Special values of zeta functions of varieties over finite fields via higher Chow groups

TL;DR

This paper introduces a new formula for special values of zeta functions of singular varieties over finite fields, utilizing a regulator morphism from higher Chow groups to weight homology, and proves the boundedness of the weight complex for such varieties.

Contribution

It constructs a novel regulator morphism linking higher Chow groups and weight homology, enabling new formulas for zeta function values over finite fields.

Findings

New formula for special zeta values using regulator

Boundedness of weight complex for any variety over perfect fields

Use of weight spectral sequence in the proof

Abstract

We study special values of zeta functions of singular varieties over finite fields. We give a new formula of special values by constructing a morphism of homology theories, which we call regulator, from higher Chow group to weight homology. Our regulator is defined by using the notion of weight complex for varieties over a perfect field, which was introduced by Gillet and Soule. The main idea of the proof of our formula of special values is to use weight spectral sequence of homology theories, whose E1 terms are homology groups for smooth projective schemes. Also, to calculate special values, we prove that the weight complex for any variety over a perfect field is bounded. This boundedness result was known by Gillet and Soule in the case that the base field admits resolution of singularities, but not in general.