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Sociedad Colombiana de Matemáticas:Publicaciones

Revista Colombiana de Matemáticas

Volumen 31 [2] (1997)Páginas 115-124

Zeta functions of singular curves over finite fields

W A Zúńiga Galindo
Universidad Autónoma de Bucaramanga, Bucaramanga, COLOMBIA

Abstract. Let X be a complete, geometrically irreducible, algebraic curve defined over a finite field Fq and let (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 (X ,t) of a non-singular curve to the functions Z(Ca(X), t) and Z(Div(X), t) by means of the singular Riemann-Roch theorem.

Palabras claves. Zeta functions, finite fields, singular curves, generalized Jacobians, compactified Jacobians.

Codigo AMS. 1991 Primary 14H25. Secondary 14H05.

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