Lower powers of elliptic units

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

S

TEFAN

B

ETTNER

R

EINHARD

S

CHERTZ

Lower powers of elliptic units

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{2 (2001),}

p. 339-351

<http://www.numdam.org/item?id=JTNB_2001__13_2_339_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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

Lower

_powers

of

_elliptic

units

par STEFAN BETTNER et REINHARD SCHERTZ

RESUME. Dans un article antérieur

_[Sch2]

il est démontré que les corps de classes de _rayon d’un _corps

_{quadratique imaginaire}

possèdent

comme

_{générateurs}

des

_{produits simples}

de valeurs

sin-gulières

de la forme de Klein defini

_plus

bas. Dans l’article

_présent

le deuxième auteur a

_{généralisé}

les résultats de

_[Sch2]

et en même

temps

corrigé

une erreur dans le Théorème 2 de

_[Sch2].

Le

_premier

auteur a

implémenté

le calcul de ces

produits

dans un _progamme

de KASH et ainsi effectué le calcul des

_exemples

en fin d’article.

Ces

_exemples,

ainsi que les cas

_particuliers

traités dans

[Sch2],

démontrent,

que les nombres définis par le Théorème 1 ont des

polynômes

minimaux à très _petits coefficients. En

_plus,

à _part des

exceptions triviales,

ces

_produits

de valeurs

_singulières

constituent même des

_{générateurs,}

ce _qui mène à la _conjecture énoncée

après

le Théorème 1.

ABSTRACT. In the

_previous

paper

[Sch2]

it has been shown that ray class fields over

_quadratic

_imaginary

number fields can be

generated by simple

products

of

_singular

values of the Klein form defined below. In the _present article the second named author has constructed more

_general

products

that are contained in ray

class fields

_{thereby correcting}

Theorem 2 of

_[Sch2].

An

_algorithm

for the

_computation

of the

_algebraic

_equations

of the numbers in Theorem 1 of _{this paper has} been

_implemented

in a KASH program

by

the first named

author,

who also calculated the list

of

_examples

at the end of this article. As in the

special

cases

treated in

_[Sch2]

these

_examples

exhibit

again

that the coefficients of the

_algebraic

equations

are rather small.

Moreover,

apart from trivial

_exceptions,

all numbers

_computed

so far turn out to be

generators of the

_{corresponding}

ray class

field, thereby suggesting

the _conjecture formulated more

_precisely

after Theorem 1.

Introduction and results

We let r be a lattice in C and _Wl, c~2 a Z-basis of r with &#x3E; 0. The normalized Klein form is then defined

_by

Here aa denotes the a-function of r and _1] the Dedekind

_1]-function.

The

number z* is defined

by

with the real coordinates zl, z2 Of Z = _{zlc.y + z2cv2} and the

quasi-periods

of the

_elliptic

Weierstrass

_(-function

of r

_belonging

_{to wl, cw2.} In what follows let K be a

quadratic imaginary

number field of

discrimi-nant

_d,

D the

_ring

of

_integers

in K

_{and f}

an

_integral

ideal of 5J. We denote

by

_Kf

the _ray class field modulo

_f

over K.

Using

this notation we now state

our main result.

Theorem 1.

_{Let ~}

be an element

of K*, f

the denominator of the ideal

(E)

and

_{f :=}

_{min(N n (f B}

We use the notation

[Wl, W2]

:=

7Gcy + Zw2

and choose an element a E

_K,

&#x3E;

_0,

so

~a,1~

and

w .

which is

_{always possible.}

Let s be some

_{integer &#x3E;}

1 and define ideals

_{bl, ..., bs}

and c of 0 of norm

b1, .., bs,

c

prime

to

_{6 f by}

where the bar denotes

_{complex conjugation.}

Note that the above second set

of

_equations

_imposes

further constraints on a.

_{..., ~S, ~}

be numbers

from

_{sJ B}

_f

and _{nl, ..., ns E} Z. We _suppose that the ideals

_{gcd((2), (ai))}

are

equal

for all i with and that

_{gcd(N(A), 6 f )}

=

1,

where

N(.)

denotes the norm.

