## Lectures on Automorphic L-functions (Google eBook)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 |

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### Contents

1 | |

5 | |

13 | |

Automorphic Representations | 21 |

Fourier Expansions and Multiplicity One Theorems | 29 |

Eulerian Integral Representations | 37 |

Local Lfunctions The NonArchimedean Case | 45 |

The Unramified Calculation | 51 |

Local Coefficients and Nonconstant Terms | 153 |

Local Langlands Correspondence | 161 |

Local Lfunctions and Functional Equations | 165 |

Normalization of Intertwining Operators | 171 |

Holomorphy and Bounded in Vertical Strips | 177 |

Langlands Functoriality Conjecture | 181 |

Converse Theorem of Cogdell and PiatetskiShapiro | 183 |

Functoriality of the Symmetric Cube | 187 |

Local Lfunctions The Archimedean Case | 59 |

Global Lfunctions | 65 |

Converse Theorems | 73 |

Functoriality | 81 |

Functoriality for the Classical Groups | 87 |

Functoriality for the Classical Groups II | 91 |

Automorphic Lfunctions | 97 |

Chevalley Groups and their Properties | 101 |

Cuspidal Representations | 113 |

Lgroups and Automorphic Lfunctions | 115 |

Induced Representations | 119 |

Eisenstein Series and Constant Terms | 129 |

Lfunctions in the Constant Terms | 137 |

Meromorphic Continuation of Lfunctions | 145 |

Generic Representations and their Whittaker Models | 147 |

Functoriality of the Symmetric Fourth | 193 |

Applications of Symmetric Power Lfunctions | 203 |

The SatoTate Conjecture | 207 |

Maass Wave Forms | 213 |

The RankinSelberg Method | 219 |

Oscillations of Fourier Coefficients of Cusp Forms | 227 |

Poincaré Series | 237 |

Kloosterman Sums and Selbergs Conjecture | 243 |

Refined Estimates for Fourier Coefficients of Cusp Forms | 247 |

Twisting and Averaging of Lserics | 253 |

The KimSarnak Theorem | 257 |

Introduction to Artin Lfunctions | 265 |

Zeros and Poles of Artin Lfunctions | 271 |

The LanglandsTunnell Theorem | 275 |

### Common terms and phrases

7-factors Ad(n admissible representation algebraic group analytic automorphic forms automorphic L-functions automorphic representation bounded in vertical central character compact constant term Converse Theorem Corollary cusp forms cuspidal automorphic representation cuspidal representation decomposition deduce defined denote eigenvalue Eisenstein series Euler product finite places finite set form of weight Fourier coefficients Fourier expansion functional equation functorial lift Galois GL(n GLn(A GLn(C grossencharacter Hence holomorphic holomorphic for Re(s I.I. Piatetski-Shapiro induced representation intertwining operator irreducible admissible representation J.W. Cogdell Jacquet Langlands correspondence Langlands functoriality Langlands-Shahidi method Lecture Lemma Let G Ls(s Maass forms Math meromorphic continuation modular form non-archimedean non-trivial non-vanishing Note number field number theory obtain parabolic subgroup pole Proof Proposition Ramanujan conjecture representation of G(A result satisfies Selberg Shahidi Shalika smooth space supercuspidal representation Suppose symmetric power tensor product unique unitary unramified vector vertical strips Whittaker functions Whittaker model zero