Cyclotomic quadratic forms

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

F

RANÇOIS

S

IGRIST

Cyclotomic quadratic forms

Journal de Théorie des Nombres de Bordeaux, tome

12, n

o

_{2 (2000),}

p. 519-530

<http://www.numdam.org/item?id=JTNB_2000__12_2_519_0>

© Université Bordeaux 1, 2000, tous droits réservés.

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

Cyclotomic quadratic

forms

par

FRANÇOIS

SIGRIST

RÉSUMÉ.

L’algorithme

de Voronoï est un

procédé

_permettant

d’obtenir la liste

_complète

des formes

_quadratiques

_positives par-faites à n variables. Sa

_{généralisation}

aux G-formes _permet de

classer les formes

_G-parfaites,

avec

l’avantage

de se dérouler dans un _espace de dimension

plus

_petite

(G

est un _sous-groupe fini de

GL (n,

Z)) .

On étudie ici la

_{représentation}

standard du _groupe

cy-clique

G = Cm _en dimension

~(m),

de

polynôme caractéristique

03A6m (x) (polynôme cyclotomique).

Une forme G-invariante est dite

forme

_{cyclotomique.}

Toute les formes

_G-parfaites

sont données

pour

~(m)

16, de même _{que pour} m = 17, où la forme

cyclo-tomique la

plus

dense est entièrement nouvelle. On obtient ainsi

une constante d’Hermite

_{cyclotomique,}

_qui s’avère être souvent

meilleure _que la constante d’Hermite habituelle. C’est le cas _pour m = 5,7,11,13,16,17,36, et vraisemblablement 32

(les

calculs

pour m = 32 sont en _{cours, et ont}

_déjà

fourni 4600 formes

C32-parfaites).

Les résultats

_complets

sont

_disponibles

à

http://www.unine.ch/math.

ABSTRACT. Voronoï ’s

_algorithm

is a method for

obtaining

the

complete

list of

_perfect

n-dimensional

_quadratic

forms. Its

gen-eralization to G-forms has the

_advantage

of _running in a

lower-dimensional space, and furnishes a

finite,

and

complete,

classifica-tion of

_G-perfect

forms

_(G

is a finite

subgroup

of

GL (n, Z)).

We

study

the standard,

~(m)-dimensioiaal

irreducible _{representation}

of the

_cyclic

group Cm of order m, and give the, often new, dens-est G-forms. Perfect

_cyclotomic

forms are

_completely

classified for

~(m)

16 and for m = 17. As a _consequence, we obtain _precise

upper bounds for the Hermite invariant of

cyclotomic

forms in this range. These bounds are often better than the known or

conjec-tural values of the Hermite constant for the

_{corresponding}

dimen-sions ; this is indeed the case for m =

5,7,11,13,16,17,36.

The

1. Introduction

Denote

_by

_~y",(x)

= 1 + _CIX + ... +

c.,t-lx" 1

+ xn the m-th

cyclotomic

polynomial,

of

_{degree n}

=

0(m).

Let G _E

GL(n, Z)

be the matrix

generating

the standard n-dimensional

_{representation}

of the

_cyclic

group

Cm.

A

_cyclotomic

_form

is a real

_quadratic

form in n

variables,

invariant under this action of

Cm:

the

_symmetric

matrix A =

(aij)

satisfies

GtAG

= A.

An _easy

_computation

shows that the matrix A has then constant

diag-onals =

ai,j),

and is

_completely

known from the first half of its first row: the _space of

_cyclotomic

forms is a linear

_subspace

of

of dimension

_n/2.

_{Nevertheless,}

for

_convenience,

we

always

shall

_give

a

cyclotomic

matrix

_by

its full first row.

A

_positive

definite real

_quadratic

form has

-

a minimum m =

min(xtAxlx

E Zn -

_{0}).

-

a set S =

I±Vl,

::I:v2, ...

+vs )of

pairs

of minimal vectors in

sat-isfying

= _m. 2s is called the

kissing

number of the form.

