A census of zeta functions of quartic K$3$ surfaces over $\mathbb{F}_{2}$. Issue Volume 19:Issue A(2016) (August 2016) - Kedlaya, Kiran S.; Sutherland, Andrew V.
Abstract : 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.