A census of zeta functions of quartic K3 surfaces over F_2

TL;DR

This paper systematically computes and compares the possible zeta functions of quartic K3 surfaces over F_2, providing numerical evidence for a Honda-Tate type classification for their transcendental zeta functions.

Contribution

It offers a comprehensive enumeration of candidate zeta functions for K3 surfaces over F_2 and identifies subsets that align, supporting conjectures on their classification.

Findings

Complete sets of candidate zeta functions for K3 surfaces over F_2

Differences and overlaps between zeta functions of quartic surfaces and K3 surfaces

Numerical evidence supporting a Honda-Tate type theorem for transcendental zeta functions

Abstract

We compute the complete set of candidates for the zeta function of a K3 surface over F_2 consistent with the Weil conjectures, as well as the complete set of zeta functions of smooth quartic surfaces over F_2. These sets differ substantially, but we do identify natural subsets which coincide. This gives some numerical evidence towards a Honda-Tate theorem for transcendental zeta functions of K3 surfaces; such a result would refine a recent theorem of Taelman, in which one must allow an uncontrolled base field extension.