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Functional equations for zeta functions of non-Gorenstein orders in global fields. (English) Zbl 0723.11058

From the Introduction: “In 1973 V. M. Galkin published a paper [Izv. Akad. Nauk. SSSR, Ser. Mat. 37, 3-19 (1973; Zbl 0261.12009), English translation in Math. USSR, Izv. 7, 1-17 (1974)], which deals with the zeta function of a non-maximal order in an algebraic number field or function field in one variable over a finite field of contants. By using the methods of Haar measure and duality he was able to establish a functional equation for non-maximal orders in a closed form involving only zeta functions together with an elementary factor, provided they were Gorenstein. He also gave examples to show that this functional equation does not hold in general in the absence of the Gorenstein condition. In the proof of the functional equation a key role is played by the dualizing module (also called the canonical module by Kunz in [J. Herzog and E. Kunz, Lect. Notes Math. 238 (1971; Zbl 0231.13009)]) for the order. For Gorenstein orders the dualizing module may be chosen to coincide with the ring and so no distinction between the two is needed. However for non-Gorenstein orders they do not coincide.
In this paper we show how the zeta function of a non-maximal order may be redefined so that in the absence of the Gorenstein condition one still obtains a functional equation. Recently much work has been done on the zeta and L-functions of arithmetic orders, particularly for orders in a finite dimensional semisimple algebra over the rational number field, by L. Solomon [Adv. Math. 26, 306-326 (1977; Zbl 0374.20007)], C. J. Bushnell and I. Reiner [Math. Z. 173, 135-161 (1980; Zbl 0438.12004), J. Reine Angew. Math. 349, 160-178 (1984; Zbl 0524.12006) and ibid. 364, 130-148 (1986; Zbl 0575.12008)]. However for nonmaximal non- Gorenstein orders in general it appears that little if any work has been done.”

MSC:

11R54 Other algebras and orders, and their zeta and \(L\)-functions
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References:

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