j E

_Gal(N+/Q).

But then the

_(p,v)l-u

are

also _squares, and it follows that is normal. Let F denote the cubic subfield

of N+.

Then

_{Gal(M/F) ~ Gal(N/F) -}

_Qs,

hence

8(v, N+ / F) =

(-1, -1, -1)

in the notation of Lemma 1 in

_[15]

_(the

entries

in

_S(p,

_N+/F)

are the elements

a 1+o,

as a runs

_through

the

_{automorphisms}

#

1 of

_Gal(N+/F)).

This

_implies

_{(+1, +1, +1),}

hence

Gal(M’/F) -

C2

x

_CZ

x

_C2,

and our claim follows. 0

We also recall a result on

A4-extensions

of local fields

(see

~11~,

Korollar

2.16):

Lemma 3. Let _p be a

prime

and assume that

K/Qp

is a normal extension

of

the

_p-adic

_field

_Qp

with Galois group

A4.

Then p =

2,

and K is the

unique

A4 -extension of

in

_particular,

2 is inert in the

_cyclic

cubic subextension F

_{= Q2}

₍₍₇₎

_of

K and

_{totally ramified}

in

_K/F.

Proof.

The inertia

_subgroup

T of is an invariant

subgroup

of

,A4

and

hence must be

_{1, V}

or

,A4.

Since unramified extensions are

_cyclic,

we have

T ~

1. In the case of tame ramification T is

_cyclic,

and since neither V nor

,A.4

are

_cyclic,

the ramification must be

_wild,

and we

have p

= 2 _or

p = 3.

If we had _p =

3,

then

T/Vl (where

Vi

is the first ramification

group)

must

be

_isomorphic

to the

_{2-Sylow subgroup V}

of

_,A4

_{contradicting}

the fact that

T/Vl

must be

_{cyclic by}

Hilbert’s

_theory.

The rest of the lemma is a well

1.4. Normal CM-fields with Galois _group

_.A4

x

_C2.

The lattice of subfields is

_given

in Subsection

_1.1,

_Diagram

2. Let

_{N; _}

i =

1, 2, 3,

denote the three

quadratic

subextensions

of N+/F;

if

we write k =

Q(B/2013m)

for the

complex quadratic

subfield of

_N,

then the

three

_non-galois

sextic CM-fields inside are

_Kj

= In the

sequel,

we fix one of the fields

_Kl,

_Kz,

_K3

and denote it

_by

K.

There is a

simple

criterion that allows us to decide whether

_{F( va.)}

is contained in an

,A,4-extension

of

_~:

Lemma 4. Let

_F/E

be a

cyclic

cubic

_extension,

y let a

generate

Gal(F/E),

and let a E

_{FBF 2.}

Then

_{aa )}

is

_{an A4 -extension}

_of

E

_if

and

_only

if

_NF/EG:

=

aaU au2

is _{a square} in E.

Proof.

This is an _easy exercise in Galois

theory.

0

Lemma 5. Let _p be an odd

prime

whose

rarraification

index in the

A4-extension

_L/Q

is even. Then _p

_splits

in the

_cyclic

cubic

_subfield

F

_of

L.

Proof.

Suppose

not;

then there is

_exactly

on

_prime

ideal p in F above _p, and since the ramification index of _p is even, it must

_ramify

in one of the

three

_quadratic

extensions of F. Since these fields are

_conjugate

over

Q,

the

_prime

_{ideal p ramifies} in all three of them. This

_implies

_{that p ramifies}

completely

in the V-extension

_L/F,

but this is

_impossible

for

_{odd p by}

Hilbert’s

_theory

of ramification. D

We next list some useful _consequences of the fact that

_hr,

is odd: Lemma 6.

_If

_hN

is

_odd,

then so is the narrow class number

hF

of

F.

Moreover,

each

_quadratic

subextension

_of

_N+/F

is

_ramified

at some

_prime

ideal above each

_{prime p}

that

_ramifies

in

_k/Q.

Proof.

Let

_(j

=

1,

2, 3)

denote the three

quadratic

subextensions

of

_N+/F,

and let _p denote _any

_prime

ramified in If

_Fl/F,

_{say, is} unram.ified _{at p,} then so are

for j

=

2, 3,

since these extensions _are

conjugate

over

_Q.

Thus

N+/F

is unramified _{at p,} hence each

_prime

above

p in

N+

ramifies in

N/N+.

