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Period Functions for Maass Wave Forms and Cohomology

R. Bruggeman Mathematisch Instituut, Universiteit Utrecht, Utrecht, The Netherlands
J. Lewis Massachusetts Institute of Technology, Cambridge, Massachusetts
D. Zagier MPI for Mathematics, Bonn, Germany and College de France, Paris, France
Available Formats:
Electronic ISBN: 978-1-4704-2503-6
Product Code: MEMO/237/1118.E
List Price: $81.00 MAA Member Price:$72.90
AMS Member Price: $48.60 Click above image for expanded view Period Functions for Maass Wave Forms and Cohomology R. Bruggeman Mathematisch Instituut, Universiteit Utrecht, Utrecht, The Netherlands J. Lewis Massachusetts Institute of Technology, Cambridge, Massachusetts D. Zagier MPI for Mathematics, Bonn, Germany and College de France, Paris, France Available Formats:  Electronic ISBN: 978-1-4704-2503-6 Product Code: MEMO/237/1118.E  List Price:$81.00 MAA Member Price: $72.90 AMS Member Price:$48.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 2372015; 132 pp
MSC: Primary 11; Secondary 22;

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 J. Lewis and D. Zagier in Ann. Math. 153 (2001), 191–258, where a bijection was given between cuspidal Maass forms and period functions.

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.

• Chapters
• Introduction
• 1. Eigenfunctions of the hyperbolic Laplace operator
• 2. Maass forms and analytic cohomology: cocompact groups
• 3. Cohomology of infinite cyclic subgroups of $\mathrm {PSL}_2(\mathbb {R})$
• 4. Maass forms and semi-analytic cohomology: groups with cusps
• 5. Maass forms and differentiable cohomology
• 6. Distribution cohomology and Petersson product
• List of notations
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2372015; 132 pp
MSC: Primary 11; Secondary 22;

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 J. Lewis and D. Zagier in Ann. Math. 153 (2001), 191–258, where a bijection was given between cuspidal Maass forms and period functions.

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.

• Chapters
• Introduction
• 1. Eigenfunctions of the hyperbolic Laplace operator
• 2. Maass forms and analytic cohomology: cocompact groups
• 3. Cohomology of infinite cyclic subgroups of $\mathrm {PSL}_2(\mathbb {R})$
• 4. Maass forms and semi-analytic cohomology: groups with cusps
• 5. Maass forms and differentiable cohomology
• 6. Distribution cohomology and Petersson product
• List of notations
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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