Zeta functions of K3 categories over finite fields

TL;DR

This paper introduces a zeta function for noncommutative K3 surfaces over finite fields, providing a new invariant for point counting and exploring its implications for geometricity and Weil polynomial restrictions.

Contribution

It defines the zeta function for noncommutative K3 categories, studies its properties, and applies it to cubic fourfolds, extending classical concepts like Honda-Tate to this setting.

Findings

Point counts can be negative, serving as obstructions to geometricity.

Point counts may fail to detect nongeometricity in certain K3 categories.

Restrictions on Weil polynomials for K3 categories of cubic fourfolds.

Abstract

We define the zeta function of a noncommutative K3 surface over a finite field, an invariant under Fourier-Mukai equivalence that can be used to define point counts in this noncommutative setting. These point counts can be negative, and can be used as an obstruction to geometricity. In particular, we study the K3 category associated to a cubic fourfold over a finite field, and show that point counts can also fail to detect nongeometricity. We also study an analogue of Honda-Tate for K3 surfaces and for K3 categories, and provide a nontrivial restriction on the possible Weil polynomials of the K3 category of a cubic fourfold.