## A Generalization of the Artin-Tate Formula for Fourfolds

J. Math. Sci. Univ. Tokyo
Vol. 17 (2010), No. 4, Page 419--453.

Kohmoto, Daichi
A Generalization of the Artin-Tate Formula for Fourfolds
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 \& Beilinson conjecture and the finite generation of the Chow group $CH^2(X)$). 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 we obtain some presentations of weight two Weil-\'etale motivic cohomology groups in terms of higher Chow groups and unramified cohomology groups under the conjectures of Lichtenbaum and Parshin.