Hilbert symbols, class groups and quaternion algebras

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

T

ED

C

HINBURG

E

DUARDO

F

RIEDMAN

Hilbert symbols, class groups and quaternion algebras

Journal de Théorie des Nombres de Bordeaux, tome

12, n

o

_{2 (2000),}

p. 367-377

<http://www.numdam.org/item?id=JTNB_2000__12_2_367_0>

© Université Bordeaux 1, 2000, tous droits réservés.

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

Hilbert

_symbols,

class

_groups

and

_quaternion

algebras

par TED CHINBURG* et EDUARDO FRIEDMAN**

To Jacques Martinet

RÉSUMÉ. Soit B une

_algèbre

de _quaternions définie sur un _corps

de nombres k. Nous associons à tout

_couple

de

_symboles

de Hilbert

_{{a, b}}

et

_{c,d}

_{pour B}

un invariant _{03C1 =}

_03C1R([D(a,

_b)],

[D(c,

d)])

dans un _quotient du _groupe des classes au sens restreint

de Cet invariant a son

origine

dans l’étude des _sous-groupes

finis d’un _groupe kleinien

_{arithmétique}

maximal. Il mesure la dis-tance entre les ordres

_{D(a, b)}

et

_D(c,

_b)

dans B associés à

_{{a, b}}

et

{c, d}.

Si a =

c, nous calculons

_{03C1R([D(a, b)], [D(c, d)])}

en termes

de

_{l’arithmétique}

du _corps

_{k (~03B1 ) .}

Le

_problème

d’etendre ce calcul au cas

_général

conduit à l’étude d’un

_graphe

fini lié aux différents

symboles

de Hilbert _pour B. Nous considérons en détail un

exem-ple

issu de la détermination de la

plus

petite variété

_hyperbolique

arithmétique

de dimension trois.

ABSTRACT. Let B be a _quaternion

_algebra

over a number field k. To a _pair of Hilbert

_symbols

_{{a, b}}

and

_{{c, d}}

for B we associate

an invariant _03C1 =

03C1R([D(a, b)], [D(c, d)])

in a _quotient of the narrow

ideal class group of k. This invariant arises from the

study

of finite

subgroups

of maximal arithmetic Kleinian groups. It measures

the distance between orders

_{D(a, b)}

and

_{D(c, d)}

in B associated to

{a, b}

and

_{c,d}.

If a =

c, we _compute

_{03C1R([D(a, b)], [D(c, d)])}

_by

means of arithmetic in the field

_k(~03B1).

The

_problem

of

_extending

this

_algorithm

to the

_general

case leads to

_studying

a finite

_graph

associated to different Hilbert

_symbols

for B. An

_example

_arising from the determination of the smallest arithmetic

_hyperbolic

3-manifold is discussed.

Manuscrit reçu le 28 f6vrier 2000.

*

Partially supported by NSF _grant DMS97-01411. **

Research _{partially supported by} FONDECYT _grant 198-1170 and _by a Chilean Presidential

1. Introduction

Two non-zero elements a and b of a number field k determine a

quater-nion

_{algebra B}

=

B(a,

b)

with center

k,

namely

the 4-dimensional

k-algebra

with k-basis

_1,

and _xy with the

_{multiplicative}

relations

Since there are _many other

_pairs

c and d in k

determining

the same

algebra

B =

B(a,

b) ,

it is natural to ask whether there is any

interesting

structure

on the set of

_pairs

_{determining isomorphic}

_quaternion

_algebras.

Motivated

_by

a

problem

in arithmetic

hyperbolic 3-orbifolds,

we consider

the

_B*-conjugacy

class of an order D =

D(a,

b)

closely

related to the

ring

y]

c B.

To define

_D,

we

replace

by

a maximal order over a Dedekind

domain

_0:-,

where

_OR

is obtained from

_C7~

_by

_inverting

a finite set R of

primes

ideals of

_{Ok . Namely,}

where

_Ramf(B)

is the set of

_prime

ideals of

_Ok

at which B

_ramifies,

the

dyadic

primes

are all those above the rational

prime

2,

and

Odd(a, b)

con-sists of the

_prime

_{ideals p} at which or

ordp(b)

is

odd,

the valuation

ordp

being

normalized so that its value _{group is} Z. Let

_Op

and

_kp

be the

p-completions

of

_C~~

and

_k,

_{respectively.}

For

_{p 0}

_R,

_ordp(a)

and

_ordp(b)

are even

_integers,

so we let

where cp and

dp

in

_kp

are such that =

ordp ( nr(~p))

=

0,

where

nr is the reduced norm. We can choose

_xp

= _x and

~p

=

y for all but

finitely

many p. Note that the subsets

_Opxp

and

Op§p

of

Bp

:= B _0k

_kp

do not

depend

on the choice of cp or

_dp.

