Dissolving a cusp form in the presence of multiplicities
type: Item
instance of: scholarly article
title: Dissolving a cusp form in the presence of multiplicities
zbMATH Open document ID: 0924.11040
DOI: 10.1215/S0012-7094-97-08810-4
published in: Duke Mathematical Journal
publication date: 1997-09-18T00:00:00Z
review text: Let \(\Gamma\) be a discrete cofinite but not cocompact group of orientation preserving motions of the hyperbolic plane \(\mathbb{H}\). Denote by \(N(x)\) the number of linearly independent \(\Gamma\)-automorphic Maaß cusp forms with eigenvalue less than \(x>0\). The group \(\Gamma\) is called essentially cuspidal if \(N(x)/x\) converges to \(\frac{1}{4\pi}\) area \((\Gamma\setminus \mathbb{H})\) as \(x\to \infty\) (Weyl's law). For a long time it was supposed that all groups \(\Gamma\) of the above type were essentially cuspidal. This conjecture has neither been proven nor disproven, but now several results indicate that the conjecture is unlikely to be true. The reason is that Maaß cusp form eigenvalues may dissolve into the continuous spectrum under a perturbation of \(\Gamma\). The proofs of these dissolution results depend on a multiplicity hypothesis requiring essentially that the eigenvalue to be dissolved is simple. Since no example of a group \(\Gamma\) with simple spectrum is known this hypothesis really is crucial. The aim of the paper under review is to relax the multiplicity conjecture. The authors show that an eigenspace of arbitrary dimension contains a Maaß cusp form that is dissolved under perturbation provided the dissolving functional does not vanish for some Maaß form belonging to the eigenspace in question. This can be used to establish the dissolution of Maaß cusp form in several cases which were considered in the literature.
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 1042483
zbMATH Keywords: Phillips-Sarnak conjecture
zbMATH Keywords: dissolution of cusp forms
zbMATH Keywords: Maaß cusp form
author: Christopher M. Judge
author: Ralph S. Phillips
reviewed by: Jürgen Elstrodt
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1042483