The size function $h^0$ for quadratic number fields

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

P

AOLO

F

RANCINI

The size function h

0

for quadratic number fields

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{1 (2001),}

p. 125-135

<http://www.numdam.org/item?id=JTNB_2001__13_1_125_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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

The size function

h°

for

_quadratic

number

fields

par PAOLO FRANCINI

RÉSUMÉ. Nous étudions le cas

_quadratique

d’une _conjecture énon-cée par Van der Geer et Schoof

_prédisant

le _comportement de certaines fonctions définies sur le _groupe des diviseurs d’Arakelov

d’un corps de nombres. Ces fonctions

correspondent

à la

fonc-tion usuelle h° relative aux diviseurs de courbes

_{algébriques.}

Nous montrons

_qu’elles

_atteignent leur maximum en les diviseurs

d’Arakelov _principaux, et nulle _part ailleurs. De

_plus,

nous

intro-duisons une fonction

k°,

_qui est

_l’analogue

de

_{exp h°}

sur le groupe

de

_classes,

et on montre que cette fonction atteint elle-aussi son

maximum en la classe triviale.

ABSTRACT. We

_study

the

_quadratic

case of a _conjecture made

_by

Van der Geer and Schoof about the behaviour of certain functions which are defined over the _group of Arakelov divisors of a number

field. These functions

_correspond

to the standard function h° for

divisors of

_algebraic

curves and we _prove that

_they

reach their maximum value for

_principal

Arakelov divisors and nowhere else.

Moreover, we consider a function which is an

analogue

of

exp h°

defined on the class _{group, and} we show it also assumes its

maximum at the trivial class.

Introduction

Given a number field

_F,

let

_RF

be the set of its real

_places

and let

_CF

be the set of its

_complex

_places,

_{i. e.,}

_CF

is

_given

_{by picking}

one of each two

distinct

_conjugate

_embeddings

F ~ C. An Arakelov divisor is a

couple

D =

(JD,

(a, 0))

where

JD

is _a fractional ideal of the

_ring

of

_{integers OF}

and

_(a,,3)

E

R RF

X We call

_JD

the

_ideal,

or even the

_finite,

_part

of

D;

the vector

_(a,

₍₃₎

is said to be the infinite

_part

of

_D,

as its

_components

correspond

to the infinite

_primes

of F. We denote

_by

_N(JD)

the

_ordinary

norm of

_JD

as a fractional

ideal,

and we define the norm of D in the

following

way:

where we write a = and

)3

= and _we define the

degree

of D as

deg(D)

=

logN(D).

One should observe that the Arakelov divisors of a number field form a _group, which will be denoted

by

Div(F),

and that

zero-degree

Arakelov divisors are a

subgroup

Div°(F)

of

Div(F).

Another

subgroup

of

_{Div(F) is PDiv(F),}

the set of

_principal

Arakelov divisors:

where

_Izl

stands for the standard absolute value of the

_complex

number z. The

_product

formula for number fields _says that the inclusion

holds. We define the Picard _group of F as ’

and then we

_put

We observe that

Pic°(F)

fits into

the exact _sequence

where T is a real torus of dimension

_#(RF)

+

_{#(CF) -}

1. More details about this sequence are

explained

in

[3].

One can view the

_ring

as well as _any fractional

ideal,

as a real lattice in

JRRF

x

COF,

via the

_{embeddings corresponding}

to the infinite

primes.

When S is a subset of a lattice in a Euclidean _space, we attach to

it the real number

- - ...1

w‘

Some

_properties

of this function have been studied in

_[1].

_Following

this

direction,

one can consider an Arakelov divisor D

just

as

JD

endowed with a metric

specified by

the infinite

_part

of

D,

in the

following

_way:

given

x G

_JD,

we

_put

.

We now define

k°(D)

as the number An

equivalent

definition is the

_following:

for x E F and D =

(JD, (a"(3)) E Div(F),

_put

/ ~ - , ,

and consider the lattice D =

JD ~.

Then _{we set}

In this _sense,

_changing

the infinite

_part

of D

_produces

different lattices

_lying

in the _same,

_fixed,

Euclidean _space x

CCF,

with the standard inner

_product

_given

_by

This will be our favourite

_point

of view. An

_important

remark to be made

is that the value of the function

ko

on the Arakelov divisor D

_depends

_just

upon the class of D in We shall be concerned with the

following

conjecture,

which is stated in

_[3]:

Conjecture.

Let F be numbers

_field

which is Galois

_{over Q}

or over an

imaginary

quadratic

number

_field.

Then the

_function

k°

on

Pico(F)

as-sumes its maximum on the trivial class

OF.