We define the

O:-order

D =

D(a, b)

C B

by

requiring

that its

_completions

_Dp

= D

~~k

Op

be

_given

_by

Then D is a maximal

OR -order

of B since the reduced discriminant

[V,

p. 24]

of

_~p]

is

_{readily computed}

to be

_Op.

The Skolem-Noether theorem

_{[R, p. 103]}

shows that x

_{and y}

in

_(1.1)

are

_uniquely

determined

_by

a and

b,

_{up to}

conjugation by

an element of

B*.

_Thus,

_starting

from a and b in k* we have obtained a

_conjugacy

class

[D(a, b)]

of maximal

_Of-orders.

Note

_{that, again}

_by

the Skolem-Noether

theorem,

the

_{OR-isomorphism}

class of an order D C B coincides with its

B*-conjugacy

class,

usually

known as its

_type.

We will assume from now on the

Then the set of

_conjugacy

_types

of maximal

_O)-orders

is

_classically

known

to be in non-canonical

_bijection

with the _group

_TR(B)

of fractional

_OR

ideals of k modulo the

_{subgroup generated by}

squares of fractional ideals

and

_by

_principal

ideals

_(a)

=

aoR

such that _{a E} k* and _{a &#x3E;} 0 at all real

places

that

_ramify

in B

_{[V, p. 89] [CF1,}

Lemma

_3.2].

Although

the _map from _a, b to the

_type

_{[D(a, b)]}

seems

quite interesting

to

us, it is hard to describe because there is no canonical

description

of

_types.

In

_particular,

it is not even clear what one would mean

by

computing

this map.

However, given

two

types

[D]

and

_[,6]

of maximal orders

_OR

orders of

_B,

there is a canonical R-distance

)

E

_TR(B),

which is

the

_image

in

_TR(B)

of the order-ideal

_[R,

_p.

_{49] p(D,,6)}

of the finite

_OR

module Under the Eichler

_condition,

it is known

_[V,

p.

89] CF1,

Lemma

_3.2

that two

_{C7k -orders}

D and.6 are

B*-conjugate

(or,

equivalently,

isomorphic)

if and

_only

if the R-distance between them is trivial. In

_fact,

the non-canonical

_bijection

mentioned above is obtained

_{by arbitrarily fixing}

some

_type

[9]

and

mapping

[D]

to

[-EI)

We can now state our

Main Problem.

_Compute

the _map that takes two

_pairs

_{(a, b)}

and

_{(c, d),}

assuming

B(a, b) = B(c, d), to

b)], [D (c,

d)])

E

_TR(B),

where R =

R(a, b) U R(c, d)

(see (1.1)

to

_{(1 .4)}

_for

_notation).

Here we

regard

the

D(a,

b)

as an

Of-order

by

inverting

ideals

in R -

_{R(a, b),}

and

_similarly

for

_{D(c, d).}

The Eichler condition is

_tacitly

assumed.

In this

_generality,

we have no idea how to solve the main

_problem.

We

present

here a solution of sorts in the

special

case a = c.

Theorem.

_Suppose

a, b and d are elements

of

a number

field

k such that

the

_quaternion

_algebras

over k

corresponding

to

(a, b)

and

(a, d)

coincide.

Suppose

also that this

_algebra

B

_satisfies

the Eichler condition and let R =

R(a, b)

U

_{R(a, d).}

If a

E

k*2,

then

_odor

=

q2

and

[D(a, d)])

is the class in

TR(B)

of

the

_fractional

_{OR -ideal}

q.

Assurne now

that a ~

1~*2,

and let K = Then

pR([D(a, b)],

D(a, d)

is the class in

_TR(B)

where the

_fractional

ideal a is

found

by

taking

_{any z} E K such that

NormK/k(z)

=

bd,

and

writing

ZOR

=

al-ac.

Here _a is the non-trivial element

of

Gal(K/k)

and

c is the extension to K

_of

a

fractional

OR-ideal

of

k.

