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,
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