NUMBER THEORETIC BACKGROUND

15

of F an additive Haar measure JLCE on E, then the function which associates with each

such E and each Ve M(WE) the number e(V, (ft • Tr^/^, /uE) is inductive in degree 0

over F in the sense of (23.2).

The unicity of such an e is clear, by (2.3.1); the problem is existence. The experi-

ence of Dwork and Langlands indicates that the local proof of existence, based on

showing that the e(%, (ft, dx) satisfy the necessary relations, is too involved to publish

completely. Deligne found a relatively short proof (see [D3, §4]; possibly also

[T2]). It has two main ingredients, one global, one local: (1) the existence of a

global e(V) coming from the global functional equation for L(V) (cf. (3.5) below),

and (2) the fact that if F is local nonarchimedean and a a wildly ramified quasi-

character of F*, there is an element y — y{a, (ft) in F* such that for all quasi-charac-

ters x of F* with a(%) ^ ia(a), we have £(%a, (ft, dx) = x~l(y)e(a, (ft, dx), a rather

harmless function of #.

Granting the existence of e(V, (ft, dx) the following properties of it are easy con-

sequences of the corresponding properties of e(%, (ft, dx), via inductivity in degree

0 and (2.3.1).

(3.4.2) s is additive in V, so makes sense for V virtual.

(3.4.3) e(V, (ft, rdx) =

rdimV

e(V, (ft, dx), for r 0. In particular, for V virtual of

degree 0, e(V, (ft, dx) = e(V, (ft) is independent ofdx.

(3.4.4) e(V, (fta, dx) = (det V)(a) \\a\\~dimV e(V, (ft, dx), for aeF* (cf. (2.3.4)).

(3.4.5) e(Vcos, (ft, dx) = e(V, (ft, dx)f(V)-d((ft)-sdimV, where:

d((ft) = qF(^ in the nonarchimedean case and is characterized in the archimedean

case by the fact that 5((fta) = Hall"15((ft), and 8((ft) = 1 for (ft as in (3.2.4) and (3.2.5).

f(V) = 1 in the archimedean case, and = q^, the absolute norm of the Artin

conductor of V'm the nonarchimedean case. This/can be characterized as the uni-

que function inductive in degree 0 such that/fy) =

qaF{t)

for quasi-characters #. For

the well-known explicit formula for a(V) in terms of higher ramification groups,

see [SI] or [D3, (4.5)].

(3.4.6) Suppose F nonarchimedean, Wunramified. Then

e(V® W, (ft, dx) = e(V, (ft, dx)dimW • det W(7ca^v)+dimVn^).

(3.4.7) Let V* denote the dual of Fand dx' the Haar measure dual to dx relative

to (ft. Then

e(V,(l9dx)e(V*al,(I{--x)9dx?) = 1.

In particular

| e( V, (ft, dx)

|2

= /(V)

(8((ft)dx/dx')dim v,

if V* = V,

i.e., if V is unitary.

(3.4.8) If E/F is a finite separable extension, VE a virtual representation of degree

0 of WE and VF the induced representation of WF, then e(VF, (ft) = e(VE, (ft o Tr).

(3.5) Global L-functions, functional equations. Let F b e a global field, (ft a non-

trivial additive character of AF/F, and dx the Haar measure on AF such that

\AF(Fdx = 1 (Tamagawa measure). Call (ftv the local component of (ft at a place v,

and let dx = T\vdxvbe any factorization of dx into a product of local measures such

that the ring of integers in Fv gets measure 1 for almost all v.