On lattice bases with special properties

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

U

LRICH

H

ALBRITTER

M

ICHAEL

E. P

OHST

On lattice bases with special properties

Journal de Théorie des Nombres de Bordeaux, tome

12, n

o

_{2 (2000),}

p. 437-453

<http://www.numdam.org/item?id=JTNB_2000__12_2_437_0>

© Université Bordeaux 1, 2000, tous droits réservés.

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

On lattice bases

with

_special

_properties

par ULRICH HALBRITTER et MICHAEL E. POHST

Dedicated to _Jacques Martinet on his 60th _birthday

RÉSUMÉ. Nous introduisons ici des réseaux

_{multiplicatifs}

de

(R&#x3E;0)r

et déterminons des réunions finies de

_simplexes

conven-ables comme domaines fondamentaux de sous-réseaux d’indices

finis. Nous définissons pour cela des bases

cycliques

positives

de réseaux arbitraires. Nous utilisons ces bases _pour calculer

les cônes de Shintani dans des corps totalement réels de

nom-bres

_{algébriques.}

Nous nous intéressons

plus paxticulierement

aux

réseaux en dimensions deux et trois

correspondants

à des _corps

cubiques

ou _quartiques.

ABSTRACT. In this _paper we introduce

_{multiplicative}

lattices in

(R&#x3E;0)r

and determine finite unions of suitable

_simplices

as

funda-mental domains for sublattices of finite index. For this we define

cyclic

non-negative bases in

_arbitrary

lattices. These bases are

then used to calculate Shintani cones in

totally

real

algebraic

num-ber fields. We

_mainly

concentrate our considerations to lattices

in two and three dimensions

_{corresponding}

to cubic and quartic

fields.

1. Introduction

Since the

_guiding

_principles

for this article are

_adopted

from

_algebraic

number

_theory

and since this is the

_topic,

where our ideas

mainly apply

(see

Section

_4),

we

_shortly

introduce some notation for

_algebraic

number fields.

Throughout

this _paper F denotes a

_totally

real

_algebraic

number field of

degree

n over the rational

numbers Q .

We assume that it is

_generated

_by

a root _p of a monic irreducible

_polynomial

Over the real numbers R the

_polynomial

_{f (t)}

_splits

into a

product

of linear

factors

We assume _p =

p~l~

for the

_conjugates

p~l~, ...,

pen)

of _p as usual. An element

a of F is called

totally

_positive

if all its

_conjugates

are

_positive

real numbers.

Arithmetical

_{problems usually}

_{require computations}

with

_algebraic

inte-gers contained in

F, i.e.,

those elements of F whose minimal

polynomials

have coefficients in Z.

_They

form a

ring

OF with a Z-basis _{wi , ... , wn}

(inte-gral

basis of

_F),

the so-called maximal order of F. Then the discriminant

dF

of F is

_given

_by

Two elements of OF are called

multiplicatively

_equivalent

( associated),

if

their

_quotient

is an invertible element

(unit)

of _oF. The units of oF form a

finitely generated

abelian _group

_UF.

For

_totally

real fields it consists of a

torsion

_part

and the direct

_product

of r = n - 1 infinite

_cyclic

groups the

generators

of

_which,

_{say Cl, ..., Cr,} are called

fundamental

units.

Equivalence

is

_usually

tested in

_logarithmic

_space. For this we consider the

mapping:

Clearly,

A :=

L(UF)

is a lattice in r-dimensional Euclidean _space with basis

_{L(sl), ..., L(ér)}

_and,

_{consequently,}

two elements _{x, y} E

_{OF B}

are associated if and

_only

iff

_L(x)

and

_L(y)

differ

_by

an element of A.

In this article we

study

the

problem

of

choosing

_appropriate

bases of lattices A in r-dimensional Euclidean _space such that so-called Shintani

cones which are of

_importance

for _ray class zeta functions become more

easily

accessible. We

_proceed

as follows. In Section 2 we define "nice"

bases of lattices

_{(respectively,}

sublattices of finite

_index)

and prove their

existence. In Section 3 we define a

generalized multiplicative equivalence

for vectors with all coordinates

_positive

in Euclidean _r-space. We show how the bases of Section 2 can be used to obtain fundamental domains for that

_{multiplicative equivalence}

for r 3 and

_conjecture

that this is true

for

_arbitray

r.

_Interpreting

them as the intersection of a

decomposition

of into Shintani cones with the

hyperplane

_zn = 1 these _can then be

used to calculate a Shintani

_{decomposition}

of where r denotes a

natural number and

_(the

field

_{degree) n}

is r + 1.

