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Generalized character sums associated to regular prehomogeneous vector spaces. (English) Zbl 1001.11053

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.

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)

Citations:

Zbl 0919.11086
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