# Period Functions for Maass Wave Forms and Cohomology

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*R. Bruggeman; J. Lewis; D. Zagier*

The authors construct explicit isomorphisms between spaces of
Maass wave forms and cohomology groups for discrete cofinite groups
\(\Gamma\subset\mathrm{PSL}_2({\mathbb{R}})\).

In the
case that \(\Gamma\) is the modular group
\(\mathrm{PSL}_2({\mathbb{Z}})\) this gives a cohomological
framework for the results in Period functions for Maass wave
forms. I, of

The authors introduce the concepts of mixed parabolic
cohomology group and semi-analytic vectors in principal
series representation. This enables them to describe cohomology groups
isomorphic to spaces of Maass cusp forms, spaces spanned by residues
of Eisenstein series, and spaces of all \(\Gamma\)-invariant
eigenfunctions of the Laplace operator.

For spaces of Maass cusp forms the authors also describe isomorphisms to
parabolic cohomology groups with smooth coefficients and standard
cohomology groups with distribution coefficients. They use the latter
correspondence to relate the Petersson scalar product to the cup
product in cohomology.

#### Table of Contents

# Table of Contents

## Period Functions for Maass Wave Forms and Cohomology

- Cover Cover11 free
- Title page i2 free
- Introduction vii8 free
- Chapter 1. Eigenfunctions of the hyperbolic Laplace operator 114 free
- Chapter 2. Maass forms and analytic cohomology: cocompact groups 2134
- Chapter 3. Cohomology of infinite cyclic subgroups of ๐๐๐ฟโ(โ) 3952
- Chapter 4. Maass forms and semi-analytic cohomology: groups with cusps 6174
- 10. Maass forms 6174
- 11. Cohomology and parabolic cohomology for groups with cusps 6477
- 12. Maass forms and cohomology 7386
- 13. Parabolic cohomology and mixed parabolic cohomology 7992
- 14. Period functions and periodlike functions for the full modular group 8497
- 15. Maass forms and holomorphic functions 91104

- Chapter 5. Maass forms and differentiable cohomology 95108
- Chapter 6. Distribution cohomology and Petersson product 115128
- Bibliography 127140
- Index 129142 free
- List of notations 131144
- Back Cover Back Cover1150