Definite quaternion orders of class number one

J

ULIUSZ

B

RZEZINSKI

Definite quaternion orders of class number one

Journal de Théorie des Nombres de Bordeaux, tome

7, n

o

_{1 (1995),}

p. 93-96

<http://www.numdam.org/item?id=JTNB_1995__7_1_93_0>

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

Definite

_Quaternion

Orders of Class Number One

par Juliusz BRZEZINSKI

, The purpose of the paper is to show how to determine all definite

quater-nion orders of class number one over the

integers.

First of

all,

let us recall

that a

_quaternion

order is a

_ring

A

_containing

the

_ring

of

_integers

Z as a

subring, finitely

generated

as a Z-module and such that A = A ₍₈₎

Q

is _a

central

_simple

four dimensional

_Q-algebra.

_By

the class number

_HA

of

_A,

we mean the number of

_isomorphism

classes of

_locally

free left

_(or

right-both numbers are

equal)

A-ideals in A. Recall that a left A-ideal I in A is

_locally

free if for each

_prime

number

_{p, Ip}

-- 1~ _Q9

Zp

is _a

principal

left

Ap

=

A Q9

Zp-ideal,

where

_Zp

denotes the

_p-adic

_integers.

Two

_locally

free left A-ideals I and I’ define the same

_isomorphism

class if I’ =

I a,

where

cxEA.

A

_quaternion

order is called definite if A Q9 R is the

_algebra

of the Hamil-tonian

_quaternions

over the real numbers R. We

want

to show that there are

_exactly

25

_isomorphism

classes of definite

_quaternion

orders of class number one over the

_integers

(an

analoguous

result,

which is much more

difhcult to _{prove, says} that there are 13 Z-orders of class number one in

imaginery

quadratic

fields over the rational

numbers).

First of

_all,

we want to

_explicity

describe all

_quaternion

orders over the

integers.

This can be done

by

means of

integral

_{ternary quadratic}

forms

where

_Z,

which will be denoted

_by

It is well known that each A can be

_given

as where

_f

is a suitable

integral ternary

quadratic

form and

_{Co (f )}

is the even Clifford

_algebra

of

_{f .}

The

_following

formulae can be considered as a definition of

Co (f ):

Co (f )

is a Z-order with a basis

1,

_{ei , e2, e3} such that:

where i, j, k

is an even

permutation

of

1, 2,

3.

Moreover,

the above construction defines a one-to-one

correspondence

between the

_isomorphism

classes of

_quaternion

orders and the

_equivalence

classes of

_positive

definite

_integral

_ternary

_quadratic

forms.

Two orders A and l~’ are in the same _genus of orders

(that

_is,

for each

prime

number _p the orders

_Ap

and

_Ap

are

isomorphic

over

Zp)

if and

_only

if the

_{corresponding}

ternary

quadratic

forms are in the same _genus

(which

means

_equivalence

of them over

_Zp

for each

_prime

number

_p).

It is well

known that the number of

non-isomorphic

orders in a _{genus is} finite. The

number of

_isomorphism

classes of orders in the _genus of A will be denoted

by

TA. TA

is called the

_type

number of A

_(two

_isomorphic

orders are said

to have the same

type).

The discriminant of A =

Co ( f )

is

d(Co(f))

= where

Co ( f )

is a Gorenstein order if and

_only

if

f

is

_primitive,

that

_is,

_SGD(aij)

= 1. Recall that A is called Gorenstein if

A#

=

Hom(A,

Z)

is

projective

_as

left

_(or

_right)

A-module

_(see

_[CR],

_p.

_778).

If

_{Co ( f )}

is not

_isomorphic

to the matrix

_ring

_M2(Z),

which

_happens

exactly

when

_{d (Co ( f ) )}

_{~ ~ 1,}

then define

modulo p is _irreducible,

modulo p is a _square of a linear _factor,

modulo p is a product of two different linear factors.

Using

the

_description

of

_quaternion

orders

_by

means of

_{ternary quadratic}

forms,

a formula for

.~~ proved

in

[B2],

(4.5)

and some results on the

THEOREM. There are 25

isomorphism

classes

of

Z-orders with class

num-ber 1 in

_definite

_quaternion

Q-algebras.

