Local and Global Zeta-Functions of Singular Algebraic Curves

Abstract

Let X be a complete singular algebraic curve defined over a finite field of q elements. To each local ring O of X there is associated a zeta-function`O(s) that encodes the numbers of ideals of given norms. It splits into a finite sum of partial zeta-functions, which are rational functions in q &s . We provide explicit formulae for the partial zeta-functions and prove that the quotient of the zeta-functions of O and its normalization O is a polynomial in q &s of degree not larger than the conductor degree of O. The global zeta-function`O X (s), defined by encoding the numbers of coherent ideal sheaves of given degrees, satisfies the global functional equation if and only if X is a Gorenstein curve. We introduce a modified zeta-function, which always satisfies the functional equation and which in the Gorenstein case coincides with`O X (s). We prove that the two global zeta-functions have the same residue at s=0, and that this residue determines the number of the rational points of the compactified Jacobian of X. Academic Press

Multi-variable Poincaré series of algebraic curve singularities over finite fields

“…By applying Theorem 2.3 to both sides of the equation of Theorem 3.3 we obtain the corresponding equation for the partial local zeta function [9], formula 2.8). This means that the Laurent series…”

“…The O-idealsb that satisfyb · O = O contain the conductor idealf (see Lemma 3.1) and therefore correspond bijectively to the sub-vector spaces of k ⊕3 that for each i = 1, 2, 3 contain a vector whose ith entry is non-zero. By putting their bases into standard forms we obtain the following list: By making direct computations we obtain: By summing up the partial local zeta functions (see [9], p. 177), we obtain the local zeta function of the rational ordinary triple point:…”

On trigonal non-Gorenstein curves with zero Maroni invariant

“…Moreover, we regard as effectives the divisors that contain the structure sheaf, and not, as usual, the converse, which is an irrelevant fact for nonsingular curves, but not in the singular case (cf. [16,Introduction]). …”

The Universal Zeta Function for Curve Singularities and its Relation with Global Zeta Functions

“…Stöhr (cf. [25], [26]) managed to attach a zeta function to X for finite k in the following manner: If O X is the structure sheaf of X, he defined the Dirichlet series Moreover, Stöhr considered local zeta functions, i.e., zeta functions attached to every local ring O P of points P at X of the form which actually establishes a link between the local and global theory. Every local factor Z(O P , T ) splits again into factors…”