Since

hN

is

odd,

p cannot

split

in

by

Proposition

1.1. But then the localization _{at p} is an

A4-extension

of

_~,

hence _p = 2

by

Lemma

3,

and 2 is inert in and

totally

ramified in

N+/F:

contradiction. Thus each

_NJIF

is ramified at some

_prime

dividing

P.

Now if

_h+

is even, then so is

_hF

_by

[1].

_{By Proposition}

_1.4,

the class

group of F contains V =

C2

x

_G2

as a

subgroup. Since, by

what we

just

have

proved,

N+/F

is

_disjoint

to the Hilbert 2-class field of

_F,

the norm _map

N :

_Cl(N+)

2013~

Cl(F)

is

_onto,

hence

Cl(N+)

has 2-rank at least 2. But

now

_Proposition

1.3(a)

shows that

_hN

must be even: contradiction. 0

Proof. Proposition

1.1 shows that any

prime

ideal p in

N+

that ramifies in

N/N+

lies above a

prime

_p that does not

split

in thus the

localiza-tion

_{Nl lo}

has

_degree

_12,

hence is an

A4-extension

of

_~.

But Lemma 3

then

_gives

_p = 2. 0

Lemma 8.

_{If hN}

is

_odd,

then

_k/~

_ramifies

at

_exactly

one

finite

prime

_po,

and this

_prime

_splits

in

_{F /Q.}

Proof.

Let _p be an odd

_prime

_ramifying

in

k/Q.

We deduce from Lemma 7

that the ramification index _e+

_{of p in}

is even. Lemma 5 then shows

that _p

_splits

in F.

Now let t denote the number of odd

_primes

_ramifying

in

_k/Q1

as we

have _seen, these

_split

in

_F,

hence there are 3t

_primes

ramified in

A/F.

_By

Proposition

1.1,

we find that

2 3t-1

I hA;

from

[24]

we

_get

that

hi )

I

_4hN

that

_{is, 23t-3}

_{1 h-.}

This

_implies

_{t 1.}

Thus there are two

_{possibilities: t}

= 0

(and

then is ramified

only

_at

200),

or t =

1;

in the last _case, there can be no ramification in at

_2,

because otherwise

_Proposition

1.1 would show that

_{8 ~ 1 hA}

and

_{2 ~ 1 hN,}

_by

[24]

again.

It remains to show

_that,

in the case t =

0,

the

prime

2

splits

in

To this

_end,

we first show that the three non-normal sextic subfields

_Nj

of

N+

are unramified outside 2. For _suppose

_not;

then there is an odd

_prime

p which ramifies in

NJIF,

and Lemma 5 shows

that p splits

in F. Hence

K/F

is ramified at at least four

_primes:

but then 2

_hN

as above.

Now assume that 2 does not

_split

in

_F/Q

and write

_Nl

= We

claim that

_Nl/F

is unramified outside 2. Since

_N11F

_only

ramifies at 2 and since 2 is inert in F

_(clearly

2 cannot

_ramify

in

_cyclic

cubic

_extensions),

we have

(a)

=

a2

_or

(a) =

(2)a2

for some ideal a of F. Let

_/j

E F be such

that

_(/3)

= We

_{get ahf}

= or

ahF

=

2,E,32

(actually,

this last case

is

_impossible

in view of Lemma

₄₎

for some e E

UF

which must be

_totally

positive.

This

_yields

e =

~2

_E

UF

(see

the

proof

of Lemma

6),

and since

hF

is

_odd,

we

_{get Nl}

=

F( JQ)

=

F( ahF)

= F and

N,

would be

abelian,

a desired contradiction. 0

""i-

-i=O

and

_using

₍₁₎

we find

,

Since wK = 2

(for

F is

clearly

the maximal abelian subfield of

K),

_z.uN =

wA and

_Q,q

= 1 for is

cyclic

(see [14,

point

5 page

352]),

_we

_get

We claim that

_QK

= 1. In

fact,

since _wK =

_2,

_we have

QK

= 2 if and

only

if we are in the situation

(i).2.(a)

of Theorem 1 in

(14~,

i.e. if and

only

if K =

F(...!i)

for some

_{totally negative}

unit e E

UF.

But since F has odd

narrow class number

(Lemma

6),

_every

totally positive

unit is a _square,

and we

_get

K =

F ( yCI)

contradicting

the fact that is non-abelian.

Therefore,

we

finally

_get

Lemma 9. Let N be a normal.

CM-field of degree

24 with Galois _group

A4

x

G2.

