Topological Hochschild homology and the Hasse-Weil zeta function

TL;DR

This paper links topological Hochschild homology and Tate cohomology of schemes over finite fields to the Hasse-Weil zeta function, providing a cohomological interpretation and insights into its periodicity.

Contribution

It introduces a novel connection between Tate cohomology of circle actions on topological Hochschild homology and the Hasse-Weil zeta function for schemes over finite fields.

Findings

Cohomological interpretation of the Hasse-Weil zeta function via regularized determinants.

Periodicity in the zeta function corresponds to periodicity in the cohomology theory.

The cohomology theory is not periodic in general, only in special cases.

Abstract

We consider the Tate cohomology of the circle group acting on the topological Hochschild homology of schemes. We show that in the case of a scheme smooth and proper over a finite field, this cohomology theory naturally gives rise to the cohomological interpretation of the Hasse-Weil zeta function by regularized determinants envisioned by Deninger. In this case, the periodicity of the zeta function is reflected by the periodicity of said cohomology theory, whereas neither is periodic in general.