Zeta functions of K3 categories over finite fields

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.