Determination of the real poles of the Igusa zeta function for curves

Zeta Functions for Analytic Mappings, Log-principalization of Ideals, and Newton Polyhedra

In this paper we provide a geometric description of the possible poles of the Igusa local zeta function ZΦ(s, f) associated to an analytic mapping f = (f1, . . . , fl) : U(⊆ K ) → K, and a locally constant function Φ, with support in U , in terms of a log-principalizaton of the K [x]−ideal If = (f1, . . . , fl). Typically our new method provides a much shorter list of possible poles compared wi...

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 ZDL(h, T ) of a quasi-ordinary power series h of arbitrary dimension over an algebraically closed field of characteristic zero from its...

An Introduction to P -adic and Motivic Zeta Functions and the Monodromy Conjecture

Introduced by Weil, the p-adic zeta function associated to a polynomial f over Zp was systematically studied by Igusa in the non-archimedean wing of his theory of local zeta functions, which also includes archimedean (real and complex) zeta functions [18][19]. The p-adic zeta function is a meromorphic function on the complex plane, and contains information about the number of solutions of the c...

The Igusa local zeta functions of elliptic curves

We determine the explicit form of the Igusa local zeta function associated to an elliptic curve. The denominator is known to be trivial. Here we determine the possible numerators and classify them according to the KodairaNéron classification of the special fibers of elliptic curves as determined by Tate’s algorithm.

Determination of the Poles of the Topological Zeta Function for Curves