Motivic zeta functions for curve singularities

Abstract

Let $X$ be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic $p$ big enough. Given a local ring ${\mathcal{O}}_{P,X}$ at a rational singular point $P$ of $X$, we attached a universal zeta function which is a rational function and admits a functional equation if ${\mathcal{O}}_{P,X}$ is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.