A generalization of the Artin-Tate formula for fourfolds

TL;DR

This paper extends the Artin-Tate formula to fourfolds, providing a new expression for the special value of the Hasse-Weil zeta function at s=2 using geometric invariants like higher Chow groups, under certain conjectural assumptions.

Contribution

It introduces a generalized formula for fourfolds' zeta functions, linking it to motivic cohomology and comparing it with Geisser's Weil-étale cohomology-based formula.

Findings

Derived a new formula for zeta(2) of fourfolds involving higher Chow groups.

Compared the new formula with Geisser's, relating Weil-étale cohomology to higher Chow groups.

Under assumptions, provided presentations of Weil-étale motivic cohomology groups.

Abstract

We give a new formula for the special value at s=2 of the Hasse-Weil zeta function for smooth projective fourfolds under some assumptions (the Tate and Beilinson conjecture, finiteness of some cohomology groups, etc.). Our formula may be considered as a generalization of the Artin-Tate(-Milne) formula for smooth surfaces, and expresses the special zeta value almost exclusively in terms of inner geometric invariants such as higher Chow groups (motivic cohomology groups). Moreover we compare our formula with Geisser's formula for the same zeta value in terms of Weil-\'etale motivic cohomology groups, and as a consequence (under additional assumptions) we obtain some presentations of weight two Weil-\'etale motivic cohomology groups in terms of higher Chow groups and unramified cohomology groups.