We recall from

_[3]

that the function

h°

=

log

ko

_can be _{seen as an}

ana-logue

of the classical function

ho

over the divisors of a curve:

_indeed,

even

in this

_setting,

a Riemann-Roch theorem holds. Under this

analogy,

the above

_conjecture

_corresponds

to the standard

_property

of

_algebraic

_curves, which says that

h°(D)

= 0 for _{any non}

principal zero-degree

divisor D and

h°(D)

=1 when D is

_principal.

We shall _prove this fact in the

_quadratic

case.

Theorem 1. Let F be a

quadratic

number

field.

The

f unction

kO

on

has a

_urtique

_global

maximum at the

_identity

element.

When F is a

complex

quadratic field,

the

proof

reduces to estimate the number of

_{representations}

of a

_{given integer}

_by

_{binary quadratic}

forms. In the real case, we look at the lattices

_D,

_recalling

that the main contributions

to

1~°

come from the shortest lattice vectors. The

_key

_point

is

_that,

for x E

OF -

_{0},

we have 2

and,

when x is not a

unit,

4.

From this we

_get

some bounds for the contributions to

_ko(D)

from distant

poins

of

_D,

which

_permit

to reduce the

_problem

to a local one. In the last

_part,

we define a function l~° on

Cl(F)

as follows:

where 7r is the

_{projection appearing}

in the _{exact sequence}

_(*),

the

_symbol

[J]

denotes the class of the ideal J and

_integration

is made with

_respect

to

the Haar measure on

~r-1 ( (J~ )

normalized to

R,

the

regulator

of F. This is a natural

analogue

of

k°

on the class _group and it should be reasonable to

expect

it still has the maximum value at the trivial class. Indeed we _prove the

_following:

Theorem 2. Let F be a

_quadratic

number

_field.

Then the

_function

k-0 on

Cl(F)

assumes its maxirraum at the trivial class and nowhere else.

About this

_subject

we also would like to mention

_[4],

where

_special

em-phasis

is

_put

on

computational

_aspects,

and

[2],

which is focused on a

harmonic

_{analysis approach.}

_Moreover,

the author wishes to thank Prof. Rene Schoof for his

_help.

1.

_{Complex quadratic}

fields

When F is a

_{complex quadratic field,}

the _group

Pic°(F)

is finite and coincides with the class _group

_Cl(F).

This follows from the _sequence

_(*).

Given a fractional ideal J of

O F,

we have

k°(J) =

Proposition

1.1. Let F be a

complex quadratic

numbers

field.

Then the

function k°

attains its maximum at the

_identity

element

_ofPicO(F)

=

Cl(F)

and nowhere else.

Proof.

As a first

_step,

observe

_that,

_given

an ideal J in

O F,

we have that

where with

_{3}

a Z-basis for

_J,

is a

representa-tive for the

_SL2(Z)-orbit

of

_quadratic

forms associated to the ideal class of J. The form cp is

primitive, positive

definite and has the same discriminant A as the field F. It

_represents

the

_integer

1 if and

only

if it is

_equivalent

to the

_principal

_{form, i. e.,}

when J is a

_principal

ideal.

° 2 ₁ 2

.H ..

Clearly,

we have

kO(OF)

&#x3E; 1 +

_En21

e 2n2

&#x3E; 1+ . Hence it is

enough

r _{en1r e1r}

to show

_that,

for a

form p

which is not

_equivalent

to the

_principal

_one, we have 1 +

e 2 . 2

_Moreover,

when

_OF

is a

_principal

ideal

&#x3E;

e 1r

domain,

the statement is

_trivial,

so we _may

assume A

-15. If we

_put

~

_{= e-21r,}

we _{may write} the above series as

where

_R(h)

stands for the number of

_{representations}

of the number h

_by

the

_{integral binary}

form _cp.

_Now,

in order to write an _upper bound for

R(h),

let

_PR(h)

denote the number of _proper

_{representations}

of h

_by

_cp,

2. e.,

those

_by

_{coprime integers.}

Since

_{OF -15,}

we have from

[5, 9.3]

that

PR(h)

2s,

where

So we have

PR(h)

2h and therefore

Hence,

we obtain that

2. Real

_quadratic

fields

Let F be a real

_quadratic

field of class number h and

_regulator

R. The connected

_component

of the

_identity

of the _group

_Pic°(F),

will be called the

_principal

_component,

which is the

_subgroup

whose

_{representatives}

in

DIVO(F)

are Arakelov divisors D with

JD

a

principal

ideal . We notice that this _group has h

_component

and that the

_principal

_{component may}

be identified with a

circle R

_Indeed,

for each of these

_elements,

there is

RZ’

a

_{unique t}

E

_{[0, R)}

such that

Et

=

(OF, (t, -t))

lies in the same

PicO(F)-class. We can summarize the above considerations

_by

_writing

the exact

sequence

(*)

for the real

quadratic

case:

