Motivic Zeta Functions for Curve Singularities

TL;DR

This paper introduces a universal motivic zeta function for curve singularities over fields of characteristic p, unifying various known zeta functions and Poincare series, with properties like rationality and functional equations.

Contribution

It defines a universal zeta function for curve singularities that generalizes and specializes to existing zeta functions and Poincare series, revealing new structural insights.

Findings

The universal zeta function is rational.

It admits a functional equation if the local ring is Gorenstein.

Specializes to known zeta functions and Poincare series.

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 Poincare 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 Poincare series introduced by Campillo, Delgado and Gusein-Zade.