We shall see that the existence of z as above follows from the

assumption

B(a, b)

~

_{B(a, d).}

_By

_OR

we mean the

integral

closure of

O)

in K.

In

_§2

we

_give

a

proof

of the Theorem. In

§3

we show how to

_apply

the Theorem to the

_computation

of torsion

_subgroups

of arithmetic Kleinian

groups. We also

give

a numerical

_example

in which k is a sextic

field,

showing

that one can sometimes avoid

having

to find z E K

by

computing

instead inside a narrow ideal class _group of K.

We now turn to a curious finite

graph

whose definition is

suggested by

the Theorem.

_Suppose

_{B(a, b) ~ B(c, d),}

and define R =

R(a, b)

U

R(c, d).

We would like to

_compute

_{[D(c, d)]).}

The Theorem does not

apply directly

if

_{(a, b)}

and

_(c,

_d)

have no common

_entry.

However,

note that

where the

_Dj

_{(1 i N)}

are _{any N} maximal

C7k -orders

of B

(cf.

[V,

ch.

II,

32]

or

[CF1, eq.(3.1)]).

Suppose

there are

_pairs

(a, e)

and

(e, d)

such

that

_{B(e, d)}

and that

_ordp(e)

is even for

p ~

R. Then

R(a, e)

and

_{R(e, d)}

are contained in R =

R(a, b)

U R(c, d).

From

(1.5)

_we obtain

_6)]

_{(D(c, d)~~}

as

where each term can be

computed

by

the Theorem.

This

_suggests

_defining

a

_graph

as follows. Fix a

_quaternion

algebra

B

and a finite set R of

_primes

of 1~

_containing

all finite

_primes

ramified in B and all

_primes

above 2. Start with all

_pairs

_{(a, b)}

E k* x 1~* such that

B E£

_{B(a, b)}

and

_{R(a, b)}

c

_R,

i. e., a and b are even

_{at p}

if

p 0

R. We

should make some further identifications

since,

as

C7 -orders,

for _any e and

f

in k*. The first

isomorphism

is clear from

(1.1)

on

_switching

x and _y, the second on

replacing

_y

by

_xy and

observing

that

(x~)2

= -ab has even valuation

_{for p 0}

R. The last

_isomorphism

in

(1.6)

follows from

(1.3),

since

_xp

=

*p

and

_~p

=

Typ.

For the _purpose of

_computing

_{[D(a, b)~,}

we _may therefore

identify

(a, b)

-(b, a) N (a, - ab) -

(e 2a, f2b)

for e and

f

in k*. We let

f a, bl

be the

equivalence

class of

_(a,

_b)

under the transitive closure of the above relation

N, with

B and R fixed.

Define the vertices of the

_graph

G =

G(B, R)

to be the set of classes

f a, bl.

This is a

finite,

possibly

_empty,

set since the extension

which determines a modulo _squares, is one of

finitely

_many extensions

K/k

unramified outside R with

_{[K : l~~}

2.

Two vertices

_b~

and

_{{c, d}}

are connected

by

an

edge

whenever these

equivalence

classes

_have,

_{respectively,}

_{representatives}

of the form

_{(a’, b’)}

and

_{(a’, d’).}

In other

_words,

we connect two vertices whenever the Theorem allows us to

_compute

_PR of the

_{corresponding}

_types.

The motivation for

_defining

G is that successive

_application

of the

The-orem and of

(1.5)

allows us to

_compute

the R-distance between any

types

corresponding

to the same connected

_component

of the

_graph

G.

When B is the matrix

_algebra

_M2(k),

G is connected for rather trivial

reasons.

Indeed,

if

B(a, b) B(c,

d) ^--’ M2(k),

then

is a

_path

in G from to

_{{c, dl.}

In some other

simple

cases G is also

connected,

but we do not see

_why

this should hold in

general.

As

long

as one cannot solve the main

_{problem above,}

it would be

_interesting

to

compute

the number of connected

_components

of G.

2. Proof of Theorem

We now _prove the first assertion in the Theorem.

_Namely,

k*2,

then =

q2

and

pR~~D(a, b)~, ~D(a, d)~~

is the class in

TR(B)

of the fractional

_OR-ideal

_q.

By

(1.2),

for

_{p g R}

we have

_ordp(b)

=

2rp

and

_ordp(b)

=

2sp

for some

rp, sp E Z. Hence

Recall that we are

_assuming

B(a, b) = B(a, d).

