Publicacions
Ma emá iques,
Vol
36
(1992),
1001-1010
.
A
bs ac
BILINEAR
FORMS
FOR
SL(2,
q),
AND
SIMILAR
GROUPS
ALEXANDR,F
TURULL
*
A icle dedica
a
la
memó ia
del
bon
amic
Pe e
Menal
The
se
o
in a ian
symme ic
bilinea
o os
on
i educible
mod-
ules
o e
ields
o cha ac e is ic
ce o
o
ce ain
g oups
is
s udied
.
Resul s
a e
ob ained unde
he
p esen e
in
a
ini e
g oup
o
ele-
nen s
o
o de
ou
whose
squa e
is
cen al
.
In
pa icula ,
we
ind
ha
he
ele an
modules
o
he
g oups
men ioned
in
he
i le
always
accep
an
in a ian
symme ic
bilinea
o m unde
which
he
module
admi s an
o hono mal
basis
.
In oduc ion
Le
G
be
a
ini e
g oup and X
some
complex
i educible
cha ac e
o
G
wi h
eal
alues
.
l
F
is
any
eal
numbe
ield
con aining
Q(X),
(he e
Q(X)
is
Q
ex ended
by
all
he alues
o X),
hen
he e
is
a
unique (up
o
iso no phism)
FG-module
M
which
a o ds
he
cha ac e
7nF(X)X,
whe e
nzF(X)
is
he
Schu
inclex
o
X
wi h
espec
o
F
.
A
basic
p oblem
in
ep esen a ion
heo y
o
ini a
g oups
is
o
desc ibe
hese
modules
.
Since
F
is
a
eal
ield,
he
s anda d
a e a
.ging
a gumen shows
ha
AI
will
a o d
some
(posi i e
de ini e)
symme ic
G-in a ian
bilinea
o m
.
Wha
can
be
said
abou
?
Synime ic
bilinea
o os
o
M
a e
classi ied
up
o iso no phism
in
GL(AI)
by
he
signa u a
o
unde
each
embedding
o
F
in o
R,
he
de e minan
o
(de ined
up
o
squa es
in
F*) and
he
1
-
lasse
in a ian
o ,
sea
o
example
Co olla y
3
.3
in p
.
168
o
[1]
.
Fo
e e y
AE
.F*,
A
will
also
be a
non-degene a
. e
G-in a ian
symme ic
bilinea
o m
on
M
and
i s
de e minan
will
be
de (A
)
=
~
din'F
(11!)
de ( )
up
o
squa es
in
F*
.
I
ollows
ha
he
p oblem
will
be
mo e
ac able
i
dimp(M)
is
*Pa ially
suppo ed
by
a
g an
om
he
NSA
.
100
2

A
.
TURULL
e en,
o
hen,
a
leas
and
A
will
hen ha e
he
same
de e minan
.
The e
a e
a ious condi ions
ha
o ce
diMF(M)
o
be
e en
.
l
he
Schu
index
o
X
wi h
espec
o
F
is
no
one,
o
example,
hen
dimF(M)
is
e en
.
This
case
is
analyzed
in
[3J
.
Ano he
example
is
ha
i
Ad'
is
ai h ul
and
G'
l
Z(G)
is
o
e en
o de
hen
X(1)
mus
be
e en
.
The
p esen
pape
analyses
si ua ions
ha
occu
equen ly
in
his
case,
and
in
pa icula
ou
esul s yield
he
answe
o
G
=
SL(2,
q),
A,,
o
S

and
X
a
ai h ul
cha ac e
o
G
.
Be o e
s a ing
ou
esul s
;
we
desc ibe
ou
no a ion
.
NVe
deno e
by
B (F)
he
B aue
g oup
o
F
.
I
a
;
b
E
F*
we
deno e
by
(a,
b)
he
elemen
o
B (F)
which
has
as a
ep esen a i e
he
qua e nion
algeb a
-,
~
-2 -2
o
dimension
4
o e
F
gene a ed
by
i
and
j
sa is ying
i

