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
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