This book based on lectures given by James Arthur discusses
the trace formula of Selberg and Arthur. The emphasis is
laid on Arthur's trace formula for GL(r), with several
examples in order to illustrate the basic concepts. The book
will be useful and stimulating reading for graduate students
in automorphic forms, analytic number theory, and
non-commutative harmonic analysis, as well as researchers in
these fields. Contents:
I. Number Theory and Automorphic Representations.1.1. Some
problems in classical number theory, 1.2. Modular forms and
automorphic representations, II. Selberg's Trace Formula
2.1. Historical Remarks, 2.2. Orbital integrals and
Selberg's trace formula, 2.3.Three examples, 2.4. A
necessary condition, 2.5. Generalizations and applications,
III. Kernel Functions and the Convergence Theorem, 3.1.
Preliminaries on GL(r), 3.2. Combinatorics and reduction
theory, 3.3. The convergence theorem, IV. The Ad lic Theory,
4.1. Basic facts, V. The Geometric Theory, 5.1. The JTO(f)
and JT(f) distributions, 5.2. A geometric I-function, 5.3.
The weight functions, VI. The Geometric Expansionof the
Trace Formula, 6.1. Weighted orbital integrals, 6.2. The
unipotent distribution, VII. The Spectral Theory, 7.1. A
review of the Eisenstein series, 7.2. Cusp forms,
truncation, the trace formula, VIII.The Invariant Trace
Formula and its Applications, 8.1. The invariant trace
formula for GL(r), 8.2. Applications and remarks