Eichler-Shimura homology and the finite generation of cusp forms by hyperbolic Poincaré series
type: Item
instance of: scholarly article
title: Eichler-Shimura homology and the finite generation of cusp forms by hyperbolic Poincaré series
zbMATH Open document ID: 0618.10021
DOI: 10.1215/S0012-7094-86-05353-6
author: William M. Goldman
author: John Millson
published in: Duke Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: Out of a quite general theorem concerning the span of decomposable classes in the first homology group of a finitely generated group with coefficients in an irreducible representation the authors obtain some new results on the finite generation of cusp forms by hyperbolic Poincaré series with respect to a discrete subgroup of \(SL_ 2({\mathbb{R}})\). Given a finitely generated group \(\Gamma\) and an absolutely irreducible nontrivial direct summand \(\sigma\) of \(Adjoint. \circ \rho\) where \(\rho\) is an absolutely irreducible finite dimensional representation of the underlying Lie group then it is proved that \(H_ 1(\Gamma,\sigma^*)\) is spanned by decomposable classes. Applying this result in the situation \(\Gamma \subset SL_ 2({\mathbb{R}})\) and \(\rho\) the m-th symmetric power of the standard representation and using Eichler-Shimura theory the authors describe a way how to pass from a set of generators for \(\Gamma\) to a finite spanning set for the space \(S_{2m}(\Gamma)\) of cusp forms with respect to \(\Gamma\).
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 4002180
zbMATH Keywords: finite generation of cusp forms
zbMATH Keywords: hyperbolic Poincaré series
zbMATH Keywords: discrete subgroup
MaRDI profile type: Publication
cites work: Q3328657
cites work: Eine Verallgemeinerung der Abelschen Integrale
cites work: Invariant functions on Lie groups and Hamiltonian flows of surface group representations
cites work: Local rigidity of discrete groups acting on complex hyperbolic space
cites work: Deformation spaces associated to compact hyperbolic manifolds
cites work: Closed geodesics, periods and arithmetic of modular forms
cites work: Eichler-Shimura Homology, Intersection Numbers and Rational Structures on Spaces of Modular Forms
cites work: Cusp forms associated to loxodromic elements of Kleinian groups
cites work: Harmonic differentials and closed geodesics on a Riemann surface
cites work: Varieties of representations of finitely generated groups
cites work: Lectures on introduction to moduli problems and orbit spaces
cites work: Ein Summationsverfahren für die Poincareschen Reihen von der Dimension -2 zu den hyperbolischen Fixpunktepaaren
cites work: The invariant theory of \(n\times n\) matrices
cites work: Q5669004
cites work: On integrals attached to automorphic forms
cites work: The Fenchel-Nielsen deformation
cites work: Monodromy groups and Poincaré series
full work available at URL: s0012-7094-86-05353-6
OpenAlex ID: W1561132476
description: scientific article; zbMATH DE number 4002180