On type numbers of split orders of definite quaternion algebras
type: Item
instance of: scholarly article
title: On type numbers of split orders of definite quaternion algebras
zbMATH Open document ID: 0851.11065
DOI: 10.1007/BF02567839
author: Yuji Hasegawa
author: Ki-ichiro Hashimoto
published in: Manuscripta Mathematica
publication date: 1996-11-25T00:00:00Z
full work available at URL: https://eudml.org/doc/156144
review text: Let \(q\) be a product of \(r\) distinct primes, where \(r\) is odd, and let \(B\) be the definite quaternion algebra over \(\mathbb{Q}\) with discriminant \(q\). Let \(N\) be a positive integer with \((q,N) = 1\) and let \(O\) be an order with level \(qN\), a so-called split order of type \((q,N)\) (for squarefree \(N\) this is an Eichler order). Let \(T_{q,N}\) denote the type number of this split order \(O\) of type \((q,N)\). Explicit formulas for \(T_{q,N}\) are known [see \textit{A. Pizer}, Acta Arith. 31, 61-89 (1976; Zbl 0348.12018)]. The authors associate \(T_{q,N}\) with the dimension of some subspace of \(S_2 (qN)\), where \(S_2 (M)\) is the space of cusp forms of weight 2 for \(\Gamma_0 (M)\): more precisely, let \(S_2 (q,N)\) be the subspace of \(S_2 (qN)\) defined by the sum \[ S_2 (q,N) = \bigoplus_{m |N} \bigoplus_{d |{N \over m}} S^0_2 (qm)^{[d]}, \] where \(S^0_2 (M)\) is the subspace of \(S_2 (M)\) spanned by newforms for \(\Gamma_0 (M)\) and \(S^0_2 (M)^{[d]} = \{f(d \tau) \mid f (\tau) \in S^0_2 (M)\}\). Then the main result is \[ T_{q,N} = 1 + \dim S_2 (q,N)^{(-, +)}, \] where \(S_2 (q,N)^{(-, +)}\) is the common \((-1, +1)\)-eigenspace for the Atkin-Lehner involutions corresponding to the prime divisors of \(q,N\) respectively.
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F25
zbMATH DE Number: 866144
zbMATH Keywords: definite quaternion algebra
zbMATH Keywords: split order
zbMATH Keywords: type number
zbMATH Keywords: cusp forms
reviewed by: Meinhard Peters
MaRDI profile type: Publication
cites work: Über die Darstellbarkeit von Modulformen durch Thetareihen.
cites work: Quadratische Formen und Modulfunktionen
cites work: Explicit formula of the traces of Hecke operators for \(\Gamma_0(N)\)
cites work: Type numbers of Eichler orders.
cites work: On the arithmetic of quaternion algebras
cites work: On the traces of Hecke operators for a normalizer of \(\Gamma_0\)(N)
OpenAlex ID: W1998512876
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description: scientific article; zbMATH DE number 866144