Families of Automorphic Forms

  • Roelof W. Bruggeman

Part of the Modern Birkhäuser Classics book series (MBC)

Table of contents

  1. Front Matter
    Pages I-X
  2. Modular introduction

    1. Roelof W. Bruggeman
      Pages 1-21
  3. General theory

    1. Front Matter
      Pages 23-23
    2. Roelof W. Bruggeman
      Pages 25-32
    3. Roelof W. Bruggeman
      Pages 33-46
    4. Roelof W. Bruggeman
      Pages 47-70
    5. Roelof W. Bruggeman
      Pages 71-84
    6. Roelof W. Bruggeman
      Pages 85-105
    7. Roelof W. Bruggeman
      Pages 107-131
    8. Roelof W. Bruggeman
      Pages 133-149
    9. Roelof W. Bruggeman
      Pages 151-176
    10. Roelof W. Bruggeman
      Pages 177-189
    11. Roelof W. Bruggeman
      Pages 191-210
    12. Roelof W. Bruggeman
      Pages 213-236
  4. Examples

    1. Front Matter
      Pages 237-237
    2. Roelof W. Bruggeman
      Pages 239-264
    3. Roelof W. Bruggeman
      Pages 265-274
    4. Roelof W. Bruggeman
      Pages 275-306
  5. Back Matter
    Pages 307-318

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)).


Analytic automorphic forms Colin de Verdiere Discrete cofinite subgroups Eigenvalue Eisenstein series Modular group Multiplier system Poincare series Singularities automorphic forms function operator review spectral theory transformation

Authors and affiliations

  • Roelof W. Bruggeman
    • 1
  1. 1.Mathematisch InstituutUniversiteit UtrechtUtrechtThe Netherland

Bibliographic information