Since a E

_{B(a, b) =}

B(a, d)

=

M2 (k).

By

(1.6),

we

might

as well assume a = 1.

Now,

since the elements x and y of B

satisfying

(1.1)

are

uniquely

determined _{up to}

B*-conjugacy by a

and b

_(as

follows from the Skolem-Noether theorem

_[R,

p.

103]),

we can assume

Let _7rp E

_C7k

_satisfy

_ordp(7rp)

= 1. Then in

(1.3),

for some local units _{up, vp}

_{E 0*.}

Recall that

D( l,

b)p

in

(1.4)

is defined as

Op

+

_Op2p

+

_Op§p

+ A short calculation

_using

_(2.1)

shows that

where we used the fact

_{that p ~}

R cannot be above

_2,

_by

_(1.2).

Similarly,

From

_(2.2)

and

_(2.3)

we obtain

Hence the finite

_{Op-module D(l,}

_b)p/(Ð(l,

_b)p

n

_D(l,

_d)p)

is

_isomorphic

to

Op/-x 0.,

the

_sign

_being

chosen so that the

_exponent

of _7rp is

non-negative.

We have thus found that

_{b)], [D(a,}

_d)])

is the class in

_TR(B)

of the ideal As

_TR(B)

has

_exponent

_2,

_{b)], [D(a,}

_d)])

is also the class in

_TR(B)

of _q =

p(Sp+rp),

which _proves the first claim in the Theorem.

We now turn to the second assertion in the Theorem. Assume that

_{a ~}

_k*2,

and let K = Then

PR ([D (a,

b) ], [D(a,

d)~~

is

the class in

_TR(B)

_{ofNormK~k(a),}

where the fractional a is found

by taking

any z E K such that

_NormK/k(z)

=

bd,

and

writing

ZOR

=

a’-’ c.

Here Q is the non-trivial element of

Gal(K/k)

and c is the extension to K

of a fractional of k.

We note that while a in the factorization =

aI-a c

is _not

unique,

the class of

_NormK/k(a)

in

_TR(B)

is

_{unambiguously}

defined.

_Indeed,

_suppose

NormK~k(z’)

= bd and

ZIOR

=

Taking

the _norm

to k,

_we find

c2

=

c’2,

where for convenience we do not

_{distinguish notationally}

between c and

cnor. k

Hence c = c’ and

(a’a-1)1-a

= Since

NormK/k(z’ / z)

=

1,

and

_K/k

is

_quadratic,

Hilbert’s Theorem 90 shows that that

_z’/z

=

71-~

for

some 7 E K*. Thus

(a’a-l1’-I)I-a =

Since R contains all the

_primes

of k which

_ramify

in

_K,

we conclude from this that

a’a-l1’-1

=

ct o) for

some

ideal

ct

of k. Hence

_{NorMK/k(a’a-1) =}

Recall that

_by

the definition of

_TR(B),

a

_principal

OR_ideal

(a)

has trivial class in

TR(B)

if a &#x3E; 0 at all real

_places

of k ramified in B. Since K =

k( va)

embeds in

B,

for _any

₀

E K* the class of is trivial in

_TR(B).

As

_TR(B)

has

_exponent

_2,

we see that the class in

TR(B)

of is trivial.

This shows that the class of

_NormK/k(a)

in

_TR(B)

is indeed well-defined since we will show below that z exists.

As we are

assuming

B(a, b)

=

B(a, d),

_we have the

equality

of local Hilbert

_symbols

_{f a, b~v}

=

la, d}v [V,

p.

74]

for all

places v

of k. Hence

~a, d/b~v

is trivial for all _v,

_showing

that

_d/b

is

_{everywhere locally}

a norm.

By

the Hasse-Minkowski theorem

_{[V, p. 75],}

_d/b

is a

global

norm.

_Thus,

there is a w E such that

_{Normk( Jä)/k( w)}

=

d/b.

We can take z = bw.

Let E B* be as in

(1.1),

so that

x 2

= _a,

y2

=

b and xy

=

-~x.

therefore

_identify

K = _C B. We have

just

_seen that there _{are e} and

f

in k such that w = _{e +}

f x

_E K satisfies

d/b

=

nr(w)

=

e2 -

a f 2.

Set

t = _{wy E} B and

_compute

since

_y2

= b is in the center k of B. As w _E

k(x),

This means that we can use _{x, y to}

_compute

D(a, b)

and _{x, t to}

_compute

D(a,

d).

