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Lectures on Automorphic $$L$$-functions
James W. Cogdell, Oklahoma State University, Stillwater, OK, Henry H. Kim, University of Toronto, ON, Canada, and M. Ram Murty, Queen's University, Kingston, ON, Canada
A co-publication of the AMS and Fields Institute.
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Fields Institute Monographs
2004; 283 pp; softcover
Volume: 20
ISBN-10: 0-8218-4800-3
ISBN-13: 978-0-8218-4800-5
List Price: US$87 Member Price: US$69.60
Order Code: FIM/20.S

This book provides a comprehensive account of the crucial role automorphic $$L$$-functions play in number theory and in the Langlands program, especially the Langlands functoriality conjecture. There has been a recent major development in the Langlands functoriality conjecture by the use of automorphic $$L$$-functions, namely, by combining converse theorems of Cogdell and Piatetski-Shapiro with the Langlands-Shahidi method. This book provides a step-by-step introduction to these developments and explains how the Langlands functoriality conjecture implies solutions to several outstanding conjectures in number theory, such as the Ramanujan conjecture, Sato-Tate conjecture, and Artin's conjecture. It would be ideal for an introductory course in the Langlands program.

Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

Graduate students and research mathematicians interested in representation theory and number theory.

James W. Cogdell, Lectures on $$L$$-functions, converse theorems, and functoriality for $$GL_n$$
• Preface
• Modular forms and their $$L$$-functions
• Automorphic forms
• Automorphic representations
• Fourier expansions and multiplicity one theorems
• Eulerian integral representations
• Local $$L$$-functions: The non-Archimedean case
• The unramified calculation
• Local $$L$$-functions: The Archimedean case
• Global $$L$$-functions
• Converse theorems
• Functoriality
• Functoriality for the classical groups
• Functoriality for the classical groups, II
Henry H. Kim, Automorphic $$L$$-functions
• Introduction
• Chevalley groups and their properties
• Cuspidal representations
• $$L$$-groups and automorphic $$L$$-functions
• Induced representations
• Eisenstein series and constant terms
• $$L$$-functions in the constant terms
• Meromorphic continuation of $$L$$-functions
• Generic representations and their Whittaker models
• Local coefficients and non-constant terms
• Local Langlands correspondence
• Local $$L$$-functions and functional equations
• Normalization of intertwining operators
• Holomorphy and bounded in vertical strips
• Langlands functoriality conjecture
• Converse theorem of Cogdell and Piatetski-Shapiro
• Functoriality of the symmetric cube
• Functoriality of the symmetric fourth
• Bibliography
M. Ram Murty, Applications of symmetric power $$L$$-functions
• Preface
• The Sato-Tate conjecture
• Maass wave forms
• The Rankin-Selberg method
• Oscillations of Fourier coefficients of cusp forms
• Poincaré series
• Kloosterman sums and Selberg's conjecture
• Refined estimates for Fourier coefficients of cusp forms
• Twisting and averaging of $$L$$-series
• The Kim-Sarnak theorem
• Introduction to Artin $$L$$-functions
• Zeros and poles of Artin $$L$$-functions
• The Langlands-Tunnell theorem
• Bibliography
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