We define

Q1 . -- .... ,.,...,.

and

_decompose

with

_fl

E N and a

primitive

ideal

f2

of norm

f2. By

f2

we denote the non

split

part

of f2

and

_by

_f2*

the

_split

_part

of

_f2.

Now we make the

following

We choose a canonical basis

_f2

=

~&#x26;, f2~

with _some other

_integral

element

a E

i7,

s(a)

&#x3E; 0. Here

_tr(a)

is

_prime

to

_f2*

and so there is a solution a of

the congruence

= 1 is _even

and ~

= 2 odd. We _{set (}

Then

and the action of the Frobenius _map

_Q(ca)

_{of Kf/K}

_belonging

to the ideal

ca is

_{given by}

where

_{e(A, a)}

is the 12-th root of

_unity

defined in

_(11).

Numerical

_experience

shows that

_apart

from trivial

_exceptions

the

num-bers defined in Theorem 1 are even

_generators

of

_Kf

over K and in the case of O

_being

a _power of a

_quotient

of two

_cp-values

this has been

_proved

in

_[Sch2]

under certain

_assumptions

about the

Ai

and

_f.

Moreover the

computations

done so far make us believe in the

_following

Conjecture.

Let the

Ai

in Theorem 1 all be

_prime

to the

_{conductor f}

so

that raised to the power

12 f

the factors in the definition of8 are

_conjugate

numbers in

_Kf.

We assume further that the

product

of these

12 f -th

_powers

is not a norm to a _proper subfield

(i.e.

the formal sum

Ei

ni[Àibi]

of _ray

classes

_{modulo f}

of the ideals with the as coefficients is not

equal

to a

multiple

of the formal sum over the elements ofa _proper

subgroup

of the _ray class _group

_{modulo f).}

Then

₍₈

is a

_generator

of

Kf/K.

Theorem 1 is a

_{generalization}

of Theorem 2 in

[Sch2],

where,

as H. Cohen

has

_pointed

out to the

_authors,

the

_hypothesis

_"gcd(f,

_f)

= I" has to be

added. So in order to correct that theorem of

_[Sch2]

we write down a

I . -.. , "I

special

case of Theorem 1 of this _paper. Herein the number I

is a

conjugate

of the number

I I

defined in

_[Sch2].

Theorem 2.

_{Let ~}

be an element

_{of K*, f}

the denominator of the ideal

(~)

and

_f

:=

min(N‘

fl

_{(f B {0})).}

We use the notation

[WI, W2]

:=

ZW2

which is

_{always possible.}

We define ideals b and of norm

_b,

c

prime

to

6f by

where the bar denotes

_{complex conjugation.}

Note that the above second

set of

_{equations imposes}

further constraints on a. Let

Ai ,

A be numbers

from D

_B

_f.

We _suppose = =

1,

where

N(.)

denotes the norm.

We define

’

, . - I

and

_{decompose f}

=

f 1 f 2

with

11 E

N and a

_primitive

ideal

_{f 2}

of norm

_{f 2.}

By

f 2

we denote the non

_split

_part

of f 2

and

_by

the

_split

_part

of f 2 .

Now we make the

following

assumptions:

1) b-1 mod4,

if 21d and 2 f

2)

b - 1 mod

_3,

_{if 3 ( d}

and 3

_f

_{f ,}

3)

bN(À¡) == 1

mod

_{,f 2 * gcd 2 f 2 * ) ’}

°

We choose a canonical basis

_{f 2 -}

f2l

with some other

_integral

element

a

_{.~s(a) &#x3E;}

0. Here

_tr(&#x26;)

is

_prime

to and so there is a solution a of

the congruence

= 1 is even

_{and J1.}

= 2 is odd. We

_{set (}

_:=

exp( 2f ).

