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 \chi 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,\chi}(s) of f. We explain the connection between the M_i(u) and the smallest real part of a pole of Z_{f,\chi}(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,\chi}(s) has no poles with real part less than -n/2. We will also consider arbitrary K-analytic functions f.