Families of Automorphic Forms

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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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

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

• Mathematisch Instituut, Universiteit Utrecht, Utrecht, The Netherland

  Roelof W. Bruggeman

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