Frobenius distributions for real quadratic orders

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

P

ETER

S

TEVENHAGEN

Frobenius distributions for real quadratic orders

Journal de Théorie des Nombres de Bordeaux, tome

7, n

o

_{1 (1995),}

p. 121-132

<http://www.numdam.org/item?id=JTNB_1995__7_1_121_0>

© Université Bordeaux 1, 1995, tous droits réservés.

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

Frobenius

distributions for real

_quadratic

orders

par Peter STEVENHAGEN

ABSTRACT. - We present a _density result for the norm of the fundamental unit in a real _quadratic order that follows from an equidistribution assump-tion for the infinite Frobenius elements in the class _groups of these orders.

1. Introduction

This _paper deals with the distribution of the Frobenius element of the

in-finite

_prime

over the class _{group in certain} families of real

_quadratic

orders.

It contains

_precise

results for these distributions that are at this moment

mostly conjectural,

but

_provide

a

satisfactory ’explanation’

for observations

that have not been understood

_otherwise,

such as those

_concerning

the fre-quency of real

quadratic

fields

having

fundamental unit of norm -1.

They

are somewhat similar in

spirit

to the

Cohen-Lenstrà

heuristics

_[3],

y which

’explain’

the _average behavior of class _groups of number fields.

_However,

our

underlying

_assumptions

are of a more

specific

nature than those in

_[3],

and weak versions of our

density

results can

actually

be

proved.

Our

con-jectures

answer an old _open

_question

that _goes back at least to

_Euler,

but does not seem to _appear

_explicitly

in the literature _any earlier than in a

1932 paper

by Nagell

[5].

Nagell’s

question

concerns the

solvability

of the well known

_negative

Pell

equation

for _non-square d E

_Z&#x3E;1

in

_integers

_{x, y} E Z. A _necessary condition for

solvability

is

_obviously

that -1 is a _square modulo all divisors of

_d,

i.e.

that d is not divisible

_by

4 or a

_prime

_{p = 3 mod} 4. This can be

_phrased

more

_{concisely by stating}

that d is the sum of two

_coprime

_squares.

_Thus,

let us denote

_by

S the set of

_integers

that can be written as the sum of

two

coprime

squares. With S-

denoting

the set of

integers

d for which

1991 Mathematics _{Subject Classification. Primary 11R11, 11R45, 11R65; Secondary}

11D09.

Mots clefs : Real _{quadratic fields, quadratic} units, Pell _equation.

x2 -

_{dy2 ~ -1}

has

_integral

_solutions,

_Nagell’s

_question

is: does S- have a

natural

_density

in

_S,

i.e. does the limit

exist? It _appears that the well known characterization of S- as the set

of

_integers

d &#x3E; 1 for which

_dl

has a continued fraction

_expansion

with odd

_period

_length

is not of _any use in

_answering

this

_question,

and it is

not even known whether the liminf and the

_limsup

of this

_expression

are in the open interval

(0, 1).

The same

applies

to the somewhat

_simpler

question

that is obtained

_by

_restricting

to

_{squarefree d}

in

_{Nagell’s problem}

or,

equivalently, by

posing

the

_problem

for real

_quadratic

fields. More

precisely,

let D be the set of real

_quadratic

fields K for which -1 is in

the norm

_image

and D- c D the subset of fields K E D for which the norm of the fundamental unit

_{equals -1.}

Then we have D =

(Q (U2) :

d E s?

and-even

_though

OK

may not be of the form

also D- =

_{(Q(v2) :}

d E

_S-}.

The

_analogue

of

_Nagell’s

_question,

which has been studied

_{extensively by}

Rédei

_[7],

can now be

phrased

as: does the

limit

, , r +, - -- , v,, , ., &#x3E; rr f r _ - A / r

exist

_and,

if it

_does,

what is its value?

_Existing

tables as those

_occurring

in

[1]

and

_[5]

show that the value of this fraction is around .860 for X

= 10~

and decreases

_slowly

to assume the value .799 for X

=107.

The considerations in this _paper make _{it very}

_plausible

that the answer to both

_questions

is

the

_following.

CONJECTURE. The limit valuers P and

_Q

exist and are

equal

to

and

with vp

denoting

the numbers

_{of factors}

2

_occurring

in

_{p -1,}

We will see in the next section

_why

the convergence to these limit values

is

_extremely

slow.

