12

J. TATE

IndE/E'

R{WE,)

is commutative for finite separable extensions E/E'/F. We say X is inductive in degree

0 over Fif the same is true with R replaced by R°.

(2.3.3)

REMARK.

By (2.3.1) a X which is inductive over F, or even only inductive

in degree 0, is uniquely determined by its value on quasi-characters% of WE(i.e.9

of CE\ for all finite separable E/F. In [D3, §1.9] there is a discussion, for finite

groups, of the relations a function A of characters of subgroups must satisfy in order

that it extend to an inductive function of representations.

(2.3.4)

EXAMPLE.

Let a € CF. Put

X(V) = (det V){rE{a)) for Ve M(WE).

Then X is inductive in degree 0 over F. This follows from property (W3) of Weil

groups and the rule.

det(Ind V) = (det V) • transfer, for V virtual of degree 0

(cf. [D3, §1]).

(2.3.5)

EXAMPLE.

Suppose v is a place of a global field F, and X is an inductive

function over F„. If we put for each finite separable E/F and each Ve M{WE)

UV) = 11 X(VW)

wplace of E; w\v

we obtain an inductive function Xv over F. If X is only inductive in degree 0, then

Xv is inductive in degree 0.

Indeed, by a standard formula for the result of inducing from a subgroup and

restricting to a different subgroup we have

QndE/FV)9^®lndEm/F9VW9

w\v

because if w0 is one place of E over v, then the map a »-* aw0 puts the set of double

cosets WE\ WFj WFv in bijection with the set of all such places, and for each a we can

identify W^with(jr WFva~-1) D WE.

3. L-series, functional equations, local constants. The £-functions considered in

this section are those associated by Weil [Wl] to representations of Weil groups.

They include as special cases the "abelian" L-series of Hecke, made with "Grds-

sencharakteren" (= quasi-character of CF\ and the "nonabelian" L-functions of

Artin, made with representations of Galois groups. Our discussion follows quite

closely that of [D3, §§3,4, 5] which we are just copying in many places.

(3.1) Local abelian L-functions. Let F be a local field.

For a quasi-character % of F* one defines L(y) e C* U {oo} as follows.

(3.1.1) F « JR. For x the embedding of F in C and N = 0 or 1,