On zeta functions of prehomogeneous vector spaces

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The main purpose of this reseach is to study an explicit formula of zeta functions of prehomogeneous vector spaces and its application to automorphic forms. On the zeta functions, we proved their convergence under the rather general assumption that the singular set is a hypersurface and gave an explicit formula for zeta fuctions in terns of local orbital zeta functions under the assumption that the Hasse principle holds for G. As applications of this formula, we calculated the global zeta functions for 4 types of prehomogeneous vector spaces, which have relative invariants of degree 4 called Freudenthal quartics, and determined the relation between the zeta functins of unsaturated prehomegeneous vector spaces and that of the prehomogeneous vector spaces containing that unsaturated prehomogenous vector spaces. By these calculations, we have determined the global zeta functions of 19 types out of 29 types of regular irreducible reduced prehomogeneous vector spaces. These result seem to suggest that the arithmetic nature of global zeta fucntions are determined by the group of the connected components of stabilizer groups of generic points.We have not made much progress on the application of zeta functions of prehomogeneous vector spaces to auttomorphic forms. But the following results were obtained. Konno proved an twisted analogue of the result by Rodier-Moeglin-Waldspurger on dimensions of degenerate Whittaker vectors and reduced the generic packet conjecture for classical groups to the twisted endoscopy of general linear groups. Ikeda proved a conjecture of Miyawaki for Siegel cusp forms of degree 3 in a generalized form and constructed many Siegel cusp forms. Nishiyama determined the relation of associated varieties in the theata correspondence and showed that in some cases the correspondence of associated cycles can be described clearly. Kato proved the uniqueness of Shintani functions for p-adic groups.