The densities in our

conjecture give

rise to absolute estimates for the

number of d E S^ and K E

_Dy,

as the

_counting

functions for the sets S

and D are known to

_satisfy

the

_asymptotic

relations

and

These relations are _consequences of results of

Rieger

[8].

2. The fundamental case .

In this section we reduce the

density problem

for real

quadratic

fields

from the introduction to a statement on Frobenius distributions and

_explain

how this

_gives

rise to the indicated value of P.

Suppose

we are

given

a

quadratic

field K E D with discriminant

_{D, ring}

of

_integers

C~ and class group Cl. Denote

by

C the narrow class _group of

K,

which _may be identified with the class group of

binary quadratic

forms

of discriminant D. There is a natural

surjection

C -~ Cl whose kernel is

generated by

the class

_~oo

E C of the

_principal

ideal

_{generated by}

m.

If C~ contains a unit E of

_negative

_norm, then the ideal

m.

O can be

_generated

by

the element

em

of norm D &#x3E; 0 and is the unit element in C. If all units in 0 have norm

_1,

then

Foe

has order 2 in C.

Thus,

we have .K E

D-if and

_only

if

_{F 00}

is the trivial element in C.

By

class field

_theory,

we can

identify

C with the Galois _group over K

of the narrow Hilbert class field H of K. The maximal real subextension

H+/K

of

_H/K

_corresponds

to the factor _group Cl of

C,

so

F 00

_generates

the

_{decomposition}

_group of the real

_prime

in and can be viewed as the Frobenius element at

_infinity.

Our

_conjecture

for real

_quadratic

fields tells us that we should

_expect

the Frobenius

Fao

for the

family

D of

heuristics on the _average behavior of class _groups

explicitly

exclude the

2-parts

of

_quadratic

class groups that we have to deal with here. The reason

for this is that these

2-parts

are not at all ’random’ in their sense of the

word,

and

_consequently

we should not

_expect

_{F 00}

to behave as a ’random

2-torsion element in a random abelian

2-group’.

What we have to take into account when

_studying

the average behavior

of the element

_{F 00 E}

C is that this Frobenius element is not

_only

in the

2-torsion

_subgroup

_C~2~

of

_ambigous

ideal

_classes,

but also in the

_principal

genus

C2

C C of squares in C. If D has

exactly

t distinct

prime divisors,

then there are t ramified

_prime

ideals _{~1, ~2, ... ,} pt in

_C~,

and it is a classical

theorem that the classes of these ideals in C

_generate

the 2-torsion

_subgroup

C[2]

C

_{C, subject}

to a

_single

non-trivial relation. More

_precisely,

if V =

F2

is a vector _space of dimension t with standard basis over the field of two

elements

_{and 1/; :}

V 2013~

_C[2]

_maps the i-th basis vector in V to the class of pj in

_{C, then o}

is

_surjective

and its kernel is 1-dimensional. As is

by

definition the ideal class of _pi in

_C,

we want to know how often

= 0 is the non-trivial relation between the classes of the ideals pi, i.e.

how often the element u =

(1)~=1

_E V

_mapping

_to

generates

the kernel of

_1/;.

We take into account that u lies in the

_subspace

V’ =

0 -’(C [21

n C2),

and that the dimension of V’

_equals

e

_{+ 1,}

where e _=,

_log2

( # ( C (2~

n

C2))

is

the 4-rank of the class group C.

This reduces the

_question

to linear

_algebra

on a vector space V’ whose

dimension does not

_depend

on the 2-rank of C

(which,

as we have _{seen, is} one less than the number t of

_prime

divisors of

D),

but

only

on the 4-rank

of C

_(which,

a

_priori,

is

merely

bounded

by

t

-1),

Thus for e &#x3E;

0,

we write

_D(e)

for the set of real

_quadratic

fields .K E D for which the 4-rank of the narrow class _group C

equals

e. For K E

_D(e),

we want the non-zero

element u in the

_{(e + 1 )-dimensional}

vector _space V’ to be the

_generator

of the 1-dimensional

_{subspace ker 0}

C V’. If we assume

(somewhat

_sloppily,

given

our

_present

_notation)

that

_{ker 0}

should behave like a ’random

1-dimensional space of

V’,

we

_expect

the

following

to hold.

2.1. CONJECTURE. For every e &#x3E;

0,

the subset

D(e)-

=

D(e)

Cl D- has

natural

_density

_{(2e+1 _ l)-l}

in

_D(e).