_odd,

then

-Now we are

_ready

to _prove

Theorem 10. Let N be a normal

CM-field of degree

24 with Galois _group

A4

x

_C2.

_If

_hN

is odd then

_exactly

one rational

prime

_po is

ramified

in

the

_{imaginary quadratic field}

_k,

_(Po)

= _PlP2P3

splits

in

F,

k =

hA -

4

_{(mod 8),}

_{hK is}

_odd,

_QN

= 2 and _{we may} choose notation such that

pic is the

_only

_prime

ideal

_of

F which is

_ramified

in

_{Kif F}

_for

1 i 3.

Proof. By

Lemma 8 there is

_exactly

one

prime

_po ramified in

k,

and _po

splits

in F. Thus at least three finite

_{primes split}

in

_A/F,

and

_Proposition

1.1

_gives

4 ~

₁

_hA.

From

_[24]

we know that

_hA

divides

_4hN,

hence we must

have

_h-

4

_{(mod 8).}

Now

₍₄₎

_implies

that

_hi

odd and that

_QN

= 2.

Since

_hK

is

_odd,

there is at most one finite

prime

ramified in

K-/F

by

Proposition

1.1,

and since F has odd narrow class number

_by

Lemma

_6,

it must

_ramify

at some finite

_prime.

Since the fields

Ki

(i

=

1, 2, 3)

are

conjugate

over

Q,

the last claim follows.

It remains to show that k = Since _po is the

only

ramified

prime

in

_k/Q,

this is clear if _{po is} odd

_(we

can even conclude that po

-3 mod 4 in this

_case).

_Ifpo

=

2, however,

we have to exclude the

_possibility

that k =

Q( .J -1).

This is done _as follows: as in the

_proof

of Lemma

_8,

we

write

Nl

= for some

_{totally positive}

a E

OF.

From Lemma 4 we

know that Na is a _{square; suppose} that a is an ideal _square. Then a =

ê{32

for some unit -

(since

F has odd class

number),

e is

_totally

_positive

(since

a

is),

hence c must be a _square

(since F

has odd narrow class

number):

Thus there is a

_prime

ideal that divides a to an odd _power, and since

Na is a _square, there are in fact two such

_prime

ideals

_(necessarily

different

by

what we

_just

have

proved),

_{say p} and

p’.

Write k = then

Ki

= and if _we had k =

Q(H),

then

Ki

=

F(~).

But

a is divisible

by

two

_prime

ideals

_{p ~}

_{p’ (to}

an odd

_power),

hence both p

and

_pt

_ramify

in

_Kl/F:

contradiction. D

1.5. _{Normal CM-fields with Galois group}

_{SL2(F3 ).}

The lattice of subfields is

_given

in subsection

_1.1,

_Diagram

1. Our first theorem shows that CM-fields

_N/Q

with Galois _group and odd relative class number

_{h Ñ}

arise as narrow Hilbert 2-class fields of their cubic

subfields F:

Theorem 11. Let N be a

_CM-field

with

_{Gal(N/Q) rr SL2(F3);}

let

N+

denote its maximal real

_subfield

and F its

_cyclic

cubic

_{subfield. If}

_hN

is

odd,

then

hp =

_{hj m}

4 mod

_8,

and

_hN+

is odd.

_Moreover,

N+

is the Hilbert 2-class

_{field of F,}

_h+(N+)

= 2 mod

_4,

and N is the narrow Hilbert

2-class

_{field of}

N+.

Conversely,

if

is a

cyclic

cubic extension with

_{hF -}

hF -

4 mod

_8,

let

N+

denote the 2-class

_{field of}

F.

_If

_hN+

is

_odd,

then there exists a

normal.

_CM-field

N with

_SL2(F3)

_containing

N+

and with odd class number

_hN¡

in

_particular,

_hN

is odd.

Proof.

Note that

_{hF = h+}

= 4 mod 8 if and

_only

if

_{C12(F) -}

C2

x

_GZ

_{by Proposition}

1.4.

Assume that

_hN

is odd. We first claim that

_N/N+

is unramified at all finite

_primes.

If at least two finite

_primes

_ramify

in

_N/N+,

then

_hN

is

even

_{by Proposition}

1.1. If

_exactly

one finite

_prime

P of N is ramified in

N/N+,

then _any

_automorphism

of

_N/Q

must leave P fixed

_(that

_is,

we

have =

24).