=
a
;

j
=
b
and
i
j
=
-
j
i
.
I
is
asymme ic
non-singula
o m on
M
we
deno e
by
Hasse( )
i s
Hasse
in a ian
.
Hasse( )
is
an
ele nen
o
B (F)
and
is
calcula ed
as ollows
.
Le
el,
. . . ;
e,,
be
an
o hogonal
basis o
M
and
se
al
=
(e¡,
ej
.
Then
Hasse( )
=
.H
.
(a2,
aj)
.
%<j
He e
he
p oduc
is
in
he
B aue
g oup
o
F
and
Hasse( )
does
no
depend
on
he
o hogonal
basis
chosen
.
Theo em
A
.
Le
G
be
a
ini e
g oup
and
x E
G
be
an
elemen
o
o de
4
such
ha
x
2
E
Z(G)
.
Le
F
be
a
eal
ield
and
X
an
i educible
cha ac e o
G
wi h
alues
in
F
.
Le
Al
be
an FG-7nodule
a o ding
mp(X)X
and
assume
ha
x
2
ac s
non- i ially
on
Al
.
Le
be
a
G-
in a ian non-ze o
symme ic
bilinea
o m
on
M
.
Then
he
ollowing
hold
.
1)
de ( )
=
1
up
o
squa es
in
F*
.
2)
Hasse( )
=
(A
o
,
-1)
o
some
A,,
E
F*,
wi h A
o
>
0
i
is
posi i o
de ini e
.
Fu he mo e,
i
dimF(M)
-
2
(mod
4),
hen
o
e e y
AE
F*
he e
is
some
p,
E
F*
wi h
Hasse(p, )
_
(A,
-1)
.
Ou
heo em
has
a
consequence
abou
local
Schu
indices
which
we
now
p oceed
o
desc ibe
.
Recall
ha ,
i
X
is
an
i educible
cha ac e
;
he
local
Schu
indices
o
X
a e he
posi i o in ege s
m,,(X)
and
m (X)
( o
p
a
a ional
p ime),
whe e
m,
> ,
>
(X)
=
mR(X)
and
m
p (X)
=
mQ,(X)
(Qn
being
he
ield
o
p-adic
numbe s)
.
The
F obenius-Schu
indica o
gi es
a
s aigh o wa d
(and
well
known)
o mula
o
n ,,,(X)
.
Namely,
i
X
has
eal alues,
m,,,,(X)
=
1
i
1

E
X(g2)
=
1,
and
m
.(X)
=
2
DGI
-qEG
BILINEAR
FORMS

1003
and
~
1

X
(g
2 )
=
-1,
o he wise
.

The e
is
no
known
simila
o -
gEG
mula
o
mp(x)
.
Fu he mo e,
knowing
m,,
(X)
p o ides
in
gene al
e y
li le
in o ma ion
abou
m,(x)
.
Fo
example,
o
e e y
ini e
subse
S
o
{oo,
2, 3,
5,
7,
11,
. .
.}
o
e en
ca dinali y
he e
is
a
a ional
alued
i -
educible
cha ac e
X
o
a
double
co e
o
some
al e na ing
g oup
such
ha
mp(X)
=
2
o
pE
S
and
mp(X)
=
1
o
p «
S, see
[2]
.
Howe e ,
some
u he
condi ions
o e
X
and
G
do
imply
some
u he
ela ionship
be ween
m,,(X) and
he
mp(X)
o
p a
a ional
p ime
.
I
F
is
a
ield
con aining
he
alues
o
X
we
deno e
by
[X]
he
elemen
o
B (F)
ep e-
sen ed
by
EndFG(M)
whe e
M
is
a e
i educible
FG-module
a o ding
he
cha ac e
MF(X)X
.
Co o111 y
1
.
Le
G
be
a
ini e
g oup
and
x E
G
be
a e
elemen
o
o de
4
such
ha
x
2
E
Z(G)
.
Le
x
be
a e
i educible
chaT ce e
o
G
which
does
no con ain
x
2
in
i s
ke nel
and
such
ha
X(
1
)
-
2
(mod
4)
andm,,
(X)
=
2
.
Le
F
be
a
eal
ield
ha
con ains
Q
(X)
.
Then
[X]
=
(A,,
-1)

