The Hecke-algebras related to the unimodular and modular group over the Hurwitz order of integral quaternions
type: Item
instance of: scholarly article
title: The Hecke-algebras related to the unimodular and modular group over the Hurwitz order of integral quaternions
zbMATH Open document ID: 0659.10027
DOI: 10.1007/BF02837824
author: Aloys Krieg
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1987-01-01T00:00:00Z
review text: The author determines the structure of the abstract Hecke algebras of \(\mathrm{Gl}_ n\) and \(\mathrm{Sp}_n\) over the Hurwitz order of integral quaternions. The main technical tool is the elementary divisor theory for the Hurwitz order as developed by the author [Linear Multilinear Algebra 21, 325--344 (1987; Zbl 0663.15003)]. In the case \(n=1\) the Hecke algebras fail to be commutative. For \(n>1\) they are commutative and isomorphic to the tensor product of their primary components. Each primary component turns out to be a polynomial ring. In the symplectic case, the relation to Siegel's \(\Phi\)-operator is investigated; the homomorphism of Hecke algebras induced by \(\Phi\) is surjective except for two exceptional weights.
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11R52
zbMATH DE Number: 4077336
zbMATH Keywords: unimodular group
zbMATH Keywords: modular group
zbMATH Keywords: Hecke algebras
zbMATH Keywords: Hurwitz order of integral quaternions
zbMATH Keywords: Siegel's \(\Phi\)-operator
MaRDI profile type: Publication
cites work: Q4101884
cites work: Hermitian modular functions
cites work: Q3963086
cites work: Q4771385
cites work: Modular forms on half-spaces of quaternions
cites work: Das Vertauschungsgesetz zwischen Hecke-Operatoren und dem Siegelschen \(\Phi\) -Operator. (The commutation property between Hecke operators and Siegel's \(\Phi\) -operator)
cites work: The elementary divisor theory over the Hurwitz order of integral quaternions
cites work: Hecke operators with respect to the modular group of quaternions
cites work: Die Primzahlen in der Theorie der Siegelschen Modulfunktionen. (The prime numbers in the theory of Siegel's modular functions)
cites work: Q5668937
cites work: ON MODULAR CORRESPONDENCES FOR SP(N, Z ) AND THEIR CONGRUENCE RELATIONS
cites work: Q5625265
cites work: Einführung in die Theorie der Modulfunktionen \(n\)-ten Grades
cites work: On the \(\zeta\)-functions of a division algebra
cites work: Arithmétique des algèbres de quaternions
cites work: On the Analogue of the Modular Group in Characteristic p
cites work: The Siegel operator and Hecke operators
description: scientific article; zbMATH DE number 4077336