On the Convergence of the Adelic Zeta Functions Associated to Irreducible Regular Prehomogeneous Vector Spaces

Abstract

Introduction. Let (G, X) be an irreducible regular prehomogeneous vector space defined over an algebraic number field k. D denotes the derived group of G. X is an affine n-space Affn. f(x) E k[xl, ... ,xn] is the relative invariant. f(gx) = v(g)f(x) for every g E G, where vi E Hom (G,Gm). Y = X -f-1(0). The subscript A denotes the adelization relative to k. The subscript k denotes the taking of k-rational points. Let AUG be a Haar measure on GA. Consider the following integral