**Memoirs of the American Mathematical Society**

2018;
167 pp;
Softcover

MSC: Primary 11;
Secondary 22

Print ISBN: 978-1-4704-2855-6

Product Code: MEMO/253/1212

List Price: $78.00

AMS Member Price: $46.80

MAA Member Price: $70.20

**Electronic ISBN: 978-1-4704-4419-8
Product Code: MEMO/253/1212.E**

List Price: $78.00

AMS Member Price: $46.80

MAA Member Price: $70.20

# Holomorphic Automorphic Forms and Cohomology

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*Roelof Bruggeman; Youngju Choie; Nikolaos Diamantis*

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

# Table of Contents

## Holomorphic Automorphic Forms and Cohomology

- Cover Cover11
- Title page i2
- Introduction 110
- Part I . Cohomology with Values in Holomorphic Functions 918
- Chapter 1. Definitions and notations 1120
- 1.1. Operators on functions on the upper and lower half-plane 1120
- 1.2. Discrete group 1120
- 1.3. Automorphic forms 1221
- 1.4. Cohomology and mixed parabolic cohomology 1322
- 1.5. Modules 1423
- 1.6. Semi-analytic vectors 1726
- 1.7. Isomorphic cohomology groups 1928
- 1.8. Harmonic lifts of holomorphic automorphic forms 2130

- Chapter 2. Modules and cocycles 2332
- Chapter 3. The image of automorphic forms in cohomology 3140
- Chapter 4. One-sided averages 4150

- Part II . Harmonic Functions 5160
- Chapter 5. Harmonic functions and cohomology 5362
- Chapter 6. Boundary germs 5766
- 6.1. Three sheaves on the real projective line 5766
- 6.2. Relation between the sheaves of harmonic boundary and analytic boundary germs 5968
- 6.3. Kernel function for the map from automorphic forms to boundary germ cohomology 6170
- 6.4. Local study of the sheaf of analytic boundary germs 6271
- 6.5. Related work 6776

- Chapter 7. Polar harmonic functions 6978

- Part III . Cohomology with values in Analytic Boundary Germs 7584
- Chapter 8. Highest weight spaces of analytic boundary germs 7786
- 8.1. Definition of highest weight space 7786
- 8.2. General properties of highest weight spaces of analytic boundary germs 7988
- 8.3. Splitting of harmonic boundary germs, Green’s form 8291
- 8.4. Periodic harmonic functions and boundary germs 8493
- 8.5. Completion of the proof of Proposition 8.4 8695
- 8.6. Related work 8897

- Chapter 9. Tesselation and cohomology 8998
- Chapter 10. Boundary germ cohomology and automorphic forms 95104
- 10.1. Spaces of global representatives for highest weight spaces 95104
- 10.2. From parabolic cocycles to automorphic forms 98107
- 10.3. Injectivity 103112
- 10.4. From analytic boundary germ cohomology to automorphic forms 106115
- 10.5. Completion of the proof of Theorem A for general weights 106115
- 10.6. Related work 107116

- Chapter 11. Automorphic forms of integral weights at least 2 and analytic boundary germ cohomology 109118
- 11.1. Image of automorphic forms in mixed parabolic cohomology 109118
- 11.2. Image of mixed parabolic cohomology classes in automorphic forms 111120
- 11.3. Exact sequences for mixed parabolic cohomology groups 114123
- 11.4. Automorphic forms and analytic boundary germ cohomology 115124
- 11.5. Comparison with classical results 117126
- 11.6. Related work 118127

- Part IV . Miscellaneous 119128
- Chapter 12. Isomorphisms between parabolic cohomology groups 121130
- Chapter 13. Cocycles and singularities 129138
- Chapter 14. Quantum automorphic forms 141150
- Chapter 15. Remarks on the literature 147156
- Appendix A. Universal covering group and representations 151160
- Bibliography 159168
- Indices 165174

- Back Cover Back Cover1182