Convergence of the zeta functions of prehomogeneous vector spaces

Abstract

Let $(G,\rho,X)$ be a prehomogeneous vector space with singular set $S$ over an algebraic number field $F$. The main result of this paper is a proof for the convergence of the zeta fucntions $Z(\Phi,s)$ associated with $(G,\rho,X)$ for large Re $s$ under the assumption that $S$ is a hypersurface. This condition is satisfied if $G$ is reductive and $(G,\rho,X)$ is regular. When the connected component $G_x^0$ of the stabilizer of a generic point $x$ is semisimple and the group $\Pi_x$ of connected components of $G_x$ is abelian, a clear estimate of the domain of convergence is given.

Moreover when $S$ is a hypersurface and the Hasse principle holds for $G$, it is shown that the zeta fucntions are sums of (usually infinite) Euler products, the local components of which are orbital local zeta functions. This result has been proved in a previous paper by the author under the more restrictive condition that $(G,\rho,X)$ is irreducible, regular, and reduced, and the zeta function is absolutely convergent.