CHAPTER I

HECKE L-FUNCTIONS

1. Abstract Harmonic Analysis: A Survey

Introduction. In this section, which is purely of a utilitarian nature, we

gather all the relevant definitions and results from abstract harmonic analysis that

are needed for the study of zeta functions on number fields. After a brief review of

some basic topological notions regarding locally compact abelian groups we recall

the notions of Haar measure, Pontrjagin duality, and Fourier transforms. The

central topic of these introductory notes is a discussion of the Poisson summation

formula. The reader who is familiar with these notions may want to proceed to

section 1.2 where the study of zeta and L-functions begins in earnest.

Basic Facts

Let {Xv}v€s be a family of topological spaces indexed by the set S. For each

v G S let Tv be the collection of open sets in Xv. The cartesian product of {Xv}ves

is denoted by

l[xv,

ves

and as usual, it is identified with the set of all functions

such that x(v) G Xv\ for simplicity, the elements of ELes ^v a r e denoted by x —

(xv)ves- A projection mapping is defined as follows:

prv : Y\XV - Xv, x = (xv), prv(x) = xv.

A topology on \\XV is desired with the property that all projection mappings

be continuous functions. Such a topology is provided by the Tychonoff topology

whereby a basis for the open sets is given by the family of subsets of the form

Y[v Uv, where Uv is an open subset in Tv and Uv — Xv for almost all v, i.e. all

v G S except a finite number; this is the weakest topology for which the projection

mappings prv are continuous. The basic result about this topology is the following:

THEOREM 1. (Tychonoff) The product space

YlvXv, ™ith

respect to the Ty-

chonoff topology, is compact if and only if each of the spaces Xv is compact.

DEFINITION 1. A topological space X is called locally compact if for each point

x G X there is an open set Ux such that x G Ux and the closure Ux is compact.

l

http://dx.doi.org/10.1090/surv/115/01