Local and global zeta-functions of singular algebraic curves
In 1973, \textit{V.~M.~Galkin} [Math. USSR, Izv. 7 (1973), 1-17 (1974); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 37, 3-19 (1973; Zbl 0261.12009)] defined a zeta-function for local rings \(\mathfrak O\) of possibly singular complete geometrically irreducible curves over finite fields. His zeta-functions are defined on all of \(\mathbb C\), and if \(\mathfrak O\) is Gorenstein then the associated zeta-function satisfies a certain functional equation. In 1989, \textit{B.~Green} [Manuscr. Math. 64, No.~4, 485-502 (1989; Zbl 0723.11058)] defined zeta-functions for the same type of local rings \(\mathfrak O\). Green's zeta-functions depend on the choice of a dualizing module for~\(\mathfrak O\), but they satisfy a functional equation even when \(\mathfrak O\) is not Gorenstein. In the paper under review, the author defines a new zeta-function for local rings \(\mathfrak O\) as above by the formula \[ \zeta({\mathfrak O},s) = \sum_{I\supseteq{\mathcal O}} \#(I/{\mathfrak O})^{-s} \text{\quad for \quad } \Re(s) > 0, \] where the sum is over all fractional ideals of \(\mathfrak O\) that contain \(\mathfrak O\); this is in contrast to Galkin's zeta-function, where the corresponding sum is over nonzero ideals contained in~\(\mathfrak O\). The author shows that this zeta-function may be extended to the entire complex plane, that it satisfies Green's functional equation, and that it is equal to Galkin's zeta-function precisely when \(\mathfrak O\) is Gorenstein. The author also defines global zeta-functions for possibly-singular complete geometrically irreducible curves \(X\) over finite fields. His global zeta-functions satisfy a functional equation, and they agree with the usual global zeta-functions when \(X\) is a Gorenstein curve, but they do not necessarily satisfy the Riemann hypothesis. The author's zeta-function and the usual zeta-function share the same residue at~\(s = 0\); this residue determines the number of rational points on the compactified Jacobian of~\(X\).
number of rational points on the compactified Jacobian
scientific article; zbMATH DE number 1236179
scientific article