We illustrate our considerations

_by

an

_example

of a fundamental domain

for a

_quartic

field. The

_application

of that

_{decomposition}

to the calculation of the values of _ray class zeta functions at

_non-negative

_integers

will be done

2. Lattice bases with

_special

_properties

In this section we consider full lattices A in r-dimensional Euclidean space for r &#x3E; 1. In later sections we shall need the existence of

special

lattice bases for A

_(or

at least for a sublattice A of A of finite

index).

Definition 2.1. A set of r non-zero vectors S

_{_ ~bl, ..., br I}

of A is called

cyclic non-negative,

if the coordinates of the i-th vector

_{b2 -}

_{(bil, ..., bir)}

satisfy

the

_inequalities

..- &#x3E; 0

_{(1 ~}

i ~

r),

where the second

_{indices j}

are considered modulo _r,

_{i.e. j}

is to be

_{replaced by}

j - r for j &#x3E; r.

We will show that _every lattice A contains

_cyclic

_non-negative

sets of

independent

vectors. In a first

_step

we construct a

_cyclic

_non-negative

set

for

_A,

where all

_inequalities

between the coordinates of the vectors are strict.

Proposition

2.1. A lattice A contains a

cyclic

non-negative

set

of

vectors

bl,

...,

br

the coordinates

of

which

satisfy

bii

&#x3E;

bi,i+l

&#x3E; ... &#x3E;

bi,i+r-1

&#x3E;

0

_{(1 ir).}

Proof.

Let _{x1, ..., xr} be _any basis of A and K an _upper bound for the absolute values of the coordinates of the basis vectors.

_Any

x E R’ has a

presentation

x =

Aixi

with coefficients

Ai

_E R. Let

m2 E Z

subject

to

_I

m2 -

Aj

1:!~- 2

( 1 i r)

and

put

y := Then we have

y = x +

~i 1 (m2 -

and therefore

Hence,

an

_appropriate

choice of x

_guarantees

&#x3E; 0. D

Next we will show that the

cyclic

non-negative

set obtained in the

propo-sition is

_linearly

_independent.

Proposition

2.2. Let

_bl,

...,

br

be a

cyclic

non-negative

set

of

vectors

of

a lattice A and denote

by

B := the matrix with rows

b1, ..., br .

Then the determinant

_of

B is

_{non-negative.}

_If

we have

b22

&#x3E;

_for

at least r - 1

of

the vectors

bl,

...,

br

the

given

cyclic

non-negative

set is

linearly

independent.

Proof.

Let

Dl

:=

det(B)

be the determinant of the matrix with rows

bl, ..., br.

We will show

Dl

&#x3E; 0

_by

induction on r. This is trivial for r =

1, 2. Hence,

_{we assume r &#x3E;} 3 in the

_sequel.

The determinant

Dl

is a linear

polynomial

in each of the entries of the

matrix B.

_Hence,

it suffices to consider the value of

_Dl

on the boundaries

of those entries. The coefficient of

_bll

is the determinant of the matrix

induction

_assumption.

As a _consequence,

Dl

becomes at most

smaller,

when we

replace

bll

_by

b12

in the

_original

matrix.

Again

we

_proceed

_by

induction on the number of

equal

entries in the

first row of the

_given

matrix

_starting

from the left. Let C be the matrix

with entries _c2~ -

bij

for 2 i _or i = 1 whereas

Clj = for

1 j

~. We conclude from K - 1 to ~.

The determinant of C is a linear

polynomial

in _cix. We subtract col-umn 1 of C from columns

_{2,..., r.}

and obtain for the coefficient of the

determinant of the matrix E with entries

for 1 i r. We add the last column of E to columns

_{1, ...,}

~ - 1. Then the rows of the new matrix F form a

_cyclic

_non-negative

_set, and we obtain 0

_according

to our induction

assumption.

This finishes the

_step

from - 1 to ~.

Hence,

the

_original

determinant

_Dl

is bounded from below

_by

the

de-terminant of the matrix

_originating

from B

_{by replacing}

all entries in the first row

_by

_bln.

_Moving

the first row to the last and the first column to

the last and

_taking

the common factor

bln

of the new last _row, we obtain

I - - - ,

Again,

the rows of the last matrix form a

cyclic

_non-negative

set.

_Carrying

out the same

procedure again,

we

_get

since

_D3

is the determinant of a matrix with 2

equal

rows with entries all one. Thus we have also shown the induction

_step

from r - 1 to r.

