A dimension formula for a certain space of automorphic forms of SU(p,1). II: The case of \(\Gamma\) (N) with N\(\geq 3\)
type: Item
instance of: scholarly article
title: A dimension formula for a certain space of automorphic forms of SU(p,1). II: The case of \(\Gamma\) (N) with N\(\geq 3\)
zbMATH Open document ID: 0582.10017
author: Suehiro Kato
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1985-01-01T00:00:00Z
review text: The author gives more explicit expressions for the dimensions of the spaces of cusp forms of \(SU(p,1)\) in the case of the congruence subgroups \(\Gamma(N)\) with \(N\geq 3\) in terms of arithmetic quantities, using his previous paper [Math. Ann. 266, 457-477 (1984; Zbl 0517.10023)]. We denote the space of such cusp forms by \(S_ m(\Gamma (N))\). Suppose \(m>2p\) and \(N\geq 3\). Then one obtains the following: \[ \begin{multlined} \dim S_ m (\Gamma (N)) = [\Gamma (1) : \Gamma (N)] (m-1)! ((m-p-1)!)^{-1} \pi^{-p} \text{vol}(\Gamma (1)\setminus G_\mathbb{R}) + \\ + [\Gamma (1) : \Gamma (N)] 2^ t \delta^ p N^{-p} \zeta (1-p) e(p,d), \end{multlined} \] where \[ e(p,d) = 2^{-t-p(p-1)/2} \pi^{-p(p-1)/2} \prod^{p-2}_{r=0}r! | D|^{p(p-1)/4} L(1,\chi) \prod^{(p-2)/2}_{r=1} L(2r+1,\chi) \zeta(2r), \] \(D\) is the discriminant of \(k\), \(\delta\) is 1 or 2 according as \(-d\equiv 1 (4)\) or \(-d\equiv 2,3 (4)\), \(t\) is the number of primes which divide the discriminant of \(k\), \(\zeta(s)\) is the Riemann zeta-function and \(\chi(p)\) is the Dirichlet character associated with the field extension \(k/\mathbb{Q}\), \(G_\mathbb{R}\) the group of real points. Notice that \(\text{Vol}(\Gamma(1) \setminus G_\mathbb{R})\) was calculated by \textit{H. Zeltinger} [Bonner Math. Schriften 136 (1981; Zbl 0501.10027)]. The author also proves a certain result concerning the number of \(\Gamma(1)\)-inequivalent cusps conjectured by Zeltinger.
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 3931111
zbMATH Keywords: dimension formula
zbMATH Keywords: spaces of cusp forms
zbMATH Keywords: SU(p,1)
zbMATH Keywords: congruence subgroups
zbMATH Keywords: number of \(\Gamma \) (1)-inequivalent cusps
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228597
OpenAlex ID: W2009336652
cites work: Q3966219
cites work: Q5615858
cites work: The dimension of spaces of automorphic forms on a certain two-dimensional complex domain
cites work: The Parabolic Contribution to the Number of Linearly Independent Automorphic Forms on a Certain Bounded Domain
cites work: A dimension formula for a certain space of automorphic forms of \(\text{SU}(p,1)\)
cites work: Q5617797
cites work: Q5628357
cites work: Theory of spherical functions on reductive algebraic groups over p-adic fields
cites work: Q3237356
cites work: Arithmetic of unitary groups
cites work: The arithmetic of automorphic forms with respect to a unitary group
cites work: Q3966221
DOI: 10.2748/TMJ/1178228597
description: scientific article; zbMATH DE number 3931111