## Hecke's Theory of Modular Forms and Dirichlet SeriesIn 1938, at the Institute for Advanced Study, E Hecke gave a series of lectures on his theory of correspondence between modular forms and Dirichlet series. Since then, the Hecke correspondence has remained an active feature of number theory and, indeed, it is more important today than it was in 1936 when Hecke published his original papers. This book is an amplified and up-to-date version of the former author''s lectures at the University of Illinois at Urbana-Champaign, based on Hecke''s notes. Providing many details omitted from Hecke''s notes, it includes various new and important developments in recent years. In particular, several generalizations and analogues of the original Hecke theory are briefly described in this concise volume. Sample Chapter(s). Chapter 1: Introduction (90 KB). Contents: The Main Correspondence Theorem; A Fundamental Region; The Case > 2; The Case |

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### Contents

1 Introduction | 1 |

2 The main correspondence theorem | 3 |

3 A fundamental region | 15 |

4 The case 2 | 23 |

5 The case 2 | 35 |

6 The case 2 | 69 |

7 Bochners generalization of the main correspondence theorem of Hecke and related results | 87 |

8 Identities equivalent to the functional equation and to the modular relation | 115 |

129 | |

135 | |

### Other editions - View all

Hecke's Theory of Modular Forms and Dirichlet Series Bruce C. Berndt,Marvin Isadore Knopp Limited preview - 2008 |

### Common terms and phrases

analytic continuation assume automorphic form Berndt Bochner bounded closed curve coeﬃcients completes the proof complex number conclude convergence Corollary 5.1 correspondence theorem cusp forms deﬁned Deﬁnition Definition 2.2 denotes the number diﬀer Dirichlet series Eisenstein series elliptic Epstein zeta function equivalent Example exists ﬁrst fixed points follows directly functional equation functional equation 7.5 fundamental region Furthermore g(ioo Guinand half-plane Hamburger’s Theorem Hence identity Im(r implies intentionally left blank ip(s linearly independent mapping Mellin transform meromorphic Mo(A modular form modular group nonconstant nontrivial notation Note one-to-one positive integer proof of Lemma proof of Theorem proved punctured neighborhood Ramanujan real axis Remark replaced respect to G(A restriction right-hand side satisﬁes satisfies 7.30 Selberg class series of signature simple pole simple zero summation Suppose tends to oo Theorem 2.1 transformation law uniformizing variable unit circle vertical strip zero of order zeta function