If the additional

_{premise bi2}

&#x3E;

_bi,2+1

holds for at least r -1 of the vectors

b1, ..., br

we can _rearrange the order of rows and columns such that this

condition is satisfied for the first r - 1 rows. Then the

proof

above

_clearly

yields

det B &#x3E; 0. D

The

_preceding

_propositions

show that we can

_compute

_cyclic

_non-negative

sets _{in every} lattice which

_generate

sublattices of finite index. Of _course,

we are interested in

_generating

sublattices of small index. For r = 2 _{we can}

prove the existence of

cyclic

non-negative

lattice bases and that

proof

also

Proposition

2.3. For r = 2

any lattice A has a

_cyclic

_non-negative

set

{bl, b2}

as a basis.

Proof.

We

_apply

a

purely geometrical

construction which is based on

Minkowski’s convex

body

theorem. We note that for

_calculating

lattice

points

in

_practice

the

_occuring

_rectangles

will be covered

_{by ellipses.}

For

algorithmic

details we refer the reader to

_~5~.

We consider the closed first

_quadrant

_(JR~O)2.

For the bisector

_Gl

:=

~(x,

y)

E =

y~

_we let

G2, G3

be the

parallel

lines _to

Gl

with distance

6 &#x3E; 0. That distance is chosen maximal

_subject

to

_M(A),

where

M(A)

denotes the minimum of the

_{lattice, i.e.,}

the norm of a shortest lattice

vector. This

_guarantees

that all lattice

_points

between

_G2

and

_G3

lie in the first and third

_quadrant.

Then we choose the smallest

0-symmetric

closed

rectangle

with two sides

on

G2

and

G3

which contains at least one lattice

_point

_c,

necessarily

on the

boundary.

We choose c in the first

_quadrant

with maximal

_distance,

_{say e,}

to

_Gl.

We need to discuss two cases

depending

on whether e is zero or not.

To

_begin

with let us assume e &#x3E; 0. We let

_G4,

_G5

be the

_parallel

lines

to

_Gl

with distance 6’. Then we choose the smallest

_0-symmetric

closed

rectangle

with two sides on

G4

and

G5

which contains at least one lattice

point

d in the first

_quadrant

which is different from c. It is easy to see that c and d lie on different sides of

Gl. Hence, putting

them into the

_right

order

_they

form a

_linearly

_{independent cyclic non-negative}

set. Also the closed

_triangle

with vertices

_0,

_c, d contains

_exactly

these vertices as lattice

points.

Therefore

_{{c, d}}

is a basis of A.

In the case e = 0 _we consider the

parallel

lines

G6, G7

with distance

q =

d(A)/lIcll,

where

_d(A)

denotes the mesh of the lattice. Let d be a lattice

_point

on

G6

or

G7

which lies in the first

quadrant.

Such a

_point

must exist

_according

to our

_assumptions.

_Ordering

_c, d

_{appropriately they}

form a

_{cyclic non-negative}

set which is a lattice basis because of the volume

of the

_{parallelogram}

_spanned

_by

them. D

We note that for r = 3 _{we can} show the existence of _a

cyclic

non-negative

set

_generating

a sublattice of index at most

_24,

_{independently}

of the lattice A. The

_proof

is

_quite

_lengthy

and therefore omitted. In

_practice

we

always

obtained sublattices of much smaller index.

3.

_{Multiplicative}

_equivalence

in

The

_concept

of

_{multiplicative}

_equivalence

for the

_integers

of an

algebraic

number field F which we introduced in Section 1 needs to be

_generalized

with

_respect

to

_{multiplicative}

_equivalence

in and Shintani cones.

For

_{multiplicatively}

_equivalent

_totally

_positive

elements E OF there

require

This is

_easily

seen to be

_equivalent

to

and therefore also to

This

_yields

the

_{following general}

definition which is

_independent

of number fields.

_(We

note that r = n -1

throughout

the

paper.)

Definition 3.1. Let

.E"

₍₁

_{i, j}

_r)

be fixed

_positive

constants.

Vec-tors x, y E are called

(multiplicatively)

_equivalent,

if there is a vector

m =

(ml,

...,

mr)

E zr such

that yj

= zj

IT~=1

From the definition it is clear that this relation is an

_equivalence

relation

and we are interested in a "nice" fundamental domain. If we

set Ei :_

(E(l) ...,E~)

(1:S i :S

r)

the vectors _x,y are

_equivalent,

if and

_only

if

L(y)

= L (x)

+

_¿~=1

_holds,

where the map L is

given

by

It is obvious that the vectors

_{L (Ei )}

should be

_linearly

_independent,

we therefore

_stipulate

this in the

_sequel.

