Softcover ISBN:  9781470428556 
Product Code:  MEMO/253/1212 
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Electronic ISBN:  9781470444198 
Product Code:  MEMO/253/1212.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 253; 2018; 167 ppMSC: Primary 11; Secondary 22;
The authors investigate the correspondence between holomorphic automorphic forms on the upper halfplane 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. Onesided 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

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The authors investigate the correspondence between holomorphic automorphic forms on the upper halfplane 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. Onesided 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