We remark

_that,

for the lattice

_Et,

we

have I

fore _

), and

there-.. -/

where - is a fundamental unit of

OF- Moreover,

we have that

B / B / B / B /

since

_quadratic

extensions are normal. This

_gives

a

_picture

of the

R-periodicity

and the

_symmetry

of the function t H

Lemma 2.1. Let

_{S #}

_{(0,0)}

be a subset

_of

a lattice in

ll82 which

_is sym-metric with

_respect

to the

_origin,

such

_that,

_for

_any

_pair

_of

distinct

_points

_

Proof.

Given r ~ 2 and J

_J2,

let us consider the set

Now,

the

_angle

between two

_points

in

_Ar,5,

seen as two vectors in

_JR2,

has to be at least

2 arcsin V

Indeed,

this

angle

corresponds

to

a

_pair

of

_{points x}

and _y such

that lix

- yll

=

V2,

with llxll

= r and

Ilyll

= _{r +} 6. As the _set S is

symmetric

with _respect to the

_origin,

and

since the

_inequality

2

elements in is at most

_{2([~] 2013 1),}

where

v - V’

the number of

_points

of

_A,,6

which are contained in a

half-plane

cannot exceed 7r divided

_by

the minimal

_angle.

In order to _{have v2,a -}

_5,

we

so that the contribution

to

_ko(S)

from the

_points

in

_{A2 ó}

is at most

8

Therefore we obtain:

,

e4,7r

1r

and,

as the function t H

~~.

decreases

for t &#x3E; 4,

To

_simplify

the

_notation,

we now fix an

_embedding

F - R and we look

at the elements of F as real numbers.

_Moreover,

for each x E

_F,

we denote

by x

its Galois

_conjugate.

Corollary

2.2. Let F be a real

_quadratic

number

field

and D E

Divo (F)

-If JD is

not a

_principal

_ideal,

then we have

ko(D)

Proof.

Let D =

(JD, (a, b))

_E

Divo(F).

The condition

deg(D)

= 0 means,

by

definition,

that =

N(JD).

Then _we have

where we

_put

and ~ ₁ For _every non-zero

x E

_JD,

we have that

(

=

2 N(JD),

since

JD

is not _a

principal

ideal.

_{Hence yx2 + yt2 &#x3E;,}

_4N(JD)

for all non-zero x E

_OF,

so that D is a lattice whose non-zero shortest vector has

length

at least 2. In

particular,

we can

_apply

lemma 2.1 and conclude that

ko(D)

1

+ 3

1 + ~ 1~°(C~F).

°

We observe that we also could have deduced

proposition

1.1 from lemma

_{2.1, just}

in the same _way as for

corollary

2.2. We

preferred

to

give

the other

_proof,

since it contains

_essentially

the same

_argument

we shall use in

_proposition

3.1.

Corollary

2.3. Let F be a real

_quadratic

number

_field,

let R be its

regu-lator and

Et

=

(OF, (t, -t))

e

_1ft

E

[!

_log

25,

2 ~,

then we have

kO(Et)

Proof.

Let

_OF

be the discriminant of F and e be the fundamental unit of

OF

larger

than 1. For s e

_{[1, ê],}

consider the set

Observe that if wx E

_Bs,

then

_1,

_{i. e., x}

E

_By

Dirichlet’s

unit

_theorem,

these element _{are, up to}

_sign,

_powers of the fundamental unit

6*.

_Hence,

to determine the set

Bs

amounts to

_finding

the solutions of the

inequality

For

_8,

we

have e # 1 +

J2,

so there exist solutions for this

only

if m E

_{10, - 11.}

The case m = 0

corresponds

to the shortest vector of the

lattice

_Et,

for small values of t.

_Moreover,

we have that w ~ E

_Bs

if and

6

only

if s &#x3E;

_{(2 -}

~)~2.

Now,

for

_5,8,13,

we

have - 1&#x3E;

2 +

B/3,

so that

(2 -

)e2 &#x3E;

_E, hence

_{BS -}

for all s E

_[1,.El.

_Therefore,

in this _case, we can

apply

lemma 2.1 to

_Et

for t E ~

2

_log

25 , 2 ~

and obtain:

6

-

-Therefore lemma 2.1

_implies

_that,

for all’

At

_last,

we deal with

Q( B1"5),

where F As a first

_step,

we

_point

out

_that,

in this case, for all s E

_[1,

~~,

we have that

_Bs

-and hence

As a second

_step,

we notice that s ~

k° (Bs )

is a

decreasing

function when

Then we obtain that for all 1

Our assertion is

_proved.

