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The fundamental theorem of prehomogeneous vector spaces modulo $$\mathcal P^m$$ (With an appendix by F. Sato). (English) Zbl 1207.11118
From the introduction: The authors prove an analogue over finite rings of the Fundamental Theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces. In general, this Fundamental Theorem expresses the Fourier transform of $$\chi(f)$$, with $$\chi$$ a multiplicative (quasi-)character and $$f$$ a relative invariant, in terms of $$\chi(f^\vee)^{-1}$$, with $$f^\vee$$ the dual relative invariant. M. Sato [Nagoya Math. J. 120, 1–34 (1990; Zbl 0715.22014)] proved the Fundamental Theorem over archimedian local fields, J. Igusa [Am. J. Math. 106, 1013–1032 (1984; Zbl 0589.14023)] over $$p$$-adic number fields, and J. Denef and A. Gyoja [Compos. Math. 113, No. 3, 273–346 (1998; Zbl 0919.11086)] over finite fields of big enough characteristic. In [D. Kazhdan and A. Polishchuk, Geom. Funct. Anal. 10, No. 6, 1487–1506 (2000; Zbl 1001.11053)], the regular finite field case is reproved. When the prehomogeneous vector space is regular and defined over a number field $$K$$ the authors prove an analogue of the Fundamental Theorem over rings of the form $$\mathcal O_K/\mathcal P^m$$, where $$\mathcal P$$ is a big enough prime ideal of the ring of integers $$\mathcal O_K$$ of $$K$$ and $$m > 1$$ (see Theorem 1.1). This result is derived from the results of Denef and Gyoja (loc. cit.) by using explicit calculations of exponential sums over the rings $$\mathcal O_K/\mathcal P^m$$.
In [ Adv. Stud. Pure Math. 15, 465–508 (1989; Zbl 0714.11053)], F. Sato introduced $$L$$-functions of Dirichlet type associated to regular prehomogeneous vector spaces. In the appendix by F. Sato to this paper, the authors’ results are used to obtain functional equations for these $$L$$-functions and, under extra conditions, their entireness.
##### MSC:
 11S90 Prehomogeneous vector spaces 11L07 Estimates on exponential sums 11M41 Other Dirichlet series and zeta functions 11T24 Other character sums and Gauss sums 11L05 Gauss and Kloosterman sums; generalizations 20G40 Linear algebraic groups over finite fields
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