-

a Hermite invariant

_1’n(A)

= _{m -}

(det

There is a familiar

dictionary

linking positive

definite forms and Euclidean

lattices,

via the Gram matrix. A Euclidean lattice has an associated

sphere

packing,

and the

_packing

_{density 6}

is related to the Hermite invariant

_by

the relation 6 =

Const(n)

-yn(A)n/2 :

a dense form means a

high

Hermite

invariant.

A

_quadratic

form is

_perfect

if it is the

_{unique quadratic}

form

_having

minimum m and set S of minimal vectors.

_Similarly,

a

_cyclotomic

form is

said to be

_Cm-perfect

if it is the

_unique

_cyclotomic

_form

with m and S as

before. A

_quadratic

form is extreme if its

_density

is

_locally

maximal in the

space of

quadratic

forms. A

cyclotomic

form is if its

density

is

locally

maximal in the _space of

_{cyclotomic forms.}

An extreme form is

_{always perfect.}

For the _converse, a theorem of

Voron6i

_requires

a condition called

_eutaxy

(cf

[Mar]).

These

_properties

carry over verbatim to the case of

Cm-forms

[B-M].

A

Cm-perfect

form is

proportional

to an

_integral

_form,

as are the usual

_perfect

forms.

There is a second useful

dictionary

linking

rational

_cyclotomic

forms to

there is a

unique

real element a E

_~(~m)

such that

xtAx

=

Trace(axx).

The

_practical

_computation

runs as follows: write d for the first column

of the matrix

_A,

and associate to a =

ao + + ... + the

column a =

(ao,

_{cxl, ... ,}

an-1)’.

Then ii = Da with matrix D =

(dij)

= The determinant of D is the discriminant of the field

_Q((m),

and is therefore nonzero: A and a determine each other. If a

cyclotomic

form is

_{positive definite,}

the element a is

_totally

_positive

_{(a » 0):}

all its

_conjugates

are

_{positive. Further,}

if u is a

unit,

the

cyclotomic

form

corresponding

to u . ii - a is

_{clearly Cm-equivalent}

to the

_original

form. This method is

_extremely

useful for

_finding,

or

checking, G-equivalences.

Taking

determinants in

xtAx

=

Trace(axx),

one

_gets

(using

the fact that

a is

real):

, ... , ... - ",. " a

It follows that the determinant of an

integral cyclotomic

form is

equal

to

where

_k(m)

is the

_square-free

_part

of the discriminant of the

cy-clotomic field

_~(~m).

_{Further, using}

the detailed

_study

of

_{decomposition}

of

_primes

in

_{cyclotomic fields,}

E.

_{Bayer-Fluckiger}

and J. Martinet obtained the

_following

useful

Proposition

( Bay-M ).

If q is a

prime

power

dividing d,

and

prime to

m,

then q’ ==:f:1

(mod

m).

Up

to

_equivalence,

there are

only

finitely

_many

perfect

quadratic

forms.

Voron6f ’s

_algorithm,

deviced in 1908

_~Vor~,

constitutes a method of

explicit

complete

classification. The densest

_quadratic

_form(s)

in each dimension

can be found with an

entirely

mechanical

_computation,

the

only

problem

being

the

_{gigantic computational}

_complexity.

For

_cyclotomic

_forms,

it was shown in

[B-M-S]

that the

orthogonal

projec-tion of Voron6f ’s

_algorithm

onto the

_(dual)

_space of

_cyclotomic

forms pro-vides an exhaustive classification.

Later,

it was

proved

[JCb]

that the

num-ber of

_{Cm-inequivalent Cm-perfect}

forms is finite. The

_practical

_advantage

lies in the dimension of the _space of forms

_(n~2

instead of

_n(n+1)~2),

bring-ing

the

_computing

time to almost

_nothing

for n

_12,

and to "tolerable" values for the _groups

_C17

and

_C32

_(n

=

16).

There are 1344

_C17-perfect

forms,

and at least 4600

_C32-perfect

forms. Both densest forms are

com-pletely

new.

As is

_customary

in

_algebraic

number

_theory,

we take m to be either odd

or divisible

_by

4.

For many values of _m, the

_conjectured

_(or

_proved)

densest form

_(D4,

E6, E8, K12,

~16)

appears as a

Cm-form.

