A vanishing result for Igusa's p-adic zeta functions with character

Abstract

Let $K$ be a $p$-adic field and let $f$ be a $K$-analytic function on an open and compact subset of $K^3$. Let $R$ be the valuation ring of $K$ and let $\chi$ be an arbitrary character of $R^{\times}$. Let $Z_{f,\chi}(s)$ be Igusa's $p$-adic zeta function. In this paper, we prove a vanishing result for candidate poles of $Z_{f,\chi}(s)$. This result implies that $Z_{f,\chi}(s)$ has no pole with real part less than $-1$ if $f$ has no point of multiplicity 2.