Quasi-ordinary power series and their zeta functions

Abstract

The main objective of this paper is to prove the monodromy conjecture for the local Igusa zeta function of a quasi-ordinary polynomial of arbitrary dimension defined over a number field. In order to do it, we compute the local Denef-Loeser motivic zeta function $Z_{\text{DL}}(h,T)$ of a quasi-ordinary power series $h$ of arbitrary dimension over an algebraically closed field of characteristic zero from its characteristic exponents without using embedded resolution of singularities. This allows us to effectively represent $Z_{\text{DL}}(h,T)=P(T)/Q(T)$ such that almost all the candidate poles given by $Q(T)$ are poles. Anyway, these candidate poles give eigenvalues of the monodromy action of the complex of nearby cycles on $h^{-1}(0).$ In particular we prove in this case the monodromy conjecture made by Denef-Loeser for the local motivic zeta function and the local topological zeta function. As a consequence, if $h$ is a quasi-ordinary polynomial defined over a number field we prove the Igusa monodromy conjecture for its local Igusa zeta function.

74 pages

References in corpus (2)

• Rationalite des series de Poincare et des fonctions Zeta motiviques
• The Nash problem on arc families of singularities

Cited by in corpus (6)

• The combinatorics of plane curve singularities. How Newton polygons blossom into lotuses
• Jet schemes of quasi-ordinary surface singularities
• Geometric motivic Poincaré series of quasi-ordinary singularities
• Igusa's p-adic local zeta function associated to a polynomial mapping and a polynomial integration measure
• Quasi-ordinary singularities and Newton trees
• Zeta Functions for Analytic Mappings, Log-principalization of Ideals, and Newton Polyhedra