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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
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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 Electronic ISBN: 978-1-4704-4419-8 Product Code: MEMO/253/1212.E List Price:$78.00
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AMS Member Price: $70.20 Click above image for expanded view 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 Available Formats:  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
 Electronic 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 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$117.00 MAA Member Price: $105.30 AMS Member Price:$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.

• 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

• Requests

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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
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