INTRODUCTION xv

Chapter II. The essential nature of Artin's reciprocity law, in Chevalley's for-

mulation, is the identification of two seemingly distinct objects of interest in the

arithmetic of number fields:

(i) The Pontrjagin dual of the Galois group of the maximal abelian extension of a

number field k,

(ii) The characters of finite order of the idele class group C£(k) = k^/kx,

together with the attendant functoriality properties describing the behavior of the

objects in (i) and (ii) under finite extensions (restriction) and the norm map (in-

duction).

As is well known, Artin constructed L-functions that generalize those of Dirich-

let in the non-abelian case. The main gluing ingredient being the Artin-Brauer

theorem, which describes how the characters of a finite group can be expressed as

linear combinations of those arising by induction from abelian groups. This, and

the fact that Hecke had constructed L-functions associated to arbitrary characters

of the idele class group, lead to the possibility of amalgamating these two types

of L-functions (Artin's and Hecke's) into a single analytic package. This challenge

was taken up by Weil, who based his construction on the existence and uniqueness

of the group extension

1 —• Cl(K) —• W{K/k) —• Gal(K/k) —• 1

associated to the fundamental class of class field theory. Weil's construction of the

Artin-Hecke L-functions associated to finite dimensional representations of W(K/k)

requires a generalization of the Artin-Brauer theorem, which works for infinite non-

abelian locally compact groups like W(K/k). This theory was complemented by

work of Tamagawa on conductors and archimedean L-factors. The main results of

this theory are explained in Chapter II which includes the functional equation

for the L-functions of representations of W{K/k) (Lemma 3 and the Main Theo-

rem in §13). We also include a brief description of the Dwork-Langlands' theorem

on the decomposition of the root number into local factors, a result of fundamen-

tal significance for the modern theory of automorphic L-functions on GL(n), and

the new non-abelian reciprocity laws of Langlands. A brief survey of the basic

Langlands functoriality principle as it applies to GL{n) is included. This principle

implies among other things that the Galois class of Artin-Hecke L-functions studied

in Chapter II coincides with a subset of the class of principal L-functions studied

in Chapter I.

Chapter III. The finite primes in a number field, that is to say, those associated

to non-archimedean valuations, are to some extent determined by the well known

theorem of Wedderburn concerning the commutativity of finite division rings. In

contrast, the infinite primes in a number field, those corresponding to archimedean

valuations, are controlled by the fundamental theorem of Gelfand, which identifies

the field of complex numbers, up to topological isomorphism, as the only com-

plex commutative Banach division ring with identity. In this context, the main

characteristic property which distinguishes number fields from function fields of

positive characteristic is the existence of archimedean primes. An old saying in