Mathematische Annalen

, Volume 336, Issue 3, pp 659–669

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


DOI: 10.1007/s00208-006-0016-8

Cite this article as:
Segers, D. Math. Ann. (2006) 336: 659. doi:10.1007/s00208-006-0016-8


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 11D7911S80Secondary 14B05

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Departement WiskundeK.U.LeuvenLeuvenBelgium