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Holomorphic Automorphic Forms and Cohomology

About this Title

Roelof Bruggeman, Mathematisch Instituut Universiteit Utrecht, Postbus 80010, 3508 TA Utrecht, Nederland, YoungJu Choie, Dept. of Mathematics, Pohang University of Science and Technology, Pohang, Korea 790–784 and Nikolaos Diamantis, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 253, Number 1212
ISBNs: 978-1-4704-2855-6 (print); 978-1-4704-4419-8 (online)
Published electronically: March 29, 2018
Keywords: Holomorphic automorphic form, Eichler integral, cohomology, mixed parabolic cohomology, period function, harmonic lift, harmonic functions, boundary germ
MSC: Primary 11F67, 11F75, secondary, 11F12, 22E40

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Table of Contents


  • 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


We 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. We use Knopp’s generalization of this integral to real weights, and apply it to complex weights that are not an integer at least $2$. We 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. We impose no condition on the growth of the automorphic forms at the cusps. Our 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.

For real weights that are not an integer at least $2$ we similarly characterize the space of cusp forms and the space of entire automorphic forms. We give a relation between the cohomology classes attached to holomorphic automorphic forms of real weight and the existence of harmonic lifts.

A tool in establishing these results is the relation to cohomology groups with values in modules of “analytic boundary germs”, which are represented by harmonic functions on subsets of the upper half-plane. It turns out that for integral weights at least $2$ the map from general holomorphic automorphic forms to cohomology with values in analytic boundary germs is injective. So cohomology with these coefficients can distinguish all holomorphic automorphic forms, unlike the classical Eichler theory.

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