The Tate conjecture for K3 surfaces over fields of odd characteristic

Using the theory of integral canonical models of
Shimura varieties (due to Faltings-Kisin-Vasiu), we extend the classical
Kuga-Satake construction for
K3 surfaces over fields of odd characteristic. This construction attaches to every polarized K3 surface X an abelian variety A,
and allows us (always when p>3; in certain cases when p=3) to identify the Picard group of X with
a certain space of endomorphisms (called 'special endomorphisms') of A.
Using new results of Kisin towards a proof of the Langlands-Rapoport
conjecture, we can now reduce the Tate conjecture
for X to the Tate conjecture
for endomorphisms of A, which is already known due to Tate and
Zarhin. Over finite fields of characteristic at least 5, the Tate conjecture for K3 surfaces is already known by work of Nygaard-Ogus, Maulik and Charles, but our proof is uniform and works also
over infinite, finitely generated fields.