Weber's class invariants revisited

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

R

EINHARD

S

CHERTZ

Weber’s class invariants revisited

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{1 (2002),}

p. 325-343

<http://www.numdam.org/item?id=JTNB_2002__14_1_325_0>

© Université Bordeaux 1, 2002, 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

Weber’s class

invariants revisited

par REINHARD SCHERTZ

RÉSUMÉ. Soit K un corps

quadratique imaginaire

de discriminant

d et

Dt

l’ordre à conducteur t ~ N dans K. L’invariant

modu-laire j(Dt)

est un nombre

algébrique

qui

génère

sur K le _corps de classes d’anneau modulo t. Les coefficients du

_polynôme

minimal

de j(Dt)

étant assez

large,

Weber considère dans

[We]

les

fonc-tions définies

_{plus bas,}

par

lesquelles

il construit

des

_{générateurs}

_{plus simples}

pour les corps de classes d’anneau.

Plus tard les valeurs

_singulières

de ces fonctions ont

_joué

un rôle central dans la solution de

_Heegner

_[He]

du célèbre

_problème

de déterminer tous les _corps

_{quadratiques imaginaires}

dont le nombre de classes est

_égal

à 1

_[He,Me2,St].

Actuellement on s’en sert en

cryptographie

pour trouver des courbes

elliptiques

sur des corps

finis avec certaines

_{jolies propriétés.}

Le but de cet article est

_i)

d’énoncer certains résultats

_déjà

con-nus de

_{[We,Bi,Me2,Schl]}

cf. Théorèmes _1,2 et 3, concernant les valeurs

_singulières

des fonctions

_{f,f1, f2,03B32,03B33,}

et

_ii)

de

_développer

une _{preuve courte} de ces résultats.

Cette méthode

_s’applique

aussi à d’autres fonctions cf. Théo-rème 4 et le tableau

_précédent

celui-ci. Les preuves des théorèmes 1 à 4 sont données en fin d’article.

Ces démonstrations résultent de la loi de

réciprocité

de Shimura

(cf.

théorème _{5, ainsi que} théorèmes 6 et

_7),

du calcul de la

racine 24-ième de l’unité

_{de ~}

= lors des transformations

unimodulaires

_(cf.

_{proposition 2,}

tirée de

_[Me1]

formules

_(4.21)

à

(4.23) p.162),

et donnent aussi via la

_proposition

3 des formules

explicites

pour les

conjugués

des valeurs

singulières,

qui

sont très utiles pour des calculs

numériques.

Certains de ceux-ci sont donnés comme

exemples juste

avant la

Bibliograpie.

ABSTRACT. Let K be a

_{quadratic imaginary}

number field of

dis-criminant d. For t ~ N let

Dt

denote the order of conductor t

in K

_{and j (Dt)}

its modular invariant which is known to _generate

the

_ring

class field modulo t over K. The coefficients of the

mini-mal

_equation

_{of j(Dt)}

_being

_quite

_large

Weber considered in

_[We]

the functions

_{f, f1, f2, 03B32, 03B33}

defined below and

_thereby

obtained

simpler

generators of the

_ring

class fields.

Later on the

_singular

values of these functions

played

a crucial

role in

_Heegner’s

solution

_[He]

of the class number one

problem

for

_quadratic

_imaginary

number fields

_[He,Me2,St].

_Actually

these numbers are used in

_cryptography

to find

_elliptic

curves over finite fields with nice

_properties.

It is the aim of this _paper

_i)

to enunciate some known results

of

_{[We,Bi,Me2,Sch1]}

cf. Theorem 1, 2 and

_{3, concerning}

_singular

values of the

_{functions f, f1, f2, 03B32, 03B33,}

and

_ii)

to

_give

a short and

easy

proof

of these results.

That method also

_applies

to other

_functions,

such as those in the table

_preceding

Theorem 4. The

_proofs

of Theorems 1 to 4

are

_given

at the end of our article.

Our

_{proofs rely}

on the

_reciprocity

law of Shimura

(cf.

Theorem

5,

and also Theorem 6 and

_7),

and on the

_knowledge

of the 24-th

root of

_unity

that

_acquires

_~ =

_by

unimodular substitution

(cf.

