the theory of automorphic L-functions on GLn(k). A more detailed presentation
can be found in A. Weil, Dirichlet Series and Automorphic Forms [210], and in
H. Jacquet and R.P. Langlands, Automorphic Forms on GL(2) [86] for the theory
on GL(2); a treatment of the higher dimensional case, GL(r),r 3, can be found
in H. Jacquet, Principal L-functions of the Linear Group [84] and in R. Godement
and H. Jacquet, Zeta Functions of Simple Algebras [69].
Chapter II. The reader is expected to be familiar with the basic theory of linear
representations of finite groups up to and including knowledge of Brauer's Theorem
as contained for instance in J.-P. Serre, Linear Representations of Finite Groups
[164], §§1-10.
For a deeper discussion of Weil and Weil-Deligne groups the reader can supple-
ment his study by consulting J. Tate, Number Theoretic Background, [187].
The ramification theory needed to understand the properties of conductors from
the point of view of the Herbrand distribution is given in C.J. Moreno, Advanced
Analytic Number Theory [127]. The definitions and elementary properties of the
absolute Weil group of a number field given in Chapter II, §2.3 are taken from the
report in A. Weil, Sur la theorie de corps de classes [211] and from the detailed
presentation in [212]. A modern exposition is also given in J. Tate's article referred
to above [187].
The descriptive survey of the local Langlands correspondence for GL{n) given
in Chapter II, §15 uses standard terminology about group representations; for these
the reader can consult A. Kirillov, Elements of the Theory of Representations [101]
or the excellent Encyclopedic Dictionary of Mathematics, second edition, published
by the Mathematical Society of Japan. For a more detailed and rigorous presen-
tation the reader can consult the excellent treatment in A. Knapp, Representation
Theory of Semisimple Groups, [100].
Chapter III. The essential requirements for this chapter have been kept to a
minimum. An acquaintance with the classical Hadamard theory of entire functions
of order 1 and their associated Weierstrass products would be sufficient. This
material is found in L. Ahlfors, Complex Analysis [1], Chapter IV, §3, and Chapter
V, §§1, 2 on harmonic and subharmonic functions. The principal results of the
chapter deal directly with the analytic properties of archimedean L-factors, also
known as gamma factors; for these the reader cannot do better than consult the
classical treatment in E. T. Whittaker and G. N. Watson, A Course of Modern
Analysis [215], particularly Chapters XII and XIII.
Chapter IV. Some acquaintance with the classical Mellin transform as well as
knowledge of the conditions under which its inverse exists is needed. The basic
theory can be deduced from that of the Fourier transform on the real line. The
latter is developed in Y. Katznelson, An Introduction to Harmonic Analysis [98],
Chapter VI (see also [115]). A more classical treatment of the Mellin transform is
in E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals [194].
The integral formulas for the Herbrand distribution used in this chapter are
discussed in great detail in the author's monograph [127] already cited above,
particularly Chapter IX.
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