in

B (F)

o
some
nega i o

A
o
E
F*
.
In
pa icula ,
i
X
is
a ional
alued hen
m
,
p(X)
=
1

o
e e y
p ime

p
-
1(mod
4)
.
Al hough
Theo em
A
ully
desc ibes
Hasse( )
whe e
dim (M)
-
2
(mod
4),
i
diMF
(M)
-
0
(mod
4)
hen,
as
we
shall
see,
Hasse(M
)
=
Hasse( )
o
all
p
E
F*
.
Hence,
Hasse( )
will
be
de e minad
by
G
and
X
a
leas
i
X(
1
)
-
0
(mod
4)
and
M
is
absolu ely
i educible
.
Howe e ,
he
conclusion
o
Theo em
A
can no
be
made
mo e
p ecise
e en
in
his
case, as
ou
nex
esul
shows
.
Theo em
B
.

Gi en
A,
E
Q*,
A
o
>
0,
hen
he e
exis
G,
x,
F
=
Q
;
X,
M
and
sa is yi7
-
¿g
he hypo heses o
Theo em
A
wi h
he
ollowing
p ope ies
:
a)
777,Q
(X)
=
1
and
X(1)
-
0
(mod
4)
.
b)
Hasse( )
=
(A,,
-1)
in
B (Q)
.
c)
E e y
G-in a ian
synáme ic non-singula
bilinea
o m
g
o e
M
sa is ies
Hasse(g)
=
(A,-1)
in
B (Q)
.
l
in
addi ion
o
assuming
ha
G
has
a
ce ain
elemen
o
o de
4
we
ass une
u he
ha
G
con ains
a
ce ain
copy
o
he
qua e nion
g oup
o
o de
8,
hen
all
Hasse
in a ian s
a e
i ial
.
100
4

A
.
TuRULL
Theo em
C
.
Le
G
be
a
ini e
g oup
and
le
Q
be a
subg oup o
G
isomo phic
o he
qua e nion
g7-oup
o
o de
8 and
such
ha
Z(Q)
C
Z(G)
.
Le
F
be a eal
ield
and X
be
an
i educible
cha ac e
o
G
wi h
alues
in
F
.
Le
M
be
an
FG-module
a o ding
MF(X)X
and
assume
ha
Z(Q)
ac s
non- i ially
on
M
.
Le
be
a
G-in a ian non-ze o
symme ic
bilinea
o m
on
UVI
.
Then
he
ollowing
hold
.
1)
de ( )
=
1
up
lo
squa es
in
F*
.
2)
Hasse( )
=
1
in
B (F)
.
This
heo em
has
a
consequence
ha
is
analogous
o
Co olla y
1,
namely
ha
wi h
he
hypo hesis
o
Theo em
C
i
m,>,(X)
=
2
and
X(1)
-
2
(mod
4)
hen
7np(X)
=
1
o
all
odd
p io es
p
.
Howe e ,
his
can
also
be
p o ed
easily
by no ing ha
mp(X)j2 and
he
i educible
ai h ul
cha ac e
o
Q
has
odd
mul iplici y
in
X1Q
.
Mo e
impo an ly,
i
should
be
no ed
ha
Theo em
C
applies
o ai h-
ul
X,
whene e
G-
SL(2,
q),
he
special
linea
g oup
o
dimension
2
o e
he
ield
o
q
elemen s,
G
-
A
n
(n
>_
4),
he
double
co e
o
an
al e na ing
g oup,
o
G-
5,,
(n
>_
4)
some
double
co e
o
a
symme ic
g oup
.
In
pa icula ,
we
ha e
he
ollowing
.
Co olla y
2
.
Le
G
be
isomo phic
o
SL(2,
q)
o
q odd,
o
some
double
co e
o
S
n
o
A
n
o
n
>_
4
.
Le
M
be
an
FG-i educible
ai h ul
G-module,
whe e
F
is
a
eal
ield
and
he
cha ac e
o
M
is
a
mul iple
o
some
i educible
cha ac e
o
G
.
Then
he e
is
a
symme ic
G-in a ian
bilinea
o m
on
1V1
unde
which
M
has
an
o hono mal
basis
.
1
.
P elimina y
Lemmas
In his
sec ion
we
e iew
some
esul s
ha
we
need
o
ou
p oo s
.
Unless
o he wise
s a ed
ou
ec o spaces
and
o ms
a e
o e
a
ixed
bu
a bi a y
eal
numbe
ield
F
.
Recall
ha
a
hype bolic
plane
is
a wo
dimensional
ec o
space
wi h
a
non-singula
symme ic
bilinea
o m
ha
has
a
non-ze o
ec o
whose
p oduc
wi h
i sel
is
ze o
.
Lemma
1
.1
.
Le ,
V
a ad,
W
be
ec o spaces
and
le
and
g
be
non-
singula
symme ic
bilinea
o ms
on
V
and
W
espec i ely
.
Then
he
ollowing
hold
.
a)
Hasse(
L
g)
=
Hasse( ) Hasse(g)
(de (
),
de (g)
),
whe e
1
g
is
he
o hogonal
suin
o
and
g
.
b)
Fo
e e y
A

