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 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 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.

Stringy E-functions of hypersurfaces and of Brieskorn singularities

“…Then we can finally state the following theorem ( [ACLM,Theorem 2.4]; compare with the second remark after Theorem 4.2 in [DH] by 'replacing' L by p and T by p −s ).…”

The monodromy conjecture for plane meromorphic germs

“…The conjecture is still open in arbitrary dimension. Particular cases are proved in [2,3,5,10,11,13,19].…”

Monodromy eigenvalues are induced by poles of zeta functions: the irreducible curve case

“…This conjecture was proved for n = 2 by Loeser (originally in the context of p-adic Igusa zeta functions) in [11]. There are by now various other partial results, for example, [3,4,10,12,14,17].…”

“…It is natural and useful to study these invariants incorporating such a more general ω; see, for example, [3,4,16]. Note, however, that there one restricts to the situation where supp(div(ω)) ⊂ f −1 {0}.…”