Conjecture

2.1 is a theorem for e =

0,

in which case the 2-Hilbert class

field of K

_equals

the genus field of

K,

and it suffices to observe that the genus field of a real

_quadratic

field is real if and

only

if we have K E D.

A serious

_problem

in

_proving

2.1 seems to be that we cannot say

anything

without

_fixing

the number of

_primes

t in D. More

_precisely,

the best

_thing

that seems to be within reach at the moment is a result for the subset

_Dt

of D

_consisting

of discriminants

_{having exactly}

t distinct

_prime

divisors. Set

Dt(e)

=

Dt

_n

D(e)

and

Dt(e)-

= Dt n

D(e)-.

Then 2.1 can be formulated

as follows for the sets

_Dt.

2.2. CONJECTURE. For every

pair

(t, e)

of

integers

satisfying

0 e

_t,

the subset

_Dt(e)~

has naturel

_density

_{(2e+1 _}

_{1)-1 in}

_D(e).

So

_far,

we can

_only

_{prove upper} and lower densities for

Dt(e)-

in

_{Dt (e)}

that are not too far from the

_{conjectured density}

_{(2e+1 _ 1)-1.}

For

_instance,

it is shown in

_[11]

_that,

for _every t &#x3E; _e, the _upper

_density

of

_Dt(e)-

in

_Dt(e)

is bounded

_by

_2-e,

so the

_probability

for the Frobenius

F 00

E C to be trivial decreases indeed

_{exponentially}

with the 4-rank e of C. For the values e = 1

and e = 2 there _are in addition the non-trivial lower bounds

1/4

and

1/32

for the lower

_density

of

_Dt(e)-

in

_Dt(e).

The

_problem

of

_{deriving density}

results for D = from results on

the subsets

Dt

also arises when we want to _prove that each subset

_D(e)

has

a natural

_density

in

_D,

which is necessary to obtain the value of P in 1.1

from our

_conjecture

2.1.

Again,

if we fix the number of

_primes

in

D,

there

is the

_following

_precise

result. For the details of the

_proof

we refer

again

to

(11~.

2.3. THEOREM. For each

_pair

_{(t, e)}

_of

_{non-negative integers}

_satisfying

t &#x3E; _e, the set

_{Dt (e)}

has a natural

_density

inside

_Dt

that is

_equal

to

...-+-1 1 1 - - -*,

In order to

_get

rid of the

_dependence

_{on t,}

one observes that the number

w(D)

of distinct

_prime

factors of D has a normal order that tends to

_infinity

with D. More

_precisely,

one can show

[4,

theorem

431]

that for _everye &#x3E;

_0,

the set of D

_satisfying

has

_density

1. It is therefore _very

_plausible

to

_expect

that

_D(e)

has a

by passing

to the limit t -~ oo in

2.3,

yielding

a value

However,

the fact that we are

_dealing

with a countable set

_D,

which does

not _carry a a-additive _measure, makes it unclear how this can be derived

from theorem 2.3.

Combination of theorem 2.3 with our weaker

_conjecture

2.2

_yields

the

following.

2.4. THEOREM.

Let t 2:

1 be an

_integer,

and _suppose that

_conjecture

2.2

holds

_{f or}

all

_pairs

_{(t, e) .}

Then the natural

_{density o f Dt}

inside

_Dt

exists and

_equals

t-1 .. ,

e==U

where the rational numbers are as

defined

in theorem 2.3.

The _{proven upper} and lower bounds for the densities in

_conjecture

2.2

lead to unconditional results for the lower

_density

_{P t}

and the _upper

_density

Pt

of

_С

inside

_Dt,

for which we refer to

[11].

This

yields

the

following

unconditional estimates for small t.

As we

_{already explained,}

the

_density

result

corresponding

to 2.4 for the full

set D is not a direct

corollary

of

2.1,

since we do not know that

_D(e)

has the

_{required density}

in D.

_However,

under the

_assumption

that this

is indeed the _case, we deduce from 2.1 that the natural

_density

of D- in D exists and is

_equal

to

This is the first half of the

_conjecture

in section 1.

We

_finally

observe that in order to observe

_numerically

that the natural

in a _range where

W ~0 ~K) )

has a

_large

_average value. In any interval of the

form

_{(1, X)}

for which it is

_{computationally}

feasible to count the number of K E

_D^,

the slow

_growth

rate of

_{log log(A(K))}

_implies

that this will

not be the case. More

precisely,

the

proportion

of

prime

discriminants,

which are all in

_D~,

will be so

high

that one finds

approximations

of the

required density

that are

considerably larger

than P.

