The Igusa local zeta function associated with the nonregular irreducible prehomogeneous vector space (Q1177321)

The purpose of this paper is to compute Igusa's local zeta function attached to the non-regular prehomogeneous vector space \((GL_ 1\times Sp_{2n}\times SO_ 3, \square\otimes\Lambda_ 1\otimes\Lambda_ 1, V(6n))\). The author gives the explicit form of the zeta function and checks that a conjecture on Igusa's local zeta function holds. The author's result is the following. Let \(Z_ p(s)\) be Igusa's local zeta function where \(P\) is the irreducible relative invariant of the reduced non-regular prehomogeneous vector space \((GL_ 1\times Sp_{2n}\times SO_ 3, \square\otimes\Lambda_ 1\otimes\Lambda_ 1, V(6n))\). Then it is given by \[ Z_ P(s)={(1-q^{-1})(1-q^{-3}t)\over (1-q^{-1}t)(1- q^{-2}t)}\times{(1-q^{-2n})\over (1-q^{-2}t^ 2)} \] where \(q\) is the order of the residue field of the \(p\)-adic field \(K\), where we are considering \(Z_ P(s)\), and \(t=q^{-s}\). This calculation shows that Igusa's conjecture holds for this non-regular irreducible case.