A census of zeta functions of quartic K3 surfaces over F2

We compute the complete set of candidates for the zeta function of a K3 surface over F-2 consistent with the Weil and Tate 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.