Zeta functions, Grothendieck groups, and the Witt ring

TL;DR

This paper investigates the relationship between zeta functions of varieties over finite fields, the big Witt ring, and motivic measures, revealing new connections and classical formulas in algebraic geometry.

Contribution

It establishes novel links between zeta functions, Grothendieck groups, and the Witt ring, expanding understanding of invariants of algebraic varieties over finite fields.

Findings

Connections between zeta functions and the Witt ring are established.

Relations with motivic measures are explored.

Classical formulas of Macdonald are extended or clarified.

Abstract

We prove some results connecting the zeta functions of varieties over finite fields with the big Witt ring over $\mathbb Z$. We explore relations with motivic measures and a classical formula of Macdonald on invariants of symmetric products of a variety.