Thus the localization

NP

is an

SLZ(IF3)-extension of Q§ ;

hence it contains an

A4-extension

of

Qp,

and Lemma 3 shows

that p

= 2.

But Weil

_[32]

_(see

also

_[11]

and

_[27])

has shown that

_Q2

does not admit

any

SL2

(F3)-extension.

Thus

_N/N+

is unramified at all finite

_primes.

We claim next that this

implies

that

_N+/F

is unramified at all finite

_primes.

In

_fact,

let P denote

a

_prime

in N and let

_NT

denote its inertia subfield in the extension

_N/F.

Then either

_NT

= N

_(and

P is unramified in

_N/F),

or

_NT

g

N+

(since

each

_{subfield 0}

N of

_N/F

is contained in

_N+).

But the second

_possibility

cannot occur since

N/N+

is unramified at all finite

_primes.

Hence

_N+/F

is an unramified abelian

V-extension,

and this

implies

that

the class number

_hF

is divisible

_by

4. We claim next that

_hN+

is

_odd;

this will

_imply

that

N+

is the Hilbert 2-class field of F and hence that

C12(F) -

V. Assume therefore that

_hN+

is even; since

C12(N+)

is

cyclic by

is unramified

_everywhere.

Since N is ramified at oo, we conclude that

M #’

N. Now M is normal over

Q,

since any

Q-automorphism

fixes N+

and maps M to a

quadratic

extension of N+ that is unramified

_everywhere,

i.e. to M itself.

Now either

_{Gal(M/Q) ~}

_A4

x

G2,

or

Gal(M/Q) ~ SL2(F3 );

in the

latter _case,

_Proposition

2 shows that the third

quadratic

extension

_M’/N+

contained in

_MN/N+

has Galois _group

_{A4 xC2;}

_moreover, since

_M/N+

and

N/N+

are unramified at the finite

_primes,

so is

M’/N+.

Let M denote this

,A4

x

_{C2-extension,}

and let _p be a

_{prime ramifying}

in the

_quadratic

subfield

k of

_M/Q:

since

_N+/F

is

_unramified,

_p must

_ramify

in

_M/N+,

and this

yields

the desired contradiction.

Finally

we know from

[2,

Thm.

2.5]

or

[5]

that is

always

even. But

the factor _group

_{Cl+(N+)/Cl(N+)}

is

_elementary

_abelian,

and

_Proposition

3.3.(c)

and

_(d)

show that either

_hN+

= or that is

_cyclic;

in both _cases, we conclude that 2 mod 4. This

_implies

that N is the

narrow Hilbert 2-class field of

N+

as claimed.

_Moreover,

we find that the narrow 2-class number of F divides 2 -

(N+ :

F)

_{= 8,}

and

_Proposition

3.4 now shows that V.

For a

_proof

of the converse we notice that

hN+

is even

_by

the results of

[2]

and

_[5].

From

_[6]

or

[29]

we know that the 2-class _group

of N+

is

cyclic;

in

particular,

there is a

unique

quadratic

extension

N/N+

that is unram.ified

at the finite

_primes.

_By

a standard

_argument,

_uniqueness

_implies

that

N/Q

is

_normal,

and

_by

_group

_theory

the Galois _group of this extension is

A4

x

_C2

or

SL2(F3).

The first case cannot _occur,

_however,

since

_N/F

would be abelian

_then;

thus

_{SL2(F3 ).}

This

_gives

_{Gal(N/F) -}

_Q,

and now

[10,

Hilfssatz

2]

shows that the class number of N is odd. 0

Since

_N+/F

is unramified we must have

_dK+

=

dF

=

f2

(see [12])

and

dK

= =

f 4

(since N/F

is unramified at all the finite

places,

the index

of ramification in of _any

_prime

ideal of N is

_equal

to 1 or

_3,

hence

is odd.

_Therefore,

the

_quadratic

extension

_K/K+

is unramified at all the finite

_places).

_Finally,

we have:

Lemma 12. Let K be one

of

the

four

non-normal octic

CM-fields of

N.

If

1 then 1.

This will

_help

us

considerably

since

hK

is much easier to

_compute

than

hN.

Our

_strategy

is first to find an _upper bound on

_f

when

_hN

= 1

(see

Theorem

_16),

second to determine all the non normal

totally

real

quartic

fields K+ with normal closure of

_degree

12 and Galois _group

_A4

and such

TABLE 1

third to

_compute

_hK

for each

_possible

_K+,

and fourth to

_compute

_hN

(where

N =

KF)

for the few K’s for which

hK =

1.