Hence,

L(E1), ..., L(Er)

are basis vectors of a

(logarithmic)

lattice A in

R’. It would be

_tempting

to take as a fundamental domain for

multiplica-tive

_equivalence

the

_preimage

of the fundamental domain of

_A,

i.e. of the

set

Even

_though

the

_exponential

function

_generally

behaves

_quite

_nicely,

such a choice does not

_yield

an

_appropriate

fundamental domain.

Namely,

its

boundary

is not contained in a finite number of

hyperplanes

which will

be

_required

at a later

_stage.

Therefore the idea comes _{up to}

deform

into a set

fl.

This

procedure

must be carried out such that the

integral

translations of the new fundamental domain still form a

_disjoint

cover of

and that the

_boundary

of is contained in a finite number of

hy-perplanes.

To achieve this in a

straightforward

manner would mean to connect the

_preimages

of the vertices

_{of 11}

_{by hyperplanes,}

and to consider the

_logarithmic

_image

of that

_{multiplicative}

"fundamental domain " as new

fundamental

_{domain 11}

for A

_applying

3.3 . In the

_sequel

we will

study

this

approach

and the

_occuring

difficulties in the

_simplest

case r = 2. The ideas

for new methods to overcome these difficulties will later form the

_guiding

principles

for

_higher

dimensions.

Let us assume that a =

(a,,

a2)

and b =

(bl, b2)

_are

linearly independent

vectors in

_logarithmic

_space

_spanning

a lattice A. In

exponential

_space the

images

of the vertices of the fundamental

_{domain rj}

of A are the

_points

We would be

_tempted

to connect 1 with

_El,

_E2

and those two with

_E12

_by

straight

lines. The

_images

of these lines in

_logarithmic

_space are

_congruent

modulo

_1,

but this does not

_{generally yield}

a fundamental domain as is

demonstrated

_by

the

_{following example.}

Example.

We choose

_El

=

(2,4),

E2

=

(4, 11).

Then the

preimage

of

_rl

has the vertices

_{(1,1), (2, 4), (4,11), (8, 44),}

and the

_straight

lines

connecting

(1,1), (4, 11),

,

respectively

P

10

(2,4), (8,44),

intersect in

_(21,

_).

₃ We therefore

_proceed

as follows. In

logarithmic

_space we assume that

Sl

and

_Tl

are

_piecewise

smooth double

_point

free curves

_connecting

the lattice

points

0 with

_L(El),

_L(E2),

_{respectively.}

We set

_S2

:=

81

+

_L(E2),T2

:=

Tl

+

_{L(Ei )}

and

_get

the

_following

lemma.

Lemma 3.1. Denote

_by

J the set

_of

_points

enclosed

_by

C :=

_SlUT2US2

U

Tl1.

If the

closed curve C is double

_point

_free

then the set

_{(JUC) B (S2}

_UT2)

is a

_fundamental

domain

_for

We note that a fundamental domain for

R /A

yields

a fundamental do-main for

_{multiplicative}

_equivalence

_upon

_applying

the

_exponential

_map.

Proof.

The

_proof

is carried out in several

_steps.

We

_{begin by simplifying}

the

_description.

For

_{simplicity’s}

sake we assume that the lattice in

logarith-mic _space is

_spanned

_by

the unit vectors

_{(1, 0), (0,1)}

and the fundamental domain is therefore

_[0,1)~.

This can be

_easily

achieved

_by

a linear

trans-formation.

In a first

_step

we introduce the

_piecewise

smooth

boundary

curves. We assume that

_{~2, ~1, 02}

are

continuously

differentiable _maps from

[0, 1]

into R

_satisfying

0 =

§2(0) =

~2(1) _ a/1 (0) =

=

and C double

_point

free.

_By

G we denote the

(closed)

area surrounded

by

C. We will show that the union of the interior of G with

_Cl

and

_C2

without the

_point

_{(0, 1)}

is also a fundamental domain.

Since we will

apply

some Fourier

theory

and the theorem of Gauss we introduce several

_auxiliary

formulae in the second

_step.

_By

the theorem of Gauss an

_integral

over G can be evaluated

by

a contour

_integral

over the

boundary

C of G.

_Using

the

_{parametrization}

of the

_boundary

of G from above we calculate Fourier coefficients:

the same result also holds for

0;

finally

we find

In the third

_step

we show that all x E

~0, 1)2

are obtained modulo 1. Let us assume that there exists x E

~0, 1)2

with

_{(m, n)}

+ x not in G for all

_{(m, n)}

E

Z2 .

Then the set

_{x

+

_{(m, n)}

_I

_{(m, n)}

E has a

_positive

distance to G which is

_larger

than a

positive

constant e.

Hence,

if y E

(0, 1~2

has distance smaller than e from x then the intersection of

{y

+

_(rrz,

_{n) I}

(m, n)

E

7G2~

with G is

_empty.

