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Lectures on Automorphic $L$-functions

Buch | Hardcover
283 Seiten
2004
American Mathematical Society (Verlag)
978-0-8218-3516-6 (ISBN)
85,95 inkl. MwSt
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Presents an account of the role automorphic $L$-functions play in number theory and in the Langlands program, especially the Langlands functoriality conjecture. This book explains how the Langlands functoriality conjecture implies solutions to several conjectures in number theory, such as the Ramanujan conjecture and Sato-Tate conjecture.
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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