Chapter V. Those properties of the gamma and related digamma functions, which
are not proved in this chapter, can be found in the treatise of Whittaker and Watson
Chapter VI. For an understanding of the geometric example in §1.1, the reader
should be acquainted with the elementary theory of zeta and L-functions of al-
gebraic curves over finite fields, particularly as it applies to elliptic curves. The
theory of these functions is explained in great detail in the author's monograph
[127], Chapters III and IV. Some background on the origin and significance of the
arithmetic example in §1.1 can be acquired in J.-P. Serre, (i) A Course in Arith-
metic, [169] Chapter VII, and (ii) Abelian t-adic Representations, [165] Chapter I,
(see also G. H. Hardy, Twelve Lectures on the Work of Ramanujan [71], Chapter
XII ). In (ii) the reader will also find a useful but brief discussion of the relations
between equipartition of conjugacy classes, L-functions, and Chebotarev's density
The theory of irreducible complex linear representations of a locally compact
group G needed in Chapter VI, §1.2, is an extension of the classical theory for
compact groups. The reader can find an elementary introduction to the theory of
linear complex representations of compact Lie groups, including the unitary group
SU(n) and the Peter-Weyl theorem, in W. Fulton and J. Harris, Representation
Theory: A First Course [55], Chapter XX. The reader will also find an elementary
treatment of the Bohr compactification in Y. Katznelson, Harmonic analysis [98],
p. 192.
The theory of Eisenstein series for maximal parabolic subgroups of GL(n) used
in the proof of the Jacquet-Shalika non-vanishing theorem is a generalization of the
Hecke-Riemann method for proving functional equations and uses the properties of
theta series developed in Chapter I. The reader who is unfamiliar with these topics
or who wishes to acquire a working knowledge of the theory may want to consult
the following treatises on the general theory of Eisenstein series: (i) T. Kubota,
Elementary Theory of Eisenstein Series [102], (ii) A. Borel, Automorphic Forms
on 5Z/2(R) [16], (hi) C. Moeglin and J.-L. Waldspurger, Spectral Decomposition
and Eisenstein Series [125], (iv) H.-Chandra, Automorphic Forms on Semisimple
Lie Groups [74]. A third alternative approach to the theory of Eisenstein series is
due to A. Selberg (for the group SX(3)) and uses the Fredholm theory of operator
equations. This has been developed independently by J. Bernstein (unpublished)
and in Shek-Tung Wong [216].
The reader who wishes to understand the technical aspects behind the powerful
Langlands-Shahidi method can consult the original exposition in F. Shahidi, Func-
tional Equation Satisfied by Certain L-Functions [178]. The approach there is quite
general and applies with minor modifications to Chevalley groups. The reader can
acquire the necessary knowledge of Whittaker models and Whittaker functions in J.
Shalika, The Multiplicity One Theorem for GLn [170] and in
H. Jacquet, Fonctions de Whittaker associees aux groupes de Chevalley [85].
Finally we must describe the method followed for cross-references. Theorems
have been numbered continuously throughout each chapter; the same is true for
lemmas, for definitions, and for the numbered formulas. The few corollaries and
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