Lectures on Automorphic $L$-functions
American Mathematical Society (Verlag)
978-0-8218-3516-6 (ISBN)
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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.
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 Poincare 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.
| Erscheint lt. Verlag | 15.4.2004 |
|---|---|
| Reihe/Serie | Fields Institute Monographs ; No. 20 |
| Verlagsort | Providence |
| Sprache | englisch |
| Gewicht | 734 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie |
| ISBN-10 | 0-8218-3516-5 / 0821835165 |
| ISBN-13 | 978-0-8218-3516-6 / 9780821835166 |
| Zustand | Neuware |
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