8 I. HECK E L-FUNCTIONS

v. This can happen only if \V(BV) is close to 1 for all v where Bv ^ Hv and

Xv(bv) = Xv(Hv) = 1 at the remaining v. This is in turn equivalent to the claim

that Xv is close to 1 in Gv for a finite number of v and \v £ H^- at the other v, i.e.

X = Ylv Xv is close to 1 in the restricted direct product of the Gv with respect to

the H^. This completes the proof.

Schwartz-Bruhat Functions, Tempered Distributions

The basic duality and structure theory of locally compact abelian groups re-

called earlier is based on the following facts (Weil [209], §26, §29):

(i) Every abelian group G, generated by a compact neighborhood of the identity is

the projective limit of groups of the form Ga — G/Ga. = R p x T ? x Z s x F , where

R is the additive group of real numbers, T is a torus, Z is the additive group of

integers and F is a finite abelian group.

(ii) Every locally compact group is the inductive limit of its subgroups, which are

generated by compact neighborhoods of the identity.

REMARKS. 1. The groups of the form R p x T 9 x Z s x F are called elementary.

2. Every locally compact abelian group is the inductive limit of projective limits of

elementary groups.

(iii) The character group of a locally compact abelian group is also locally compact.

(iv) Z and T are the character groups of one another. The character group of the

cyclic group Z/iVZ is isomorphic to itself. R is isomorphic to its dual.

From (iv) it is clear that the character group of an elementary group is also ele-

mentary.

(v) Every locally compact abelian group G is of the form R p x Gi, where G\ is a

group containing a compact subgroup H such that G\/H is discrete.

The Space S{G) of Schwartz-Bruhat Functions on a Locally Compact

Abelian Group G

We consider first the case of an elementary abelian group G = R

p

x T

9

x Z

r

x F ,

where F is a finite group. Recall that a polynomial function on G is, by definition,

a polynomial on the coordinates of the factors Hp and Z r with coefficients which

are complex valued functions on the group T 9 x F ; the space S(G) will then be

defined as the set of functions I, infinitely differentiable on G, such that P • D $ is

bounded on G for any translation invariant differential operator D and polynomial

function P; the topology on 5(G) is that induced by the sequence of seminorms

i/PtD($) = Sup\P-D$\.

To define S(G) for an arbitrary locally compact abelian group we introduce the

family of pairs of subgroups (H, H') of G with the following properties: