The Igusa Local Zeta Function Associated with the Singular Cases of the Determinant and the Pfaffian

Abstract

This paper describes the theory of the Igusa local zeta function associated with a polynomial f (x) with coefficients in a p-adic local field K. Results are given in two cases where f (x) is the determinant of a Hermitian matrix of degree m with coefficients in: (1) a ramified quadratic extension of K; and (2) the unique quaternion division algebra over K.

The local zeta function for the non-trivial characters associated with the singular Jordan algebras

“…In [8,11] the local zeta functions for the trivial character are given for all composition algebras. In this paper, we quickly show in the quaternion case that Z(t, χ) = 0 for all nontrivial characters and then compute the local zeta function in the ramified quadratic extension case for χ equal to the quadratic character.…”

“…As mentioned above, the local zeta function for the trivial character χ 0 is known [11]. Adopting the notation (a) = (…”

“…We will outline the proof of this theorem but note that it follows closely the procedure used in [11] …”

“…Using the Gauss identity [3] and Lemma 3 in [11] with x = q −2 , we can show that the inner sum above is precisely q 2k 2 t k . For completeness, we state Lemma 3 in [11] without proof.…”

A new generating function for calculating the Igusa local zeta function

“…Numerous other works study the poles of the local zeta function [25,34,20,35,3,36,4,7,5,44,45,30,47,38,39,23,24]. Many authors have labored on the calculation of local zeta functions in various situations [28,37,1,5,42,43,21,29,30,9], and many works either use local zeta functions or else apply the methods developed for obtaining them [8,46,48,31,17,18,49,40,33,50]. Nonetheless, certain classes of polynomials have proved forbidding to those who wish to obtain general results.…”