These classes are

represented by

the orders

_{Co ( f ),}

where

_f

is one

of

the

following forms

(the

index

of

the

matrix

_{corresponding}

to a

quadratic form f

is the discriminant

of

the order

Proof.

Let A be a

quaternion

Z-order with class number

HA

=1.. Then

_... ,

-(see ~K~,

Thm. 1 or

[B2], (4.6)).

Denoting

by 0

the Euler totient

function,

we have

where _pi and

_p~

are all

prime

factors of

d(A

such that epi

(A)

= 1 and = 0. This

_{inequality implies}

that

~(d(A))

12 and if

4)(d(A))

= 12,

then for each

_prime

factor _p

_{of d(A),}

_{ep(l~) _ -1.}

The condition

_~(d(A))

12 _says that 2

_d(A)

16 or

d(A)

_{= 18, 20, 21, 22,}

_24,

_{26, 28, 30, 36,}

42.

Assume now that A is a Gorenstein Z-order. Then A =

Co ( f ),

where

f

is a

_primitive

_integral

_ternary

_quadratic

form with

_only

one class in its

genus, since

TA ~I~

(see

[V],

p.

88).

Thus, using

the tables

[BI],

we can

next test is

_given

_by

the

_inequality

_(*),

which can be checked for the _genera

with

_TA

= 1

using

the reduction modulo the

primes

p~d(A).

In that _way,

we

_get

ep(A)

and choose those classes for which

(*)

is valid.

Finally,

using

a

_general

formula for

HA

_proved

in

[B2], (4.5),

we

_compute

HA

for the

remaining

classes and obtain our list of 23

isomorphism

classes of orders

having

class number 1.

The case of non-Gorenstein orders

unexpecteddly

_gives

two more classes.

Assume that A is a non-Gorenstein order with class number 1. Then

d(A)

must be divisible

_by

a third _power of an

_{integer &#x3E; 1,}

since A = Z ₊ dA’ for an

_integer

d &#x3E; 1

_according

to

_{~B 1~,}

_(1.4).

The

_{only possibility}

is d = 2

by

0(d(A)) :!~

12. Thus

_d(A)

E

_{8,16,24}.

But

_d(A)

= 8 is

impossible,

since

then

_d(A’)

=

1,

which can not

_happen

for an order in a definite

algebra.

If

_{d(A) =}

_24,

then

_{d(A’) =}

3,

which

_gives

_{e3 (A) = - 1 (and,}

of _course,

e2 (A)

=

0).

Using

the formula of

[B2], (4.5),

an _easy

_computation

shows

that

HA

= 1. If

d(A)

=

16,

then A = Z +

2A’,

where

d(A’)

= 2

_and,

in _a

similar _way, we

get H~

=1.

Finally

notice that 10

_isomorphism

classes with class number 1

cor-responding

to Eichler orders

_(that

_is,

those orders whose discriminant is

square-free)

were determined

_by

M.-F.

_Vigneras

(see

[V] ,

_p.

155).

REFERENCES

[BI]

H. _BRANDT, O. INTRAU, Tabellen reduzirter positiver ternärer quadratischer Formen, Abh. der Sächsischen Akad. der Wissenschaften zu _{Leipzig 45 (1958).}

[B1]

J. BRZEZINSKI, On orders in quaternion algebras, Comm. _Alg. 11 _(1983),

501-522.

[B2]

J. _BRZEZINSKI, On _{automorphisms of quaternion orders,} J. Reine _angew. Math.

403 _(1990), 166-186.

[CW]

C.W CURTIS, I. REINER, Methods _{of Representation Theory,} vol. _I, New-York 1981.

[K]

O.

_KÖRNER,

Traces of Eichler-Brandt matrices and type numbers _of quaternion orders, Proc. Indian Acad. Sci. (Math. Sci.) 97

_{(1987), 189-199.}

[V]

M.-F.

_VIGNÉRAS,

_{Arithmétique} des _Algèbres de Quaternions, Lect. Notes in Math.

800, Berlin-Heidelberg-New York 1980.

Juliusz BRZEZINSKI

Department of Mathematics

University of _G6teborg and Chalmers _University of Technology

S-412 96 _G6teborg, SWEDEN e-mail: _{jub@math.chalmers.se}