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Lectures on Automorphic $L$-functions

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James W. Cogdell, Oklahoma State University, Stillwater, OK, Henry H. Kim, University of Toronto, Toronto, ON, Canada and M. Ram Murty, Queen’s University, Kingston, ON, Canada

Publication: Fields Institute Monographs
Publication Year: 2004; Volume 20
ISBNs: 978-0-8218-4800-5 (print); 978-1-4704-3147-1 (online)
DOI: https://doi.org/10.1090/fim/020
MathSciNet review: MR2071722
MSC: Primary 11F70; Secondary 22E55

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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