On zeta functions of prehomogeneous vector spaces

本文主要研究预齐次向量空间的zeta函数的一个显式公式及其在自守型中的应用。对于zeta函数，我们在奇异集是超曲面的一般假设下证明了它们的收敛性，并在Hasse原理对G成立的假设下，给出了局部轨道zeta函数的π中zeta函数的一个显式公式.作为该公式的应用，我们计算了4类具有4次相对不变量的预齐次向量空间的整体zeta函数，并确定了非饱和预齐次向量空间的zeta函数与包含该非饱和预齐次向量空间的预齐次向量空间的zeta函数之间的关系.通过这些计算，我们确定了29类正则不可约预齐次向量空间中19类的整体zeta函数。这些结果似乎表明，整体zeta函数的算术性质是由一般点的稳定子群的连通分支群所决定的。但获得了以下结果。Konno证明了Rodier-Moeglin-Waldspurger关于退化Whittaker向量维数的结果的扭曲模拟，并将经典群的通用包猜想减少到一般线性群的扭曲内窥镜。Ikeda以推广的形式证明了Miyawaki关于3次Siegel尖点型的猜想，并构造了许多Siegel尖点型.西山在Theata对应中确定了相关变种的关系，并表明在某些情况下，相关旋回的对应可以清楚地描述。加藤证明了p-adic群的Shintani函数的唯一性。

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.