Now,

while

Thus,

as

_Op-modules,

A

_prime

_{p ~ R}

cannot

_ramify

in

_k(x)/k.

Assume first

_{that p}

is not

split

in the

_quadratic

extension Then F :=

kp(x)

= is _a

quadratic

field extension of

_l~p

_containing

the elements

_xp

and

_wp.

_However,

Normp/kp

(wp)

=

nr(wp)

E

_0*,

_implies

_wp

E

_{As p ~}

R is not above 2

and £§

we have

_Op

+ =

OF.

It follows that

so from

(2.4)

we see that there is no contribution

_{at p}

to

_PR([D(a,

_b)],

[D(a,

d)])

in the

_non-split

case.

Now suppose p

splits

as =

pip2.

Then x2

= a =

r~

for some

rp E

kp,

so we can

assume £)

= l. Hence there is a

_kp-algebra

_isomorphism

fp : kp(xp)

H

_{kp x}

_kp

_mapping

1 to

_(1,1)

and

_~p

to

_{(1, -l~.}

Thus

_fp(Op

+

where 7rp E

_7~~

satisfies

_ordp(7rp)

= 1. Recall that

ordp(d/b)

is _even

for p 0

R and

Hence in

_(2.5),

_ap =

ordp2(w)

is an

_integer.

We order pi

and p2 so that

_{ap &#x3E;}

0. From

_(2.4)

and

_(2.5)

we find that

fp

induces an

isomorphism

of

_C7p-modules

where ap is

_given

_by

_(2.5)

and

_"split"

refers to the

_quadratic

extension

K

If is the

_piece

_{above p}

of the factorization of we can write

where Q is the non-trivial element of

Gal(K/k).

Thus

WOR

=

al-af,

where

and f

is the extension to K of a fractional

OR_ideal

in k. As

NormKk(a)

=

q in

_(2.6),

the

_proof

is done on

letting z

= bw and c =

bf.

3.

_Application

to 3-orbifolds

We now describe our

_geometric

motivation for the main

_{problem posed}

in

_{§ 1.}

In

_[CF2]

we studied torsion in maximal arithmetic Kleinian _{groups r}

and found that

_computing

dihedral

_subgroups

of r is

_closely

connected with the _map

_{(a, b)}

H

[D(a, b)]

described _in

_§1.

The connection with Hilbert

_symbols

in the case of a

_4-group

Z/2Z

x

_Z/2Z

is

_{straight-forward:}

If B ^--’

_B(a,

_b)

is

_generated

as a

_{k-algebra by x}

and _y as in

(1.1),

then

the

_subgroup

H =

H(a,

b)

C

_B*/k*,

_{generated by}

the natural

_projective

images 7

and

_y,

is in fact a

4-group. Conversely,

_any

4-group H

C

is

_conjugate

to a

_subgroup

of the form

H(a, b),

where a and b are

_unique

modulo

_k*2,

_except

that one can switch a and b or

_replace

b

_by

-ab

[CF2,

Lemma

_2.4].

The list of

_conjugacy

classes of

_4-subgroups

H c

_B*/k*

is therefore in

bijection

with the list of

_pairs

of

_{(a, b)}

such

_{B(a, b),}

where we

again

take a and b modulo

k*2

and allow the same trivial modifications.

A

_problem

left unresolved in

_[CF2]

is how to

_actually

find which

(con-jugacy

classes

_of)

maximal discrete

_subgroups

of contain a

_given

4-subgroup

H(a, b).

We now

_explain

how this

_problem

is

_equivalent

to that of

_computing

the _map

_{(a, b)}

~

[D(a, b)~.

Borel

_[Bo]

showed _{that any} arithmetic Kleinian _{group is}

_conjugate

in

PGL(2, C)

to a discrete

subgroup

r C C

_PGL(2,C),

where B is a

quaternion

algebra

over a number field k

having exactly

one

_pair

of

_complex

conjugate

embeddings,

and B ramifies at all of the real

_places

of 1~. This

complex place

is used to embed in

_PGL(2,

_C~).

Borel constructed a

list of discrete

_subgroups

_rS,D

c such that _any maximal

(with

respect

to

_inclusion)

discrete

_subgroup

T c

_{B* /k*}

is

_conjugate

to

some

_rs,,D.

