Kazhdan, D.; Polishchuk, A. Generalized character sums associated to regular prehomogeneous vector spaces. (English) Zbl 1001.11053 Geom. Funct. Anal. 10, No. 6, 1487-1506 (2000). In the theory of prehomogeneous vector spaces, the fundamental theorem is the functional equation of the zeta function associated to a relative invariant of a prehomogeneous vector space. This is due to M. Sato in the real and complex cases, and to J. Igusa in the \(p\)-adic case. The character sum is nothing but an analogue of such a zeta function in the case of a finite field. Its functional equation, which corresponds to the above ‘fundamental theorem’, is investigated by J. Denef and A. Gyoja [Compos. Math. 113, 273-346 (1998; Zbl 0919.11086)] using a lift of a prehomogeneous space to the characteristic zero. On the other hand the authors of this paper give this functional equation by using the Picard-Lefshetz formula in \(l\)-adic cohomology. Reviewer: Hiroshi Hosokawa (Yokohama) Cited in 1 ReviewCited in 1 Document MSC: 11T24 Other character sums and Gauss sums 11S90 Prehomogeneous vector spaces 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Keywords:prehomogeneous vector spaces; functional equation; character sum; Picard-Lefshetz formula; \(l\)-adic cohomology Citations:Zbl 0919.11086 PDFBibTeX XMLCite \textit{D. Kazhdan} and \textit{A. Polishchuk}, Geom. Funct. Anal. 10, No. 6, 1487--1506 (2000; Zbl 1001.11053) Full Text: DOI arXiv