Zeta functions of K3 categories over finite fields

Abstract

We study the arithmetic of the K3 category associated to a cubic fourfold over a non-algebraically closed field k, specifically, the Galois representation on its ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document}-adic Mukai realization. For k a finite field, we define the zeta function of an arbitrary K3 category, an invariant under Fourier–Mukai equivalence that can be used to study its geometricity. We show both how the zeta function can obstruct the geometricity of the K3 category of a cubic fourfold, as well as fail to detect nongeometricity. Finally, we 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.