[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
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.
1 1
Motive Mv over Fv
with ex. multn. by E
3°(4 8 4) t h e 0 r y Ji-rchi
Hodge structure Hdg(AQ
over Fv with ex. multn. by E
System (HX(MV)) of jl-adic
representations of GFp
v nonarchimedean;
cf. (4.2)
(cf. (4.4))
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
||$ || =
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
p e c ^ ,
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