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Families of Automorphic Forms

  • Book
  • © 1994

Overview

Authors:
  1. Roelof W. Bruggeman

    1. Mathematisch Instituut, Universiteit Utrecht, Utrecht, The Netherland

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  • New material so far mostly available in articles
  • Includes supplementary material: sn.pub/extras

Part of the book series: Modern Birkhäuser Classics (MBC)

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Table of contents (15 chapters)

  1. Front Matter

    Pages I-X
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  2. Modular introduction

    1. Modular introduction

      • Roelof W. Bruggeman
      Pages 1-21

  3. General theory

    1. Front Matter

      Pages 23-23
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    2. Automorphic forms on the universal covering group

      • Roelof W. Bruggeman
      Pages 25-32

    3. Discrete subgroups

      • Roelof W. Bruggeman
      Pages 33-46

    4. Automorphic forms

      • Roelof W. Bruggeman
      Pages 47-70

    5. Poincaré series

      • Roelof W. Bruggeman
      Pages 71-84

    6. Selfadjoint extension of the Casimir operator

      • Roelof W. Bruggeman
      Pages 85-105

    7. Families of automorphic forms

      • Roelof W. Bruggeman
      Pages 107-131

    8. Transformation and truncation

      • Roelof W. Bruggeman
      Pages 133-149

    9. Pseudo Casimir operator

      • Roelof W. Bruggeman
      Pages 151-176

    10. Meromorphic continuation of Poincaré series

      • Roelof W. Bruggeman
      Pages 177-189

    11. Poincaré families along vertical lines

      • Roelof W. Bruggeman
      Pages 191-210

    12. Singularities of Poincaré families

      • Roelof W. Bruggeman
      Pages 213-236

  4. Examples

    1. Front Matter

      Pages 237-237
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    2. Automorphic forms for the modular group

      • Roelof W. Bruggeman
      Pages 239-264

    3. Automorphic forms for the theta group

      • Roelof W. Bruggeman
      Pages 265-274

    4. Automorphic forms for the commutator subgroup

      • Roelof W. Bruggeman
      Pages 275-306

  5. Back Matter

    Pages 307-318
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Keywords

About this book

Automorphic forms on the upper half plane have been studied for a long time. Most attention has gone to the holomorphic automorphic forms, with numerous applications to number theory. Maass, [34], started a systematic study of real analytic automorphic forms. He extended Hecke’s relation between automorphic forms and Dirichlet series to real analytic automorphic forms. The names Selberg and Roelcke are connected to the spectral theory of real analytic automorphic forms, see, e. g. , [50], [51]. This culminates in the trace formula of Selberg, see, e. g. , Hejhal, [21]. Automorphicformsarefunctionsontheupperhalfplanewithaspecialtra- formation behavior under a discontinuous group of non-euclidean motions in the upper half plane. One may ask how automorphic forms change if one perturbs this group of motions. This question is discussed by, e. g. , Hejhal, [22], and Phillips and Sarnak, [46]. Hejhal also discusses the e?ect of variation of the multiplier s- tem (a function on the discontinuous group that occurs in the description of the transformation behavior of automorphic forms). In [5]–[7] I considered variation of automorphic forms for the full modular group under perturbation of the m- tiplier system. A method based on ideas of Colin de Verdi` ere, [11], [12], gave the meromorphic continuation of Eisenstein and Poincar´ e series as functions of the eigenvalue and the multiplier system jointly. The present study arose from a plan to extend these results to much more general groups (discrete co?nite subgroups of SL (R)).

Reviews

From reviews:

"It is made abundantly clear that this viewpoint, of families of automorphic functions depending on varying eigenvalue and multiplier systems, is both deep and fruitful." - MathSciNet

Authors and Affiliations

  • Mathematisch Instituut, Universiteit Utrecht, Utrecht, The Netherland

    Roelof W. Bruggeman

Bibliographic Information

  • Book Title: Families of Automorphic Forms

  • Authors: Roelof W. Bruggeman

  • Series Title: Modern Birkhäuser Classics

  • DOI: https://doi.org/10.1007/978-3-0346-0336-2

  • Publisher: Birkhäuser Basel

  • eBook Packages: Springer Book Archive

  • Copyright Information: Birkhäuser Basel 1994

  • Softcover ISBN: 978-3-0346-0335-5Published: 23 November 2009

  • eBook ISBN: 978-3-0346-0336-2Published: 28 February 2010

  • Series ISSN: 2197-1803

  • Series E-ISSN: 2197-1811

  • Edition Number: 1

  • Number of Pages: X, 318

  • Additional Information: Originally published in the series: Monographs in Mathematics Vol 88

  • Topics: Real Functions

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