Mathematische Annalen

, Volume 336, Issue 3, pp 659–669 | Cite as

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

Article

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 Mi(u) be the number of solutions of f  =  u in (R/Pi)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 Zf(s) of f. We explain the connection between the Mi(u) and the smallest real part of a pole of Zf(s). We also prove that Mi(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: Zf(s) has no poles with real part less than  − n/2. We will also consider arbitrary K-analytic functions f.

Mathematics Subject Classification (2000)

Primary 11D79 11S80 Secondary 14B05 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Departement WiskundeK.U.LeuvenLeuvenBelgium

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