2. NORMAL CM-FIELDS WITH GALOIS GROUP

_A4

X

_C2

AND RELATIVE

CLASS NUMBER ONE. The

_following

result is

_proved

in

_[26]:

Theorem 13. Let A be an

_imaginary

_cyclic

sextic

field.

Then A is the

compositum

of

an

_imaginary

_quadratic

number

field

k = and _a real

cyclic

cubic

_field

F whose conductor we denote

by f .

Assume that

exactly

one rational

_prime

_{po is}

rarrcified

in the

_quadratic

extension

k/Q

and that

po

splits

in F.

Then,

h Ã

= 4

if

and

only if

we are in one

of

the

_following

ten cases:

In ddl these cases we have

h+ =

1.

Gauss has shown how to find a

generating polynomial

for

cyclic

cubic

fields with

_prime

_power conductor

_(see

_e.g.

_[3,

Theorem 6.4.6 and

_Corollary

6.4.12]);

the results are

( f, PF(X)) _

(19, X3 -

X2 -

6X +

_7),

_(31,

X3

-X2-lOX+8), (43,X3-X2-14X-8), (61,X3-X2-20X+9),

(67,

X3

-X2 -

22X -

_5),

_(73,

X3 - X2 -

24X +

₂₇₎

and

_(9,

X3 -

3X +

_1).

The two cases with D = -4 are excluded

_by

Theorem

_10;

for each of

the other

_{eight possibilities}

for A we now find a

_{totally positive}

_{aF E}

OF

such that

_NF/Q(aF)

= _Po

(it

follows from the

proof

of Theorem 10 that

such an element

_always

_exists).

We then have K =

F( -aF ).

According

to Table

_1,

there are

only

two normal CM-fields of

degree

24 with Galois

Note that we have

_hK

=

hK

in all cases.

_Moreover,

it would have been

sufficient to

_compute

_only

the four class numbers written in bold letters since we know that _aK must be

_primary,

_i.e.,

_congruent

to the _square of

an element of

_OF

_(the

_ring

of

_{algebraic integers}

of

_F)

modulo the ideal

40F. However,

the

_{multiplicative}

_group

_(OF/40F)*

is

_isomorphic

either to

(Z/2Z)3)

or to

_(Z/7Z)

x

(Z/2Z)3) ,

_according

as the

_prime

2

_splits

in F or

is inert in F. In

_{particular, g}

E

_(OF/40F)*

is a _{square in} this _group if and

only

if

_g7

= 1 in this _group,

i.e.,

_-aK is

primary

if and

only

if

a7

+ 1 is in

40F.

The reader will

_easily

check that aK is

primary

if

(D, f )

=

(-8, 31),

(-3, 61), (-7, 73)

and

_{(-11, 43).}

Theorem 14. There are

exactly

two normal

CM-fields

with Galois _groups

isomorphic

to

_.A4

x

_C2

and relative class number one.

_Moreover,

both

_fields

have class number one.

F(7+)

from which Pari GP

_[25]

_{readily yields}

=

2 18

31g

and

hN+ -

1

if D = -8 and

f

=

31,

and

dN+ =

36 ~ 618

and

hN+ =

1 if D = -3 and

f

= 61. D

3. NORMAL CM-FIELDS WITH GALOIS GROUP

_SL2(F3)

AND CLASS

NUMBER ONE.

From now on we assume that N is a normal CM-field of

degree

24 with

Galois _group

_isomorphic

to

_{SL2 (F3 ).}

We

_give

lower bounds on its relative

class number.

3.1. Factorizations of some Dedekind zeta functions.

Let j

=

exp(2Jri13) =

(-1

₊

H)/2

denote _a cube _root of

unity;

with

the notation of subsection

_1.1,

Table 2

_gives

the irreducible characters of

,,44

(see

[8,

page

181]):

Since X4 =

0*

is induced

by

_a linear

chaxacter 0

of the abelian

subgroup

V of

_,A4,

Artin’s L function s e

_L(s,

_X4,

N+ /Q)

is entire and

where X

is one of the two

_conjugate

Dirichlet characters associated to the

cyclic

cubic field F. In

_{particular, s}

_E~O,1~

_implies

_{(F(8) = ((8 )L(8, X)L(8, X)}

=

~(s)~L(s, X)~z

0.