Proposition 2,

and

_[Mel]

_p.162);

_they

also

_give

via

_Proposition

3

_explicit

formulas for the

_conjugates

of the

_singular

values

_(of

the above

_functions),

that are

_quite

useful for numerical calculations.

Examples

of such calculations are to be found

immediately

be-fore the References.

Weber’s Class Invariants

The Schllifli functions Weber used in

_[We]

are defined

by

where

denotes the Dedekind

_q-function.

_They

are related

by

the

_identity

Herein j

is the modular invariant and _{g2, g3} are the Eisenstein series

They

have the

_q-expansions

with

The Schlafli functions

_{f , f 2}

and the modular

_{invaxiant j}

are related

_by

the

formulas

Some basic results

_(cf.

_[De]

or

[La]).

In what follows let K be a

quadratic

imaginary

number field of discriminant d and for some

_integer

t we let

Dt

denote the order of conductor t in K.

_Using

the notation :=

7Gw1

+

_ZW2

the order

_Dt

is

_{explicitly given by}

A Z-module a of rank 2 in K is called a _proper ideal of

Dt,

if

The set

t

of _proper

_idt-ideals

is a _{group under}

multiplication

and the

quotient

is called the ideal class _group of For

we set

which is well

_{defined, because j}

is invariant under all unimodular transfor-mations of

_{E j (t)}

is called the modular invariant or the modular

invariant of a. For _any class t E

_9tt

the

_{value j (t)}

is an

algebraic

number

Qt

is the abelian extension of K

_belonging

to the

_subgroup

_lit

of the ideal group of K that is

generated

by

all ideals of the form

(a),

A

integral, prime

to t and A z r mod t for some r E Z. For

_{t, t’}

E we have

and the E

_91t,

form a

complete

_system

of

conjugate

numbers over K. It is even a

_complete

_system

of

_conjugates

over

Q.

In

_particular

this

_implies

We have the

_explicit

formula

where the

_product

is over all

_primes

_p

dividing

_t,

and

d

is the

Legendre

p

symbol

(with

d the discriminant of

_K).

Here

_hK

:= is the class

number of

_K,

and e the index of the _group of all roots of

_unity

in

_K,

that are

_congruent

to 1 mod

_t,

in _{the group of all} roots of

_unity

in K.

More

_precisely

there is an

isomorphism

where

_G(f2t/K)

denotes the Galois _group of

_{Qt /K}

such that the action on

the

_j-invariants

is

This

_isomorphism

is in close connection to the

_description

of

_{G(Ot/ K)}

_given

by

class field

_theory.

Let c be an

_integral

ideal of

.01, prime

to t. Then ct := c n

_,~t

is a _proper

Dt-ideal

and

where in abuse of notation

_{a(c) = a(cut)}

denotes the Frobenius _map

asso-ciated to

_cUt

by

class field

_theory

and on the

_right

side

a(ct)

=

In what follows H will denote the _upper half

_plane.

A

_quadratic

imagi-nary number a E Hn K is the root of a

_{quadratic equation}

AX 2 +BX +C

=

0 which is

_uniquely

determined

_{by a}

if normalized

_by

We call such an

equation

primitive.

The discriminant

for some This

implies

~a, l~

E

_{Jt, and,}

_conversely,

if

_{[al, a2l}

is

in

_3‘t

then the

_quotient

a

_{= 2}

is the root of some

_{primitive equation}

of

discriminant

t2d.

So for a E 1HI with this

_property

the field

Q(j(a»

is

conjugate

to the maximal real subfield of

_f2t.

If g

is one of the functions

f , f l, f 2,’Y2,1’3

the above formulas tell us that

And

_following

Weber we call

g(a)

a class invariant if

To describe the

_conjugates

of

_g(a)

we make the

following

definition.

Definition. Let N be a natural number and _{al, ... , aht} E ~I such that

is a

_system

_{of representatives}

for

_91t

and that further the

_primitive

_equations

AiX 2

+

_BiX

+

_C,

= 0 of the _ai

satisfy

gcd(Ai,

N)

=1 and

_Bi

=

_Bj

mod

_2N,

1

_{i, j}

_ht.

Then we call _{al, ... , aht} a

N-system

mod t.

As we shall _prove later in

_Proposition

_3,

there

always

exists

_{a N-system}

mod t for _every natural number N.

