Zeta functions of K3 categories over finite fields
Asher Auel, Jack Petok
TL;DR
The paper defines a zeta function and point counts for noncommutative K3 surfaces (K3 categories) over finite fields, proving that these invariants are preserved under Fourier–Mukai equivalence and can obstruct geometric realizations as (twisted) K3 surfaces. It then specializes to the K3 category of a cubic fourfold, giving explicit relations between the category’s zeta function and the fourfold’s cohomology, and showing that negative point counts or growth failures provide obstructions to geometricity, with connections to the Fano variety of lines and Hilbert schemes. A Honda–Tate–style program for K3 surfaces and noncommutative K3s is outlined, including necessary arithmetic constraints on Weil polynomials and the status of existing results (Kedlaya–Sutherland census; Taelman/Ito). The paper presents a Noether–Lefschetz type and reduction argument yielding a special cubic fourfold over $\mathbb{Q}$ whose K3 category is nongeometric, yet whose reduction to $\mathbb{F}_2$ has a zeta function of K3 type, illustrating that zeta data alone may fail to detect nongeometricity and motivating a noncommutative Honda–Tate framework.
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.