Motivoc zeta functions for curve singularities

Julio José Moyano-Fernández , Wilson A. Zúñiga-Galindo

MSC 2000

14H20 Singularities, local rings

14G10 Zeta-functions and related questions

32S40 Monodromy; relations with differential equations and $D$-modules

11S40 Zeta functions and $L$-functions

Abstract
Let X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring $O_{P;X}$ 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 $O_{P;X}$ 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.