, Volume 336, Issue 3, pp 659-669
Date: 21 Jun 2006

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

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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(s) of f. We explain the connection between the M i (u) and the smallest real part of a pole of Z f(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(s) has no poles with real part less than  − n/2. We will also consider arbitrary K-analytic functions f.