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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.
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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