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Lectures on Automorphic $L$-functions
 
James W. Cogdell Oklahoma State University, Stillwater, OK
Henry H. Kim University of Toronto, Toronto, ON, Canada
M. Ram Murty Queen’s University, Kingston, ON, Canada
A co-publication of the AMS and Fields Institute
Lectures on Automorphic $L$-functions
eBook ISBN:  978-1-4704-3147-1
Product Code:  FIM/20.E
List Price: $97.00
MAA Member Price: $87.30
AMS Member Price: $77.60
Lectures on Automorphic $L$-functions
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Lectures on Automorphic $L$-functions
James W. Cogdell Oklahoma State University, Stillwater, OK
Henry H. Kim University of Toronto, Toronto, ON, Canada
M. Ram Murty Queen’s University, Kingston, ON, Canada
A co-publication of the AMS and Fields Institute
eBook ISBN:  978-1-4704-3147-1
Product Code:  FIM/20.E
List Price: $97.00
MAA Member Price: $87.30
AMS Member Price: $77.60
  • Book Details
     
     
    Fields Institute Monographs
    Volume: 202004; 283 pp
    MSC: Primary 11; 22;

    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
     
     
    • Lectures on $L$-functions, converse theorems, and functoriality for $GL_n$, by James W. Cogdell
    • Preface
    • Lecture 1. Modular forms and their $L$-functions
    • Lecture 2. Automorphic forms
    • Lecture 3. Automorphic representations
    • Lecture 4. Fourier expansions and multiplicity one theorems
    • Lecture 5. Eulerian integral representations
    • Lecture 6. Local $L$-functions: The non-Archimedean case
    • Lecture 7. The unramified calculation
    • Lecture 8. Local $L$-functions: The Archimedean case
    • Lecture 9. Global $L$-functions
    • Lecture 10. Converse theorems
    • Lecture 11. Functoriality
    • Lecture 12. Functoriality for the classical groups
    • Lecture 13. Functoriality for the classical groups, II
    • Automorphic $L$-functions, by Henry H. Kim
    • Introduction
    • Chapter 1. Chevalley groups and their properties
    • Chapter 2. Cuspidal representations
    • Chapter 3. $L$-groups and automorphic $L$-functions
    • Chapter 4. Induced representations
    • Chapter 5. Eisenstein series and constant terms
    • Chapter 6. $L$-functions in the constant terms
    • Chapter 7. Meromorphic continuation of $L$-functions
    • Chapter 8. Generic representations and their Whittaker models
    • Chapter 9. Local coefficients and non-constant terms
    • Chapter 10. Local Langlands correspondence
    • Chapter 11. Local $L$-functions and functional equations
    • Chapter 12. Normalization of intertwining operators
    • Chapter 13. Holomorphy and bounded in vertical strips
    • Chapter 14. Langlands functoriality conjecture
    • Chapter 15. Converse theorem of Cogdell and Piatetski-Shapiro
    • Chapter 16. Functoriality of the symmetric cube
    • Chapter 17. Functoriality of the symmetric fourth
    • Bibliography
    • Applications of symmetric power $L$-functions, by M. Ram Murty
    • Preface
    • Lecture 1. The Sato-Tate conjecture
    • Lecture 2. Maass wave forms
    • Lecture 3. The Rankin-Selberg method
    • Lecture 4. Oscillations of Fourier coefficients of cusp forms
    • Lecture 5. Poincaré series
    • Lecture 6. Kloosterman sums and Selberg’s conjecture
    • Lecture 7. Refined estimates for Fourier coefficients of cusp forms
    • Lecture 8. Twisting and averaging of $L$-series
    • Lecture 9. The Kim-Sarnak theorem
    • Lecture 10. Introduction to Artin $L$-functions
    • Lecture 11. Zeros and poles of Artin $L$-functions
    • Lecture 12. The Langlands-Tunnell theorem
    • Bibliography
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 202004; 283 pp
MSC: Primary 11; 22;

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.

  • Lectures on $L$-functions, converse theorems, and functoriality for $GL_n$, by James W. Cogdell
  • Preface
  • Lecture 1. Modular forms and their $L$-functions
  • Lecture 2. Automorphic forms
  • Lecture 3. Automorphic representations
  • Lecture 4. Fourier expansions and multiplicity one theorems
  • Lecture 5. Eulerian integral representations
  • Lecture 6. Local $L$-functions: The non-Archimedean case
  • Lecture 7. The unramified calculation
  • Lecture 8. Local $L$-functions: The Archimedean case
  • Lecture 9. Global $L$-functions
  • Lecture 10. Converse theorems
  • Lecture 11. Functoriality
  • Lecture 12. Functoriality for the classical groups
  • Lecture 13. Functoriality for the classical groups, II
  • Automorphic $L$-functions, by Henry H. Kim
  • Introduction
  • Chapter 1. Chevalley groups and their properties
  • Chapter 2. Cuspidal representations
  • Chapter 3. $L$-groups and automorphic $L$-functions
  • Chapter 4. Induced representations
  • Chapter 5. Eisenstein series and constant terms
  • Chapter 6. $L$-functions in the constant terms
  • Chapter 7. Meromorphic continuation of $L$-functions
  • Chapter 8. Generic representations and their Whittaker models
  • Chapter 9. Local coefficients and non-constant terms
  • Chapter 10. Local Langlands correspondence
  • Chapter 11. Local $L$-functions and functional equations
  • Chapter 12. Normalization of intertwining operators
  • Chapter 13. Holomorphy and bounded in vertical strips
  • Chapter 14. Langlands functoriality conjecture
  • Chapter 15. Converse theorem of Cogdell and Piatetski-Shapiro
  • Chapter 16. Functoriality of the symmetric cube
  • Chapter 17. Functoriality of the symmetric fourth
  • Bibliography
  • Applications of symmetric power $L$-functions, by M. Ram Murty
  • Preface
  • Lecture 1. The Sato-Tate conjecture
  • Lecture 2. Maass wave forms
  • Lecture 3. The Rankin-Selberg method
  • Lecture 4. Oscillations of Fourier coefficients of cusp forms
  • Lecture 5. Poincaré series
  • Lecture 6. Kloosterman sums and Selberg’s conjecture
  • Lecture 7. Refined estimates for Fourier coefficients of cusp forms
  • Lecture 8. Twisting and averaging of $L$-series
  • Lecture 9. The Kim-Sarnak theorem
  • Lecture 10. Introduction to Artin $L$-functions
  • Lecture 11. Zeros and poles of Artin $L$-functions
  • Lecture 12. The Langlands-Tunnell theorem
  • Bibliography
Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
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