However,

even when

working

with small discriminants

_only

one can check the numerical ade-quacy of the basic

hypotheses

2.1 and 2.2

directly

rather than from formal

corollaries as theorem

2.4,

see

[11].

Extensive numerical evidence for our

basic

_assumptions

can be found in

[2].

3. The

_general

case

In order to answer

Nagell’s original

_question,

we need to extend the

analysis

of the

_previous

section to

_arbitrary

real

_quadratic

orders. We have seen that for the subset E C S of

squarefree

numbers d without

prime

divisors

_congruent

to 3 mod

4,

the

_density

of E- = E n S- in E is

_{conjecturally equal}

to the number P defined in the introduction. As S

consists

_by

definition of all non-square

integers

D &#x3E; 1 that are not divisible

by

4 or a

_prime

_congruent

to 3 mod

_4,

standard

_arguments

show that the subset E of

_squarefree

elements in S has natural

_density

in S. Now _every element D E S can

_uniquely

be written as D = with

d and

_{f &#x3E;}

_1,

and there is the obvious

_implication

D = _E

d E ~-. What we _propose to do in this section is to define a function

:

Z~i 2013~

[0, 1]

such

that,

at least

_{heuristically,}

is the fraction of d E E- for which the converse

_holds,

i.e. the

_density

in E- of those d

satisfying

E S ~ . In

_particular,

we will

have ib( I)

=1

and 1b( f)

= 0 if

f

is divisible

_by

2 or a

prime

_congruent

to 3 mod 4.

The derivation of the heuristical

_{density Q}

of S~ in S is then

straight-forward. As the

_counting

function for S _grows like the

den-sity

in S of the numbers D =

f 2d

_E S with

_given

_{’square part’ f equals}

If we

_require

in addition that the

_squarefree

_part

d be

_{in E- ,}

this

_density

_gets

_{multiplied by}

_P,

and if we

finally

want

_{f 2 d}

E S-there is

_by

definition

of V)

an additional factor

o (f ). Summing

over

f ,

we

We will see in a moment

_{that V)}

is a

multiplicative

function defined

by

ib( f) =

_{Epi f}

prime

’ 1)

so the Euler

product

expansion

shows that our

_expression

for

Q

is the same as the one

_occurring

in the

introduction.

In order to define the function

_0,

we need to determine for _every

_{f &#x3E;}

1

the of those d in E - that

_satisfy

_{f 2 d}

E S- . A basic

obser-vation,

due to R6dei

_[6],

is that this

_{only depends}

on the set of

_primes

dividing f .

He _proves the

_following.

3.1. LEMMA. Let d in E- and

_{f &#x3E;}

1 be

_given,.

Then

_{f 2d is in}

S-

_if

and

only

if p2d 2s

in S-

for

every

prime

number In

addition,

we have the

following.

(1)

if p =

2

_{or p = 3 mod 4,}

then

_{p2d is}

not in

_{57 ;}

(2)

if p is

odd and divides

_d,

then

_{p2d is}

in

_s-;

(3)

if p -1 mod 4

and

_{(p) _ -1,}

_{then p2d is 2n s-.}

We are still left with the case that _p = 1 mod 4 is a

_prime

that

_splits

completely

in

_{Q(U2) .}

This is the most difficult _case, and there is no

crite-rion for

_solvability

of the

_negative

Pell

_equation

for

_p2d

that is similar to

those

_given

in lemma 3.1.

_However,

we can _prove the

_following.

3.2. LEMMA. Let d zn E- and a

_prime

p -1 mod 4 that

splits completely

in be

_given.

Let Ed be a

fondamental

unit in 0 =

and pip

_a

prime

in 0. Then

_p2d

is in S-

_if

and

_only

_if

the order

_{(0/1’)* its}

congruent

to 4 mod 8.

Proof. Note first that the condition that the order of Ed be

congruent

to

4 mod 8 does not

_depend

on the choice of the fundamental unit.

