UC San Diego

We compute the complete set of candidates for the zeta function of a K$3$surface over$\mathbb{F}_{2}$consistent with the Weil and Tate conjectures, as well as the complete set of zeta functions of smooth quartic surfaces over$\mathbb{F}_{2}$. 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 K$3$surfaces; such a result would refine a recent theorem of Taelman, in which one must allow an uncontrolled base field extension.