Extended Epstein's zeta functions over $CM$-fields

Lectures on Zeta Functions over Finite Fields

These are the notes from the summer school in Göttingen sponsored by NATO Advanced Study Institute on Higher-Dimensional Geometry over Finite Fields that took place in 2007. The aim was to give a short introduction on zeta functions over finite fields, focusing on moment zeta functions and zeta functions of affine toric hypersurfaces. Along the way, both concrete examples and open problems are ...

Computing Zeta Functions Over Finite Fields

In this report, we discuss the problem of computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo pm of the zeta function of a hypersurface, where p is the characteristic of the finite field. 1991 Mathematics Subject Classification: 11Y16, 11T99, 14Q15.

New Non-Abelian Zeta Functions for Curves over Finite Fields

In this paper, we introduce and study two new types of non-abelian zeta functions for curves over finite fields, which are defined by using (moduli spaces of) semi-stable vector bundles and non-stable bundles. A Riemann-Weil type hypothesis is formulated for zeta functions associated to semi-stable bundles, which we think is more canonical than the other one. All this is motivated by (and hence...

Zeta Functions of Artin - Schreiercurves over Finite Fields

Zeta functions of equivalence relations over finite fields

We prove the rationality of the generating function associated to the number of equivalence classes of Fqk -points of a constructible equivalence relation defined over the finite field Fq . This is a consequence of the rationality of Weil zeta functions and of first-order formulas, together with the existence of a suitable parameter space for constructible families of constructible sets.