e

F,

Hasse(A
)

=

Hasse( )(A,
(-1)
n
2
1
d"-1)
whe e
n=
di nF(M) and d
=
de (
)
.
P oo
.
See
Theo e n
B
o
[3]
.
BILINGAR
ORNIS

1005
c)
I
W
is
a
hype bolic
plan(
hen
de (g)
=
-1
and
Hasse(g)
=
:1
.
P oo
::
These
ac s
a e
easily
e i ied
.
They
can
also
be
ound
in
[1]
;
o
example
b)
appea s
as
Fxe cise
8
on
page
140
.
Le nma
1 .2
.
Le
C
be
a
cyclic
g oup
o o de
4,
and
M
be
an
i e-
ducible
FC-module
ai h ul
o7
-
C
.
Le
be
a non-singula
C-in aHan
sy7nnie ic
bilinea
o o
on
Al
.
Then
de ( )
=
1
and
Hasse( )
o
sume
AE
F*
.
P oo
.
Since
F
is
eal,
he
cha ac e
a o ded
by
Al
will
be
he
sum
o
he
wo
ai h ul
i educible
cha ac e s
o
C
.
Any
C-in a ian
bilinea
o m
g
en
Al
®
O
sa is ies
g(eL,
el)
=
g(iei,
¡el)
=-
g(ei
;
el)
=
0
=
g(e2,
e2),
whe e
e
l
,
e2
E
M
®
O
a e
eigen ec o s
o
a
gene a o
o
C
co espond-
ing o
i
and
-i
espec i ely
.
I
ollows
ha
he
O-space
o
C-in a ian
symme ic
bilinea o os
on
AJO
O
is
one
dimensional
.
This,
in
u n,
implies
ha
he
F-space
o
C-in a ian
symme ic
bilinea
o os
en
111
is
one
dimensional
.
Hence,
by
Le n na
1 .1,
b),
and
he
well
known
ac s
ha
(u,
1)
=
1
a ld
(a,
0)
(c ',
3=
(CYLY~,,~)
in
13
(F)
o
cY,
c ',
E
F
*
,
i is
enough
o
show
ha
Lenuna
1 .2
holds
o
so ne
.
Le
W
be
he
one
dimensional
ai h ul
module
o
he
cyclic
sub-
g oup
o
o de
2 o
C
o e
F
.
Then
W
a o ds
a symme ic
in a i-
an
bilinea
o m
b
a ad
a
basis
ec o
e
such
ha
b(e,
e)
=
1
.
AY
is
iso no pliic
o
he
C-module
induced
om
W
;
and
i
ollows
ha
M
a o ds
a
C-in a ian
synune ic
bilinea
o o
and a
oasis
el,
e2
such
ha
(el,
eI)
=
(e2,
e2)
=
1
and
(el,
e2)
=
0
.