We have

_p2d

E s- if and

_only

if the order

_{Op =}

of index

p in C? contains units of

_negative

_norm, and this

_happens

if and

_only

if d is in ~~ and the

_quotient

of _{unit groups}

_C~*/C~p,

which is finite and

generated

by

Ed mod

_0;,

has odd order. We use the natural

_embedding

C~*~C~p ~

In our _case, the

_ring

is

_isomorphic

to

along

the

_{diagonal. Writing R :}

() ~ for the reduction modulo a

_prime

p of C~

lying

over _p, we have a natural

_isomorphism

It follows that the

_image

of ed

for d E E- has odd order in

_{(0/pO)*/(Z/pZ)*}

if and

_only

if has odd order in

_(0/p)*.

As the order of is

divisible

_by

_4,

we arrive at the conclusion of the lemma. 0

It follows from the

_preceding

two lemmas that the

_probability

with which

we have

f 2d

E S- for fixed

_{f depends strongly}

on the residue class of

d E E- modulo

_{f .}

We now invoke a result of

Rieger

[8]

that tells us how

the elements of E are distributed over the residue classes modulo a

_given

number

_{f .}

3.3. THEOREM. Let

_f

&#x3E; 1 be a

product of

distinct

_prirraes

_congruent

to

1 mod

_4,

and

_define

Then the set

_{E f(a) _}

_{x

E ~ : x - a mod

f }

has a natural

density

inside

E

_for

each

_integer

_a, and this

_{density equals}

In

_deriving

the correct value of

_{~{ f ),}

we need two ’reasonable’ assump-tions that appear to be correct in

_practice,

but are

_probably

not so _easy to

prove. The first

assumption

is that the distribution result in

_3.3,

which is

proved

for E

_only,

remains correct if we

replace E by

~-. This is a natural

assumption

as no relations are known to exist between the residue class of

the

_{discriminant A(K)}

of a real

_quadratic

field K modulo an odd

_prime

p ~ 1 mod 4 and the

_sign

of the norm of the fundamental unit of K.

With this

_assumption,

we

_try

to establish for a

_given

_prime

number _p

the natural

_density

of the set of d E E- that

_satisfy

_p2d

E S- inside

lemma 3.1

_(1).

For p == 1 mod

4,

we know

_by

the same lemma that

p2 d

is in S~ if the

Legendre symbol

_(~)

is

_equal

to 0 or -1. These d contribute

to

_by

3.3 and our

_assumption.

For the

remaining d,

i.e. those d

satisfying

₍₄₎

= 1,

we have to determine in view of lemma 3.2 whether the order of the fundamental unit Ed modulo

(a

prime

over)

p is

congruent

to

4 mod 8. At this

_point

we need a second

assumption,

on which we will

comment in a moment: the

elements Ed

for d E E-

_satisfying

₍₄

= 1 _are

P)

randomly

distributed over the non-zero residue classes modulo _p, so

they

have order

_congruent

to 4 mod 8 with

_probability

21-v

for _p a

_prime

with

v ~

2 factors 2 in the factorization of

_~~Z~pZ)*

=

p -1. Thus, writing

vp for the number of factors 2 in

_{p -1,}

we

_expect

a contribution

to

_{0 (p)}

from the d E ~- that lie in the

_{P 21}

_residue

classes modulo _p

satisfying

_(.4

= 1.

Summarizing,

we see that

0(p)

for _p

_prime

has to be defined

_by

with vp

denoting

the number of factors 2

_occurring

in

_{p -1.}

We now

com-bine lemma 3.1 and the

_independence

of the distributions modulo different

primes

p

implied by

3.3 to arrive at the definition

for

_{arbitrary f &#x3E;}

1. Note that this is the

_{multiplicative}

definition

_{of 0}

we

used in our Euler

product

expansion.

The second

_assumption

in the

_preceding

heuristic derivation involves the distribution of the residue class of the fundamental

_{unit éd}

modulo a

_prime

ideal p

of fixed index p in C~ = when d _{ranges over} certain subsets

of E or E-. We assumed that this residue

class,

which is of course

only

’equidistributed

over

(O /p) * ’

if all these

rings

are identified with

~Z/pZ)*.

In certain situations similar to ours such

_assumptions

have been

proved,

and numerical evidence for it exists in others. We refer to

_[10]

for

equidis-tribution results when d ranges over the

_{primes congruent}

to 1 mod 4 and p is a _power of the

prime

over 2. In

[9]

there are numerical data related to

a

_problem

of

_Eisenstein,

which deals with the case that d is a

squarefree

number

_congruent

to 5 mod 8

and p

is the

_prime

20. For such

_d,

the

_ring

C~/2C~

is the field of 4

_{elements, and Ed}

turns out to be the unit element in

the _group

_{(C~~2C~) * ^--’ Z/3Z}

in

_{approximately}

1 out of 3 cases. It is shown in

_[12]

that _Ed is in the unit class for

_infinitely

many

d,

and that the upper

density

of the set of such d inside the set of all

_squarefree

d = 5 mod 8 is at most

_1/2.