D

At this

_point,

in order to finish the

_proof

of theorem

_1,

it is

_enough

to

show the

_following:

Lemma 2.4. Let F be a real

quadratic

number

field and, for

every t

E

_If,

set

_Et

=

(OF, t,

-t) E

Div°(F).

The

function

k°(Et) is

strictly

decreasing

when t E

Proof. Define,

for s E

_{R+ ,}

the function

Our next task is to see that

f

is

decreasing

on the interval

[1,

i] .

Observe

that,

since

_{f (s)}

=

f ( s ),

we have

f’(1 )

=

0,

hence it is

_enough

to check that

f"(s)

0 for all s E

_[1,

25~.

_Therefore,

we have to prove that

that amounts to

_showing

that

which _may be rewritten as:

Here,

the left hand side _may also be viewed be as

where

_Ls

is the lattice As in the

_proof

of lemma

_2.3,

we have

2 Wx

E Bs

only

when x E

_(9~.

Now,

for

_{F ~}

_Q(V5),

we have

Bs =

for each s E

[1,~~].

Thus,

since we

clearly

have

),

it is

enough

to see

that,

for all s

which is

_readily

checked.

When F =

Q(v,’5-),

there _are minor

complications

due _to the fact

that,

in this _case,

_Bs

= Hence _we have

e E

where e = is the fundamental unit of

Q( V5).

Thus,

it is sufficient to

see that

for all s E

_[1,

25 ~’

Such

_inequality

can be obtained from direct

computation,

for instance

_{by splitting}

_(1,

_{2~ ~}

_{as [1,}

_{~] U}

_100’

_{50 U ( ~o ~ 2~ ~}

_and,

for each

interval,

evaluating

maxima of terms on the left and minima of terms on

the

_right.

D

3. A function on

Cl(F)

When F is a

complex quadratic field,

Cl (F) _

Pic°(F) ,

so the function k° coincides with

_{ko. Indeed,}

the

_properties

of this function are _very similar

in both the real and the

_complex

case.

Let F be a real

quadratic

field

having

discriminant A and let J be an ideal in the

_ring

of

_{integers OF.}

For h E

_Z,

we define the set

which is either

_empty

or infinite. In this second _case, the _group

at

of units with

_positive

norm acts on

S(h)

by multiplication

and the number

Or(h)

of

orbits is finite. Elements in

_S(h)

_correspond

to the various

_{representations}

of h

_by

the

_{integral quadratic}

form = where

is _any 7G-basis of J. To write an _upper bound for

Or(h),

we first consider the number

_POr(h)

of _proper

_orbits,

_i.e.,

those which do not contain elements

of the kind _mx, with x E J and 1 m E Z. This amounts to

_considering

the orbits of _proper solutions of the

_equation

_n)

= h.

Following

[5,

9.3],

each orbit is characterized

_by

an

integer

x which is a solution to the

congruence x2

= ~

_(mod

_41hl)

such that

_{0 x}

_{2 ~ h ~ .}

Therefore we have

POr(h) x Ihl

and

,

We can now _prove theorem 2

by showing

the

following.

Proposition

3.1. Let F be a real

_{quadratic field.}

Then the

f unction

io

attains its maximum at the trivial class and nowhere else.

Proof.

Let 6 be the fundamental unit of

_OF.

_{Set -y}

= 1 if

N(é)

= 1 and ~ = 2 if = -1. Let J be an ideal of

OF.

We have that:

Hence, setting 1

J

and

_using

the same notation as in the above

_discussion,

we obtain:

11

When J is not a

_{principal ideal,}

from the discussion

_above,

we have that

-

-The last

_inequality

comes from

_writing

The fact that

1-

concludes the

proof.

’.r-References

[1] W. _BANASZCZYK, New bounds in some _transference theorems in the geometry of numbers. Math. Ann. 296 _(1993), 625-635.

[2] A. BORISOV, Convolution structures and arithmetic cohomology. Math. _AG/9807151 at http://xxx.lanl.gov, 1998.

[3] G. VAN DER _GEER, R. SCHOOF, Effectivity of Arakelov divisors and the theta divisor of a

number field. Math. _AG/9802121 at _{http://xxx.lanl.gov,} 1999.

[4] R.P. GROENEWEGEN, The size _{function for} number fields. Doctoraalscriptie, Universiteit van

Amsterdam, 1999.

[5] H.E. ROSE, A course in number theory. Oxford _{University Press,} 1988. Paolo FRANCINI

Dipartimento di Matematica "G.Castelnuovo" Universita "La _Sapienza"

Roma Italia