NUMBER THEORETIC BACKGROUND

23

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