Zeta functions of singular curves over finite fields

Abstract

Let X be a complete, geometrically irreducible, algebraic curve defined over a finite field Fq and let ς (X,t) be its zeta function [Ser1], If X is a singular curve, two other zeta functions exist. The first is the Dirichlet series Z(Ca(X), t) associated to the effective Cartier divisors on X; the second is the Dirichlet series Z(Div(X),t) associated to the effective divisors on X, In this paper we generalize F. K. Schmidt's results on the rationality and functional equation of the zeta function ς(X, t) of a non-singular curve to the functions Z(Ca(X), t) and Z(Div(X), t) by means ofthe singular Riemann-Roch theorem.