Then

The action of the Frobenius _map of

_Kf/K

_belonging

to the ideal cÀ

is

_{given by}

where e(A,a)

is the 12-th root of

_unity

defined in

_(11).

According

to Theorem 3 in

_[Sch2]

all _powers of the numbers defined in Theorem 2 of this _paper are

_generators

for

Kf/K

under certain conditions.

For these conditions to be satisfied _{it is necessary} that the ideal

_Aib

is

not in the

_principle

_ray class

_{modulo f,}

a condition

_which,

_according

to

our

_Conjecture

is also believed to be sufficient. However there are _cases,

where such a

_pair

_{À1, b}

cannot be found.

_Generalizing

an idea that the

authors have been told

_by

H. Cohen it then follows from class field

_theory

that

_Kf

g

_K(~12f).

Moreover the above inclusion

_implies

that the Hilbert class field of K is Abelian over

Q.

Thus K is one of the finite number of

quadratic imaginary

fields with

_only

one class _{per genus.} Further we can

conclude

_{that f}

=

f,

and it follows more

_precisely

_{C Kf}

C

_K((12f).

In

_particular

this

_implies

that

_Kf

can be constructed

_using

roots of

_unity,

when no

_{pair À1, b}

_satisfying

the

_hypothesis

of Theorem 2 exists.

Proof of Theorem 1

The

_proof

of Theorem 1

_requires

the

_reciprocity

law of

_complex

multi-plication

which we are now

_going

to

_explain

_(see

_{[La, St]).}

For a natural

number N let

_Fnr

be the field of modular functions

_belonging

to the group

that have at _{every cusp}

_q-expansion

coefficients in the N-th

_cyclotomic

field.

_FN

is

_generated

over

Fl by

a

primitive

N-th roots

_unity

_(N

and the functions

where T denotes Weber’s T-function. The extension

FN/Fl

is Galois and

its Galois group is

isomorphic

to

(10) 1.

For an

_integral

matrix A of determinant a

_prime

to N the action of the

_{corresponding}

automorphism

of is

_{given by}

To

_compute

the action of A on an

arbitrary

function

f

on

FN

we observe

that

where in abuse of notation we write

M(w)

instead of

~ .

Looking

at the

q-expansion

of _7x

_(see

_[De])

we see

_{that Tx}

o

( § [ )

is obtained

_{by applying}

the

_automorphism

aa =

((N e

~~,)

of

~

to the

q-coefficients

of _Tx. So

for an

_arbitrary

function

f

E

FN

with

_q-expansion

we also have

and

For an

arbitrary integral

matrix A of determinant a

prime

to N this action

can then be

computed

via a

decomposition

which

_always

exists.

Reciprocity

law. Let a =

~al, a2~

be an ideal in K and a :=

~

with

&#x3E; 0. Then for

_f

E

_FN

with

_{/(~) 7~}

oo we have

and the action of the Frobenius _map

_Q(c)

_belonging

to an

integral

ideal c

prime

to N is

_{given by}

where C is an

_integral

matrix of determinant c &#x3E; 0 such that

_{C ( a2 )}

is a

basis of 6.

We remark that the

_reciprocity

law is

_proved

for

_{primitive prime}

ideals c

first.

_{By multiplicativity}

it is then

_generalized

to

_primitive

ideals c and it is

in fact also valid for

_integral

ideals c. This can

_easily

be derived

observing

on the one hand that

f o A only depends

on A modulo N and that on the

other hand

_given

an

integral

ideal c

_prime

to N there is a relation

with

_integral

numbers

_{Aj m}

1 mod N and a

_primitive

ideal _co from the _ray

class modulo N of c.

To

_apply

the

_reciprocity

law to the functions of Theorem 1 we start

_by

collecting

some transformation formulas and

q-expansions.

For a

complex

lattice r =

[WI,

W2], ~(~)

&#x3E;

_0,

and w E r we obtain from the

transforma-tion formula of the a-functransforma-tion

where

_{the zi}

have been defined in

_(1).

