Electronic ISBN:  9781470425036 
Product Code:  MEMO/237/1118.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 237; 2015; 132 ppMSC: 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, ofJ. Lewis andD. 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 semianalytic 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

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 semianalytic cohomology: groups with cusps

5. Maass forms and differentiable cohomology

6. Distribution cohomology and Petersson product

List of notations


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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 semianalytic 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 semianalytic cohomology: groups with cusps

5. Maass forms and differentiable cohomology

6. Distribution cohomology and Petersson product

List of notations