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

Abstract

Let O be the local ring at a singular point of a geometrically integral algebraic curve defined over a finite field, and let m be the number of branches centered at the curve singularity. By encoding cardinalities of certain finite sets of ideals, we associate to each pair of ideal classes of O a power series in m variables with integer coefficients, which can be represented by an integral within the framework of harmonic analysis. We prove that partial local zeta functions can be expressed in terms of these multi-variable power series. The main objective of this paper is to investigate the properties of these series, and to provide in this way a deeper insight into the nature of local zeta functions.

Fractional ideals and integration with respect to the generalised Euler characteristic

“…We describe now the functional equations for the series P g (b, t, L). First of all, we state the following two results, due to Stöhr (see [12] and [13]). We include the proofs by the sack of completeness.…”

Poincaré series for curve singularities and its behaviour under projections

“…Some related results in the flavour of that theorem were proven later on (e.g. [20], [15], [23], [21]).…”

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

“…In the spirit of the preceding paragraphs, the author showed in his thesis [17] (see also the joint paper with his advisor Delgado [11]) that the factors Z(O P , O P , T ) coincide essentially with the generalized Poincaré series of Campillo, Delgado and Gusein-Zade, under a suitable specialization for finite fields (see §3.7 below). These ideas have also provided some feedback: for instance Stöhr achieved a deeper insight into the nature of the local zeta functions (see [27], and [16] together with his student J.J. Mira).…”