The

_following

Theorem contains Weber’s results on

f

and

fi

and also

includes the assertions

_{conjectured by Weber,}

and

_proved

in the meantime in

_{[Bi,Me2,Schl].}

Theorem 1. Let a E 1Hf be the root

_of

the

_{primitzve equation}

of

discriminant

_D(a)

=

B2 -

4AC = -4m =

t2d.

Herein the

_factor

_(j)

denotes the

_Legendre

_symbol

which is _necessary

_for

the

_following

to hold.

If

a = _{al, ... ,}

aht is a

_16-system

mod t the above

_singular

values

_g(ai)

form a complete

set

_{of conjugates}

over

Q.

Thus the minimal

equation

over

Q

is

_{given by}

and has

_coefficients

in

_Z,

because

_from

_[De]

we know that

g(a)

is

integral.

For discriminants not divisible

by

3 the result of Theorem 1 can be

improved

using

the above relation between

_f

and _y2

_together

with the

following

Theorem. It then turns out that the assertions of Theorem 1

even hold without the outer

_exponent

3 if the

_{primitive equation}

of a also

satisfies the conditions

_3f

A and The

_conjugates

are then described

by

a 3 -

_16-system

modulo t.

_Indeed,

we have for _{q2 :}

Theorem 2. Let a E H be the root

_of

the

_{primitive equation}

of

discriminant.

_D(a)

=

B2 -

4AC =

t2d.

Then

I

Herein

_Q(j(3a))

is

_conjugate

to the maximal real

_{subfield of}

_Q3t

which is

of degree

3 over

Q(j(a))

when

_31D(a)

and

_{D(a) =1=}

-3.

Moreover,

in the case

3f t2 d ,

if

a = _{a1,... ,}

aht is a

_3-system

mod

_t,

then

the

_singular

values

_’Y2(ai)

_form

a

completes

set

of

conjugates

over

Q.

Thus

the minimal

_equation

_{over Q}

is

_{given by}

and has

_coefficients

in Z. A similar result holds for _y3:

Theorem 3. Let a E H be the root

_of

the

_{primitive equation}

Then

Herein

_{Q(j (2a))}

is

_conjugate

to the maxilnal real

_{subfield of S12t}

which is

of

degree

2 over

Q(j(a))

when

21D(a), D(a) =,4

-4.

Moreover,

in the case

2 f

t2d,

_if

a = _a1,...,

aht is a

_2-system

mod

_t,

the above

_singular

values

_1’3(ai)

_form

a

corraPlete

set

_of

_conjugates

over K.

Thus the minimal

_equation

over K is

given by

and has

_coefficients

in

_.1.

The method of

_proof

described in the next section also

_applies

to other functions as for

_example

Table 1

From

_[Sch4],

Theorem

_5,

we know that the last function without the

q2-and

_13-factors

_{is very} useful for the numerical construction of

_ring

class fields.

_They

lead to

_generating

_equations

for all

_ring

class

_fields,

whereas the Schliifli functions

_{only apply}

to the cases when

t2d

is even. Other useful

functions for the construction of

_ring

class fields have been defined in

_[Mo].

In

_{fact, by Proposition}

2

_below,

_{it is easy} to show that the functions

_(of

the above Table

₁₎

all do

_satisfy

the

_hypothesis

of the

_following

Theorem.

It refers to the field

FN

of modular functions of level

N,

N E

_N,

whose

q-expansion

at every cusp has coefficients in the N-th

cyclotomic

field.

Theorem 4.

_{Let g}

E

FN.

We assume

g(z)

and to have a rational

q-expansion

and to be invariant under all unimodular

transf ormations

M

-( * ° )

mod N. Let a e H be the root

o f a

_{primitive equation}

AX 2+BX +C

=

Then,

if

oo, we have

Moreover,

if

a = _al,

... , 7 aht is

a N-system

mod

t,

then the numbers

g(ai)

run

_through

the

_images

_of

g(a)

under the

different automorphisrrcs of

Ot/ K.