Here S is any finite set of

prime

ideals of

C~~

disjoint

from

Ram f( B) ,

and D is _any maximal

_Ok-order

of B. When S is

_empty,

_rs,D

is

_just

the normalizer of

_D,

i. _e., the

_image

in of

_{those y}

E B* for which

_yDy-1

= D. We denote this _group

by

rD.

The definition of for a

_general

S is

_similar,

_except

for a local twist

at

_primes

in

_S,

as we now

explain.

For

each p

E S choose a maximal order

Ep

C

_Bp

such that

_{[Dp :}

_Ðp

n

_{Ep] =}

_Say

that 7 E

Up

to

_{B*/k*-conjugacy,}

does not

_depend

on the choice of

Ep

nor on

the

_completions

_Dp

at

_places

in

A

_given

_{non-cyclic subgroup}

of

_B*/k*

is contained in

_only

a few

_rs,D.

More

_precisely:

([CF2,

Theorem

_5.1])

Let B be a

_quaternion

_{algebra satisfying}

the Eichler

condition over a number field

k,

and let H be a

non-cyclic

finite

subgroup

of

B*/k*.

Then there are two finite sets s =

s(H)

and t =

t(H)

consisting

of

prime

ideals of k not in

_RamAB),

and a

_type

T(H)

of maximal

O:t

-orders

of B with the

_following

_property:

A

_{B*/k*-conjugate}

of H is contained in

rS,Ð

if and

_{only if s}

C S c t

and DRt

E

_{T (H) .}

Here E)R,

:=

O:t0okV

C B is the extension of D to

O:t.

°

The sets s and t are

easily computed

and

given

explicitly

in

[CF2, §5],

but the

_type

_T(H)

is subtler. When H =

H(a, b)

is a

4-group,

we have

in the notation of

_(1.2).

Moreover a review of the

_proof

(see

the local

computations

in the

_proof

of Lemma 4.1 of

_[CF2])

shows that

_{T(H(a, b)) =}

[D(a, b)]

as

_types

of maximal This was our

_original

motivation

for

_{investigating}

the

_{problem posed}

in

_31.

We conclude with a numerical

example

which we

_hope

illustrates the kind

example

while

_tracking

down the smallest arithmetic

_hyperbolic

3-manifold

[CFJR].

There we treated this case

_by

_geometric

methods.

Let k =

Q(,8),

where

,86 - ,85 - 204 -

203 + 02

₊

30

_{+ 1} = 0. PARI shows that k is a sextic field of discriminant -215811 =

-33 7993,

having

exactly

one

_{complex place,}

2

_stays

_prime

in but

30k =

_p~7’

where p27 has norm 27.

Also, k

has class number one and narrow class number

2,

where narrow is taken in the strictest sense,

involving

all real

places

of k.

_Thus,

modulo _squares, there are four distinct

_totally

_positive

units

61

= 1,

62, C3, E4 = _6263. The

_prime

_p27

splits

in

k(vl---3)/k,

_so the _narrow Hilbert class field of k is

_{k( 3)}

=

l~ ( -~2 ),

say.

Let B be the

_quaternion

_algebra

ramified

_only

at the four real

_places

of k. The

_problem

faced in

_[CFJR]

was:

Given _any maximal

_Ok-order

D of

_B,

does

_rD

contain a

4-subgroup?

Since S and

_Ramf(B)

are

_empty

and 2 remains

_prime

in we find

Rt =

_{Rt(H) =}

_~2C~~~

for _any

_4-group

H. As k has narrow class

num-ber

_2,

B has

_exactly

2

_types

of maximal

_{Ok-orders, say E}

and ,~’. We

can

_distinguish

these maximal orders

_{intrinsically. Namely,}

from

[CF1,

Theorem

_3.3]

we find that one of these maximal

orders,

saye,

contains a

primitive

cube root

₍₃

of

_unity

and the other one does not

₍₍₃

"selects"

one of the

types).

Likewise,

only 9

contains a _square root of -C2. Since

B =

B ( -1, -1 )

is

_isomorphic

to the standard Hamilton

_quaternion

al-gebra

over

k,

we

recognize 9

as the order

_containing

the Hurwitz order

Ok

+

_Oki

+ + + i

_{+ j}

+

_ij)/2,

where

i2

=

j2

= -1 and

ij

=

-ji

(since (1 +i+ j +ij)/2

is a

_primitive

cubic root of

_unity).

