24
J. TATE
of 0. Let (p: XQ - XQ be the Frobenius morphism, and a: k ~+ k the Frobenius au-
tomorphism. The composition p x a acts on X0 = Z0 x k by fixing points and by
mapping/^ fv in the structure sheaf. This map induces (a morphism canonically
isomorphic to) the identity on the site (X0)et, so the action of the Frobenius mor-
phism (p on
Hr(X0,
Qi) is the same as that of
a"1,
which is the one corresponding to
our 0 under the isomorphism (*). That is why Deligne calls 0 the geometric Fro-
benius.
Deligne [D4] has proved Weil's conjecture, that the characteristic polynomial of
p acting on Hr(X0, Qi) has coefficients in Z, is independent of /, and that its com-
plex roots have absolute value
qr/z.
From the independence of / it follows in this
case of good reduction that the 0-semisimplifications of the Ht(My$ form a com-
patible system; and the ///(M)'s are known to be 0-semisimple for r = l.
It is natural to say that a motive M over F has good reduction, or is unramified if
and only if Ht(M) = Ht(My, i.e., if V(M) = V(MyN. In case M = HX{A\ A an
abelian variety, this is equivalent to A having good reduction (criterion of Neron-
Ogg-Shafaryevitch in [ST]).
Similarly we say M has potential good reduction o N = 0, and M has semi-
stable reduction if V(M) = V(My. Clearly this latter can always be achieved by a
finite extension of the ground field.
(4.4) F archimedean. Let now M, E, n, m be as in (4.2.4), but take F to be archi-
medean, instead of nonarchimedean. Let z: F C be the embedding of F in C
if Fis real, or one of the two isomorphisms of Fon Cif Fis complex. Such a z gives
us a motive Mz over C and Mz has a "Betti realization" HB{MZ) which is an «-
dimensional vector space over Q whose complexification HB(MZ) ® C = ® HP*{MZ)
is doubly graded in such a way that the map 1 ® c (c = complex conjugation)
takes H* to H*P. (For example, if M = Hr(X) as in (4.3), then HB(MZ) =
Hr(X™,
Q), where X™ is the complex analytic variety underlying the scheme
X x
FtZ
C, and the complexification of this space,
Hr(Xln,
C), is doubly graded by
Hodge theory.)
Let z = c0z:F-~*Cbe the map conjugate to z. By transport of structure, there is
an isomorphism
T: HB(MZ)
-* HB(MZ) such that t ® c preserves the bigrading on
the complexifications; hence v ® 1 carries HP^(MZ) onto i/^(AQ. The field E of
complex multiplications acts on HB(MZ) preserving the bigradation on the com-
plexification, and t is an E-homomorphism. Let a: F-» C. Putting Vz(Mff) =
HB{Mt) ®BtffC we obtain a bigraded complex vector space of dimension m and
a linear isomorphism r ® 1: Vz(Mff) ^(M,) taking K|» to Vj*.
There is a natural action of the Weil group WF on these spaces as follows:
F complex, z: F « C an isomorphism, WF = F*, and WF acts on V^ by scalar
multiplication via the character z~P{z)~q- Clearly, r ® 1 is JFF-equivariant, so the
two representations Vz(Mff) and Vz(Ma) are isomorphic. We let V(Mff) denote their
isomorphism class.
F real, z ~ z: F -* C is the embedding, and WF = C* [} jC*. This time Mz =
M2, so we have only one space, Vz(Ma) = Vz{Ma), and r ® 1 is an automorphism
of it. The action of WF on it is as follows:
u e C* acts as multiplication by u~P(u)~~^ on V**.
j acts as iP+i(v ® 1) on Vp.
Again, let V(Mff) denote the equivalence class of this representation.
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