continuous automorphism of C, then Va is again a representation of WF, and pa
an additive character, but eL(Va, pa) is not in general equal to SL(K fY (not is
£L(0, V«, {;«) = eL(0, F, j))«). The trouble is that the absolute value (in (3.6.3)) may
not be preserved by a, and/or that the self-dual measure dx^ in (3.6.1) may involve
s/p, and hence may not be preserved by a. If one does wish to eliminate the measure
dx, it is probably preferable to define, say,
(3.6.10) £X(F, if,) = eD(V, f9 dxx\
where dxx is the measure for which gets measure 1 in the nonarchimedean case,
and is the measure described in (3.2.4) and (3.2.5) in the archimedean case. This
convention has the minor disadvantage that the e(V) in the global functional equa-
tion is not equal to the product of the local e\(Vm ^„)'s, but is, rather, arl times that
product, where a is the square root of the discriminant for a number field, and is
qg-\ for
function field of genus g with q elements in its constant field. But the
ej(F, (j)) has the advantage that in the nonarchimedean case we do have
e\(Vff9 ])a)
£i(F, ])Y for all automorphisms a of C This is clear, by unicity (2.3.3) and the
e i
, 0) = %{c)q»^ 2 X(")(v)
which follows from ( and ( where the notation is explained. Thus
in the nonarchimedean case we can define, for Fand ]) over any field E of charac-
teristic 0 (an open subgroup of I acting trivially on F, and ff trivial on some icn®X
an $i(V, j)) E*, in a unique way such that £i(Fa, pa) = ei(F, $)a for any homo-
morphism a: E -* E' and such that ex is the old ej, given by (3.6.10), when E =
C. So defined, ei( F£, ^ TrE/ F) is inductive in degree 0 (2.3.2) for every field of scalars
E of characteristic 0, and e^F, /;) will be given by (3.6.11) if F corresponds to a
quasi-character #: F* -• E*.
In writing these notes I was tempted to shorten things a bit by using only ei(F, $)
instead of eD{V, fi, dx% but decided against it because (1) the e^-system avoids all
choices and is the most general and flexible—any other system, like eL or $\ can be
immediately described as a special case of eD;(2) the dependence of e on dx shows
"why" s is inductive only in degree 0, and (3) in case our local field F is nonarchi-
medean, the e^-system, like the eu works over any field E of characteristic 0, as
soon as one defines the notion of Haar measure on F with values in E (cf. [D3,
4. The Weil-Deiigne group, A-adic representations, £-funetions of motives. The
representations considered in §3 are just the beginning of the story. Those of Galois
type are effective motives of degree 0—which Deligne calls Artin motives in his
article [D6, §6] in these PROCEEDINGS—with coefficients in C We cannot discuss
the notion of motive here (cf., e.g., [Dl] and [D6] for this) but we do want to discuss
the way in which L-functions and e-functions are attached to motives of any degree.
Only very special motives of degree # 0 correspond to the representations of WF
considered in §3, namely, those of type A$, i.e., those which, after a finite extension
E/Fy correspond to direct sums of Hecke characters of type A0 over JE". (A candidate
for a "motivic Galois group" for these is constructed by Langlands in these
Previous Page Next Page