Ob iously
;
o
his
,
de ( )
=
1
and
Hasse( )
=
1
.
Hence,
he
Lemma
holds
.
Theo em
1 .3
.
Le
G
be
a
ini e
g oup,
F
some
eal
ield
and
X
some
i7- educible
cha ac e
o
G
wi h
alu,es
in
F
such
ha
m"
.(x)=?~
l
.
.
Le
M
be
an
FG-module
o o- ding
he
cha ac e
mF(X)h
and
be
so ne
G-in a ian
non-singula
symme ic
bilinea
o na
on Al
.
Then
he
ol-
loiaing
hold
.
1)
de ( )
=
1
(up
o
squa es
in
F*)
.
2)
Hasse( )
=
(-1,-1)[Y]
i
4 ~
;Y(
1
),
and
Hasse( )
=
1
i
4X(1)
.

100
6

A
.
TURULL
Lemma
1 .4
.
Le
X
be
an
i educible
cha ac e
o
some
ini e
g oup
G
and assume
ha
he
alues
o
X
a e
in
F
.
Suppose
[XI
=
(A,
-1)
in
B (F)
o
some
A E
F*
.
Then
m,,,,(X)
=
2
i
and
only
i
A
is
nega i e
.
Fu he mo e,
i
F=
Q,
hen
mp(X)
=
1
o
e e y
a ional
ini e
p ime
p
such
ha
p
-
1
(mod
4)
.
P oo
:
The
local
Schu
indices o
X
a e
1
o 2
depending
on
whe he
o
no he enso
o
(A,
-1)
wi h
he
comple ion
o
F
spli s
.
Fo
example,
m
<
,
(x)
=
1 i
and
only
i
(A
;-1)
=
1 in
B (IF8)
.
Howe e ,
he
la e
condi ion
holds
i
and
only
i
A
is
posi i e,
so
he
i s
asse ion
o
he
lemma
holds
.
I
p
is
any
ini e
p ime
and
F
=Q
hen
7
np(X)
=
1
i
and
only
i
(A,
-1)
=
1
in
B (Q,), whe e
Q,
is
he
ield
o
p-adic
numbe s
.
Now
(A,
-1)
=
1
in
B (Q
p
) i
and
only
i
Ax
2
-
y
2
=
z
2
has
a
solu ion
wi h
x,
y,
z
E
Q
7
,
and
z
7~
0
.

Suppose p
is
a
ini e
a ional
p ime and
p
-
1
(mod
4)
.
Then
p
is
he
sum
o
wo
a ional
squa es
.
So, in
his
case
we
may
assume
ha
p
is
no
in ol ed
in
he
p ime
ac o iza ion
o
A
.
Bu
hen
(A,
-1)
=
1
in
B (Q
p
)
by, o
example,
Exe cise
10 in p
.
186
o
[1]
.
This
comple es he
p oo
o
he
lemma
.
Lemma
1
.5
.
Le
n >
1
be
an
in ege ,
le
G=
S
n
be
he
symme ic
g oup
o
deg ee
n
.
Then
he e
is
an
absolu ely
i educible
QG-module
W
o
dimension
n
-
1
and
a
G-in a ian
symme ic
bilinea
o m
on
W
such
ha
de ( )
=n
up
lo
squa es
in
Q*
.
P oo
.
Le
N
be
he
na u al
pe mu a ion
module
o
S
n
o e
Q
.
Then
dimQ(N)
=
n
and
he
.pe mu a ion basis
el,
. .
.,
e n o
N
can
be
aken
o
be
an
o hono mal
basis
o
an
S
n
-in a ian
symme ic
bilinea
o a g
on
N
.
Then
de (g)
=
1
.
The
ec o
=
el
+
...
+
e n
is
S
n
-in a ian
and
g( ,
)
=n
.
The
space
1
o
ec o s
in
N
which
a e
o hogonal
o
is
an
S
n
-submodule
o
N
o
dimension
n
-
1
.
We
se
W
=
1
.
Tüen
N
=
W
1G
>
.
I
ollows
ha
i
we
se
o
be
he
es ic ion o g
o
W
hen de ( )
=
n,
since
de (
)n
=
1
up
o
squa es
in
Q*
.
Since
S
n ac s
doubly
ansi i ely
on
e
l
, . .
.,
e
n
,
W
is
absolu ely
i educible,
and
he
lemma
holds
.
Lemma
1
.6
.
le
G
be a
ini e
g oup
and
M
an
FG-module
.
Suppose
M
is
endowed
wi h
a
non-singula
G-in a ian
symme ic
bilinea
oTin
.
Then
we
can
w i e
M=M
.®M
I
®M
2
as
FG-modules
whe e
he ollowing
hold
:
1)
M
o
is
an
o hogonal
sum
o i educible
submodules
on
which
is
non-singula
.
BILINGAR
FORMS

