On Poincaré series of singularities of algebraic curves over finite fields

Abstract

Let $${\mathcal{O}}$$ be the local ring at a singularity of a geometrically integral algebraic curve defined over a finite field $${\mathbb{F}_q}$$ , and let m be the number of branches centered at the singularity. In a previous paper the second author extended the notion of partial local zeta-functions, by considering for each pair of $${\mathcal O}$$ -ideals $${\mathfrak{a}}$$ and $${\mathfrak{b}}$$ a Poincare series $${P(\mathfrak{a},\mathfrak{b},t_{1},\ldots ,t_{m})}$$ in m variables, which encodes cardinalities of certain finite sets of ideals. To study the behavior of these power series under blow-ups, we generalize the theory by allowing that $${\mathcal{O}}$$ is a semilocal ring of the curve. In this context we establish an Euler product identity, which provides the connection between the local and semilocal theory. We further present a procedure to compute the Poincare series, and illustrate the method by some examples of local rings. Another purpose of this paper is to study the reduction $${\mod q-1}$$ of $${P(\mathfrak{a},\mathfrak{b},t_{1},\ldots,t_{m}),}$$ which becomes a polynomial if m > 1.