32. TRAC E FORM S WIT H VANISHIN G STIEFEL-WHITNE Y CLASSE S 81
(Note tha t (2b ) implie s tha t i s a produc t o f fields whos e degree s ove r
k ar e power s o f 2. )
Here also , wha t w e hav e don e show s tha t thi s i s tru e fo r n 8 .
One ma y b e eve n mor e optimistic , an d gues s tha t ther e i s a highe r ran k
version o f Mestre' s theor y [Me s 90] whic h change s E int o b y a trace -
form preservin g deformation . Thi s i s know n fo r n 4 , bu t no t (yet ) fo r
n = 4 .
(3) Wha t abou t n 8? Som e necessar y condition s fo r a quadrati c for m q o f
rank 8 t o b e a trac e for m are :
(i) q 9* (2) -((1) 0 Q) fo r som e Q o f ran k 7 , se e 31.7.
(ii) Wi(Q) = 0 fo r i 5 , se e 25.6 .
(hi) X l(Q - 3 ) = 0 fo r i 5 , se e 29.4 .
(iv) I f d d(q) = d(Q), th e for m Q 0 (d) represent s 1.
To se e tha t (iv ) i s necessary , on e make s a n odd-degre e extensio n s o tha t
the Galoi s grou p o f k i s a 2-group . I n tha t case , w e hav e a filtration
E = E% D E4 D E2 D Ei = k
1
wher e th e subscrip t i s th e ran k an d
each algebr a i s etal e o f ran k 2 ove r th e nex t one . W e ma y writ e E2 a s
a quadrati c algebr a k[X]/(X 2 a). Henc e q contain s (2)-(l,a) , i.e. , Q
contains (a) ; writ e Q a s (a) 0 Q f. Th e discriminan t d i s a nor m o f a n
element o f E
2
(namely , th e nor m o f th e discriminan t o f E/E
2
)j henc e i s
such tha t
(a)-(d) = 0 i n H 2(k),
which i s equivalent t o (a , d) = (1, ad). W e then hav e Q(B(d) = Q ; 0(a , d) =
Q' 0 (1, ad), whic h show s tha t Q 0 (d) represent s 1.
It i s doubtfu l tha t thes e condition s ar e sufficient .
32. Trac e form s wit h vanishin g Stiefel-Whitne y classe s
In thi s section , w e conside r etal e algebra s E o f ran k n whic h hav e th e sam e
trace for m a s th e spli t algebr a k x x k, i.e. , suc h tha t qE = ( 1 , . . . , 1).
A necessar y conditio n fo r thi s i s th e vanishin g o f th e Stiefel-Whitne y classe s
^I(QE) r a U * 0 . Thi s i s sufficien t fo r n 6 . Mor e precisely :
P R O P O S I T I ON 32.1. Ifn6 and Wi(q
E
) = 0fori = 1,2, then q
E
= ( 1 , . . . , 1).
P R O O F . I f n = 5 , b y 31.6 w e hav e q
E
^ ( 1 , 1 ) 0/ wit h rank(/ ) = 3 . Th e for m
/ ha s th e sam e w\ an d w
2
a s th e uni t for m (1,1,1) , henc e i s isomorphi c t o it . Thi s
shows tha t q
E
^ (1,1,1,1,1).
The cas e n 5 follow s b y replacin g E by kx---xkxE, s o tha t i t become s
of ran k 5 .
32.2. T H E CAS E n = 6 . W e no w assum e tha t n = 6 . (Th e cas e n 7 is similar ,
see 32.28.) Not e that , i n that case , the propertie s Wi(qE) = 0 and w
2
(qE) = 0 impl y
that ws(qE) 0 sinc e W3 = w\ -w
2
, an d the y als o impl y Wi(qE) = 0 fo r i 3 b y
Th. 25.13. Henc e th e vanishin g o f th e Wi(qE) fo r i 0 ca n b e restate d as :
(32.3) Wi(q
E
) = 0 an d w
2
(qE) = 0 ,
or equivalent s (cf . Th . 25.10):
(32.4) wf\E)=0 fo r i = 1,2.
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