## Automorphic Forms and RepresentationsThis book covers both the classical and representation theoretic views of automorphic forms in a style that is accessible to graduate students entering the field. The treatment is based on complete proofs, which reveal the uniqueness principles underlying the basic constructions. The book features extensive foundational material on the representation theory of GL(1) and GL(2) over local fields, the theory of automorphic representations, L-functions and advanced topics such as the Langlands conjectures, the Weil representation, the Rankin-Selberg method and the triple L-function, and examines this subject matter from many different and complementary viewpoints. Researchers as well as students in algebra and number theory will find this a valuable guide to a notoriously difficult subject. |

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### Contents

V | 1 |

VI | 3 |

VII | 17 |

VIII | 26 |

IX | 41 |

X | 54 |

XI | 65 |

XII | 76 |

XXVIII | 278 |

XXIX | 291 |

XXX | 307 |

XXXI | 321 |

XXXII | 341 |

XXXIII | 346 |

XXXIV | 368 |

XXXV | 375 |

### Common terms and phrases

Abelian adele ring admissible representation analytic continuation Archimedean assume automorphic forms automorphic representation Borel coefficients commutative compact subgroup compactly supported complex number conjecture constant cusp form decomposition defined definition denote dimensional direct sum double coset eigenvalues Eisenstein series element equals Exercise exists finite finite-dimensional follows formula functional equation GL(n Haar measure Hecke algebra Hecke character hence Hilbert space homomorphism identity implies induced inner product invariant subspace irreducible admissible representation irreducible representations isomorphism Jacquet L-function Langlands Lemma Let F Let G Lie algebra linear functional locally compact Maass forms matrix modular forms module multiple non-Archimedean nonramified nonzero notation obtain orthogonal poles Proof Let Proposition prove quadratic quasicharacter re(s representation of G representation of GL(2 satisfying Eq Section sheaf side of Eq smooth spherical Suppose Theorem theory topology trivial unique unitary representation vector space Whittaker functional Whittaker model zero