Hence,

if

(N+(80)

_&#x3E; 0 for some _so

E]O, 1[

then the

en-tire function s e

L(s,

_X4,

N+ /Q)

has a real zero on and

_’N+

has at

least three real zeros on

]so,

1[.

The

_special

linear _group

_SL2(F3)

has order

_24,

contains seven

conjugacy

classes,

two of them

_(namely

_Cil

=

{Id}

and

Cl2

=

{2013Id})

consisting

of

only

one element. Next

Cll

U

_C12

= =

Z(SL2(IF3)),

and Table 3

gives

the irreducible characters of

_{SL2(F3) (see [8,}

_pages

_403-404,

solution

to exercise

_27.1]).

TABLE 3

According

to

₍₂₎

and Table

_{3, if}

N is a normal CM-field of

degree

24 and

Galois _group G =

SL2(F3),

then

3.2. Lower bounds on residues.

Lemma 15. 1.

_If

the absolute value

_of

the discriminant

_of

a number

then the Dedekind zeta

_function

s H

_(M(S)

_of

M has at most m real

zeros in the _range

In

_particular,

_Cm

has at most two real zeros in the _range

1

2. Let M be a normal

field of degree

12 with Galois _group

,A4.

Then,

s

e]0, 1[

and

Cm(s)

&#x3E; 0

_implies

that

_Cm

has at least three real zeros

in the _range

_{~s,1 (.}

In

_particular,

1 -

_{(1/ log dM)}

s 1

implies

0.

3. Let N be a normal

CM-field of degree

24 with Galois _group

isomorphic

to

_SL2(F3).

Set

Proof.

The first

_part

of

_point

2 was

proved

in the

previous

subsection. The

second

_part

of

_point

2 follows from

_point

1. The first

_part

of Point 3 follows from

_point

2 and

_(5),

and the second

_part

of

_point

2 follows from

_[20].

Let

us now _prove

_point

1. Assume

_~M

has at least m + 1 real zeros in the _range

According

to the

_proof

of

_[28,

Lemma

_3]

for _{any s} &#x3E; 1 we have

and since

_h(ti)

_h(2)

_0,

we have a contradiction.

Indeed,

let q

=

0.577 - -’ denote Euler’s constant. Since

_h(s)

&#x3E; 0 for s &#x3E; 0 we do have

3.3. Lower bounds on

h-From now on we assume that

hN

= 1.

By

Theorem

_11,

_{we see} that

N/F

is unramified at all the finite

_places,

therefore

_dN

= = _EN =

E f,

and

_according

to Lemma 15.3 we have

The field

N+

is

_totally

real of

_degree

12 and unramified over its

cyclic

cubic

subfield F of conductor

_{f ;}

therefore we have

_dN+

= Since

N+/F

is an unramified abelian

_quartic

extension,

we have

(see

[21]

or

[23]):

Combining

(7)

and

₍₈₎

_gives

the

_following

Theorem:

Theorem 16. Let N be a normal

CM-field of degree

24 mith

Gal(N/Q)

isomorphic

to

_SL2(F3).

_If

_hly

is odd then

In

_particular,

_hN

= 1

implies f

83000.

We remark that the bound of

_f

obtained

_through

₍₈₎

is about ten times

smaller than the bound

_f

106

obtained from

_general

_upper bounds on

residues of Dedekind zeta functions.

According

to numerical

_computations

of bounds on

_Ress=1

((N+)

for all

the F’s of

_prime

conductors

_f

83000 and

_according

to

₍₇₎

we

proved

in

[23):

Proposition

17.

_(See

_(23)).

Let N be a normal

_{CM-field of degree}

24 with

Galois _group

_isomorphic

to

_SLZ(IFg).

Assume that the class nurraber

_of

N is 1.

_Then,

1. The class nnmber

hF

and narrow class number

/t~

_{o f F}

are

_{equal to}

4,

which

_implies

that the conductor

_{f of}

F is a

_prime

_congruent

to

1 mod 6.

2.

N+

is the narrow Hilbert 2-class

field of F,

is

2,

and N is the

second narrow Hilbert 2-class

field of

F.

3.

_{Finally, f}

is one

of

the

following

23

prime

values:

f

=163, 277, 349,

397, 547, 607, 709, 853, 937,1399,1789, 2131, 2689, 2803, 3271, 4567,

5197,

6079,

8011, 10957, 11149,

14407 or 18307.

