22

J. TATE

sets up a bijection between the set ofX-adic representations (px, Vx) of WF and the set

of representations (p, N, V) of WF over Ex. The corresponding bijection between iso-

morphism classes of each is independent of the choice oftl and 0.

To show that every px is of this form one uses

(4.2.2)

COROLLARY (GROTHENDIECK).

Let (px, Vx) be a l-adic representation of WF.

There exists a nilpotent endomorphism NofVxsuch that px(a) = exp(ti(a)N)for a

in an open subgroup of J.

A proof of the corollary can be found in the appendix of [ST]. Here is a sketch.

Since /is compact, px(I) stabilizes a "lattice" L in Vx. Replacing F by a finite exten-

sion we can assume that px(I) fixes L (mod

I2).

Then px{I) is a pro-/-group, so is a

homomorphic image of tt{I\ since Ker tt is prime to /. Choose c e Qt such that

cti(I) = Z/. Then there is an a € GL(F^) fixing L (mod I2) such that

Px(a) =

actl{ff)

= Qxp(t{(a)N)

for all a e /, where N = c log a. Conjugating by px(0) we find

Px($)Npx(0)~l

=

q~lN.

Thus the set of eigenvalues of N is stable under multiplication by

q~l.

Since q

is not a root of unity in characteristic 0, it follows that the only eigenvalue of N is

zero, i.e., JVis nilpotent.

(4.2.3)

COROLLARY.

If Vx is a semisimple l-adic representation of WF then some

open subgroup of I acts trivially on Vx, so Vx can be viewed as an "ordinary" represen-

tation of WF.

For any Vx the kernel of N is stable under WF because px(w)Npi(w)~l = || w\\N.

So if Vx is irreducible, then N=0, and the statement follows from (4.2.2). A semi-

simple Vx is a direct sum of irreducible subrepresentations.

(4.2.4) In view of (4.2.3), e{Vx) and a (Vx) have meaning if Vx is semisimple. For

arbitrary VXi if fe, Vx) and (p, N, V) correspond as in (4.2.1), we define the L- and

e-factors associated to Vx to be those associated to V. These can be expressed di-

rectly in terms of Vx as follows:

Z(F, 0 = det(l -0\V{) = Z(Vht),

a(V) = a(Vf) + dim(Ff)' - dim Vx = a(Vxl

det( - 0\(VfY)

and

e(V9 t) = e(Vx)t°n = e(Vx, t)9

where Vf is the semisimplification of Vx in the ordinary sense. One can define a

"0-semisimplification,,

of VX, analogous to that of V (4.1.3). The quantities on the

right do not change if we replace Vx by its 0-semisimplification, but they are not

additive in Vx, because V\ is not.

(4.2.4) Motives. Suppose now E is a finite extension of Q. Let M be a motive with

complex multiplication by E, defined over our nonarchimedean local field F ([Dl],

[D6]). Let n be the rank of M. Attached to M will be /-adic representations Ht{M),