Lower Bound for the Poles of Igusa’s p-adic Zeta Functions

Abstract

Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P. Let f be a polynomial over R in n >1 variables and let χ be a character of $R^{\times}$. Let M _i (u) be the number of solutions of f  =  u in (R/P ⁱ)ⁿ for $i \in \mathbb{Z}_{\geq 0}$ and$u \in R/P^i$. These numbers are related with Igusa’s p-adic zeta function Z _f,χ(s) of f. We explain the connection between the M _i (u) and the smallest real part of a pole of Z _f,χ(s). We also prove that M _i (u) is divisible by $q^{\ulcorner (n/2)(i-1)\urcorner}$, where the corners indicate that we have to round up. This will imply our main result: Z _f,χ(s) has no poles with real part less than  − n/2. We will also consider arbitrary K-analytic functions f.