Without loss of

_generality

we can assume

that x E

(0, 1)2

and that the _open

_{e/2-neighborhood}

of x is contained in

(0,1)2.

Consequently,

there exists a function h :

~0,1~2

-&#x3E; R with the

proper-ties :

(i) h

is

_non-negative

and

_h(z)

&#x3E; 0 for all z in the _open

_{e/2-neighborhood}

of x;

(ii)

h(z)

= 0 for all _{z E}

~0,1~2

having

_a distance of at least _e from

x;

(iii)

h is

_arbitrarily

often differentiable.

The

_periodic

continuation of h to all

of IEg2

is denoted H. Also H is

arbitrar-ily

often differentiable and has therefore an

_absolutely

_convergent Fourier

our

assumption

on x the function H vanishes on G.

Hence,

we

_get

a

con-tradiction as follows:

In the fourth and last

_step

we show that no

_point

x can occur several

times modulo 1. We start to prove this for x in the interior of G. We

assume that also x +

_{(m, n)}

E G for some non-zero

(m, n)

E

Z~.

_{Let p}

denote the

_Lebesgue

measure. We obtain

Hence,

we must have

_{equality everywhere}

and translations of G

_by

different

integral

vectors cannot have common interior

_points.

A

_slightly

modified

_argument

also shows that a

boundary

_point

of one

translate of G cannot be an interior

_point

of another one.

_Eventually,

con-sidering

two

_boundary

_points

in different

_translates,

we _argue

by deforming

the

_boundary

curves. D

Remark. The last lemma is

_proved

in a much more

_general

version than

needed.

_Namely,

in our case the curves

,S’1

and

_Tl

_{are just images}

of

_straight

lines under L. But the latter does not lead to any considerable

simplifica-tion of the

_proof

for r = 2.

However,

for r &#x3E; 2 we indeed use the fact that the

multiplicative

funda-mental domain in

_exponential

_{space is} a

_2r-gon.

The

_application

of Gauss’

theorem in that situation becomes much easier since the outer normal is

almost

_everywhere

well defined on the

_boundary.

The

_proof

for r = 2

given

above can therefore

_easily

be

_adopted

to r &#x3E; 2 and so we omit it.

Clearly,

the

_premises

of the lemma are not satisfied in the

_example

above if we choose the

_logarithmic

_images

of the

_straight

lines as curves. We will

see,

however,

that the lemma becomes

applicable,

if

L(El), L(E2)

form a

linearly independent cyclic

non-negative

set in

_logarithmic

_space.

Example

(cont.).

Clearly,

the

_original

vectors

do not form a

cyclic

_non-negative

set. The reduction

_{procedure developed}

in Section 2 for r = 2

yields

_{as new}

(and

cyclic

non-negative)

basis

The

_quadrangle

with vertices

,, 4

is convex and the last lemma becomes

applicable.

This

_phenomenon

is

_{explained by}

the next

_proposition.

Proposition

3.1.

_{Any cyclic}

_non-negative

basis

_of

vectors

_{bl, b2}

_of

a 2-d2mensional lattice A

_(in

_logarithmic

_space)

_yields

a convex

_fundamental

domain

_{f or multiplicative equivalence.}

Proof.

Let x2

=

exp(bil),

yi =

exp(bi2)

for i =

l, 2. Hence,

_we have 1

yl zi ,

1

x2 y2. We first show that the

slope

of the

straight

line

through

(1,1)

and

_{(ziz2 ,}

_YlY2)

is

_larger

than that

_through

_(1,1)

and

_{(zi ,}

_yl)

and smaller than that

_through

_(1,1)

and

_{(X2, Y2).}

From our

_assumptions

we obtain

_successively

as well as

hence,

in _{case x2} &#x3E; 1

_(i

_{=1, 2),}

Finally,

we need to show that the

point

(xl x2,

is not in the interior

of the

_triangle

with vertices

_{1,1),

_{(X2, y2).}

This is

_again

obvious from the size conditions.

0

Although

in

_principle

the same ideas can be used in dimensions r &#x3E; 2 the

description

of the fundamental domain in

_exponential

_space becomes much

more

_complicated.

The reason for this is that for vertices of a maximal

bounding

r - 1-dimensional

parallelotope

of the fundamental domain of

A the

_preimages

of these vertices in

_exponential

_space will not

_necessarily

lie in a

hyperplane

_anymore.

Hence,

we need to

_decompose

each of these surfaces in an

_appropriate

_way. This will be

_explained

in the

_sequel.