Note that

£Rt

and

remain in distinct

_ORt-tvpes

and that

SRI

and can be

_{intrinsically}

distinguished

from each other

_exactly

as 9 and ,~’.

A short calculation shows that there are

(up

to

_{B*/k*-conjugation)}

ex-actly

five

_4-groups

_{H(a, b)}

contained in

_Fs

or

_{F Jr.}

These are

In this

_computation

one uses that a and b must be

_totally

_negative,

H(a, b)

=

H(a,

-ab)

=

H(b, a),

that

_s(H(a,

_b))

=

Odda,b - Ramf(B)

_must be

_empty

_(since

S is

_empty),

and that

_{B i B(-1, -1)}

with the sextic field k

_having

one

place

above 2 and

having

class number 1.

Our earlier

_question

can now be

_rephrased

as:

In which

_Dire

or is each one of the five

4-groups

listed above?

Since a and b are

_units,

we have

~~

=

x,

~p = y

in

(1.3)

for all p.

Thus,

the

_ORt-order

_{D[a, b]}

in

_(1.4)

_{is just}

_{ORt [X,}

_{) .}

As we saw

_above,

the maximal

oRt -order

ERt

is characterized _{up to}

as the first two orders contain

_-s2

and the third one

_{(3. Hence,}

if

_r~

contains a

_4-group,

it must be

_H(-1,

_-s3)

or

H(-1,

-E4),

in which case at

least one of

[D(-1, -E3)]

or

[D(-l, -ê4)]

would

_equal

To show that in fact does contain a

4-group,

we turn to the Theorem with a = b = -1 and d = with i determined _as follows. PARI tells _us that K :=

k( I)

has class number 2

(a

priori

this class number is

only

divisible

_by

_2).

_Thus,

all ideals classes are fixed

by

Gal(K/k)

and the norm

map induces an

isomorphism

from the class _group of K to the narrow class

group of k.

Let a be _any ideal of K with non-trivial class. Then

_(z)

:=

al-a

for

some z E

_K,

since the class _{group is} fixed

_by

the Galois _group.

_Thus,

ê :=

NormK/k(z)

=

sib2

for _some 1 i 4 and _some b _E

Applying

the

_Theorem,

we find that

pRc(~D(-1, -1)~, [D(- I , -ei)])

is

non-trivial,

as

it

_equals

the class of

_NormK~k(a)

in

_TRt

_{(B) =}

_Z/2Z.

_Thus,

_(D(-1,

[E,(- 1, - 1)] =

since otherwise the

Rt-distance

would be trivial. From

(3.2)

we see that

~D(-1, -êi)] =

and that i = 3 _or 4. Hence all

rD’s

contain a

_4-group.

One can

actually

show

(applying

(3.2),

E4 =

ê2ê3 and the Theorem to

_{(-1, -1)}

and

_(-1,-))

that

_{[Ð(-1,-ê3)] = [Ð(-1,-ê4)] =}

[TRt

_.

References

[B] A. _{BOREL, Commensumbility} classes and volumes of hyperbolic 3-manifolds. Ann. Scuola Norm. Sup. Pisa 8 _(1981), 1-33. Also in Borel’s _Oeuvres, Berlin _{Springer (1983).} [CF1] T. CHINBURG, E. _FRIEDMAN, An embedding theorem for quaternion algebras. J. London

Math. Soc. ₍₂₎ 60 _(1999), 33-44.

[CF2] T. CHINBURG, E. FRIEDMAN, The finite subgroups of maximal arithmetic Kleinian

groups. Ann. Institut Fourier 50 _(2000), 1765-1798.

[CFJR] T. _CHINBURG, E. FRIEDMAN, K.N. JONES , A.W. _REID, The arithmetic hyperbolic

3-manifold of smallest volume. Ann. Scuola Norm. Sup. Pisa _{(to appear).}

[R] I. _REINER, Maximal Orders. Acad. Press London _(1975).

[V] M.-F. VIGNGRAS, Arithmétique des algèbres de Quaternions. Lecture Notes in Math.

800, Springer-Verlag Berlin _(1980).

Ted CHINBURG

The University of _Pennsylvania

Philadelphia PA 19104 U. S. A. E-mail : _{[email protected]} Eduardo FRIEDMAN Facultad de Ciencias Universidad de Chile Casilla 653 Santiago 1 Chile E-mail : friedman~uchile. cl