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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®, 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®, 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®, 8.2. Applications and remarks