Proofs

First we state Shimura’s Theorem

_(see

_{[Sh,La] ) .}

The extension is Galois and there is a natural

isomorphism

between

via the

_following

action of

_integral

2 x 2 matrices B over Z

_having

deter-minant

_prime

to N on

functions g

E

FN:

with b

_prime

to N is obtained

_{from g by}

_applying

the

_isomorphism

_{((N e}

the N-th

_cyclotomic

field to the coefficients of the

_q-expansion

of _g. An

_arbitrary

_integral

matrix B of determinant

_prime

to N has a

decompo-sition of the form

with b

_prime

to N and unimodular matrices

_{Ml, M2.}

The action of B on

g is then

given

according

to the above rules

by

We recall that for an

_integral

ideal b of

Ù1 prime

to t the intersection

bt

:= b n

.ot

is a _proper

Dt-ideal.

We can now state the

Theorem 5

_(Reciprocity

law of

_Shimura).

Let _g be in

_FN

and a E

Jt

with Z-basis al, a2 and a

= 2

E oo. Let b be an

_integral

ideal

_of

~71

of

norm b

_prime

to tN and let B be an

integral

matrix

of

determinant

b such that I -- ..

Then

1)

g(a)

is in

_KtN,

the _ray class

_field

modulo tN over

K,

2)

the action

_of

the Frobenius _map

_Q(6)

_belonging

to b is

_given

_by

For _many functions

_occurring

in

_{complex multiplication}

the

_singular

value in Theorem 5 in fact is contained in a much smaller field. We observe

Theorem 6.

_{Let 9}

E

_FN

have a

q-expansion

with rational

coefficients

and

we assume

that 9

o M _{= g}

for

all unimodular matrices

~o ~~

mod N

and let a be as in Theorem 5 with 00.

Then

_g(a)

E

_ONt.

Proof.

The Galois _group of the extension

_(i-e

ray

class/ring

class

field modulo

_tN)

is the set of Frobenius maps

Q(r)

belonging

to

integral

ideals b =

(r)

generated by

natural numbers r

_prime

to tN. The matrix B

in Theorem 5 is then B

_{= ( § § )}

and one can find a unimodular matrix M

satisfying

with a natural number

r’,

rr’ - 1 mod N. Theorem 5 now

implies

g(a).

Hence

_g(a)

E

_QtN.

0

For the

_computation

of the

_conjugates

of the numbers

_g(a)

in Theorem 6 we derive from the

reciprocity

law

Theorem 7. Let 01, ... , aht E H be

a N-system

modulo t with

primitive

equation

AiX 2

+ +

_Ci

= 0.

For g

E

FN

we set

-

-and we assume oo.

Then

and there exist

_{automorphisms}

~1, ... , Qht

of

KtN/K

with

such that the restrictions

_of

the a i on

f2t

are

which constitute the

_{different automorphisms of}

_S2t/K.

In

_{particular, if}

91 (a1)

E

_Qt,

then

-Proof. By Proposition

3 which will be

_proved

_later,

there exist unimodu-lar transformations

_Mi

E

_r(N),

so that the

ai

:=

Mi(ai)

have

_primitive

equations

satisfying

As

_g(ai)

= and

~(~ai,1))

= it suffices to consider the case

when the

_Ai

are

prime

to tN.

_(Note

that = _oo because

Aiai -

A1a

mod Then the _proper

_17t-ideals

are contained in

Ot

and

_prime

to tN and so for

we have

Now let Q2 :=

o,(ci)

be the

corresponding

Frobenius _maps of Then

their restrictions to

_f2t

are

which constitute the different

automorphisms

of

_{f2t/ K.}

We set

This is a

_quotient

of a basis of

Dt,

and as

_{by assumption}

g(ao) =,4

_00, the

reciprocity

law

_implies

, I

--Using

B1

=

_Bi

mod 2N we can write

/ 2013B

so

by

the

reciprocity

law we obtain

This

_implies

the assertion of the Theorem.

~

0 We now consider a more

special

situation.

Proposition

1.

_{Let g}

be as in Theorem 6 and

Dt

= with

Q

_prime

_to

Nt and &#x3E; 0. Let A E N be

_prime

to Nt with the

_property

that A is

the norm

of

a

_primitive

ideal a and that

_at

=

A~.

