Automorphic forms are present in almost every area of modern number
theory. They also appear in other areas of mathematics and in physics. I
have lectured on these topics many times at Rutgers University with only
slight overlapping of content, and I still have new material to teach that is
important. It is indeed a vast territory which cannot be grasped by any
one person. While research publications on automorphic forms are rapidly
increasing in quantity and quality, the demand for textbooks, particularly
on the graduate level, is also growing. There are fine books on this subject
such as [Lan], [Miy], [Sh2], but still more are needed, especially those which
favor analytic methods.
The present book is based on my lecture notes (almost verbatim except
for Section 5.5) from a graduate course which I delivered in the Fall of 1994
and in the Spring of 1995 at Rutgers. The course was formulated, as the
title implies, to acquaint our new students with the subject matter from
various perspectives. Thus I have not followed direct or traditional paths,
but rather I have frequently ventured into areas where different ideas and
methods mix and interact. To cover a lot in a limited time, some material
is necessarily presented as a survey. For example, the numerous connections
of automorphic forms with L-functions of number fields are discussed in
Chapter 12 without details. However we do provide complete proofs of the
most basic results in the earlier sections.
An experienced reader will find some of our arguments to be nonstan-
dard. It would be pointless to argue which approach is better, since our
choice was made simply for the purpose of showing different possibilities.
For example, our presentation of the theory of Hecke operators in Chapter 6
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