1007
2)
is
o ally
iso opic
on
bo h
M
l
and
M
2
.
3)
As
quad a ic
spaces
M
=
M,,
1
(A11
+
M2)
and
A1
1
+
M2
is
he
o lioyonal
sum
o dime,
(M )
=
dim (Nl2)
copies
o
he
hype bolic
plane
.
P oo
::
Suppose
he
le nma
is
also,
and
pick
a
coun e example
wi h
dim (M)
as
small
as
possible
.
Le
N
be an
i educible
submodule
o
M
.
Suppose
is
non-singula
on
N1
.
Then
111
=
N
1N¡
and
Nl
is
a
G-submodule
on
which
is
non-singula
.
Hence,
by
he
minimali y
o
ou
coun e example,
he
le ima
holds
o
NI
L,
and
i
ollows
ha
i
also
holds
o
A4l,
a
con adic ion
.
The e o e,
we
mus
assu ne
ha
is
singula
on
N
1
and on
e e y
o he
i educible
G-sub odule
o
M
.
Since
is
G-in a ian ,
i
ollows
ha
is
o ally
iso opic
on
N
and on
e e y
o he
i educible
G-submodule
o
A1
.
In his
case
;
N
C_
N
iL
.
By
Maschke's
Theo em,
hese
is
al]
i educible
sub odule
N
2 o
NI such
ha
N2
n
Ni
=
0
.
Now
is
o ally
iso opic
on bo h
N
1
and
N2,
and
(Ni
+N2)n(Ni+N2)
1
=(Ni+N2)nNi
n
N2
=Ni
nN2=0,
so
is
non-singula
on
N
+
N
2
.
The
bilinea o a
p o ides
an
iso-
mo phis n
be ween
he
dual o
N
and
N2
-
N
+
N2/Ni
.
Hence
;
i
we
choose
e
l
, . .
.,
e,,
o
be
a
basis
o
NI,
we
can
hen
choose
o
N
2
he
co esponding
"dual"
basis
e*,
. .
.,
e*,
Le
.

dim1
:
(N1)
=
dimp(N
2
)
and
(e
;,
e~)
=
S
i
s
whe e
b2j
is
K onecke 's
del a
.
Hence,
as
quad a ic
spaces
,
he
<
e¡,
e?
>
a, e
iso no phic
o
he
hype bolic
plano
and
N
+
N2
=<
e1,
el
>l<
e2
:
e2
>1
.
...
1<
e,
e*
>
.
Hence,
induc ion applied
o
<
N
+
N2
>1
comple es
he
p oo
o
he
lemma
.
2
.
P oo s
o
he
main
esul s
P oo
o
Theo em
A
:
Le
C=<
x
>
be
he
subg oup
o
G
gene -
a ed
by x
.
Then
C
is
a
cyclic
g oup
o
o de
4
and
x
2
ixes
no
non-
ze o
ec o
in
A1
.
I
ollows
ha ,
as an
FC-module,
M
is
he
di ec
sum
o
ai h ul
i educible
FC-modules
.
Apply
Lemma
1.6
o
M
un-
de he
ac ion
o
C
.
Then,
as
a
quad a ic
space,
A1
is
he
o hogonal
sum
o
non-singula quad a ic
spaces
on
i educible
C-submodules
o
M
and
diMF
(AJII)
copies
o
he
hype bolic
plano,
whe e
M
1 is
some
FC-
sub odule
o
M
.
Since
F
is
eal,
dimF(1M
1 )
is
o en
and
i
ollows
om
Le nma
1 .1
ha
he
sum
o
all
hese
hype bolic
planes
o ms
a
subspace
wi h
de e minan
1
and
Hasse
in a ian
ei he
1 o
(-1,
-1)
in
B (F)
.
The
o he
o hogonal
summands
o
M
all
na e
de e minan
1
and
Hasse
in a ian
(A,
-1)
o
a ious
A
E F* by
Le nma
1
.2
.
Hence,
by
Lemma
100
8

