Quadratic enrichment of the logarithmic derivative of the zeta function
Year: 2024
Type: article
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.
DOI: 10.1090/btran/201
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,750
Funder: National Science Foundation
Awards: Motivic Homotopy Theory and Applications to Enumerative Geometry, CAREER: Etale and Motivic Homotopy Theory and Applications to Arithmetic Geometry, PostDoctoral Research Fellowship, G6671297155, CAREER: Theory, Heuristics, and Data for Arithmetic Invariants, Geometry and Arithmetic of Brill--Noether Loci and Brill--Noether curves
OpenAlex ID: W4403057043