Then,

oo, we have

-Proof. Keeping

in mind

_{that g}

has rational

_q-expansion

coefficients the first and third assertion follow

_directly

from Theorem 5 and 6. To prove

the second assertion we first observe that

i7t

=

[/~1]

= whence

00t

=

[0, b]

with b

=,8p.

Theorem 5 now

implies

g(P)u(P)

= =

9~ ~ ~

which

_completes

the

_proof.

D

To

_{apply Proposition}

1 to the Schläfli functions we

_quote

from

[Mel],

_p.

162 formulas

_{(4.21), (4.22)}

and

_(4.23):

Proposition

2. Let be a unimodular matrix which we assume

to be normalized

_by

We

_{de fi ne}

ci and A in Z

by

Then we have the

transformation formula

and

"1

Herein

_{( a ) is}

the

_{Legendre symbol}

and

₍₂₄

=

Proo f .

The formula is

_easily

derived from

_{~Mel~ .}

In fact there are two formulas in

_[Mel],

one in the case

2 ~

c and another in the case

2 f

a. The

above formula is obtained

_by

_applying

the

_{quadratic reciprocity}

law to the

Legendre symbol

( ~ )

in front of the second formula in

_[Me1]

which then in the case

2 f

c coincides with the first formula. L7

Remark. A

multiplicative

interpretation

of the above values

_e(M),

with M

_unimodular,

can be found in the _paper of Farshid

_Hajir

[Ha].

Proposition

3. Let N be a natural

number,

ao E H the

_quotient

o f

a basis

o f

an ideal _ao

f rom

Jt

and

AoX2

+

_BoX

+

Co

= 0 its

primitive equation.

We assume that

gcd(Ao, N)

= 1.

(this

last

exigency

puts

no restriction

on the above ideal

ao).

Then in any class

o f

9~

there exists an ideal a

and a

_of

a such that its

_quotient

a E 1HI has a

_{primitive equation}

For any such a there exists a unimodular

transformation

M -

mod N such that the

_{coefficients of}

the

_{primitive equation}

+

B’ X

+

C’

= 0

of

M(a)

sdtisfy

Proo, f.

Let a E H be the

_quotient

of a basis of an ideal a E

3’t.

Then

a’

E H is the

_quotient

of a basis of an ideal from the same class as a if and

only

if

a’ =

_M(a)

with a unimodular transformation M. So we must show

that there exists a unimodular transformation M such that the

_primitive

equation

of

_M(a)

has the desired

_properties.

We do this

_by

induction on

the number of

_{primes dividing}

N. For N = 1 there is

_nothing

to be shown. Note that

B2 - t2d

mod 4 which

_implies

the desired _congruence for B in the case N = 1. So we assume the assertion to be shown for some N E N and we contend that it is also true for N’ =

Nps, s

_E

N,

with _a

prime

p

not

_dividing

N. To construct the unimodular transformation

_needed,

we

define

If w E H is a root of the

_primitive

_equation

then

_Mp(w)

resp.

Np.(w)

are roots of the

equations

where the

_gcd

of the coefficients is

_{again equal}

to 1. Now we assume that a is a root of the

primitive quadratic equation

AX 2

+ BX + C = 0 with

gcd(A, N)

=1, A

_&#x3E; 0 and B -

Bo

mod N.

If a is transformed

by

some

_M~

or

N,

with p

divisible

by

N,

these

condi-tions are conserved for _p

sufficiently

large.

Note that

B2 -

4AC 0 and

A &#x3E; 0

_implies

C &#x3E; 0. Further

_{by applying}

a

_product

of

_M~’s

and

_N,’s

we can achieve that A becomes

_prime

to _p. For 2 or

(p

= 2 and

2~B)

this

becomes clear

_{by writing}

_I-iC).

If

_(p

= 2 and

2 f B)

we must first achieve that

_{2 ~ C}

_by

_applying

some

_M~

and then we

get

gcd(A, 2)

= 1

by applying

_some

Np..

In this way we end _up with an

equation

where A is

_prime

to N’. In order to

_get

an

_equation

in which B is

congruent

to

_Bo

mod N’ we

apply again

some

_M~.

Then B is transformed

and

_keeping

in mind that B -

_Bo

mod

_2,

we then see that

by

a suitable

choice of a

number it

= 0 mod N the _congruence B -

_Bo

mod 2N’ can be

satisfied.

To _prove the second assertion of

_Proposition

3 we continue

applying

this

construction to the

_primes

_p

_dividing

t and not

_dividing

N. Then the

parameters

p are divisible

_by

N. So the unimodular transformation M

satisfies

_{M - ~ ( o ° )}

mod N because it is a

_product

of

_M~’s

and

_Np.’s

with

It = 0 mod N. D

Proof of

Theorem 1.

_{Using Proposition}

2 and the relation

_f

= fl f2 ,

we find

that

_{f 3}

is invariant under unimodular Transformations M =

_(~~)

mod 16. Further we can see from the definition that

f 3

has rational

_q-expansion

coefficients and

_{by Proposition}

2 it follows that

_{f 3}

is in

_F16.

Let m be

the natural number from Theorem 1 and t be the conductor of the order

[.B/--m, 1].

For _{p E} Z we set

Then

_by

Theorem 6 we have

Here

_{[Q16t :}

= 16 for -4 and

[Q16t : Qt]

= 8 for

t2d

= -4.

Choosing

the numbers _p mod 8 with the

_property

that the

_/3J1.

are

_prime

to t we find that

By

Proposition

1 we obtain the Galois action

keeping

in mind that

_by

class field

_{theory (8}

is in

_S2lst

_(or

_by

_concluding

(8

=

f (,30)3f (pl)-3

_E

016t).

Herein _we have

~’8 «’‘~ _ ~’8’‘

with the

_complex

norm _{n IA} of

(3p..

Further from

_Proposition

2 we

_get

the

_identity

=

f (z).

Whence our Galois action becomes

Using

the relation

_{~16 f 3(z}

+

₁₎

we further obtain

Discussing

cases we now obtain that for the

_special

_arguments

a =

the numbers of Theorem 1 are in

f2t.

Moreover these numbers are real as can be seen from the

q-expansions.

So we can conclude that

they

are

contained

_{in Ot}

n R =

They

are even

_generators

for this

_field,

because of the

_{relation j}

=

f24_16 3

=

( f24+163

.

1

The assertions of Theorem 1 now follow from Theorem 7

keeping

in mind

that the action of the

_{automorphism oi}

in Theorem 7 on

.J2

is

given

by

Proof of

Theorem 2.

_Using

_Proposition

2 it can be seen that _~2 is

invari-ant under unimodular Transformations

_{M - ~ o ° ~}

mod 3. Further q2 has

rational

_q-expansion

coefficients. As in the

_proof

of Theorem 1 we first assume a to be of the form

with a natural number m and conclude

by

Theorem 6

where t denotes the conductor of the order =

[a, 1].

For _a =

9+2

the

assertion is trivial because

_{q2 (a)}

= 0. Otherwise _we find

, , , ~

We

_compute

the Galois action on

q2 (a)

by Proposition

1:

u-rp,

Here _n, is the norm of a

_+/~

which is

_congruent

to m

+ /-t2

modulo 3.

Using

further the transformation formula

_{-y2( zl)}

=

-y2(z)

the Galois action

becomes

The rest of the

proof

now follows

_again

from Theorem 7. 0

Proof of

Theorem 3. As in the

_{preceding proofs}

it suffices to consider the

case when a is of the form

with a natural number m. satisfies the

_hypothesis

of

Proposition

1

with N = 2 we have

where t is the conductor of

_Dt

=

[a, 1].

Here

D(a) =

_-m, and

In the case 2

f

D(a)

this

implies

13(a)

E

_Slt,

because the square of

y3(a)

is in

_f2t.

Here is real and

_by

the usual

_arguments

it follows

In the case

2ID(a)

the Galois _group of

Q2t/f2t

is

generated by

Q(a + 1)

and, using

the identities

_73(z

+

_{1) = -q3(z)}

and

_{73( Zl) _}

_-q3(z),

we

get

-/3(a).

This

_implies

_{~(’Ys~a)) _}

_n2t

n R =

_{Q(j (2a))}

because in this case

y3(a)

is real. The last assertion of Theorem 3 follows

again

from Theorem 7. 0

Proof of

Theorem

_4.

From Theorem 5 we know that

g(a)

is in

KtN.

So we are left with the

proof

of

g(a)

_being

invariant under the

_{automorphisms}

of

_KtNlQt.