A
.
TURULL
1
.1,
de ( )
=
1
up
o
squa es
in
F*
and
Hasse( )
is
a
p oduc
o
elemen s
o
he
o m
(A,
-1)
o
a ious
A
E
F*
.
Since
(A,
-1)
(p,
-1)
=
(AM,
;
-1)
in
B (F),
i
ollows
ha
Hasse( )
=
(A,
-1)
o
some
A
o
E
F*
.
I
is
posi i o
de ini e,
hen Hasse( )
is
i ial
o e
R, so
A
o
>
0
in
his
case
.
Hence
1)
and
2)
o
he
heo em
hold
.
Suppose
now
ha
dimF(M)
-
2(mod
4)
.
Then
by
Lemma
l
.l,
b),
Hasse(p
)
=
(A
.,
-1)(p,
-1)
=
(Aop,
-1)
.
I
ollows
ha
Hasse(p
)
_
(A,
-1)
o
e e y
A
E F*
i
we
se
p,
=
AoA
.
P oo
o
Co olla y
1
:
Since
F
D_
Q(X),
we
can
ake
M
o
be an
i -
educible
FG-module
a o ding he cha ac e
MF(X)X
.
Since
F
is
eal,
he e
is
a
posi i e
de ini e
G-in a ian
symme ic
bilinea o o
on
M
.
The
Hasse
in a ian
o
can
be
calcula ed
in
wo ways
.
On
he
one
hand,
Theo em
A
ells
us
ha
Hasse( )
=
(A,
-1)

in
B (F)
o
some
A
o
E
F*
.
On
he
o he
hand,
Theo em
1
.3
ells
us
ha
Hasse( )
=
(-1,-1)[X]
.
lld=V®W
.
Sol ing
o
[X]
we
ob ain
[X]
=
(
-A,
-1)
.
By
Lemma
1
.4,
-A,
is
neg-
a i e,
Since
moo(X)
=
2
.
Fu he mo e,
i
F
=
Q,
hen
m (X)
=
1
o
e e y
a ional
ini e
p ime p
such
ha
p
-
1
(mod
4),
by
Lemma
1 .4
.
Hence
Co olla y
1
holds
.
P oo
o
Theo em
B
:
Since
o
e e y
a
E
Q*,
(A, -1)
=
(a
2
~Xa,
-1),
we
assume
wi hou
loss
ha
a o
is
a
posi i e in ege
di isible
by
9
.
Fu -
he mo e,
(2,
-1)
=
1
in
B (Q),
so
we
u he
assume
wi hou
loss
ha
A
o
is
odd
.
Now
A
o
>
1
and
we
se
n
=
a o
and
G=
D8
xS,
o be
he
di ec
p oduc
o
he
dihed al
g oup
o
o de
8
and
he
sym ne -
ic
g oup
o deg ee
n
.
We
ake
x
o
be
any
elemen
o
o de
4 in
D8
.
Then
1
0
x
2
E
Z(G)
.
Le
V
be
a
quad a ic
F-space
o
dimension
2
wi h
o hono mal
basis
el,
e2
.
We
endow
V
wi h
he s uc u e
o a
D8-
module
by
le ing
wo
gene a o s
o
o de
2 o
D8
ac
on
V
as
linea
ans o ma ions
which
ha e
he
ollowing ma ices,
wi h
espec
o
he
basis
el
;
e2
.
I is
clea
ha
D8
s abilizes
he
quad a ic
o m
on
V
.
Le
W
be
he
S,,
module
gi en
by
Lemma
1 .5
.
Se
BILINEAI2
.
FORMS