This

gives

new matrix represen-tations of these well-known

_forms,

and is

_particularly

suitable for further

computation

of their

_algebraic

invariants.

Even with a

long

reference

list,

I cannot do

_justice

to those who

_helped

Bordeaux

_colleagues

Anne-Marie

_Berge

and

_Jacques

Martinet,

for their

help

with _many crucial

_points.

For

_general

results on

quadratic

forms and

lattices,

we refer to the books

by

Conway-Sloane (C-S~,

and Martinet

_[Mar].

2. The Vorondf

_algorithm

Start with a

_quadratic

form

q(x)

=

xtAx,

with minimum _m, and minimal

vectors

_{f ±vi 1, (1 i s).}

Endow the _space of

_symmetric

ma-trices with the Euclidean structure

_given

_by

A. .B =

Trace(AB).

Consider the s

matrices Y

=

viv2

as elements of the dual _space

The

_simple

observation: m = = = A ~

Y

exhibits

the matrix coefficients _aij as a solution of the

_system

of linear

equations

=m.

The Voronoi domain of the matrix

_A,

denoted

_by

_D(A),

is the convex

hull,

in

_8ymn(JR)*,

of the _rays

_{aY, ~ &#x3E; 0;}

it is a convex

polyhedral

cone. Its

dimension in is called the

_{perf ection}

rank of

_A,

its codimension the

_{de f ault of perf ection}

of A. A

_perfect

form has default zero

by

definition:

the

_system

of

_{equations X - Vi}

= _m has X = A _as its

unique

solution. The dual of the Voron6i domain

is,

by

construction,

the subset =

vi X vi

&#x3E;

₀₎

of

_8ymn(JR).

Suppose

first that A is not

_perfect.

Then there is a matrix

F #

0

vanishing

on all _vi. If

101

is small

_enough,

all forms

A (0 )

= have _same minimum m and same set of minimal vectors. There are two values _{pl, p2} &#x3E;

0

_(possibly

_infinite,

but not

_both)

such that

_A(-pl)

and

_A(p2)

still have minimum m, but a

strictly

larger

set of minimal vectors.

_{Consequently,}

after a finite number of

_steps,

one reaches a

perfect

form.

If A is

_perfect,

the domain

_D(A)

is of maximal

_dimension,

and is bounded

by

faces which are

hyperplanes

in Given such a face

_,~,

we

denote

_{by F}

the

_(uniquely

_defined,

_{up to} a

_positive

scalar

factor)

matrix which is the inner normal to .~.

It is

_convenient,

at this

_stage,

to

_give

the

_practical

method for

_finding

all faces of

_D(A).

A face matrix F has the

_{following algorithmic description:}

If S = is the set of minimal vectors of

A,

there is _a non-trivial

partition

S =

Sl

U

S2

satisfying:

- F is

unique,

up to a

_positive

scalar factor.

The

_important

fact here is that this

_computation

runs

entirely

in

8ymn(JR),

though, clearly,

the

_geometric

_description

in

_{Symn (R)*}

is more convenient for

_{understanding}

the _process.

Take now a

perfect

form

A,

F one of its face

_matrices,

and consider the

has minimum m, but has default of

perfection

1,

because,

by

construction

above,

its set of minimal vectors is

_Sl.

_{Define p}

=

sup

(0 1 A (0)

has

minimum

_m).

If _{p is}

_infinite,

the matrix F is

_{semi-positive definite,}

and the face T is said to be a "cul-de-sac". As was observed in

[JCa] ,

and

implicitely

used in

_[Vor],

this does not

_happen

in the _case, but

should be

_kept

in mind in

_generalizing

this process. If p is

finite,

A(p)

is

again

a

perfect

form: the

neighbouring

form of

A

alone

~’.

The

_neighbouring

relation

_gives

the set of

_perfect

forms the structure

of a

graph,

and Voronors fundamental result

[Vor]

asserts that this

graph

is corcnected.

_Moreover,

the

_{GL(n, Z)-equivalence}

of forms preserves the

neighbouring

relation,

so a

_quotient

_graph

can be defined.

_By

standard

arguments

(e.g.

Hermite

_reduction),

the

_quotient

_graph

is

_fcnite:

this is Voron6f ’s

_algorithm.

