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 on the complex \(R\psi_h\) 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.

Readership

Graduate students and research mathematicians interested in analysis and number theory.

Table of Contents

• Introduction
• Motivic integration
• Generating functions and Newton polyhedra
• Quasi-ordinary power series
• Denef-Loeser motivic zeta function under the Newton maps
• Consequences of the main theorems
• Monodromy conjecture for quasi-ordinary power series
• Bibliography