_{Unfortunately,}

the

_general

case of our

assumption

does not

appear to be

_easily

accessible at the moment.

4. Generalizations

The

_approach

we have

given

to examine the distribution of the class of a

fixed

_prime

over the class _{group in} a

family

of

quadratic

fields can

_readily

be extended to more

general

situations.

Already

in the real

_quadratic

case, one can focus attention on the

_precise

relation in the class _group that exists between the classes of the

_ramifying

primes

rather than

_restricting

one’s attention to the

_question

whether

_{F 00 =}

0 is that relation. For a

description

of the

equidistribution phenomenon

that

should then be

_expected

we refer to

[11].

For abelian fields of

_{higher degree,}

say of

prime

degree

p, the

p-part

of

the class _group can be studied in a _way that is

_highly

similar to the

_study

of the 2-class _{group in} the

_quadratic

case. One then studies the

p-class

group as a module over the

complete

cyclotomic

ring

R = In this

situation,

there is

_again

an

_essentially

_unique

relation between the classes of

the

_ramifying

_primes

whose form can be examined. Its distribution should

depend

on the

m2-rank

of the class _group, where m is the maximal ideal of R. We leave it to the reader to extend the heuristics from the

_quadratic

case to this more

general

situation. Unlike in the

_quadratic

_case, there is

to _my

_knowledge

no numerical material available that could be used to see

whether such heuristics

_give

a

_description

that is matched

_by

the behavior

REFERENCES

[1]

B. D. Beach and H. C. _Williams, A numerical _{investigation of} the Diophantine equation x2 -

dy2

= _-1, Proc. 3rd Southeastern Conf. on Combinatorics, Graph

Theory and _Computing, 1972, pp. 37-52.

[2]

W. Bosma and P. Stevenhagen, Density computations for real _quadratic units, Math.

Comp., to _appear

_(1995).

[3] H. Cohen and H. W. _{Lenstra, Jr.,} Heuristics on class _{groups of} number fields,

Number _{Theory Noordwijkerhout} 1983 _{(H. Jager, ed.), Springer} LNM _1068, 1984.

[4]

G. H. _Hardy and E. M. _Wright, An introduction to the _{theory of numbers,} Oxford

University Press, 1938.

[5]

T. _Nagell, Über die Lösbarkeit der Gleichung x2 -

_Dy2

_{= -1,} Arkiv för _{Mat., Astr.,} o. _Fysik 23 (1932), no. B/6, 1-5.

[6]

L. Rédei, Über die Pellsche

_{Gleichung t2 -}

du2 = _-1, J. _{reine angew.} Math. 173

(1935), 193-221.

[7]

L. Rédei, Über einige Mittelwertfragen im _{quadratischen Zahlkörper,} J. reine angew.

Math. 174

_(1936),

131-148.

[8]

G. J. _Rieger, Über die Anzahl der als Summe von zwei _Quadraten darstellbaren und in einer _primen Restklasse _gelegenen Zahlen unterhalb einer _positiven Schranke. _II, J. reine _angew. Math. 217

_(1965),

200-216.

[9]

A. J. _Stephens and H. C. Williams, Some _{computational} results on a _{problem of}

Eisenstein, Théorie des Nombres - Number Theory (J. W. M. de Koninck and C.

Levesque, eds. ), de _{Gruyter, 1992,} pp. 869-886.

[10]

P. Stevenhagen, On the 2-power divisibility of certain quadratic class numbers, J. of Number _Theory 43 _(1993), no. (1), 1-19.

[11]

P. _Stevenhagen, The number of real quadratic fields having units _{of negative norm,}

Exp. Math. 2 _(1993), no.

(2), 121-136.

[12]

P. _Stevenhagen, On a problem of Eisenstein, Acta Arith., (to _appear, 1995).

Peter STEVENHAGEN

Faculteit Wiskunde en Informatica Universiteit van Amsterdam

Plantage Muidergracht 24

1018 TV _{Amsterdam, Pays-Bas} e-mail: psh©fwi.uva,nl