Describing the special values of zeta functions of arithmetic schemes is a classical problem in number theory which has led to several outstanding conjectures. For smooth projective varieties over finite fields, the first major progress in this direction was achieved by Milne. Later, Lichtenbaum (and Geisser for arbitrary varieties over finite fields) introduced modifications of the étale topology with respect to which motivic cohomology has better properties, and yields a more systematic description of Milne’s formula.

More generally, for a variety over a finite field equipped with an action of a finite group, one can ask for an explicit description of the values of the equivariant zeta function, in the spirit of the ETNC. In this talk we report on a work in progress which gives a conjectural formula for the values of this function at all non-negative integers. This provides an equivariant refinement of Geisser’s formula.