Local Zeta Function for Curves, Non Degeneracy Conditions and Newton Polygons

Abstract

This paper is dedicated to the description of the poles of the Igusa local zeta functions $Z(s,f,v)$ when $f(x,y)$ satisfies a new non degeneracy condition, that we have called arithmetic non degeneracy. More precisely, we attach to each polynomial $f(x,y)$, a collection of convex sets $Γ^{A}$ called the arithmetic Newton polygon of $f(x,y)$, and introduce the notion of arithmetic non degeneracy with respect to $Γ^{A}(f)$. The set of degenerate polynomials, with respect to a fixed geometric Newton polygon, contains an open subset, for the Zariski topology, formed by non degenerate polynomials with respect to some arithmetic Newton polygon.If $L$ is a number field, our main result asserts that for almost all non-archimedean valuations $v$ of $L$, the poles of $Z(s,f,v),$ with $f(x,y)\in L[x,y]$, can be described explicitly in terms of the equations of the straight segments that conform the boundaries of the convex sets that belong to $Γ^{A}(f)$. Moreover, our proof gives an effective procedure to compute $Z(s,f,v)$.