The R-linear function 1 is alternate. For any basis wl , w2 of r such that

W2

&#x3E;

_0,

one has

and for the calculation of I the rule

for

_{complex conjugate}

numbers

_{~, ~}

_(see

_[Ro])

_{is very} useful.

For x =

_{(XI, X2)}

_{E Q}

_{x Q}

we consider the function

for all M E

_SL2(Z).

The factor

E(M)2

is a 12-th root of

_unity

that comes

from the transformation formula of the

_1]-function:

If c &#x3E; 0 and d &#x3E; 0 if c = 0 we have the

_explicit

formula

Here ci = _c if _c is odd and

ci = 1 if c is _even, and we use the notation

(12

= This formula is

easily

derived from

[Me].

In fact there are

two formulas in

_[Me],

one in the case

2 t

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

_[Me],

which

then in the case

2 t

c coincides with the first formula. To evaluate

E(M)2

for

_arbitrary

M E

_{SL2 (Z)}

we have to use the relation

9x has the

q-expansion

(see

[La])

with

Now let a =

a2~

be _an ideal of D with _a =

~j

in the _upper half

_plane

and 6 E where

_f

is defined as in Theorem 1. Then we can write

and

(5)

and

₍₈₎

_imply

that _9x is in

_{F12 f2}

and the

_reciprocity

law tells us that

To

_compute

the Galois action let c be a

primitive

ideal of sJ

prime

to

6 f

and let

_Q(c)

denote the Frobenius

_automorphism

of

_{K12 fz/K}

_belonging

to

where C is a rational matrix of determinant c =

N(c)

that transforms the

basis of a into a basis of aë. With a

_{decomposition}

of the matrix

cC-1,

we then

_compute

using

the transformation formula

(5)

- , .. - . .. I’t....- _ _

By

homogeneity

of _p we find

So the Galois action becomes as

already

shown in

[Ro, Schl]

and

_{herein ~ C}

( a2 )

is a basis of

ac-1.

Now we consider two

_special

cases:

(i)

First we assume that a =

[a, 1]

and ac =

[a, c~.

Then C =

( o ° ),

whence

cc-,

_{= (?}

_{( o o ) ~ ol}

_1),

and

_by

the formulas

_(6),(7)

we

_get

(ii)

Next we assume a = 0 =

[a, 1]

and _c =

(A)

_to be _a

principal

ideal

prime

to

_{12 f .}

We write

As

_Q(a)

_depends

on A

_only

mod

_{12 f 2}

we can

_change

A modulo

12 f 2

thereby

achieving

that

, - .. , ,

-Then

with

_{a, b}

E Z such that

_det(M2)

= 1. Moreover

as v 0

0 _{we can} choose a &#x3E; 0. So

_using

the

_explicit

formula

₍₆₎

for

_e(Mi)

we can now evaluate the

factor

By

the formula

_(9),

it does not

_depend

on the

_auxiliary

choice of a and

b. We omit the

_boring

case

by

case calculation

_using

the fact that c =

u2 +

uvs +

v2N

is

_prime

to 6. The result is

Moreover,

in case

_(ii),

assume that S =

tr(a)

satisfies the congruences

stated in Theorem 1. Then it can be worked out that

Observing

further that for c =

(À)

_we

have )C

C a2 )

= a ~ i ~ ,

the above formula

₍₉₎

becomes

where

_{by homogeneity}

of _cp the

_{factor a}

in the second

_argument

is

_replaced

by

the factor A in the first

_argument.

Now we use

(10)

and

(13)

to prove the assertions of Theorem 1. The relative

_{automorphisms}

of

are of the form A = 1

+ w, w E

_f.

_By

₍₁₃₎

we

_compute

_using

the

transformation formula

₍₂₎

and the

_properties

₍₃₎

and

₍₄₎

of 1

To evaluate

_l(l,w)

we choose

in f

a canonical

basis, f

=

/i[o~ /2J?

s(a)

_&#x3E;

_0,

and write

Then,

keeping

in mind that a - a mod

Z,

we

_get

l(l, w) =

_-2.7rifly.

