Local and Global Zeta-Functions of Singular Algebraic Curves

Abstract

LetXbe a complete singular algebraic curve defined over a finite field ofqelements. To each local ring O ofXthere is associated a zeta-functionζO(s) that encodes the numbers of ideals of given norms. It splits into a finite sum of partial zeta-functions, which are rational functions inq−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 inq−sof degree not larger than the conductor degree of O. The global zeta-functionζOX(s), defined by encoding the numbers of coherent ideal sheaves of given degrees, satisfies the global functional equation if and only ifXis a Gorenstein curve. We introduce a modified zeta-function, which always satisfies the functional equation and which in the Gorenstein case coincides withζOX(s). We prove that the two global zeta-functions have the same residue ats=0, and that this residue determines the number of the rational points of the compactified Jacobian ofX.