Addendum to: Milne, Values of zeta functions of varieties over finite fields, Amer. J. Math. 108, (1986), 297-360

TL;DR

This addendum revisits the special values of zeta functions of varieties over finite fields, incorporating recent progress on Lichtenbaum's conjecture to refine the cohomological interpretation of these values.

Contribution

It updates the original results by integrating new insights from motivic cohomology, providing a clearer understanding of zeta function values in terms of Z-cohomology groups.

Findings

Restates main theorem using Z-cohomology groups

Connects zeta values with motivic cohomology progress

Enhances cohomological framework for finite field varieties

Abstract

The original article expressed the special values of the zeta function of a variety over a finite field in terms of the $\hat{Z}$-cohomology of the variety. As the article was being completed, Lichtenbaum conjectured the existence of certain motivic cohomology groups. Progress on his conjecture allows one to give a beautiful restatement of the main theorem of the article in terms of $Z$-cohomology groups.