Quadratic enrichment of the logarithmic derivative of the zeta function

TL;DR

This paper introduces a new enriched version of the logarithmic derivative of the zeta function for varieties over finite fields, linking it to topology and motivic measures, and proves rationality results for cellular schemes.

Contribution

It defines an enrichment of the logarithmic derivative of the zeta function with coefficients in the Grothendieck--Witt group and establishes its rationality for cellular schemes.

Findings

Enrichment relates to the topology of real points of a lift.

Proves rationality of the enriched zeta function for cellular schemes.

Computes examples including toric varieties, showing it as a motivic measure.

Abstract

We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck--Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure.