94 A.A LETTER FROM M. ROST TO J-P. SERRE
extension ove r whic h ip is isotropic, on e see s that a^f(ip) = bj((p) an d {(—a))~f(ip) =
«-&»7M-
Hence thes e consideration s giv e ne w definition s o f a , f3
1
7 withou t chain -
equivalence o f diagonalizations .
In an y cas e /3((f) can b e viewed a s a subspace o f A(f o f half dimensio n whic h i s
related t o th e *-operator,
c
o r wit h th e self-dualit y A : AV^ g AV A
nV.
Thi s ca n
be mad e mor e precis e i n th e cas e n = 0 (mo d 4 ) a s follows .
Identify AV wit h the Cliffor d algebr a C(ip) in the natural way by v\ A Avk 1—•
vi . . .Vk fo r pairwis e orthogona l vi. The n th e *-operato r o n AV i s give n b y th e
action o f Z = F 0 A nV = C(A n(p) o n C(p) sa y fro m th e left . I f n = 0 (mo d 4 )
then Z ~F[t]/(t 2 - 5). Le t
E = {x e AV ®
F
Z \ (1®
UJ)X
=
(UJ®
l)x fo r u G A nV}
be th e "eigenspac e o f th e sta r operator" . Considerin g /?(/? ) a s a n elemen t o f
W(F)/((5)) C W(Z) on e finds
(2).p(p) = [Ap®
F
Z\E]e W(Z).
Hence for n ^ 2 (mo d 4 ) ther e i s a natural constructio n o f /?(^), which implies e.g .
that / ? can als o be defined fo r quadrati c form s ove r schemes. I don't kno w whethe r
this i s possible fo r n = 2 (mo d 4) , n 2.
II) Relatio n wit h trac e form s o f algebra s
If y/^1 G F* on e ma y interpre t a , /? , 7 als o a s trac e form s o f algebras : Le t t r
be th e trac e o f th e eve n Cliffor d algebr a Co (if) ove r it s cente r an d le t n : C(ip) =
Co((p) 0 Ci(ip) * Co((p) be th e projection . The n
(1) I f n i s odd, the n /?(/? ) is given b y th e trac e for m Co(^f) —• F , £ •—• tr(x 2).
(2) I f n i s even, the n 7(v? ) i s given by the trac e for m Co(p) Z , # 1— tr(x 2).
(3) I f n = 0 (mo d 4 ) pu t fi: C(^ ) Z , x - » tr7r(xxt). The n 0 an d 0|c
o
(y)
are form s ove r Z representin g (3((p) an d 7(/? ) respectively .
Similar remark s hol d fo r a(p).
Hence i f y/^1 G F * th e invariant s ar e closel y relate d wit h th e Hasse-Wit t
invariant. O n th e othe r extreme , i f F = R the n (th e cohomologica l version s of ) a ,
/?, 7 evaluate d o n th e anisotropi c par t o f a for m giv e a complet e se t o f invariants .
Maybe th e assumptio n y/^1 G F* i s not ver y goo d i n considerin g a , /? , 7.
III) Cohomologica l interpretatio n o f th e trac e for m
Suppose A/^ T G F*. The n x
n
= 0 fo r x G K^F/2 an d n 1, bu t ther e ar e
operations
T
n
: X
2
M
F / 2 ^ K
2
^ F / 2
c The *-operato r i s define d a s follows , cf . [KMRT98 , 10.22]: Le t V b e a k- vector spac e o f
dimension n endowe d wit h a nondegenerat e quadrati c for m q o f discriminan t 1. Fi x a nonzer o
element e in A n V suc h tha t (A nq)(e) = 1; the wedg e produc t determine s a nondegenerat e pairin g
A%V x A n~zV—• k. Th e for m q give s a nondegenerat e for m /\ %q o n A lV, henc e w e hav e A %V =
(A*V)*. Combinin g thes e tw o fact s give s a n isomorphis m A %V = A n~lV. Sinc e (A nq)(e) 1,
this i s a n isometr y o f AV; i t i s know n a s th e *-operato r an d i s o f muc h us e i n Hodg e theory .
When n = 2r a i s even, th e automorphis m o f A raV tha t w e get ha s squar e 1 if m i s even, an d
square 1 if m i s odd. I n the first cas e (i.e. , n divisibl e by 4) the eigenspaces of the *-automorphis m
give a n orthogona l decompositio n o f A
n
/
2
V a s tw o subspace s o f th e sam e dimension ; th e corre -
sponding quadrati c form s ar e isomorphi c t o th e for m £(q) o f 27.19.
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