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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(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
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