The

_{general procedure}

is

_roughly

as follows. The

original

fundamental

domain in

_logarithmic

_{space is}

_decomposed

into a finite number

_{of simplices.}

For each such

_simplex

we consider the

_images

of its vertices in

exponential

space.

They

form the vertices of a

_simplex

there. What we then need to

show is that the

_logarithmic

_image

of the union of the

_{exponential simplices}

becomes a fundamental domain for the lattice in

_logarithmic

_space.

We

_decompose

the fundamental domain of the lattice A with basis

bi , ... , br

i

For o, E S,

we set

r

and

_consequently

obtain

_(we

_{put xu(O)}

:= 0 Via E

_Sr)

The accent at the last union indicates that sets

_occuring

several times are considered

_only

once. Then all sets of the fourfold union are

_disjoint

and and all

_components

are _open

simplices

in their

corresponding

dimension.

(Without

the accent we had a union over 2rr! sets. With the accent this number decreases from 48 to 26 for r =

3.)

The vertices of _every such

_simplex

are

mapped

into

exponential

_space

and the

_images

become the vertices of a

corresponding simplex

there. The

simplex

defined

_by

_{P( a)}

in this _{way in}

_exponential

_space will be denoted

_by

Pe(a).

We will show that for _{a, T} E

_Sr

the closures intersect

in the closed

_simplex

which is

_spanned

_by

the

_preimages

of the vertices of

_{P( a) n P(T)}

thus

_{exactly mirroring}

the situation in

_logarithmic

_space.

Because of the definition of

_{Pe ( a)}

it suffices to show that r the sets

Q

and

_{Pe (T)}

are

separated by

a

hyperplane

in

_exponential

_space which contains the

_preimage

of

_P(a)np(r).

Therefore we can

_glue

the

_exponential

simplices

Pe (a)

in the same manner as the fundamental domain of A is

glued

as indicated

_by

the

_{decomposition}

of

_fl

into the

_simplices

_P(~).

Thus we

get

a

_2’’-gon

(which

is not

_necessarily

_convex)

whose

_image

in

_logarithmic

space has

boundary

sets which are

_congruent

modulo 1.

Again

we can

apply

Gauss’ theorem _{to prove} that we indeed have a fundamental domain.

To avoid

_{complications}

at first it should be assumed that the

_exponential

simplices

Pe(oa)

contain interior

_points.

The other cases are then obtained

by

performing

limits. We omit details since all

_important

_steps

were

already

explained

in the

_proof

of Lemma 3.1 and the

_subsequent

remark.

Of _course, the

_{practical problem}

remains to show that the union of

ex-ponential

simplices

constructed as above indeed satisfies the

_requirements

of the

_{preceding paragraph.}

We

_conjecture

that this is in

_general

the case

when we choose a

_cyclic

_non-negative

basis for A in the

_beginning.

A

For

_proving

that different _open

_exponential

_simplices

do not have

common

points

it suffices to show that their vertices are

separated by

a

hyperplane.

For a better

understanding

we enumerate the vertices of the

exponential

simplices

from 0 to 7 in

_binary

_{representation,}

e.g., 6 =

(o, l,1)

denotes the vertex whose coordinates are the

_exponential

values of the

coordinates of the vector

b2

+

b3.

In the

_sequel

we

only

consider the six closed three-dimensional

simplices.

We order them as follows:

For

_{example, concerning}

the intersection of

_simplices

₍₁₎

and

₍₂₎

we show

that the

_point

1 is

_separated

from

_{points 2,6 by}

the

_plane

determined

_by

the three

_{points 0,3,7 .}

We do this

_{by considering}

the

_sign

of the determinant whose first two rows are the vectors

spanning

the

separating plane

and the

last row is the vector from 0 to the

_point

under

_{consideration, i.e.,}

1 or

2,6.

In this _way we need to discuss a total of 15 intersections. Because of

occuring symmetries

it suffices to consider 6 different determinants.

We recall that

_{bi, b2, b3}

is a

_cyclic

_non-negative

basis. The coordinates of these vectors are denoted

by

bi

=

(b21,

bi2,

for 1 i 3. In

exponential

space we write

_Bij

:=

(1 ~

_{i, j}

3)

for abbreviation. Then the

plane

through

0,3,7

is

_{spanned by}

the vectors

_{B12 B22 B32}

-1, B13 B23 B33 -

1 ) (BIIB21 -

1, B12 B22 - 1 B13 B23 -

1)

and for the

_point

1 we need to determine the

sign

of the determinant

We do this

_similarly

as in Section 2. We write the determinant as a

polyno-mial

_{812, 813, B21, B22 , B23 , B31, B32 ,}

_{B33 ) ~}

This

_polynomial

is lin-ear in each of its 9 variables.

