NUMBER THEORETIC BACKGROUND
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vector spaces of dimension n over Ql on which GF acts continuously, one for each
/ T* char(F). The field E will act on these, and for each / we get a decomposition
H{(M) = @X\i
HJHM\
where for each place X of E above /, we put HX{M) =
Ex ®E®Qi Hi(M), a vector space of dimension m over E, where m is the rank of M
over E, given by n = mfl?: (?].
For each / ^ /? and each A above / let H'x(M) be the representation of W'F over Isa
corresponding to Hx(M) by (4.2.1). If our motive M lives up to expectations, the
system of A-adic representations HX(M) will be compatible over E in the sense that
the system H[{M) is compatible over E in the following naive sense: for any two
finite places X, pL of E not over/? and every commutative diagram
the m-dimensional representations of WF over C, H[(M) ®El C and H'M(M) ®E/l C,
are isomorphic (or at the very least, have isomorphic 0-semisimplifications).
If so, then the isomorphism class of (the 0-semisimpliflcations of) these representa-
tions depends only on the embedding E a Cm the diagram above.
We denote this isomorphism class by V(M0), where a denotes the embedding of
E in C and Ma = M ®E, a C is the motive of rank m with coefficients in C deduced
from the original M, the action of E on it, and the embedding.*? of Em C, cf. [D6,
2.1]. Associated to V(Mff) as explained in (4.1.6) are the local quantities a, L, and
s which we shall denote by L(Mffi s), etc.
(4.3) Reduction. Let r be an integer ^ 0 and X a projective nonsingular variety
over F. In this paragraph we shall restrict our attention to the special motive M =
Hr(X)
given by the r-dimensional cohomology of X, and we shall ignore any com-
plex multiplication. For the moment F can be any field. Put X = X x
F
F, the
scheme obtained by extending scalars from F to F. For each prime / ^ char(F)the
/-adic etale cohomology group Hr(Xet, Qi) is defined, and gives an /-adic representa-
tion of GF = Gal(F/F) (by functoriality, GF acting on X through F). In the nota-
tion of the previous paragraph we have now E Q.,1 /, Ht{M)
Hr(Xet,
£?,).
I do not know to what extent the compatibility of the Ht{My% is known (assuming
now again that F is local nonarchimedean), but the compatibility at least of their
0-semisimplifications is known in one very important case^—that of
(4.3.1) Good reduction. Let (9 be the ring of integers in F, and k = OjicOtht re-
sidue field. The scheme X is said to have good reduction if there exists a scheme
X projective and smooth over 0 such that X X x
G
F. Choosing such an X, one
calls X x 0 k the reduction of X. Let us denote this reduction by X0. Putting X0 =
X0 x
k
k, where k is the residue field of F, the base-change theorem gives a canonical
isomorphism
(*) H£M) = Hr& Qd * H'(XQ9 Qd
compatible with the action of the Galois groups. Hence Ht{M) is unramified, i.e.,
fixed by /, and the structure of Ht(M) as representation of WF is given by the action
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