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Period Functions for Maass Wave Forms and Cohomology

About this Title

R. Bruggeman, Mathematisch Instituut Universiteit Utrecht, 3508 TA Utrecht, Nederland, J. Lewis, Massachusetts Institute of Technology, Cambridge, MA 02139, USA and D. Zagier, Max-Planck-Institut für Mathematik, 53111 Bonn, Deutschland; and Collège de France, 75005 Paris, France

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 237, Number 1118
ISBNs: 978-1-4704-1407-8 (print); 978-1-4704-2503-6 (online)
Published electronically: January 8, 2015
Keywords: Maass form, period function, cohomology group, parabolic cohomology, principal series, Petersson scalar product, cup product
MSC: Primary 11F37, 11F67, 11F75, 22E40

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Table of Contents


  • 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


We 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.

We introduce the concepts of mixed parabolic cohomology group and semi-analytic vectors in principal series representation. This enables us 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 we also describe isomorphisms to parabolic cohomology groups with smooth coefficients and standard cohomology groups with distribution coefficients. We use the latter correspondence to relate the Petersson scalar product to the cup product in cohomology.

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