Finiteness of K3 surfaces and the Tate conjecture
  Given a finite field k of characteristic at least 5, we show that the Tate conjecture holds for K3 surfaces defined over the algebraic closure of k if and only if there are only finitely many K3 surfaces over each finite extension of k. 
Max Lieblich, Davesh Maulik, and Andrew Snowden
math.AG math.NT