Certain identities of zeta functions on a quaternion algebra (Q2856648)
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scientific article; zbMATH DE number 6220979
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Certain identities of zeta functions on a quaternion algebra |
scientific article; zbMATH DE number 6220979 |
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30 October 2013
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quaternion algebra
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\(\zeta\)-function
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order
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Hecke operator
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Brandt matrix
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Certain identities of zeta functions on a quaternion algebra (English)
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\(\zeta\)-functions of orders in quaternion algebras were introduced around 1960 by \textit{R. Godement} [The \(\zeta\)-functions of simple algebras. I, II. (French) Sémin. Bourbaki 11 (1958/59), Exp. No. 171, 23 p.; Exp. No. 176, 20 p. (1959; Zbl 0213.33704)], \textit{G. Shimura} [Introduction to the arithmetic theory of automorphic functions, Princeton, NJ: Princeton Univ. Press (1994; Zbl 0872.11023)], and \textit{T. Tamagawa} [Ann. Math. (2) 77, 387--405 (1963; Zbl 0222.12018)]. Identities between such \(\zeta\)-functions were found by \textit{H. Shimizu} [Ann. Math. (2) 81, 166--193 (1965; Zbl 0201.37903)], using the trace formula of Hecke operators. The authors extend these results. They define \(\zeta\)-functions in terms of Brandt matrices as they represent Hecke operators on spaces of modular forms associated to orders in a quaternion algebra.
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