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The Hecke-algebras related to the unimodular and modular group over the Hurwitz order of integral quaternions

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Abstract

In the present paper the elementary divisor theory over the Hurwitz order of integral quaternions is applied in order to determine the structure of the Hecke-algebras related to the attached unimodular and modular group of degreen. In the casen = 1 the Hecke-algebras fail to be commutative. Ifn > 1 the Hecke-algebras prove to be commutative and coincide with the tensor product of their primary components. Each primary component turns out to be a polynomial ring inn resp.n + 1 resp. 2n resp. 2n+1 algebraically independent elements. In the case of the modular group of degreen, the law of interchange with the Siegel ϕ-operator is described. The induced homomorphism of the Hecke-algebras is surjective except for the weightsr = 4n-4 andr = 4n-2.

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  1. Mathematisches Institut, Westfälische Wilhelms-Universität, Einsteinstr. 62, D-4400, Münster, Federal Republic of Germany

    Aloys Krieg

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  1. Aloys Krieg
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Krieg, A. The Hecke-algebras related to the unimodular and modular group over the Hurwitz order of integral quaternions. Proc. Indian Acad. Sci. (Math. Sci.) 97, 201–229 (1987). https://doi.org/10.1007/BF02837824

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