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Holomorphic Automorphic Forms and Cohomology
 
Roelof Bruggeman Universiteit Utrecht, Utrecht, The Netherlands
Youngju Choie Pohang University of Science and Technology, Pohang, South Korea
Nikolaos Diamantis University of Nottingham, Nottingham, United Kingdom
Holomorphic Automorphic Forms and Cohomology
Softcover ISBN:  978-1-4704-2855-6
Product Code:  MEMO/253/1212
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $46.80
eBook ISBN:  978-1-4704-4419-8
Product Code:  MEMO/253/1212.E
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $46.80
Softcover ISBN:  978-1-4704-2855-6
eBook: ISBN:  978-1-4704-4419-8
Product Code:  MEMO/253/1212.B
List Price: $156.00 $117.00
MAA Member Price: $140.40 $105.30
AMS Member Price: $93.60 $70.20
Holomorphic Automorphic Forms and Cohomology
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Holomorphic Automorphic Forms and Cohomology
Roelof Bruggeman Universiteit Utrecht, Utrecht, The Netherlands
Youngju Choie Pohang University of Science and Technology, Pohang, South Korea
Nikolaos Diamantis University of Nottingham, Nottingham, United Kingdom
Softcover ISBN:  978-1-4704-2855-6
Product Code:  MEMO/253/1212
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $46.80
eBook ISBN:  978-1-4704-4419-8
Product Code:  MEMO/253/1212.E
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $46.80
Softcover ISBN:  978-1-4704-2855-6
eBook ISBN:  978-1-4704-4419-8
Product Code:  MEMO/253/1212.B
List Price: $156.00 $117.00
MAA Member Price: $140.40 $105.30
AMS Member Price: $93.60 $70.20
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2532018; 167 pp
    MSC: Primary 11; Secondary 22

    The authors investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least \(2\) this correspondence is given by the Eichler integral. The authors use Knopp's generalization of this integral to real weights, and apply it to complex weights that are not an integer at least \(2\). They show that for these weights the generalized Eichler integral gives an injection into the first cohomology group with values in a module of holomorphic functions, and characterize the image. The authors impose no condition on the growth of the automorphic forms at the cusps. Their result concerns arbitrary cofinite discrete groups with cusps, and covers exponentially growing automorphic forms, like those studied by Borcherds, and like those in the theory of mock automorphic forms.

  • Table of Contents
     
     
    • Chapters
    • Introduction
    • 1. Cohomology with Values in Holomorphic Functions
    • 1. Definitions and notations
    • 2. Modules and cocycles
    • 3. The image of automorphic forms in cohomology
    • 4. One-sided averages
    • 2. Harmonic Functions
    • 5. Harmonic functions and cohomology
    • 6. Boundary germs
    • 7. Polar harmonic functions
    • 3. \redefinepart
    • 4. Cohomology with values in Analytic Boundary Germs
    • 5. \oldpart
    • 8. Highest weight spaces of analytic boundary germs
    • 9. Tesselation and cohomology
    • 10. Boundary germ cohomology and automorphic forms
    • 11. Automorphic forms of integral weights at least $2$ and analytic boundary germ cohomology
    • 6. \redefinepart
    • 7. Miscellaneous
    • 8. \oldpart
    • 12. Isomorphisms between parabolic cohomology groups
    • 13. Cocycles and singularities
    • 14. Quantum automorphic forms
    • 15. Remarks on the literature
    • A. Universal covering group and representations
    • Indices
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2532018; 167 pp
MSC: Primary 11; Secondary 22

The authors investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least \(2\) this correspondence is given by the Eichler integral. The authors use Knopp's generalization of this integral to real weights, and apply it to complex weights that are not an integer at least \(2\). They show that for these weights the generalized Eichler integral gives an injection into the first cohomology group with values in a module of holomorphic functions, and characterize the image. The authors impose no condition on the growth of the automorphic forms at the cusps. Their result concerns arbitrary cofinite discrete groups with cusps, and covers exponentially growing automorphic forms, like those studied by Borcherds, and like those in the theory of mock automorphic forms.

  • Chapters
  • Introduction
  • 1. Cohomology with Values in Holomorphic Functions
  • 1. Definitions and notations
  • 2. Modules and cocycles
  • 3. The image of automorphic forms in cohomology
  • 4. One-sided averages
  • 2. Harmonic Functions
  • 5. Harmonic functions and cohomology
  • 6. Boundary germs
  • 7. Polar harmonic functions
  • 3. \redefinepart
  • 4. Cohomology with values in Analytic Boundary Germs
  • 5. \oldpart
  • 8. Highest weight spaces of analytic boundary germs
  • 9. Tesselation and cohomology
  • 10. Boundary germ cohomology and automorphic forms
  • 11. Automorphic forms of integral weights at least $2$ and analytic boundary germ cohomology
  • 6. \redefinepart
  • 7. Miscellaneous
  • 8. \oldpart
  • 12. Isomorphisms between parabolic cohomology groups
  • 13. Cocycles and singularities
  • 14. Quantum automorphic forms
  • 15. Remarks on the literature
  • A. Universal covering group and representations
  • Indices
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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