Now

The action of

_a(A)

on 8 is then

given by

Further we find

The

_assumptions

about the

_{Aj and nj}

_together

with the

_property

₍₁₂₎

of

c(A,

a)

now

imply

that

(8

is invariant under the action of the Galois _group

of

_{K12 f2 /Kf}

and so is contained in

_Kf.

The

_computation

of the

_conjugates

as described in the Theorem follows from

(10)

and

(13).

Examples

By

the

_following

_examples

it is shown on the one

_hand,

that the

min-imal

_polynomials

of the numbers constructed in Theorem 1

_mostly

have rather small coefficients. On the other hand all numbers considered in the

following

are

_generators

for

Kf/K

thereby

confirming

our

_conjecture.

Using

Theorem 1 we determine O and its

_conjugates

as well as the root

of

_{unity ( by}

which we

_get

the minimal

_polynomial

of

₍₀

over K. In all

cases

CO

turns out to be a

_generator

of

Kf/K,

though

in the

examples

5.-7.

the conditions of our

conjecture

are not satisfied.

In the

_following

_{mg K denotes the minimal}

_polynomial

of 0 over K.

Fur-ther we denote

by

_{p2, p3, p5, P7}

prime

ideals

dividing

2, 3, 5, 7

and the

num-ber a is assumed to be of the form a =

9

with the discriminant d of K

and u =

tr(a)

_E Z.

where

where

4. Let d =

-31, f

=

p3

and

where 1 and

The denominator of O can be obtained

by

the known factorisation of

and

j , .

The

_special

form of

_shows,

that

05

does not

_generate

_KTIK.

This is not in contrast to our

_conjecture,

because

5

is not

_prime

to

f .

Rather one can write

05

with _

as 10

and Theorem 1

_supplies

05

E

_Kf

with f

=

(2)5.

So this

_example

also

shows,

that in

_general

the

_exponents

of the numbers

_given

in Theorem 1

can not be reduced. 6. Let d =

_{-52, f}

= _{p7 and}

Here

₍₈

is a

_generator

of

_Kf/K,

_though

it is a relative norm in the sense of our

_conjecture.

Also

((8)2

_generates

_Kf/K

with

Considering

we _see, that

((8)4

_generates

_only

a _proper

sub-field of

_Kf

over K.

_Kf

itself is

generated by

(~O)4

over

_K(l)

at least. 7. With "small"

_{denominators f}

the Hilbert classfield

_K(l)

can often be

constructed

_by

the

_products

in Theorem 1. This does not follow from

where and

with t ₃ and

References

[De] M. _DEURING, Die Klassenkörper der _{komplexen Multiplikation. Enzykl.} d. math. Wiss. I/2, 2. Aufl., Heft _{10, Stuttgart,} 1958.

[La] S. _{LANG, Elliptic functions.} Addison _Wesley, 1973.

[Me] C. MEYER, Über einige Anwendungen Dedekindscher Summen. Journal Reine Angew.

Math. 198 _(1957), 143-203.

[Ro] G. ROBERT, La racine 12-ième canonique de

0394(L)[L:L]/0394(L)

Sém. de th. des nombres Paris,

1989-90.

[Sch1] R. SCHERTZ, Niedere Potenzen elliptischer Einheiten. Proc. of the International Con-ference on Class Numbers and Fundamental Units of Algebraic Number Fields, Katata,

Japan (1986), 67-87.

[Sch2] R. SCHERTZ, Construction _{of Ray} Class Fields by Elliptic Units. J. Théor. Nombres Bordeaux 9 _(1997), 383-394.

[St] H. _{STARK, L-functions} at s = _1, IV. Adv. Math. 35 (1980), 197-235.

Stefan BETTNER, Reinhard SCHERTZ

Institut fiir Mathematik der Universitat _Augsburg Universitatsstraf3e 8

86159 _Augsburg

Germany

-. stefan.bettnerQMath.Uni-Augsburg.DE

°