_Hence,

we

only

need to discuss its

sign

on the

boundary

of each variable. Instead of

_just

_considering

29

cases we reduce

the number of variables

_by

_{3, i.e.,}

we discuss

polynomials

in 6 variables

We therefore start to consider the

_sign

of the coefficient

Pi

of

_Bll

in P: P =

Pl Bll

₊

P2. Repeating

this

procedure

_we obtain

successively:

At this

_stage

the coefhcient

_polynomial

Plll

is a

polynomial

in

only

6

variables

_{Bij (1 j}

_{3 ,}

i =

2, 3).

Writing

it _as

polynomial

in

B22,

then

the coehicients as

polynomials

in

_B23

the

_sign

of

_Plil

(and

similarly

for any such

polynomial

in the same 6

_variables)

can be

_computed

_{fairly easily.}

Next we set

_Bl3

= 1 in

_Pll

and determine the

_sign

of that

polynomial.

These two

_signs

take care of the coefficient

PI1

of

B12

(in

the coefficient

Pl

of

_Bll

in

_P),

_i.e.,

_they

determine the

_sign

of

_P,

for

_B12

_large.

This is followed

_by

a discussion of the case

B12

=

813.

We need _to consider _a

polynomial

P,

=

PllBl3

₊

Pl2

in the _new main variable

Bl3

which can be

treated

_similarly.

’

We note that

_considering

a coefficient of one variable or

_setting

one

vari-able to 1

_yields

a

polynomial

with one variable

_less,

but the

_degrees

in the other variables

_stay

invariant.

_However,

if we need to

_replace

one variable

by

another _{one, say} A

_by

_B,

the

_degree

of of B in the

_{resulting polynomial}

can increase.

_{Fortunately, only quadratic polynomials}

occur. For those we need to consider the coefhcient of the

quadratic

_term,

the value at a

potential

extremal

_point

_{(derivative vanishing),}

and the value obtained

_by

replacing

the variable

_by

the next smaller one. For all

quadratic

polyno-mials it turned out that their extremal values were outside the considered

interval of the main variable.

For

_discussing

the different cases we wrote a small

_Maple

_program

[2]

which made these

_{investigations}

a lot easier. As we

conjectured

it turned

out that the

_signs

for any of these

parametrized

determinants were the same for all

potential

values of the

_parameters

and did indeed

_separate

the

considered

_simplices.

We therefore

_proved

the next theorem. Theorem 3.1. Let r 3

_{and fl}

=

Uo-ESr

P(a)

be the standard

decompo-sition 3.7

_of

a

fundamental

domain

given

by

a

_cyclic

_non-negative

basis in

logarithmic

space. Let

0,

v2 (a~),

...,

Vr+l(a)

be the r + 1 vertices

of

P(a)

and

1,

_E(vj(a))

(2 j

r +

₁₎

be their

_preimages

in

_exponential

_space. Let

Then ,~’ is a

fundamental

domain

for multiplicative

equivalence as

intro-duced in

_Definition

3.1.

Remark.

_(i)

If the

_cyclic

_non-negative

basis in the theorem is denoted

bl, ..., br,

then we have

_Ei

=

exp(bi) (1 i

r)

in Definition

3.1,

where

the

_exponential

map is

applied

coordinatewise.

(ii)

The amount of work for

_proving

a similar theorem in

higher

dimensions grows

exponentially,

but the cases r =

4, 5, 6

_are

easily

within the

scope of

these methods. Also we

emphasize

that for

explicitly

_given

lattices it is no

problem

to check

_numerically

whether the

_{corresponding}

_simplices

can be

_separated

as demonstrated for r = 3.

4. Shintani

_{decomposition}

Let f be a module of the

totally

real

algebraic

number field

F,

i.e. f is an

_integral

ideal of F combined with a subset of the infinite

places.

In the

sequel,

we

only

need the ideal

_part,

which we also denote

_by

f. Let b be a

(fractional)

ideal

_coprime

to

_f,

hence

_representing

a narrow _ray class b

modulo f. The

_{corresponding}

_ray class zeta function is defined

_by

where the summation is over all

integral

ideals of b modulo f. We recall Shintani’s ideas

_[7]

for

_evaluating

that zeta function at

_non-positive

_integers.

Besides the full _{unit group}

UF

of F we need the

_subgroup

of

_totally

positive

units

_ut,

_respectively

the

_subgroup

_{ut (f)}

of

_totally

_positive

_ray class units. Both

_subgroups

are of finite index in

UF.

A full set of

_generators

for them is

_easily

obtained from fundamental units of

_UF.

Example.

For

_determining

_UF

we

_just

need to find

_{representatives}

of those classes of

_UF/UF,

which contain

_{only totally}

_positive

units. This is

straightforward

from the

_signs

of the

_conjugates

of the fundamental units

of

UF

and

_requires

_{only binary}

arithmetic.

The group

_ut

(respectively

acts on as described in the

previous

section. Shintani showed that there exists a finite

_number,

_{say s,}

of open

simplicial

cones

_Cj

(1 j s)

such that

The union on the

_right-hand

side is

_disjoint.

The _open

_simplicial

cones are

Following

Shintani we set for S C

and

_note,

that

_R(j,8)

is

_finite,

when S is a subset of a fractional ideal. In

the

_sequel

we use the notation x E S + 1 for x -1 E S. For m e N Shintani obtains:

where the

_Aj

are matrices the rows of which _span one of the s

simpli-cial cones and the

Bm

are

generalized

Bernoulli

polynomials.

Once the

simplicial

cones are

known,

all

_remaining

calculations are

_quite

straightfor-ward. This will be demonstrated in a

forthcoming

_paper

by

the authors.

In the remainder of this section we show how to

_apply

the

_{general theory}

of

multiplicative

fundamental domains

_developed

in Section 3 in this

_special

situation.

It is of

_importance

that we need

multiplicative

(in)dependence

only

for non-zero

integers

of F of the same norm. This is used to eliminate one

dimension. Whereas Shintani

_[7]

and Okazaki

_[4]

do this

_by

_requiring

the

trace of the

_integers

to be one, we rather follow Reidemeister’s

precedent

[6]

and normalize the last coordinate to one.

In the

_sequel

we assume that _{~1, ..., are} a basis of the unit group under

consideration

_(consisting

_only

of

_totally

_positive

_units).

Then the lattice A

in

_logarithmic

_space has the basis

From these vectors we determine a

cyclic

_non-negative

basis

bl, ...,

br

which

generates

a lattice A of finite index t in A. We

_just

have to

_keep

in mind that the elements which need to be determined in the

_simplicial

cones

coming

from

A

will occur with

_multiplicity

t. As outlined in Section 3 the fundamental domain of A with

_respect

to

_{bi, ..., br}

is

_decomposed

in

the standard _way into r!

_simplices

and the

_{corresponding simplices}

for the

multiplicative

fundamental domain are calculated.

Example.

Let F =

Q( p)

for a root p of the

polynomial x4+x3-4x2-4x+1.

The discriminant of F is 1125.

_(Up

to

_isomorphy

we could also

_{put F}

=

15

+

6V5),

but in that case the

_equation

order would not be

_maximal.)

The _{unit group}

UF

of F is

_{generated by -1}

and the three fundamental units

el =

3 p -

p3,

e2 = 1 +

p, e3 = 3 -

p2.

None of the fundamental units is

totally

positive,

only

the

_product

of all 3.

_Hence,

the index of

_UF

in

_UF

is 4.

_Choosing

the order of the

_conjugates

of _p

_{appropriately,}

we obtain a

cyclic non-negative

basis of A from the

_totally

_positive

units

This kind of calculations is very easy with a software

_package

like KANT

~1~.

References

[1] M. DABERKOW ET _{AL., KANT V4.} J. Symb. Comp. 24 _(1997), 267-283.

[2] D. REDFERN, The Maple handbook: Maple V release 3. Springer Verlag, New York _(1994).

[3] R. _OKAZAKI, On an _effective determination of a Shintani’s _{decomposition of the} cone _Rn+.

J. Math. _Kyoto Univ. _(1993), 1057-1070.

[4] R. _OKAZAKI, An _{elementary proof for} a theorem of Thomas and Vasquez. J. Number Th. 55 _{(1995), 197-208.}

[5] M. _POHST, H. _{ZASSENHAUS, Algorithmic Algebraic} Number Theory. Cambridge University

Press 1989.

[6] K. _{REIDEMEISTER,} Über die Relativklassenzahl _gewisser relativ-quadratischer Zahlkörper.

Abh. Math. Sem. Univ. _Hamburg 1 _(1922), 27-48.

[7] T. _SHINTANI, On evaluation _of zeta functions of totally real algebraic number fields at

non-positive integers. J. Fac. Sci. Univ. Tokyo IA 23 _{(1976),393-417.}

Ulrich HALBRITTER Mathematisches Institut Universitat zu K61n Weyertal 86-90 50931 K61n Germany E-mail : uhalb(Qmath.uni-koeln.de Michael E. POHST Fachbereich 3 _Mathematik, MA 8-1

Technische Universitat Berlin

Straiie des 17. Juni 136 10623 Berlin

Germany