1009
Then
AI
is
an
absolu ely
i educible
G-module
and
x 2
does
no
ac
i -
ially
on AI
.
Le
x
be
he cha ac e
a, o ded
by
M
and
be
he
sym-
me ic
bilinea o n
ob ained
by
enso ing
hose
o
V
and
14
7
.
Clea ly
ITIQ(X)
=
7
.
and
X(1)
=
2(A,
-
1)
-
0
(mod
4)
.
As
a
quad a ic
space
AI
-
W
-L
1 , 17,
so ha
Hasse( )
=
(de (W),
de (W))
=
(,
o
,,
o
)
by
Lemma
1 .1
and
Lemma
1
.5
.
Since
(,
o
,
A
j
=
-1)
in
B (Q),
his
shows
a)
and
b)
o
Theo em
B
.
Since
M
is
absolu ely
i educible
all
G-in a ian
bilinea
o ms
on AI
a e
mul iples
o
.
Since
de ( )
=
1
;
c)
ollows
om
Le nma
1 .1,
b)
.
P oo
o
Theo em
C
:
No ice
ha
as
F
is
eal,
Q
has
only
one
isomo -
phisin
class
o
ai h ul
i educible
.FQ-modules
and
hey
ha e dimension
4
.
Since
Z(Q)
ixes
no non-ze o
ec o
o
M,
M
is
a
su i
o
i educible
ai h ul
FQ-modules
.
Applying
Le n na
1
.
.6, i
ollows
ha
as
quad a ic
Q-modules
M
=
M,
1
(11
1
+
112)
whe e
M,,
is
he
o hogonal
sum
o
non-singula
i educible
FQ-modules
and
All,
+M2
is
an
FQ-module
and
as
o hogonal
space
i is
he
o hogonal
su i
o a
mul iple
o
4
copies
o
a
hype bolic
plane
.
I
ollows
ha
de (Al
+
AI2)
=
1
and
Hasse(M
I
+
M
2
)
=
1,
by
.Lemma
1 .1
.
Le
0be
he
ai h ul
i educible
complex
cha ac e
o
Q
.
I
is
well
known
ha
[0]
=
(-1,
-1)
in
B (F)
.
I
hen
ollows
om
Theo em
1
.3
ha
i
N
is
any
o
he
o hogonal
summands
o
AI,,,
hen
de (N)
=
1
and
Hasse(N)
=
(-1,
-1)(-1,
-1)
=
1
.
Hence,
by
Lemma
1 .1,
de ( )
=
1
and
Hasse( )
=
1,
as
desi ed
.
a
P oo
o
Co olla y
2
:
Le
o
be
a
sym ne ic
bilinea o n
on
M
unde
which
M
admi s
an
o hono mal
basis
.
Since
F
is
a eal
ield
o
is
posi i e
de ini e,
a.nd in
ac ,
i is
posi i e
de ini e
unde
each
imbedding
o
F
in o
R
.
De ine
:AllxA/1-->F
( ,
w)
=
1
~
sé
o(9 ,9w)
o
, .w
E
M
.
Then
is
a
G-in a ian
symme ic
bilinea
o m
.
Fu he -
mo e
;
is
posi i e
de ini e
unde
e e y
imbedding
o
F
in o
R
.
Fu he -
mo e
;
by
Theo ein
C,
de ( )
=
1
and
Hasse( )
=
1
.
I
ollows
ha
AI
admi s
an
o hono mal
basis
unde
.
This
comple es he
p oo
o
he
Co olla y
.