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),
Previous Page Next Page