Local zeta function for curves, non-degeneracy conditions and Newton polygons

Abstract:

This paper is dedicated to a description of the poles of the Igusa local zeta function $Z(s,f,v)$ when $f(x,y)$ satisfies a new non-degeneracy condition called arithmetic non-degeneracy. More precisely, we attach to each polynomial $f(x,y)$ a collection of convex sets $\Gamma ^{A}(f)=\left \{ \Gamma _{f,1},\dots ,\Gamma _{f,l_{0}}\right \}$ called the arithmetic Newton polygon of $f(x,y)$, and introduce the notion of arithmetic non-degeneracy with respect to $\Gamma ^{A}(f)$. If $L_{v}$ is a $p$-adic field, and $f(x,y)\in L_{v}\left [ x,y \right ]$ is arithmetically non-degenerate, then the poles of $Z(s,f,v)$ can be described explicitly in terms of the equations of the straight segments that form the boundaries of the convex sets $\Gamma _{f,1},\dots , \Gamma _{f,l_{0}}$. Moreover, the proof of the main result gives an effective procedure for computing $Z(s,f,v)$.