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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
Available Formats:
Electronic ISBN: 978-1-4704-3147-1
Product Code: FIM/20.E
List Price: $92.00 MAA Member Price:$82.80
AMS Member Price: $73.60 Click above image for expanded view Lectures on Automorphic$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 Available Formats:  Electronic ISBN: 978-1-4704-3147-1 Product Code: FIM/20.E  List Price:$92.00 MAA Member Price: $82.80 AMS Member Price:$73.60
• 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.

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

• Requests

Review Copy – for reviewers who would like to review an AMS book
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.

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 reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
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