NUMBER THEORETIC BACKGROUND

19

PROCEEDINGS

[L3].) The simplest motives not of this type are those given by elliptic

curves with no complex multiplication; their L-functions are the "Hasse-Weil

zeta-functions" which are not expressible in terms of Hecke's L-functions.

The procedure for attaching L-functions to motives in the form given it by

Deligne [D3], [D6] can be outlined schematically as follows:

F is a global field.

v is a place of F.

E is a field of finite degree over Q.

X runs through the finite places of E whose residue characteristic is prime to

charCF).

a is an embedding of E in C.

Motive M over F

with ex. multn. by E

System (HX(M)) of A-adic

representations of GF

gd. fld.

extension

1 1

/-adic

coho.

JI

restriction

Motive Mv over Fv

with ex. multn. by E

3°(4 8 4) t h e 0 r y Ji-rchi

m

edean

Hodge structure Hdg(AQ

over Fv with ex. multn. by E

System (HX(MV)) of jl-adic

representations of GFp

a

v nonarchimedean;

cf. (4.2)

(cf. (4.4))

it

Equiv. class V(Mvff) of repns.

over C of the Weil group WFv

cf. (3.3), (3.4) J|

L- (i.e. T7-) and e-factors

at the archimedean place v

Equiv. class V(MVtff) of

repns. of the Weil-Deligne

group WFp over C, invariant

under Aut(C/E)

cf. (4.1)

L- and e-factors at the

nonarchimedean place v

In the next sections we discuss some of the steps and concepts indicated in the

above chart. We begin with the Weil-Deligne group. This is a group scheme over

Q, but what counts, its points in and representations over fields of characteristic

0, can be described naively with no reference to schemes.

(4*.l) The Weil-Deligne group and its representations. Let Fbe a nonarchimedean

local field and let F, GF = Gal(F/F), WF (Weil group), and / (inertia group) have

their usual meaning. For w e WFJ let ||w|| denote the power of q to which w raises

elements of the residue field, as in (1.4.6). Thus we have ||w|| = 1 for we I, and

||$ || =

q~l

for a geometric Frobenius element 0. We view WF as a group scheme

over Q as follows: for each open normal subgroup / of /, we view WFjJ as a

"discrete" scheme, and we put WF = proj lim (WF/J), the limit taken over all J.

In other words, we have

^

F =

j j 0 « /

=

{ j

S

p e c ^ ,