These are all Frobenius _maps

Q(c)

belonging

to some

principal

primitive prime

ideal c =

ÀD1

of _{norm c}

prime

to tN with A E

Ot.

To

compute

g(a)"(")

we observe that

By

adding

to a a suitable number from NZ we can further achieve that

Under this

_change

of a the value

g(a)

remains the same

because g

has the

period

N and also the

_assumptions

about the coefficients of the

_primitive

equation

remain valid. The

_reciprocity

law now

implies

, ,

where u, v

E Z come from the

_{representation}

A = u + vAa E

_0~

=

Further we have

So there is a unimodular matrix M

satisfying

Comparing

coefficients,

we now

_get

the

_identity

which tells us that

because

_by

_assumption

we have

NIG

and

gcd(N,

c)

= 1. From

(+)

we now

conclude that 2 =

M(a)

and the Galois action therefore becomes

...

This invariance

_implies

_g(a)

E

l1t.

0

Examples.

The

_polynomials

in the

_{following examples}

are the minimal

_polynomials

A

_{slightly simpler generating}

element for the same field is obtained

_by

the

_singular

value

which is a

_quotient

of the form

ai/a2,

for the ideal a =

(a, 5],

in the _sense

of

_[Ha-Vi]

_pp.518-519.

For the above a the values

q2(a)

and

_q3(a)

are

class invariants. So Theorem 4

_implies

that

ê

is in

_K(j(a))

=

524

and

from

_{[Sch2, Sch3],}

we know that

6

is a

_generator

of

S24/K.

It also is a

References

[Bi] B. J. _BIRCH, Weber’s Class Invariants. Mathematika 16 _(1969), 283-294.

[De] M. _DEURING, Die _{Klassenkörper} der _{komplexen Multiplikation. Enzykl.} d. math. Wiss.

I/2, 2.Auflage, Heft _{10, Stuttgart,} 1958.

[Ha-Vi] F. _HAJIR, F. R. _{VILLEGAS, Explicit elliptic units,} I. Duke Math. J. 90 _(1997), 495-521.

[He] K. HEEGNER, Diophantische Analysis und Modulfunktionen. Math. Zeitschrift 56 _(1952), 227-253.

[La] S. LANG, Elliptic functions. Addison Wesley, 1973.

[Mel] C. _MEYER, Über _{einige Anwendungen} Dedekindscher Summen. J. Reine Angew. Math. 198 _(1957), 143-203.

[Me2] C. _{MEYER, Bemerkungen} zum Satz von _{Heegner-Stark} über die imaginär-quadratischen Zahlkörper mit der Klassenzahl Eins. J. Reine _Angew. Math. 242 _(1970), 179-214.

[Mo] F. _MORAIN, Modular Curves, Class Invariants and Applications, preprint.

[Schl] R. SCHERTZ, Die singulären Werte der Weberschen Funktionen J. Reine

Angew. Math. _{286/287 (1976),} 46-74.

[Sch2] R. SCHERTZ, Zur Theorie der _{Ringklassenkörper} über _{imaginär-quadratischen} Zahl-körpern. J. Number _Theory 10 _(1978), 70-82.

[Sch3] R. SCHERTZ, Zur _{expliziten Berechnung} von Ganzheitsbasen in Strahiklassenkörpern über

einem _{imaginär-quadratischen Zahlkörper.} J. Number _Theory 34 _(1990), 41-53.

[Sch4] R. _SCHERTZ, Construction _{of Ray} Class Fields _{by Elliptic} Units. J. Théor. Nombres Bordeaux 9 _(1997), 383-394.

[Sh] G. _SHIMURA, Introduction to the Arithmetic _{Theory of Automorphic Functions,} Iwanami Shoten and Princeton _{University Press,} 1971.

[St] H. _STARK, On the "Gap" in a Theorem _{of Heegner.} J. Number _Theory 1 (1969),16-27.

[We] H. _WEBER, Lehrbuch der _Algebra, Bd 3, 2. Aufl. _{Braunschweig,} 1908; Neudruck, New

York, 1962.

Reinhard SCHERTZ

Institut f3r Mathematik der Universitat _Augsburg Universitatsstralie 8

86159 _Augsburg, Germany