Besides the

_finding

of all faces of a

domain,

a

time-consuming

task,

there are three other

_practical

_steps

needed. We list them for

_completeness

(More

on this in

[Vor], [JCa], [Mar]):

- Find all minimal vectors. - Given

a

face,

find the

neighbouring

form

(i.e.

find

p).

- Decide

GL(n, Z)-equivalence

between two

_perfect

forms.

If G is a finite

subgroup

of

GL(n, Z)),

it is

possible

to run Voron6f ’s

algorithm

in the linear

_subspace

T of G-invariant forms in If A is a

G-form,

its relative Voron6f domain

DG (A)

is then the

orthogonal

projection

of

_D(A)

onto the dual

_subspace

T* of

_5ymn(JR)*.

It is

_proved

in

[B-M-S]

that the

_resulting

_{neighbourhood graph}

is connected. The finiteness of the

_quotient

_graph

is more delicate to obtain

(G-equivalences

have to be taken in the

_subgroup

of

_{GL(n, Z)}

_preserving

_T),

and has been

_proved

for G-forms in

_[JCb].

"Culs-de-sac" can _appear

(see

[B-M-S]),

and it also

happens

that the

_complete

_graph

is

_already

finite.

_However,

for

_cyclotomic

forms,

one has the

Proposition.

If

G is a

_{Q-irreducible subgroup}

_of

_{GL(n, Z)),}

there are no

"culs-de-sac" in

_running

G- Voronoï ’s

_algorithm.

Proo f . Irreducibility

implies

that the orbit of _any minimal _{vector spans} the whole _space. A

_positive

semi-definite form

_vanishing

on a minimal vector

must

_consequently

be the zero form. D

3. Raw data

We shall

_separate

the lists of

_cyclotomic

forms into three families. In

family A,

the critical form

_(D4,

_{E6, Eg, K12,}

_A16)

is

_{cyclotomic. Family}

B describes

_Cp-perfect

_forms,

with _p a

_prime.

The

_remaining

cases are listed

For each

_family,

the first table

_gives

_successively

_m,

_0(m),

GP = number of

_Cm-perfect

_forms,

P = number of

Cm-perfect

forms which _are also

perfect

in the

_ordinary

_sense, H=Hermite invariant of the densest

form,

with its

Name,

if known. The second table

_gives

first _m, First _row, then

_min,

det =

minimum,

_resp. determinant of the densest

form,

in its smallest

integral

representative, E,

F = number of

edges,

_resp.

faces,

of its Voron6i domain.

Theorem A. The data

_{for family}

A are

Comment. In the cases with

A16,

the number of

_edges

of the domain

makes the

_finding

of the faces

_{practically impossible.}

Before

_giving

the data on

_family

_B,

let’s

_briefly

discuss the

_Craig

lattices

~4~.

These famous lattices are

thoroughly

investigated

in

[C-S]

and

[Mar].

Their direct construction as difference lattices

_gives

them matrices

_(cf.(C-S~

having

characteristic

_polynomial

1-I- x -f- ...

+ x’- 1

+ xn. The

_diagonals

are

constant, and the first row is filled from left and from

_right

with binomial coefhcients:

This

_description

_{is very}

_convenient,

but does not show the value of the

minimum,

whose determination is a difhcult combinatorial

problem

(cf.

[Mar]

p.

139).

When n + 1 is a

prime

_p, the

Craig

lattices are

_cyclotomic.

In

P-1

our _range of

_{investigation,}

their minimum is

_2r,

by [Ba-Ba],

and their

density

is

_pretty

_high.

We shall

_evidently

meet these forms below. In

_high

dimensions,

Craig

lattices are sometimes the densest known

(e.g.

_p=151,

cf

[C-S]

p.

43),

although

it is

generally accepted

that the

Cp-Hermite

constant

is lower than the absolute constant. Theorem B below shows

_that,

_{up to}

a

single,

and

_{surprising exception,}

the densest

cyclotomic

form is a

Craig

form.

Theorem B. The

_{data for family B are}

/oB

Comment. The

_Craig

form

A13

₁₆ is the densest

_{Craig form,}

with minimum 6 and determinant

175.