3.4. The determination.

Our first

_job

is the construction of the 2-class fields of these 23 cubic fields. To this

_end,

we first construct a

_quadratic

unramified extension of

F and then use the formulas in

[12]

to

_compute

the

_quartic

_polynomial

PK+(X)

with

_integer

coefficients whose roots

_generate

N+.

The construction of the

_quadratic

unramified extension of F is

straight-forward from the theoretical

_point

of

_view;

we did our

_computations

_using

PARI,

and since the latest versions of KANT and PARI

_(which

we did

not

_yet

have at our

disposal)

allow the direct

_computation

of Hilbert class

fields,

we _may

skip

the details here. Table 4

gives

the results of our

com-putations,

including

the class number of a

_quartic

field

K+

_{generated by}

a

root of

_PK+(X).

We can check that the fields with

_hK+

= 2

(these

class numbers were

computed using

PARI)

in this table

_really

have even class number

by

explic-itly constructing

the unramified

_quadratic

extension. In each case where

the class number of

K+

in the wide sense is odd we now

_compute

the

quadratic

extension

_{K/ K+}

which is unramified at all finite

_primes

_(since

N/N+

is unramified outside oo and

N+ ~K+

is

cyclic

of

degree

_3,

K/K+

must also be unramified at all finite

_primes)

and

_give

an octic

polynomial

such that K is

_{generated by}

a

(suitable)

root of

PK (x).

Note that

the fact that this construction works

implies

that the class number of

K+

is

odd -

_{again by}

_[2].

_Finally

we

_compute

the class number of

K;

the results are collected in Table 5.

This leaves us with the

problem

of

computing

the relative class numbers

of the fields N for

_f

_{=163, 349, 397,}

853 and 937. Florian Hess

_(Berlin)

first checked that

_hK

= 1 for these fields

using

KASH

(see [9])

and then

computed

Cl(N):

The

_computation

for

_f

= 163 holds under the

_assumption

of the

_validity

of

_GRH,

and for the other conductors the groups

given

in this table are

subgroups

of

_Cl(N)

without any

assumption.

It

is, however,

reasonable to

TABLE 4

Theorem 18. There is at most one

CM-field

N with

Gal(N/Q) - SL2 (F3)

and class number

_1,

_namely

the maximal solvable extension

_of

the cubic

_field

of

corcductor 163 that is

unramified

at all

_finite

_prirraes.

It is the

_splitting

field of

the

_polynomials

x8

+

9x 6

+

23~

+

14X2

+ 1.

_If

GRH holds then N has class numbers 1.

Jiirgen

K13ners has called our attention to the fact that the field in Theorem 18 is the one with minimal discriminant and Galois _group

SL2(IF3 )

(see

[4]).

ACKNOWLEDGEMENT

The authors thank Florian Hess for

_computing

the class _groups of the fields of

_degree

24 and the referee for his comments.

TABLE 5

-I - I ,

Added in

_proof.

_Meanwhile,

the second author succeeded in

_removing

the GRH condition from the result of Theorem 18. See his articles

[Compu-tation

_of

relative class numbers

_{of CM-fields by}

_using

Hecke

_L-functions,

Math.

_Comp.

69

_(2000),

_371-393]

and

_[Computation

_of

_L(O,

_X)

and

_of

relative class numbers

_{of CM-fields,}

_preprint

Univ. Caen

_(1998)].

REFERENCES

[1] J.V. _Armitage, A. _Frohlich, Classnumbers and unit _signatures. Mathematika 14 _(1967), 94-98.

[2] C. _Bachoc, S.-H. Kwon, Sur les extensions de groupe de Galois Ã4. Acta Arith. 62 _(1992), 1-10.

[3] H. Cohen A course in _{computational algebraic} number theory. Grad. Texts Math. _138,

Springer-Verlag, Berlin, Heidelberg, New York _(1993).

[4] H. Cohen, F. Diaz y Diaz, M. Olivier, Tables of octic fields with a _quartic subfield. Math.

Comp., à _paraître.

[5] A. _{Derhem, Capitulation} dans les extensions _quadratiques non ramifiées de corps de nombres

cubiques cycliques. thèse, Université _{Laval, Québec (1988).}

[6] Ph. Furtwängler, Über das Verhalten der Ideale des Grundkörpers im _{Klassenkörper.}

Monatsh. Math. _Phys. 27 (1916), 1-15.

[7] H. _Heilbronn, Zeta-functions and L-functions. Algebraic Number _{Theory, Chapter} VIII,

J. W.S. Cassels and A. _Fröhlich, Academic _{Press, (1967).}

[8] G. _James, M. Liebeck, Representations and Characters of groups. Cambridge University Press, (1993).

