Quasi-ordinary Power Series and Their Zeta Functions

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(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(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.