Our new form is

_denser,

because

24

17. The

proportion

of

_perfect

forms is

_high,

and will be

_explained

below.

For the last

_family

of

_forms,

with the

_exceptions

of

D8

(for Cls )

and

D16

_{(for C32),}

there are no otherwise known

_Cm-perfect

forms. The

densest form has

_already

been found

_(dense

forms tend to have

_{high kissing}

numbers,

and

_consequently

to _appear

_early

in the

_search).

Theorem C. The data

_{for family}

C are

We close this

_chapter

with a few

general facts, gathered

from our

_present

sample

of ca. 6000

_{Cm-perfect cyclotomic}

forms.

_First,

in all _cases, there is a

_spanning

orbit of minimal

_vectors,

bringing

the minimum of the form

on the

_diagonal

of the matrix. This seems to be a

general

fact,

but the low

dimension,

and the

_triviality

of the ideal class _group, could contribute to

this.

_{Nevertheless,}

let’s formulate this for

_perfect

_cyclotomic

lattices:

Conjecture

1.

_{Ang perfect cyclotomic}

lattice has a

_spanning

orbit

of

min-imal vectors.

Being

more

_pessimistic

leads to:

Conjecture

2.

_{Any perfect cyclotomic}

lattice is

_{generated by}

its minimal

vectors.

Another

_intriguing

fact in our

_sample

is

Observation 1. All

_{perfect cyclotomic forms}

are even.

Comment. In the case the

_prime

2 is unramified in the

_{cyclotomic field,}

then _any

_{cyclotomic form, perfect}

or _not, is

even .

In the ramified _case,

one has to assume

C~-perfection,

but the fact is

_unexplained.

1 This result is due to J. Martinet. The _{representative} element a, being real, has an even

4. Two detailed

_samples

We

_begin

with a case

_appearing

in Theorem A. Notations

_{"H" ,}

"E" , "F" , "min" ,

"det" are as above. "No" is the

_numbering

used

_by

the

Voron6f

_algorithm,

and is

_simply

used here to locate

_neighbours.

"s" is the half

_kissing

_number,

and "i" the default of

_perfection.

Nbr

_gives

the list of

_neighbours.

These 13 forms are listed

_{by decreasing}

_H,

_beginning

with

.K12 .

Theorem A28. The

_{C28 -perfect}

_forms

are

The next

_example

is

_C13,

contained in Theorem B. The

_Craig

forms appear, and one can

observe,

_contrary

to the

_preceeding

_case, the

_high

proportion

of

_perfect

forms.

Comment. There are two

_non-perfect

forms in this

_list,

both with a default

of

_perfection

of

_12,

and both with a minimum divisible

_by

13.

_Similarly,

in the list of 1344

_C17-perfect

_forms,

one finds 11

non-perfect forms,

all with

a default of

_perfection

of

_16,

and a minimum divisible

_by

17. This

_suggests

Observation 2. The

_{default of perfection of}

a

Cp-perfect

form

is divisible

by (~ - 1).

Observation 3. The minimum

_of

an

integral

Cp-perfect,

non-perfect

Cp-form

is divisible

_by

_p.

Observation 2 was first

proved

by

J. Martinet

(unpublished).

Here is a more

_general

result

(proof

of _my

own) settling

the claim:

Lemma. Let A be the matrix

_of

a

_positive

cyclotomic form.

Call

6(A),

resp.

6p(A)

its

default of perfection,

resp.

default of

Cp-perfection.

Then

6(A) -

6p(A)

is divisible

_by

_{(p -}

_1).

Proof.

The _group

_Cp

acts on

D(A),

_fixing

Dp(A).

As _any

_{fixpoint-free}

ratio-nal

_{representation}

of

_Cp

has dimension divisible

_by

_(p-1),

the codimension of in

_D(A)

is divisible

_by

_{(p - 1).}

Hence

(p(p- 1)/2 - 6 (A)) -

((p-1)/2 - 6p (A))

is divisible

_by

_{(p -}

_1).

The lemma

follows. D

Observation 3

remains,

at

_present,

at the botanical

_stage.

