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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
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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).

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 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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