[9] KANT _{V4, by} M. Daberkow, C. Fieker, J. Klüners, M. _Pohst, K. _Roegner, M. Schörnig,

and K. Wildanger, J. Symbolic Computation 24 _(1997), 267-283.

[10] H. Koch, Über den _{2-Klassenkörperturm} eines quadratischen Zahlkörpers. I, J. Reine Angew.

[11] P. Kolvenbach, Zur arithmetischen Theorie der _{SL(2, 3)-Erweiterungen.} Diss. Köln, 1982.

[12] S.-H. Kwon, Corps de nombres de _degré 4 de type alterné. C. R. Acad. Sci. Paris 299

(1984), 41-43.

[13] S. _Lang, Cyclotomic Fields II. _{Springer-Verlag} 1980

[14] F. Lemmermeyer, Ideal class groups of cyclotomic number fields I. Acta Arith. 72 _(1995), 347-359.

[15] F. Lemmermeyer, Unramified quaternion extensions of _quadratic number fields. J. Théor.

N. Bordeaux 9 _(1997), 51-68.

[16] Y. _{Lefeuvre, Corps} diédraux à multiplication complexe principaux. Ann. Inst. Fourier, à

paraitre.

[17] S. Louboutin, R. Okazaki, Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one. Acta Arith. 67 _(1994), 47-62.

[18] S. _Louboutin, R. Okazaki, The class number one _problem for some non-abelian normal CM-fields of _{2-power degrees.} Proc. London Math. Soc. _76, No.3 _(1998), 523-548.

[19] S. _Louboutin, R. Okazaki, M. _Olivier, The class number one _problem for some non-abelian normal CM-fields. Trans. Amer. Math. Soc. 349 _(1997), 3657-3678.

[20] S. Louboutin, Lower bounds for relative class numbers of CM-fields. Proc. Amer. Math. Soc. 120 _(1994), 425-434.

[21] S. _{Louboutin, Majoration} du résidu au _point 1 des fonctions zêta de certains corps de nombres. J. Math. Soc. _Japan 50 _(1998), 57-69.

[22] S. Louboutin, The class number one _problem for the non-abelian normal CM-fields of degree

16. Acta Arith. 82 _(1997), 173-196.

[23] S. Louboutin, Upper bounds on _{|L(1, ~)| and applications.} Canad. J. _Math., 50 _{(4), (1998),}

794-815.

[24] R. Okazaki, Inclusion of CM-fields and _divisibility of class numbers. Acta _Arith., à paraître.

[25] User’s Guide to PARI-GP, by C. _Batut, K. _Belabas, D. _Bernardi, H. Cohen and M. Olivier,

last _updated Nov. _{14, 1997;} Laboratoire _A2X, Univ. Bordeaux I.

[26] Y.-H. _Park, S.-H. _Kwon, Determination of all _imaginary abelian sextic number fields with class number 11. Acta Arith. 82 _(1997), 27-43 .

[27] A. _{Rio, Dyadic} exercises for octahedral extensions. preprint 1997;

URL: http://www-ma2. upc. as/~rio/papers. html

[28] H.M. Stark, Some effective cases of the _{Brauer-Siegel} theorem. Invent. Math. 23 _(1974), 135-152.

[29] O. _Taussky, A remark on the class field tower. J. London Math. Soc. 12 _(1937), 82-85.

[30] A. D. _Thomas, G. V. _{Wood, Group Tables,} Shiva Publishing Ltd, Kent, UK 1980.

[31] L.C. _Washington, Introduction to _Cyclotomic Fields. Grad. Texts Math. _{83, Springer-Verlag}

1982; 2nd edition 1997.

[32] A. Weil, Exercices _dyadiques. Invent. Math. 27 _(1974), 1-22.

Franz LEMMERMEYER

Max-Planck-Institut fur Mathematik

Vivatsgasse 7 53111 Bonn _{/ Germany} E-mail: _{lemmermompim-bonn.mpg. de} Ryotaro OKAZAKI Department of Mathematics Faculty of _Engineering Doshisha _University

Kyotanabe, Kyoto, 610-03 _Japan

E-mail : _{rokazakiodd. iij4u. or. jp}

St6phane LOUBOUTIN Université de Caen

Dept de _Mathematique et _Mecanique BP 5186

14032 Caen _Cedex, France