Fields Institute Monographs 2004; 283 pp; softcover Volume: 20 ISBN10: 0821848003 ISBN13: 9780821848005 List Price: US$92 Member Price: US$73.60 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 PiatetskiShapiro with the LanglandsShahidi method. This book provides a stepbystep 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, SatoTate conjecture, and Artin's conjecture. It would be ideal for an introductory course in the Langlands program. Titles in this series are copublished with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada). Readership Graduate students and research mathematicians interested in representation theory and number theory. Table of Contents 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 nonArchimedean 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 nonconstant 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 PiatetskiShapiro
 Functoriality of the symmetric cube
 Functoriality of the symmetric fourth
 Bibliography
M. Ram Murty, Applications of symmetric power \(L\)functions  Preface
 The SatoTate conjecture
 Maass wave forms
 The RankinSelberg 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 KimSarnak theorem
 Introduction to Artin \(L\)functions
 Zeros and poles of Artin \(L\)functions
 The LanglandsTunnell theorem
 Bibliography
