Advice to the Reader

It is necessary for the reader of this survey to be familiar with the following

topics:

(a) He must have a good elementary knowledge of the theory of analytic functions

of one complex variable, as contained for instance in L. Ahlfors, Complex Analysis

[!]•

(b) He must know the basic definitions in the theory of functions of a real variable

at the level of H. Royden, Real Analysis [150] (see also Rudin [151]).

(c) He must have taken a graduate level course in number theory at a level com-

parable to that in E. Hecke, Lectures in the Theory of Algebraic Numbers [76] (see

also [104],[105],[207]). In particular, he must know the three fundamental results

of algebraic number theory: the unique factorization of ideals, the finiteness of the

class group, and the Dirichlet unit theorem.

If he wishes to profit from the reading of the proofs in the individual chap-

ters, each written in an increasing order of sophistication, he must also have an

acquaintance with certain concepts and results which unfortunately appear scat-

tered throughout the mathematical literature. In the following we suggest a road

map that can facilitate his reading of the relevant literature and prepare him for

further study and research of the relevant topics.

Chapter I. The rudimentary knowledge of abstract harmonic analysis needed can

be acquired by selectively reading those chapters in L. Loomis, An Introduction

to Abstract Harmonic Analysis [115] or in the short and elegant monograph by G.

Bachman, Elements of Abstract Harmonic Analysis [8], which deal specifically with

topological groups, Haar measure, character and dual groups, and Fourier analysis

on locally compact abelian groups. An exposition of these topics that is still worth

reading from a historical point of view can be found in A. Weil, L 'integration dans

les groupes topologiques et ses applications [209] and in L. Pontrjagin, Topological

Groups [144].

An excellent introduction to the basic theory of distributions can be gleaned

from the first two chapters in L. Hormander, The Analysis of Linear Partial Dif-

ferential Operators I [80].

The Appendix in Chapter I, §3 on principal L-functions on GL(n) is meant

to serve only as an outline of how the Hecke-Tate theory on GL\(k) generalizes

to GLn(k), and requires some concepts from representation theory not covered in

the earlier sections, but which are essential ingredients in the understanding of

ix