An

_argument

in favour is the fact that the

_orthogonal

_projection

of

_D(A)

on

_Dp(A)

is the _average on

_orbits,

i.e. division

_by

_p.

5. Final remarks

Our definition of

_cyclotomic

forms is not as

general

as is used in

alge-braic number

_theory,

where it means

Trace(az3f)

on an ideal I of rank one in

Q((m).

This

general description

is very

convenient,

as is illustrated

with the

_Craig

forms which are obtained with c~

= 1

and I = per

(7~ _

( 1- ~p ) ) .

Further,

our densest

C17-form

was identified

by

J. Martinet:

c = 1 and I =

p-7Q,

where Q

is _an ideal such

that 66 =

(2).

It is denser

p

than

A3

₁₆ because

28

172:

this

_ultimately

boils down to the fact that 17 is a Fermat

prime !

i Voronoi’s

algorithm

does not fit well in this

_context,

because it runs under a

fixed

representation

of the _group In _presence of

exotic

_cyclotomic

_{representations,}

one has to rerun the

algorithm entirely,

only changing

the matrix G

_generating

the

_{representation}

at the

_beginning

of the _program. There is therefore a

cyclotomic

Hermite constant for each

pair

of

_conjugate

ideals in the ideal class _group of

_Q((m).

_Twisting

forms with an ideal can

_only

increase the Hermite

_constant,

and

looking

at the

results of

_[Ba-Ba]

on

C23,

one deduces for

example:

for the standard

cyclo-tomic

_{representation,}

the

_presumably

densest form is

A22~

with a Hermite

invariant of

_{2.94..; twisting}

with one of the exotic

representations

has the

effect of

_doubling

the minimum 2 of

A22

to

_{4, giving}

it an invariant of

3.46... As the

_presumed

_q22 is

_3.57..,

this shows that

_{Cp-cyclotomic}

forms

can become _very dense.

However,

all cases treated in this _paper have a

trivial ideal class group, and the Hermite invariant of the densest

can

_safely

be called the

cyclotomic

Hermite constant.

We did not include results on

_eutaxy

in this _paper. One reason is that

eutaxy

is an absolute notion

(a

cyclotomic

form is

Cm-eutactic

iff it is

eutac-tic).

But in view of the _very

_interesting

recent results on the classification

of eutactic forms

_(cf

_[Mar],

_chap.

_IX,4),

there is no doubt that the

study

of

_cyclotomic

eutactic forms will _prove fruitful. References

[Ba-Ba] C. _BACHOC, C. _BATUT, Etude algorithmique de réseaux construits avec la forme trace. J. _Exp. Math. 1 _(1992), 183-190.

[Bay-M] E. BAYER-FLUCKIGER, J. MARTINET, Formes quadratiques liées aux _algèbres

semi-simples. J. reine angew. Math. 451 _(1994), 51-69.

[B-M] A.-M. _BERGÉ, J. _MARTINET, Réseaux extrêmes pour un groupe d’automorphismes.

Astérisque 198-200 _(1992), 41-66.

[B-M-S] A.-M. _BERGÉ, J. MARTINET, F. _SIGRIST, Une généralisation de l’algorithme de Voronoï pour les formes quadratiques. Astérisque 209 _(1992), 137-158.

[C-S] J.H. CONWAY, N.J.A. SLOANE, Sphere Packings, Lattices, and Groups. Springer-Verlag (1992).

[JCa] D.-O. _{JAQUET-CHIFFELLE,} Enumération complète des classes de formes parfaites en dimension 7. Ann. Inst. Fourier 43 _(1993), 21-55.

[JCb] D.-O. _{JAQUET-CHIFFELLE,} Trois théorèmes de finitude pour les G-formes. J. Théor. Nombres Bordeaux 7 _(1995), 165-176.

[Mar] J. MARTINET, Les réseaux _parfaits des espaces euclidiens. Masson (1996).

[Vor] G. VORONOÏ, Sur quelques propriétés des _{formes quadratiques} positives parfaites. J.

reine angew. Math. 133 _(1908), 97-178.

François SIGRIST

Institut de Mathématiques

Université de Neuchgtel Rue Emile _Argand 11

CH-2007 Neuchitel Switzerland